Image Reconstruction and Discrimination
at Low Light Levels
by
Petros Zerom
Submitted in Partial Fulfillment of
the Requirements for the Degree
Doctor of Philosophy
Supervised by
Professor Robert W. Boyd
The Institute of OpticsArts, Sciences and Engineering
Edmund A. Hajim School of Engineering and Applied Sciences
University of RochesterRochester, New York
2013
ii
Dedicated To My Parents
iii
Biographical Sketch
The author was born in Asmara, Eritrea. In 1997 he obtained a Bachelors of
Science in Physics from the University of Asmara, graduating with great distinction.
In the fall of 2000, he joined the Masters program in Physics at Washington State
University in Pullman, Washington and obtained his M.Sc. degree in 2002. He con-
tinued his graduate studies at the Institute of Optics at the University of Rochester.
His doctoral research in nonlinear and quantum optics was supervised by Professor
Robert W. Boyd.
Publications
1. P. Zerom, Z. Shi, M. N. O’Sullivan, K. W. C. Chan, M. Krogstad, J. H.Shapiro, and R. W. Boyd, “Thermal ghost imaging with averaged specklepatterns,” Phys. Rev. A 86, 063817 (2012).
2. P. Zerom, K. W. C. Chan, J. C. Howell, and R. W. Boyd, “Entangled-photoncompressive ghost imaging,” Phys. Rev. A 84, 061804(R) (2011).
3. C. J. Broadbent, P. Zerom, H. Shin, J. C. Howell, and R. W. Boyd,“Discriminating orthogonal single-photon images,” Phys. Rev. A 79, 033802(2009). [March 2009 issue of Virtual Journal of Quantum Information]
4. M. Malik, H. Shin, M. O’Sullivan, P. Zerom, and R. W. Boyd, “Quantumghost image identification with correlated photon pairs,” Phys. Rev. Lett.104, 163602 (2010). [May 2010 issue of Virtual Journal of QuantumInformation]
5. P. Zerom and R. W. Boyd, “Self-focusing, conical emission, and otherself-action effects in atomic vapors,” (book chapter) in Self-focusing: Past andPresent - Fundamental and Prospects, Eds. R. W. Boyd, S. G. Lukishova andY. Shen, Topics In Applied Physics, Volume 114, 231-251 (2009)
iv
6. R. W. Boyd, N. N. Lepeshkin, and P. Zerom, “Slow light in a collection ofcollisionally broadened two-level atoms,” Laser Physics, 15, 1389 (2005)
7. G. S. He, C. G. Lu, Q. D. Zheng, P. N. Prasad, P. Zerom, R. W. Boyd, andM. Samoc, “Stimulated Rayleigh-Bragg scattering in two-photon absorbingmedia,” Phys. Rev. A. 71, 063810 (2005)
8. M. S. Bigelow, P. Zerom, and R. W. Boyd, “Breakup of ring beams carryingorbital angular momentum in sodium vapor,” Phys. Rev. Lett., 92, 083902(2004)
Conference Papers
1. J. Howell, G. Howland, R. Boyd, P. Zerom, and J. Schneeloch, “Entropy,information and compressive sensing in the quantum domain,” in Research inOptical Sciences (Optical Society of America, 2012), p. QT4B.5.
2. G. Howland, P. Zerom, R. W. Boyd, and J. C. Howell, “Compressive sensingLIDAR for 3D imaging,” CLEO - Laser Applications to Photonic Applicationsp. CMG3 (2011).
3. P. Zerom, K. W. C. Chan, J. C. Howell, and R. Boyd, “Compressive quantumghost imaging,” in International Conference on Quantum Information(Optical Society of America, 2011), p. QTuF3.
4. P. Zerom, G. Piredda, R. Boyd, and J. Shapiro, “Optical coherencetomography based on intensity correlations of quasi-thermal light,”, inConference on Lasers and Electro-Optics/International Quantum ElectronicsConference (Optical Society of America, 2009), p. JWA48.
5. R. W. Boyd, K. W. C. Chan, A. Jha, M. Malik, C. O’Sullivan, H. Shin, and P.Zerom, “Quantum imaging: enhanced image formation using quantum statesof light,” in Proc. of SPIE Vol. 7342, 73420B (2009).
6. R. W. Boyd, G. M. Gehring, G. Piredda, A. Schweinsberg, Z. Shi, H. Shin, J.Vornehm, and P. Zerom, “Slow, fast, and backwards light propagation inerbium-doped optical fibers,” in Nonlinear Optics: Materials, Fundamentalsand Applications (Optical Society of America, 2007), p. WB1.
7. Y. Chen, Z. Shi, P. Zerom, and R. W. Boyd, “Slow light with gain induced bythree photon effect in strongly driven two-level atoms,” in Slow and Fast Light(Optical Society of America, 2006), p. ME1.
v
8. Y. Chen, P. Zerom, Z. Shi, and R. W. Boyd, “Slow light using thethree-photon effect in a dressed two-level atomic system,” in Frontiers inOptics (Optical Society of America, 2006), p. JWD29.
9. R. W. Boyd, N. Lepeshkin, A. Schweinsberg, P. Zerom, G. Gehring, G.Piredda, Z. Shi, H. Shin and Q.-H. Park, “What are the limits to the timedelay achievable using ”slow-light” methods?,” in Proceedings of SPIE Vol.5924, 592402 (2005).
10. R. W. Boyd, M. S. Bigelow, N. Lepeshkin, A. Schweinsberg, and P. Zerom,“Ultraslow and superluminal light propagation in room temperature solids,”in Nonlinear Optics: Materials, Fundamentals and Applications (OpticalSociety of America, 2004), p. FA5.
vi
Acknowledgments
This thesis would not have come to fruition without the assistance of many people
whose contributions I would like to gratefully acknowledge here.
First of all, I would like to express my sincere appreciation of the help, both
scientific and otherwise, I have received through the years from my thesis supervisor
Prof. Robert W. Boyd, without whom this thesis would not have been possible.
I am thankful to Professors Nicholas Bigelow and Carlos R. Stroud for agreeing
to serve on my Ph. D. committee. I appreciate their feedback and comments.
I would like to thank current and former members of Prof. Boyd’s and Prof.
Stroud’s research groups for their constant support and collaborations. I would like
to specifically thank Dr. Matt Bigelow (for work on spatial solitons), Dr. Giovanni
Piredda, Colin O’Sullivan, Dr. Kam Wai Clifford Chan, Dr. Anand Jha (for many
discussions on quantum related subjects and for introducing me to the games squash
and badminton), Dr. George Gehring (for the coherence propagation work), Dr. Luke
Bissell (for single photon sources using NV centers), Dr. Zhimin Shi and Dr. Heedeuk
Shin for the wonderful collaborations and great discussions. I would like to thank
Colin O’Sullivan for constantly answering all my questions and his collaborations
on the ghost imaging work. I would like to thank Dr. Kam Wai Clifford Chan for
collaborations on the compressive sensing and speckle averaging projects. I would like
to thank Dr. Zhimin Shi for all his efforts and collaborations on the ghost imaging
and slow light related projects. I would be remiss if I don’t thank Dr. Heedeuk
vii
Shin for his collaborations on the image discrimination projects. I specially extend
my gratitude to Dr. Giovanni Piredda for collaborations on coherence tomography
work, for being a wonderful roommate for two years and for educating me about the
Italian culture. I would also like to thank Dr. Svetlana Lukishova for her help and
collaborations on the single photon sources projects.
I would like to thank all the staff of the Institute for their invaluable support and
for being available whenever I needed their help: Maria Schnitzler, Joan Christian,
Lissa Cotter, Noelene Votens, Betsy Benedict, Gina Kern, Lori Russell, Marie Banach,
and Per Adamson.
I would like to thank my sisters Harnet (and her family), Lidia and Senait for
their love and moral support. A special thanks goes to Tsega K. for all her love and
support. Finally, I would like to dedicate this thesis to my parents Hiwet Mosazghi
and Zerom Tesfayesus. I would be eternally grateful for your constant love, support
and encouragement.
viii
Abstract
Quantum imaging is a recent and promising branch of quantum optics that exploits
the quantum nature of light. Improving the limitations imposed by classical sources
of light in optical imaging techniques or overcoming the classical boundaries of image
formation is one of the key motivations in quantum imaging. In this thesis, I describe
certain aspects of both quantum and thermal ghost imaging and I also study image
discrimination with high fidelity at low light levels.
First of all, I present a theoretical and experimental study of entangled-photon
compressive ghost imaging. In quantum ghost imaging using entangled photon pairs,
the brightness of readily available sources is rather weak. The usual technique of
image acquisition in this imaging modality is to raster scan a single-pixel single-photon
sensitive detector in one arm of a ghost imaging setup. In most imaging modalities,
the number of measurements required to fully resolve an object is dependent on
the measurement’s Nyquist limit. In the first part of the thesis, I propose a ghost
imaging (GI) configuration that uses bucket detectors (as opposed to a raster scanning
detector) in both arms of the GI setup. High resolution image reconstruction using
only 27% of the measurement’s Nyquist limit using compressed sensing algorithms
are presented.
The second part of my thesis deals with thermal ghost imaging. Unlike in quantum
GI, bright and spatially correlated classical sources of radiation are used in thermal
ix
GI. Usually high-contrast speckle patterns are used as sources of the correlated beams
of radiation. I study the effect of the field statistics of the illuminating source on the
quality of ghost images. I show theoretically and experimentally that a thermal GI
setup can produce high quality images even when low-contrast (intensity-averaged)
speckle patterns are used as an illuminating source, as long as the collected signal is
mainly caused by the random fluctuation of the incident speckle field, as opposed to
other noise sources.
In addition, I describe transverse image discrimination and recognition using holo-
graphic matched filtering techniques using heralded single photons from a spontaneous
parametric downconversion source. Heralded single photons are used for encoding and
discriminating images from our predefined orthogonal basis set. Our basis set consti-
tutes two locally spatially orthogonal objects. We show that if the object is a member
of a predefined set, we can discriminate the objects in the set with high confidence
levels.
x
Contributors and Funding Sources
This thesis is a result of a collaboration with many colleagues. All work is done
under the supervision of my thesis advisor Prof. R. W. Boyd. If no affiliation of a
collaborator is mentioned below, the University of Rochester is assumed.
The research in chapter 2 was performed in collaboration with Prof. R. W. Boyd,
Dr. K. W. C. Chan of Rochester Optical Manufacturing Company and Prof. John
C. Howell. Equation 2.12 in section 2.3.1 was derived by Dr. K. W. C. Chan. I
carried out all the experiments. I created all the figures, except Figures 2.3 and 2.4
(by Dr. K. W. C. Chan). I wrote the paper with help from Dr. Chan, Prof. Boyd and
Prof. Howell. Most of this work was published in Physical Review A 84, 061804(R)
(2011). This work was supported through a quantum imaging MURI grant and the
DARPA/ARO InPho grant.
The research in chapter 3 was performed in collaboration with Prof. R. W. Boyd,
Dr. Z. Shi, M. N. O’Sullivan, Dr. K. W. C. Chan of Rochester Optical Manufacturing
Company, M. Krogstad of the University of Colorado at Boulder and Prof. J. H.
Shapiro of Massachusetts Institute of Technology. I carried out all the experiments
with help from Dr. Z. Shi and M. N. O’Sullivan. I carried out all the analysis and
created all figures except for Figs. 3.1 and 3.2 (by Z. Shi). Equation 3.18 in section
3.2 was derived by M. N. O’Sullivan. I wrote the first draft of the paper. Z. Shi
took over after that, with input from the rest of the coauthors. Most of this work
was published in Physical Review A 86, 063817 (2012). This work was supported
xi
by the DARPA/DSO InPho Program and by the Canada Excellence Research Chairs
Program.
The research in chapter 4 was performed in collaboration with Prof. R. W. Boyd,
Dr. C. J. Broadbent, Dr. H. Shin and Prof. John C. Howell. Prof. R. W. Boyd
designed the project. I conducted the initial experiment using highly attenuated
coherent light in our lab, together with H. Shin. The experiment described in this
thesis was conducted using spontaneous parametric downconversion source in Prof.
Howell’s lab. The experiment was mainly carried out by Dr. C. J. Broadbent, with
help from both myself and H. Shin. I was also involved in developing the multiplexed
holograms used in the experiment. All analysis was carried out together with Dr.
C. J. Broadbent and Dr. H. Shin. I created all the figures with the help of Dr. H.
Shin and Prof. R. W. Boyd, except for Figs. 4.2 and 4.3 (created by Dr. C. J.
Broadbent). Prof. R. W. Boyd wrote the paper with help from the rest of coauthors.
C. J. Broadbent wrote the experimental part of the paper. Most of this work was
published in Physical Review A 79, 033802 (2009). Note that I am the second author
of the published paper. This work was supported by the U.S. Army Research Office
through a MURI grant.
Contents
Acknowledgments vii
Abstract viii
Contributors and Funding Sources x
List of Tables xv
List of Figures xx
1 Background 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Ghost Imaging–Introduction . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Compressive Sensing–Introduction . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Sparse and Compressible Signals . . . . . . . . . . . . . . . . 14
1.3.2 Incoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.3 Restricted Isometry Property . . . . . . . . . . . . . . . . . . 18
1.3.4 Signal Reconstruction Algorithms . . . . . . . . . . . . . . . . 20
xii
CONTENTS xiii
1.3.5 Sparse Signal Reconstruction – A Numerical Example . . . . . 21
1.4 Image recognition – Introduction . . . . . . . . . . . . . . . . . . . . 23
1.5 Summary and Outline of Thesis . . . . . . . . . . . . . . . . . . . . . 24
2 Compressive Quantum Ghost Imaging 26
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Single-Pixel Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Compressive Quantum Imaging: Theory . . . . . . . . . . . . . . . . 34
2.3.1 Entangled-Photon Compressive Ghost Imaging . . . . . . . . . 35
2.3.2 Single-Photon Single-Pixel Compressive Imaging . . . . . . . . 38
2.4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5 Image Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.6 Photon Efficiency Comparison . . . . . . . . . . . . . . . . . . . . . . 50
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 Speckle Averaging Effects in Thermal Ghost Imaging 52
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.1 Fast Detection Speed . . . . . . . . . . . . . . . . . . . . . . . 62
3.2.2 Slow Detection Speed . . . . . . . . . . . . . . . . . . . . . . . 63
3.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
CONTENTS xiv
4 Discriminating Orthogonal Single-Photon Images 73
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 Hologram Characterization . . . . . . . . . . . . . . . . . . . . . . . . 84
4.5 Single-Photon Image Discrimination . . . . . . . . . . . . . . . . . . . 85
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5 Conclusions and Discussion 92
Bibliography 97
A Algorithms for Compressed Sensing 125
List of Tables
4.1 Image-discrimination results showing the total number of raw (C) and
accidental (A) coincidences and the C/A ratio. We also show singles
rate for the hearlding and image-discrimation channels . . . . . . . . 88
xv
List of Figures
1.1 Schematic of quantum and thermal ghost imaging setups . . . . . . . 8
1.2 Cameraman image and transform coding (signal compression) using
the Discrete Cosine Transform (DCT) basis . . . . . . . . . . . . . . 16
1.3 Compressive sensing at work: The original signal (spikes) is represented
by the red dots. A random sensing matrix was used in the reconstruc-
tion of the sparse signal using (a) ℓ1 and (b) ℓ2 minimizations. The
recovered signal is represented by blue circles. Exact reconstruction of
the sparse signal was achieved when the number of measurements was
set at four times the sparsity level of the signal as can be seen in (a).
The method of least squares (ℓ2 minimization) fails to approximate the
original sparse signal as can be seen in (b). . . . . . . . . . . . . . . . 22
1.4 Matched filtering technique . . . . . . . . . . . . . . . . . . . . . . . 23
xvi
LIST OF FIGURES xvii
2.1 Schematics for (a) the single-pixel camera [78] and (b) a computa-
tional ghost imaging [82] setup. In both cases, a single-pixel (bucket)
detector collects the signal. In (a), random patterns are impressed on
the amplitude-only spatial light modulator (SLM) working in reflective
mode. In (b), the field distribution at the object plane is computation-
ally determined for each controllably impressed random phase patterns
on the SLM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 Experimental demonstration of single-pixel imaging. In (a), we show a
direct reconstruction of the ghost image of the logo of the University of
Rochester (UR), using computational ghost imaging techniques. In (b),
we use a transform basis (discrete cosine transform) and compressive
sensing techniques to reconstruct the object using fewer realizations
(measurements) than in (a). . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Schematics for compressive quantum ghost imaging. The object and
spatial light modulator (SLM) planes are conjugate to each other. In-
set: the corresponding unfolded Klyshko picture. . . . . . . . . . . . . 36
2.4 Schematics for compressive single-photon single-pixel imaging. The
object is imaged onto the plane of the spatial light modulator (SLM).
Inset: the corresponding unfolded Klyskho picture. . . . . . . . . . . 39
LIST OF FIGURES xviii
2.5 Setup for entangled photon compressive ghost imaging. PBS, polariz-
ing beam splitter; SLM, spatial light modulator; L, imaging lens; HWP,
half-wave plate; BBO, β-Barium Borate crystal. A and B represent
bucket detectors used for coincidence measurement. Inset: example of
a two-dimensional random binary pattern impressed onto the SLM. . 43
2.6 Experimental image reconstruction using compressive sensing algo-
rithms. Reconstructed ghost image of (a) the Greek letter Ψ and (b)
the University of Rochester (UR) logo. The insets show the masks
used in the test arm of the ghost imaging setup. (c, d) The absolute
value of the calculated two-dimensional discrete cosine transforms of
the insets in (a) and (b), respectively. . . . . . . . . . . . . . . . . . . 45
2.7 The calculated mean-squared error of the reconstructed ghost images
of the logo of the University of Rochester (UR) (●) and the Greek
letter Ψ (�) as functions of the number of measurements M . . . . . 46
2.8 The calculated signal-to-noise ratio of the reconstructed ghost images
of the University of Rochester (UR) logo (●) and the Greek letter Ψ
(�) as functions of the number of measurements M . . . . . . . . . . 47
LIST OF FIGURES xix
3.1 Normalized second- and fourth-order moments about the mean (a) and
kurtosis (γIM/σIM )4 (b) as functions of the speckle averaging factorM .
Here the lines are the theory [cf. Eqns. 3.25 and 3.26], and symbols
are the calculated results from one typical numerical simulation real-
ization [83, 118]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2 Schematics of our thermal ghost imaging setup. The spatial light mod-
ulator (SLM) is used to impress a sequence of random phase distribu-
tion on the laser field. BS: beam splitter; CCD: charge coupled device 68
3.3 (a) Representative speckle pattern of the sort used in our experiments
and (b) the intensity average of 25 patterns of the sort shown in (a).
The statistics of the two patterns are very different, as described in
the text. Nonetheless, ghost images obtained under the two conditions
are essentially identical. (c) A ghost image of a double slit mask (1.2
mm long, 100 µm wide, and with 40 µm gap in between) taken using
individual speckles and (d-f) a ghost image taken using the intensity
average of M = 5, 15 and 25 individual speckle patterns, respectively.
In each case, N = 10 000 measurements were used to obtain the ghost
image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
LIST OF FIGURES xx
3.4 CNR as a function of the number of measurements N for the ghost
imaging system that responds to different numbers M of averaged
speckle patterns for each measurement. Here the symbols are experi-
mental results, and the lines are simulation realizations. . . . . . . . 70
4.1 Concept for the single photon image discrimination experiment . . . . 79
4.2 Laboratory setup for writing the multiplexed hologram. Biplex holo-
grams are exposed using a HeNe laser and a pair of object-reference
beam combinations sequentially. A shutter is used to electronically
control the exposure time. For each exposure, the reference-object
pair is selected using a rotation stage and a translation stage. NPBS,
nonpolarizing beamsplitter . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3 Laboratory setup for the single-photon image readout. Heralded sin-
gle photons are sent through either object A or B, during the image-
discrimination phase of the experiment, and are then detected at either
detector A or B. TCSPC, time-correlated single-photon counter; BiBO,
Bismuth Borate crystal; NPBS, nonpolarizing beamsplitter . . . . . . 83
4.4 Object reconstruction using a plane wave read beam. . . . . . . . . . 85
4.5 Hologram readout with an image carrying read beam. . . . . . . . . . 86
4.6 Single-photon image-discrimination results. Total number of raw co-
incidences (C), accidental coincidences (A), and C/A ratio for each
object-detector combination. . . . . . . . . . . . . . . . . . . . . . . . 89
Chapter 1
Background
1.1 Introduction
Quantum imaging is a recent and promising branch of quantum optics that exploits
the quantum nature of light. Unlike optical imaging techniques that use classical
light sources, quantum imaging uses spatially multimode non-classical states of light.
The key motivation in this field is to improve limitations imposed by classical sources
of light in optical imaging techniques or overcome the classical boundaries of im-
age formation. Some of the possible applications that fall under quantum imaging
range from detection of small displacements with a precision beyond the Rayleigh
limit [1–8], subwavelength phase measurements [9, 10], noiseless amplification of op-
tical images [1, 11], quantum lithography [12–15], remote ghost spectroscopy [16],
entangled two-photon microscopy [17], high sensitivity imaging for the detection of
weak amplitude (or phase) objects beyond the standard quantum limit [18], quantum
1
1.1 Introduction 2
teleportation of optical images [19–21], quantum holography [22], to quantum imag-
ing using entangled-photon pairs, i.e., ghost (coincidence) imaging [6, 7, 23–29, and
references therein].
Non-classical states of light that display strong spatial (temporal) correlations can
be used as a source for most of the above applications. In this thesis, we are interested
in some aspects of ghost (coincidence) imaging. The source of radiation in ghost
imaging, in the quantum domain, is entangled photons created through the process of
spontaneous parametric down-conversion (SPDC) in a nonlinear optical medium [30–
32]. In SPDC, a strong pump photon of higher frequency ωp interacts with a non-
centrosymmetric crystal and results in the annihilation of a pump photon and the
creation of two down-converted photons of lower frequencies (usually called the signal,
of frequency ωs and idler, of frequency ωi). The nonlinear mixing results in signal
and idler photons that are entangled in frequency and momentum. Depending on the
phase-matching condition, the down-converted photons could be collinear (with the
direction of the pump) or noncollinear; same polarization (for type-I phase matching
condition) or orthogonal polarization (for type-II phase matching).
The first ghost imaging experiment used entangled photon pairs generated via
SPDC [23]. The inherent difficulties associated with quantum ghost imaging are two
fold: source brightness and detection inefficiencies. First, SPDC in bulk crystals pro-
duces a rather weak light source, as it is an inefficient process (depending on the
crystal length and nonlinearity, the generation efficiency is typical in the range of
1.1 Introduction 3
10−12 to 10−8). Second, the usual technique of raster scanning a single-element de-
tector requires long integration times to acquire an image with high resolution and in
addition to this, transverse single photon sensitive detector arrays are an expensive
and cumbersome resource. In the first part of the thesis, we discuss methods that ad-
dress and solve the detector inefficiency problem using single-pixel (bucket) detectors
(as opposed to a raster scanning detector) and improved (shorter) integration times
using compressive sensing methods.
Ghost (coincidence) imaging has also been demonstrated using bright, classically
correlated beams of light. In the literature (and for reasons that will be clear later),
this imaging modality is known as thermal ghost imaging. In thermal GI, unlike
ghost imaging using quantum entangled photon pairs that requires coincidence mea-
surement using single photon sensitive detectors, intensity correlation measurements
are carried out to form the ghost image. Though there are similarities and differences
between quantum and thermal GI, the photon-limited aspects of the source (bright-
ness problem) in quantum GI can be addressed and easily be overcome using brighter,
spatially correlated classical sources of radiation. In thermal GI, two copies of speckle
patterns (splitted thermal-like beams or spatially correlated beams) are used in each
arm of the optical path. Intensity correlation measurements of the photocurrents of
the two detectors (one spatially resolving and the other bucket) results in ghost image
formation. In the second part of the thesis, we study how the quality of ghost image
formation is affected when using intensity averaged speckle patterns with arbitrary
1.2 Ghost Imaging–Introduction 4
statistical properties.
The third part of the thesis deals with transverse image discrimination and recog-
nition in photon-limited situations. Ghost imaging using quantum entangled photons
have been shown to provide superior performance (better signal-to-noise) as com-
pared to thermal ghost imaging in photon-limited (low-light-level) situations [33].
Low-light-level imaging arises in many applications, such as radar, astronomy, medi-
cal imaging, to mention a few. Optical image recognition with photon-limited images
is also one such application [34–37]. It has been shown that random quantum fluc-
tuations of light at such low-light levels set the ultimate performance limit for such
optical recognition [5, 37]. We study transverse image discrimination using holo-
graphic matched filtering techniques using heralded single photons produced by an
SPDC source.
1.2 Ghost Imaging–Introduction
Ghost imaging (GI), also known as two-photon coincidence imaging, has attracted
tremendous attention in the quantum optics community since its first experimental
demonstration by Pittman et al. [23] using entangled photons generated via sponta-
neous parametric down-conversion [24–26,38–48]. The theoretical foundation of ghost
imaging was first laid out by Klyshko [49, 50]. Ghost imaging is a novel transverse
imaging technique that uses two spatially correlated beams that travel through two
separate optical paths, usually called the test and reference arms. In the test arm,
1.2 Ghost Imaging–Introduction 5
one of the two correlated beams impinges on the object, whose image we want to de-
termine, and the transmitted or reflected light is collected by a single pixel (bucket)
detector that does not provide any spatial information about the object. In the ref-
erence arm, we have a spatially resolving detector. The beam in this arm does not
interact with the object. No information about the object can be inferred from one
of the correlated beams alone. A cross correlation between the signals of the two
spatially separated detectors results in the image of the object.
The first experiment on ghost imaging used orthogonally polarized entangled pho-
tons generated from a type-II phase-matched spontaneous parametric down-conversion
source [23]. Because of the nature of the source used, it was claimed that ghost imag-
ing was a quantum effect and shortly thereafter it was argued theoretically that
entanglement is a prerequisite for achieving distributed quantum imaging [24,47,48].
However, Bennink et. al [25, 38] experimentally demonstrated ghost imaging using
classically correlated beams of light. They used a pair of collimated laser beams that
produced angularly correlated pulses (analogous to momentum-correlated photons
produced by an entangled source) and showed that a ghost image can be formed
using this classical source of light and argued that entanglement is not required for
ghost (coincidence) imaging. Since then other classical sources of radiation (thermal
or pseudothermal) have been used in ghost imaging experiments [26, 51–53].
The image acquisition of a generic ghost imaging system is described next (see
Fig. 1.1). In Fig. 1.1(a) a nonlinear crystal (NLC) is used as a source of entangled
1.2 Ghost Imaging–Introduction 6
photon pairs (commonly referred to as signal and idler photons). One of the photon
pairs (say, the signal) propagates through the test arm (say, arm 1) and the idler
photon traverses through the reference arm (arm 2). The unknown object is located
in the test arm. The impulse responses (optical transfer functions) of the test and
reference arms are given as h1(x1,x′1) and h2(x2,x
′2), respectively. The impulse re-
sponses describe the field propagation from the transverse plane x′i to xi, for i = 1, 2,
where x′i is the plane of the output face of the nonlinear crystal and xi is the plane
of the detectors. In the test arm, we have a fixed, spatially non-resolving (“bucket”)
detector (labeled D1 in Fig. 1.1). It collects the total intensity falling on plane x1 and
therefore no spatial information about the object can be retrieved using the test arm
alone. However, a spatially resolving detector is used in the reference arm to collect
light. In the case of ghost image formation using entangled photon pairs, usually
referred to as quantum ghost imaging, a scanning single-pixel detector (labeled D2
in Fig. 1.1) is used to scan the location of the idler photon in the transverse plane
x2. The image of the object is retrieved by measuring coincident events between the
signal and idler photon pairs as detector 2 is scanned in plane x2.
When the source of radiation is not quantum entangled photons, but rather ther-
mal or pseudothermal (usually produced by passing laser light through a rotating
ground glass), we refer to it as thermal ghost imaging [43, 54–58]. In Fig. 1.1(b) we
show a schematic for a thermal GI setup. Here two copies of spatially correlated
beams are created by splitting the incoming chaotic light with thermal statistics us-
1.2 Ghost Imaging–Introduction 7
ing a beam splitter. Similar to the quantum GI, the detection in the test arm uses
a spatially non-resolving detector. In the reference arm, however, spatially resolving
array of detectors (e.g. CCD) are usually employed. The intensity distribution of
the object (ghost image) is retrieved by measuring the correlation function of the
intensity fluctuations in both arms as a function of the pixel position of the array
detector in arm 2.
Before we describe some of the similarities and differences between quantum and
thermal ghost imaging techniques, we present a description of the theory of ghost
imaging when the source of radiation is entangled photons (quantum GI) and when
using spatially correlated thermal beams (thermal GI). In both cases, information
about the object is retrieved by measuring the fourth-order spatial correlation func-
tion of the intensities detected by the bucket (D1) and spatially resolving (D2) detec-
tors.
Let us first consider quantum ghost imaging. We will take the source of our entan-
gled photon pairs to be from a type-II phase matched χ(2) nonlinear crystal through
the process of spontaneous parametric down-conversion. In spontaneous parametric
down-conversion, the photons of a high-intensity pump field (of central frequency ω0)
are split into pairs of lower energy photons, usually called signal and idler (of central
frequency ω1 and ω2) through a nonlinear interaction with the medium. For a type-II
phase matching condition, the signal and idler photons are orthogonally polarized.
We briefly describe a parametric down conversion (PDC) process using a three-
1.2 Ghost Imaging–Introduction 8
(a)
(b)
x2
x1
x2
x1
D2
D1
D2
D1
source of
entangled
photons
source of
thermal
radiation
coincidences
,
h2 (x
2 , x2 )
,
h2 (x
2 , x2 )
,
h 1(x 1
, x 1)
,
h 1(x 1
, x 1)
,
h 1(x 1
, x 1)
<<
I2(x
2)I
1(x
1)
Figure 1.1: (a) Quantum ghost imaging setup. Entangled photons from a sponta-neous parametric down-conversion (SPDC) are used as a source of radiation. Thesignal and idler photons traverse the test and reference arms. Due to the low-fluxnature of such SPDC sources, photon-coincidence detection is carried out for ghostimage reconstruction. (b) Classical ghost imaging setup: Here a beam splitter isused to create two classical (thermal, pseudothermal) beams with a strong spatialcorrelation. Usually a laser beam passing through a rotating ground glass diffuser isused as a source of a pseudothermal radiation. Intensity cross correlation between thephotocurrents of the two detectors is carried out to reconstruct the object’s intensitytransmission function.
1.2 Ghost Imaging–Introduction 9
wave (pump, signal and idler) interactions inside a nonlinear crystal. Let the field
envelope operators for the signal and idler photons at the output face of a χ(2)
nonlinear crystal of length lc be b1(x) and b2(x), where x is the position in the
transverse plane. The Fourier transform of the field envelopes bi(x) are given by
bi(q) =∫(dx/2π) e−iq·x bi(x). For PDC initiated by vacuum fluctuations, no signal
(a1(x)) and idler (a2(x)) fields are present at the crystal input face. In the plane wave
pump approximation limit, the field envelope operators at the output of the crystal
are related to the input operators through [5, 59–61]:
b1(q) = U1(q)a1(q) + V1(q)a†2(−q) (1.1)
b2(q) = U2(q)a2(q) + V2(q)a†1(−q) (1.2)
where Ui(Vi) (i = 1, 2) are gain functions and exact expressions are given in [5,59,60].
The fields at the detection planes (x1 and x2) are related to the fields at the exit face
of the crystal (source planes) through ci(xi) =∫dx′
i hi(xi,x′i) bi(x
′i), (i = 1, 2) for a
lossless system and hi(xi,x′i) are the impulse response functions of the system. The
correlation function for the quantum GI is thus given by [7]
G(x1,x2) =
∣∣∣∣∫ ∫
dx′1dx
′2 h1(x1,x
′1) h2(x2,x
′2)〈b1(x′
1)b2(x′2)〉∣∣∣∣2
. (1.3)
For the thermal GI, we use a thermal field a(x) characterized by a Gaussian field
statistics as the source of our radiation and a beam splitter to create two spatially
1.2 Ghost Imaging–Introduction 10
correlated beams bi(x) (i = 1, 2). Let the complex transmission and reflection coef-
ficients of the beam splitter are t and r, respectively. The fields bi(x) (i = 1, 2) are
given interms of the input field a(x) using the standard beam splitter input-output re-
lations. Following similar formulation as in the quantum GI, the correlation function
is given by
G(x1,x2) = |tr|2∣∣∣∣∫ ∫
dx′1dx
′2 h
∗1(x1,x
′1) h2(x2,x
′2)〈a†(x′
1)a(x′2)〉∣∣∣∣2
. (1.4)
Here 〈a†(x′1)a(x
′2)〉 is the thermal second-order correlation in the source plane. We
see the analogy between the correlation functions for the quantum and thermal ghost
imaging techniques from Eqns. 1.3 and 1.4, where the signal-idler correlation function
〈b1(x′1)b2(x
′2)〉 takes the place of the thermal second-order correlation function. In
both cases, the transverse coherence length is determined by the correlation functions
(〈a†(x′1)a(x
′2)〉 and 〈b1(x′
1)b2(x′2)〉). In both configurations (thermal and quantum
GI), a bucket detector (with no spatial resolution) is used in the test arm (arm 1
in Fig. 1.1). All the object information is thus contained in the measured quantity
∫dx1G(x1,x2).
O’Sullivan et al. and Erkmen et al. have performed detailed comparison of dif-
ferent aspects, such as resolution, noise characteristics, image acquisition times, of
the two imaging modalities [33, 62]. For photon-limited applications, quantum GI
performs better (higher signal-to-noise ratio) than thermal GI, although thermal GI
needs only a slightly higher average illumination intensity to outperform quantum
1.3 Compressive Sensing–Introduction 11
GI. Image acquisition times for thermal GI are much shorter than those for quantum
GI since the readily available source of entangled photons are rather weak.
Some of the concepts described in this section will be used in chapters 2 and 3.
In chapter 2, we present a detailed theory and experiment on a quantum ghost image
reconstruction technique using compressive sensing methods. In chapter 3, the effects
of using intensity averaged speckle patterns on the quality of thermal ghost image
is studied, both theoretically and experimentally. In the next section, we present an
introduction to compressive sensing.
1.3 Compressive Sensing–Introduction
Compressive sensing (compressive sampling or sparse recovery) is a new and novel
sampling and signal reconstruction method that requires far less samples (measure-
ments) than that would be deemed necessary by the Nyquist-Shannon criterion [63–
66]. It exploits the fact that many natural signals and images are sparse or compress-
ible in certain basis, such as Fourier, wavelets and discrete cosine transforms (DCT),
to mention a few. That is, if a signal or image is sparse in some transform basis,
it can be expressed or approximated by a linear combination of a small set of basis
vectors. Compressive sensing provides a way of reconstructing the signal or image
from a small (highly incomplete) measurements of the signal.
The conventional approach for full signal recovery is such that the sampling rate
must be at least twice the bandwidth (the maximum frequency) of the signal (i.e., the
1.3 Compressive Sensing–Introduction 12
Nyquist rate). This approach inherently requires that we make as many observations
(measurements) as there are unknowns. This sampling technique does not exploit the
sparsity or compressibility of signals. For example, the traditional approach to many
lossy signal compression techniques, such as in music (MP3 standard), video (MPEG)
and images (JPEG, JPEG-2000 standards), is first to acquire the full information on
the signal. The transform coefficients of the acquired signal are then calculated in a
suitable basis where the signal is sparse or compressible. Compression is possible since
the largest coefficients and their locations are stored and the rest are set to zero. This
resource inefficient approach throws away most of the acquired information content.
In the compressive sensing paradigm, however, we sample not at the Nyquist rate,
but at the “information rate” and directly acquire the “compressed” data.
Consider the linear set of equations,
yi = 〈x, Ai〉 = ATi x, i = 1 . . .M (1.5)
which is the inner product between x and Ai and (·)T denotes transposition. . Here
x ∈ RN is the signal that we want to reconstruct (an N × 1 column vector), yi are
the measurements, and {Ai}Mi=1 are a collection of measurement (sensing) vectors.
Rearranging the sensing vectors ATi as columns in an M ×N matrix A, Eqn. 1.5 can
be re-written, in a matrix form, as
y = Ax, (1.6)
1.3 Compressive Sensing–Introduction 13
where y ∈ RM is an M × 1 column vector and A is a known M × N measurement
matrix. In order to ensure reconstruction of the signal x from the measurement y,
the fundamental theorem of linear algebra dictates that M ≥ N . The linear set of
equations in Eqn. 1.6 is underdetermined for M < N and no unique solution exists.
In compressive sensing, we are generally interested in the cases where M ≪ N .
The question that is addressed in the compressive sensing field is: if we know
that a signal or image x (of length N) is sparse or nearly sparse (compressible) in
a certain basis, that is, if x depends on a small number of unknown parameters,
can we recover x exactly (or with high probability) using M ≪ N measurements?
Rephrasing the question: can we reconstruct a signal x (of length N) with no or little
information loss by acquiring a condensed representation (of length M) through a
dimensionality reduction? It has been shown in the compressive sensing (sampling)
field that a signal that is sparse in one basis can be reconstructed using a small
set of projections onto a second measurement basis that is incoherent with the first.
The number of measurements required for full signal recovery is dependent on two
factors: (1) the sparsity level of the signal in the sparsifying transform basis and (2)
the degree of coherence between the sparsifying and the measurement bases. Below
a brief description of the two fundamental tenets of compressive sensing, namely,
sparsity and incoherence is presented.
1.3 Compressive Sensing–Introduction 14
1.3.1 Sparse and Compressible Signals
Any signal x ∈ RN can be expressed in terms of a basis of N × 1 vectors {φi}Ni=1 as
x =N∑
i=1
θiφi, (1.7)
or, in matrix notation, as x = ΦΘ. Here the weighting coefficients θi = 〈x, φi〉. A
signal x is k-sparse1 in the basis Φ, i.e., it has k non-zero components, if it can be
approximated by a linear combination of k vectors from the sparsifying transform
basis Φ. A signal x is compressible2 if it can be represented by a few large coefficients
θi in the representation given by Eqn. 1.7. The fact that k such non-zero or large
coefficients (and their locations) can be used to approximate (with a minimum error)
a vector of size N , where k ≪ N , is the basis for many efficient fundamental signal
processing methods such as transform coding (data compression). However such a
process requires that we make N measurements of the signal x as the locations of
the most significant coefficients θi are not known in advance. Consider the example
given in Fig. 1.2. The discrete cosine transform (DCT) of the original image given in
Fig. 1.2(a) was first calculated. The locations and coefficients of the transform were
stored. We then discard 50% of the smallest coefficients and inverse transform back
1A signal x ∈ RN is called k-sparse if
‖x‖0 = #{1 ≤ i ≤ N : xi 6= 0} ≤ k.
2A signal x ∈ Rn is called compressible if the sorted magnitude of the coefficients {|xi|}ni=1
decayrapidly.
1.3 Compressive Sensing–Introduction 15
into the canonical basis where the image was first acquired. Comparing Fig. 1.2(b)
with the original, we almost see no discernible differences. Generally speaking, the
disadvantage in such transform coding (data compression) techniques is that we have
wasted valuable resources in acquiring all N coefficients.
If a signal is k-sparse, i.e., it has k degrees of freedom, in the canonical basis (in
which case Φ is the identity matrix I) or is sparse in a different basis, using the signal
sparsifying basis, Eqn. 1.6 can be rewritten as
y = AΦΘ. (1.8)
Since we want to reconstruct a sparse signal from the measurements, a natural ap-
proach would be to solve the following ℓ0 optimization problem3:
Θ = minΘ
‖Θ‖0 subject to AΦΘ = y (1.9)
The ℓ0 norm counts the number of non-zero entries of the signal we want to recon-
struct. Solving Eqn. 1.9, however, is a hard combinatorial problem, is numerically
unstable and is computationally intractable as there are(Nk
)possible combinations
for the locations of the entries of the sparse signal Θ. For image acquisition, N can
easily be on the order of 106 or bigger. Due to the computational intractability of
3The ℓp norm of v is defined as ‖v‖p :=
(N∑
i=1
|vi|p)1/p
for 0 < p < ∞. The ℓ0 norm counts the
number of non-zero entries, the ℓ1 norm of v gives the sum of the absolute values of the elements of v,
i.e., ‖v‖1 =∑N
i=1|vi|, and the ℓ2 norm of v gives the Euclidean norm, i.e., ‖v‖2 =
(∑Ni=1
|vi|2)1/2
.
1.3 Compressive Sensing–Introduction 16
the ℓ0 optimization problem, Eqn. 1.9 is usually recast interms of the ℓ1 norm. The
equivalence of the above ℓ0 optimization problem with the computationally tractable
ℓ1 minimization problem will be shown below using a numerical example.
The number of measurements required to acquire a signal nonadaptively depends
on how sparse (or compressible) the signal is. The relationship between the number
of measurements and the sparsity level will be given after we review the concept of
incoherence in compressive sensing.
(a) (b)
Figure 1.2: (a) Image of a Cameraman. The discrete cosine transform (DCT) of theimage in (a) was calculated. Although not shown, most of the coefficients in thetransform basis are small. After taking the DCT of the image in (a), we set about50% of the coefficients to zero and transform it back into the measurement basis. Theresulting compressed image is shown in (b). It shows the effect of the sparse nature ofthe image in the chosen basis. We see that there is no discernible difference betweenthe original and the compressed image.
1.3 Compressive Sensing–Introduction 17
1.3.2 Incoherence
The mutual coherence between two orthonormal bases A and Φ is defined as [67]
µ(A,Φ) =√N max
i,j| 〈Ai, φj〉 | (1.10)
where Ai, φj ∈ RN denote the column vector of A and Φ, respectively, and N is the
length of the column vector. The mutual coherence measures the maximum correla-
tion between elements of the measurement (sensing) basis A and the representation
basis Φ. For highly incoherent basis pairs, there is no sparse representation of the
elements of one basis in terms of the other.
The mutual coherence takes values between 1 and√N , where the lower (upper)
bound is for completely incoherent (coherent) bases. For reasons that will be clear
later, highly incoherent basis pairs are preferable in compressed sensing applications.
One such example of maximally incoherent basis pairs is between canonical or spike
basis Ai(t) = δ(t − i) and the Fourier basis φj(t) = N−1/2ei2πjt/N . The maximum
of the inner product of the two basis pairs is N−1/2. Another example is random
orthonormal basis which are largely incoherent with any fixed basis Φ, as such they
are a universal measurement basis [67].
The main reason for requiring maximally incoherent basis pairs is that if our mea-
surement basis (A) is maximally coherent with the sparsifying basis (Φ), sampling
the transform coefficients will result in values very close to zero (for a compressible
signal) or zero (for a sparse signal) most of the time, since Θ is already sparse (com-
1.3 Compressive Sensing–Introduction 18
pressible) in the Φ basis. However, if the measurement basis is highly incoherent with
the sparsifying basis, every measurement returns a little bit of information about the
sparse signal.
The mutual coherence determines the number of measurements required for signal
recovery. Suppose we take m measurements uniformly at random from the measure-
ment basis A. If x is k-sparse in the sparsity basis Φ, then
m ≥ C · µ2(A,Φ) · k · logN (1.11)
measurements (for some positive constant C) guarantees exact recovery of the signal
x with high probability [67, 68]. Note that for highly (or maximally) incoherent
basis pairs, the number of samples is on the order of k logN instead of N . This is
crucial since we can exactly recover a k-sparse signal (of length N) using m ≪ N
measurements.
1.3.3 Restricted Isometry Property
The above discussion about the mutual coherence between the measurement and the
sparsifying bases does not address how we design a stable measurement matrix A.
The notion on constructing such a matrix, called restricted isometry property (RIP),
was put forward by Candes and Tao [69]. Let A be anM×N matrix (where as above
M < N) and K be a positive number. Then, the isometry constant δK is defined as
1.3 Compressive Sensing–Introduction 19
the smallest number such that, for each integer K = 1, 2, . . .
(1− δK)‖x‖22 ≤ ‖Ax‖22 ≤ (1 + δK)‖x‖22 (1.12)
for all K-sparse vectors x. The restricted isometry constant δK quantifies how close
to isometrically the measurement matrix A acts on the K-sparse vectors x. Note
that we want to reconstruct a length-N signal (x) from M < N measurements. The
above condition states that for any K-sparse signal x, if the K locations are known,
then the problem is not ill-conditioned provided that M ≥ K.
Designing a measurement matrix that satisfies Eqn. 1.12 requires checking all
(NK
)combinations for a K-sparse signal x of dimensions N . As mentioned above,
random matrices (for example, with elements Ai,j that are independent and identically
distributed (iid) random variables) are generally incoherent with any fixed basis and
are usually chosen in different applications of compressed sensing as measurement
matrices. For example, if the sparse signal representation basis is the spike basis, an
iid Gaussian matrix is shown to have the RIP with high probability provided that we
make M & O[K log(N/K)] measurements [65, 69, 70].
Given the above definition for the restricted isometry constant, Candes and Tao [69]
have shown that if the measurement matrix A satisfies:
δ3K + 3δ4K < 2, (1.13)
1.3 Compressive Sensing–Introduction 20
then the K-sparse vector x is the unique minimizer of the following ℓ1 optimization
problem:
x = minx
‖x‖1 subject to Ax = y (1.14)
1.3.4 Signal Reconstruction Algorithms
The traditional approach to determine the signal x from such a linear set of equations
(Eqn. 1.6) is to use the least-squared method, i.e., ℓ2 minimization. That is minimizing
the sparse signal x through
x = minx
‖x‖2 subject to y = Ax. (1.15)
Unlike the ℓ0 minimization problem (Eqn. 1.9) which is computationally intractable,
the method of least squares (Eqn. 1.15) is easy to solve. In fact, there exists a close-
form solution x = AT (AAT )−1y, where T denotes transposition. However, for a
sparse signal x, this minimizer does not necessarily guarantee a sparse solution. We
elaborate this statement by giving a sample numerical example below.
The most common algorithm used in finding sparse or compressible solution in
compressed sensing is the ℓ1 norm. Minimizing the ℓ1 norm encourages small com-
ponents of a sparse signal to become exactly zero, thus promoting sparse solutions.
In fact, Eqn. 1.14 can be recast as a linear program, for which efficient algorithms
exist. For example, when the measurement matrix A is taken to be a random matrix
(such as random matrices whose entries are independent and identically distributed
1.3 Compressive Sensing–Introduction 21
Gaussian), then a K-sparse signal can be recovered exactly using the ℓ1 optimization
if M & O[K log(N/K)] measurements are taken [70].
1.3.5 Sparse Signal Reconstruction – A Numerical Example
We next give a numerical example of sparse signal reconstruction from undersampled
measurements (see Appendix for code used to generate Fig. 1.3). Consider a signal x
(represented as red dots in Fig. 1.3 of length N = 512 that is sparse in the canonical
basis (spikes), with a sparsity level of K ≡ ‖x‖0 = 30. The measurement matrix
A is formed by sampling i.i.d entries from the normal distribution, with zero mean
and 1/M variance. Here the number of measurements M is taken to be 4 times
the sparsity level of the signal, i.e., M = 120. We can see from Fig. 1.3(a) that we
have exact recovery of the sparse signal when using ℓ1 minimization (blue circles in
the Figure). The results of reconstruction of a sparse signal using methods of least
squares (ℓ2 minimization) is also given for comparison purposes in Fig. 1.3(b), where
we see that the minimization fails to reasonably approximate the original signal (red
dots).
In chapter 2, we describe a theoretical and experimental work on entangled-photon
compressive ghost imaging. Concepts introduced in sections 1.2 and 1.3 will be
used. We will describe high-resolution compressive ghost imaging at the single photon
level using entangled photons produced by a spontaneous parametric down-conversion
(SPDC) source and using single-pixel (bucket) detectors.
1.3 Compressive Sensing–Introduction 22
100 200 300 400 500
−2
−1
0
1
2
(a) Sparse signal reconstruction using ℓ1 minimization
100 200 300 400 500
−2
−1
0
1
2
(b) Sparse signal reconstruction using ℓ2 minimization
Figure 1.3: Compressive sensing at work: The original signal (spikes) is representedby the red dots. A random sensing matrix was used in the reconstruction of the sparsesignal using (a) ℓ1 and (b) ℓ2 minimizations. The recovered signal is represented byblue circles. Exact reconstruction of the sparse signal was achieved when the numberof measurements was set at four times the sparsity level of the signal as can be seen in(a). The method of least squares (ℓ2 minimization) fails to approximate the originalsparse signal as can be seen in (b).
1.4 Image recognition – Introduction 23
1.4 Image recognition – Introduction
In chapter 4 of the thesis, we describe transverse image discrimination and recog-
nition using heralded single photons produced by a spontaneous parametric down-
conversion. We have used holographic matched filtering methods to achieve image
discrimination with high confidence level. Below we present a brief description of
matched filtering [71–73].
ζξ η
input
plane
output
plane
frequency
planeL
1L
2
Figure 1.4: Matched filtering technique.
Consider the Vander Lugt optical correlator schematically shown in Fig. 1.4. Let
the input signal, in plane ξ, be given by f(x, y) and let L1 and L2 be the Fourier
transforming lenses, i.e., plane η (ζ) is the Fourier plane of ξ (η), respectively. The
Fourier transform of the input signal, f(x, y), is given by [73]
F (fX , fY ) =
∫ ∫dx dy f(x, y) exp[−i2π(fXx+ fY y)]. (1.16)
1.5 Summary and Outline of Thesis 24
We say that the filter (in plane η) is matched to the input signal f(x, y) if its transfer
function is
H(fX , fY ) = F ∗(fX , fY ) (1.17)
as this maximizes the cross correlation function of h(x, y) and the input signal f(x, y).
For a signal centered in the input plane ξ, when the matched filtering condition is
satisfied, the signal after the mask in the frequency plane η is transformed into a
bright spot at the origin of the output plane ζ by the second Fourier transforming
lens L2. For any other input signal, the transmitted light will not be brought into a
bright spot by L2.
1.5 Summary and Outline of Thesis
In this chapter, I have reviewed ghost imaging (both quantum and thermal), compres-
sive sensing, and image discrimination using matched filtering technique. In chapter
2, I will describe a theoretical and experimental work on high-resolution compres-
sive ghost imaging at the single-photon level using entangled photons produced by a
spontaneous parametric down-conversion source and using single-pixel detectors. In
section 1.2, I have described both quantum and thermal ghost imaging. The raster-
scanning of single-photon sensitive detectors in one arm of the quantum GI setup in
order to retrieve information about the object was described. In our experiment, we
only use single-pixel detectors in both arms and compressive sensing algorithms re-
viewed in section 1.3 are used to reconstruct the ghost image. In chapter 3, I present
1.5 Summary and Outline of Thesis 25
a theoretical description and experimental results on how intensity-averaged (that
is, “blurred”) speckle patterns affect the quality of ghost image formation. I show
theoretically how the contrast-to-noise ratio depends on the speckle-averaging factor.
I also describe an experimental study in support of the theoretical work. In chapter
4, I show how to discriminate images using holographic-matched filtering technique
(section 1.4) using heralded single photon sources with high confidence level. I close
by presenting general discussions and conclusions in chapter 5.
Chapter 2
Compressive Quantum Ghost
Imaging
In this chapter, we present the results of a theoretical work and an experimental
demonstration of quantum compressive ghost imaging protocol at the single-photon
level using biphotons generated by spontaneous parametric down-conversion (SPDC)
and using spatially non-resolving (single-pixel) detectors only [74]. We show that
compressive sensing (CS) can be usefully implemented at the level of few-photon
imaging. A single-pixel compressive sensing setup using heralded single photons from
a SPDC source is also described. We also show that for a given mean-squared er-
ror of the reconstructed two-dimensional, high-resolution quantum ghost image, the
number of measurements and the number of photons needed by the compressive sens-
ing algorithm is much smaller than quantum ghost imaging experiments employing a
26
2.1 Introduction 27
raster scan. This implies both an improvement in the acquisition time of the ghost
image and a more economical use of photons for low-light-level imaging. The success
of this demonstration suggests that compressive sensing methods are likely to prove
useful much more generally in applications involving quantum light fields.
2.1 Introduction
One of the key goals of many imaging protocols is to form an image using as small a
number of photons as possible. Such strategies are especially useful for applications
in quantum information, where the quantum nature of the light field is a key aspect
of the problem at hand, or in other applications where photons are “expensive,” such
as in image formation at unusual wavelengths [75].
Due to the inherently weak sources of entangled photons, quantum imaging with
entangled photons suffers from low photon flux and resource-inefficient transverse de-
tection. Owing to the need for gating to achieve high temporal resolution, transverse
arrays are expensive and require intensive electronics even for low- to moderate-
resolution images [76, 77]. The most commonly employed technique used is to raster
scan a single-pixel detector to acquire the image. However, to obtain images with high
resolution and high signal-to-noise ratio, this cheaper and simpler method requires
long integration times. In this chapter we show that compressive sensing (sampling)
can be usefully implemented to address these problems at the level of few-photon
imaging. We describe an implementation of CS algorithm in the context of a specific
2.1 Introduction 28
quantum imaging protocol, that of single-photon ghost imaging, where the primary
goal is to transfer or acquire an image using the absolute minimum number of trans-
mitted photons.
As we have described in Sec. 1.3, compressed sensing (CS) is a novel sampling
and signal reconstruction method that requires far less data than would be deemed
necessary by the Nyquist-Shannon criterion [63–65]. As a resource-efficient sensing
paradigm, CS has proven to be extremely useful in the context of classical image for-
mation, the first application of which is the single-pixel camera developed by Duarte
and co-workers [78]. The method has also been recently applied to quantum state
tomography [79] and quantum process tomography [80].
The configuration of the classical single-pixel camera [78,81] is conceptually equiv-
alent to that of the computational ghost imaging, first described by Shapiro [82] the-
oretically and later verified experimentally by Bromberg et al. [83]. Conventionally,
a ghost imaging (quantum or thermal) setup involves two beams of light, which are
termed object and reference beams [23, 25, 39, 42, 43, 84] (see Sec. 1.2). The object
beam illuminates the object and the transmitted or reflected light is monitored by
a spatially non-resolving (bucket) detector. The light in the reference arm is moni-
tored by a spatially resolving detector. The image of the object is then formed by
a coincidence measurement (in the quantum case) or intensity correlation (for the
thermal case) between the object and reference signals, as described in section 1.2.
In computational ghost imaging [82, 83] a single-beam and a single spatially non-
2.2 Single-Pixel Imaging 29
resolving (single-pixel) detector are used. In both cases, the intensity distribution
of the reference beam is determined computationally. This flexibility in determining
the field distribution computationally lends itself to applications in three-dimensional
(3D) imaging [83].
This chapter is organized as follows. In section 2.2, we review classical single-pixel
imaging protocols. First the single-pixel camera developed by Duarte et al. [78] and
computational ghost imaging (CGI) put forward by Shapiro [82] are described. We
also present experimental results based on CGI and its compressive counterpart [85].
A detailed theoretical description of entangled-photon compressive ghost imaging and
heralded single-photon compressive imaging is laid out in section 2.3. In section 2.4,
we present details of the experiment on entangled-photon compressive ghost imag-
ing. This is followed by analysis of image reconstruction using compressive sensing
algorithms in section 2.5 and photon efficiency comparison between entangled-photon
compressive ghost imaging and GI based on raster scanning in section 2.6. A summary
is presented in section 2.7.
2.2 Single-Pixel Imaging
The theory of compressive sensing [63–65], introduced in section 1.3, can potentially
has a number of practical applications in astronomy, medicine such as in MRI [86,87],
computational biology [88], radar analysis [89–94], metrology [95], robotics, seismol-
ogy, generally in signal processing [96–98] and imaging (e.g. single-pixel terahertz
2.2 Single-Pixel Imaging 30
imaging system by Chan et al. [99]) and many others [100, a website dedicated to
compressive sensing, both theory and applications].
The classical single-pixel camera developed by Duarte and co-workers [78, 81] is
the first application of the theory of compressive sensing. In the experiment of [78],
a schematic of which is shown in Fig. 2.1(a), a coherent laser beam was used as
a light source and the image of the object was formed on the plane of a reflective
spatial light modulator (SLM) using a lens (not shown in Fig. 2.1(a)). In [78], a
digital micromirror device (DMD) of 1024 × 768 pixels is used as a reflective SLM.
Bucket
detector
source
ζ
x1
object
SLM
Bucket
detector
object
SLML
ζ x1
Laser
(a)
(b)
Figure 2.1: Schematics for (a) the single-pixel camera [78] and (b) a computationalghost imaging [82] setup. In both cases, a single-pixel (bucket) detector collects thesignal. In (a), random patterns are impressed on the amplitude-only spatial lightmodulator (SLM) working in reflective mode. In (b), the field distribution at theobject plane is computationally determined for each controllably impressed randomphase patterns on the SLM.
2.2 Single-Pixel Imaging 31
Each mirror of the DMD can be positioned into one of two states (the ’on’ and ’off’
positions). No light is collected from a DMD pixel in the ’off’ position and is deflected
away from the bucket detector. The light falling on the DMD can thus be reflected in
two directions depending on the orientation of the mirror. A bucket detector is used
to collect light reflected from the DMD from one of the two directions. The image of
the object can be reconstructed following the discussion used in section 1.3.
Consider the schematics of a computational ghost imaging setup shown in Fig. 2.1(b).
A phase-only spatial light modulator is used to impress a controllable random phase
φ(ξ, η) to the input field E0 incident onto the SLM. That is, the field distribution
right after the SLM is given by E(ξ, η) = E0 eiφ(ξ,η). The field distribution E(x, y) in
the object-plane, a distance L from the SLM-plane, can be easily calculated as
E(x, y) =
∫ ∫dξ dη E(x− ξ, y − η) h(ξ, η) (2.1)
where the Huygens-Fresnel free-space propagation Green’s function
h(ξ, η) =eikL
iλLexp[i
k
2L(ξ2 + η2)], (2.2)
where λ is the wavelength of the light source and the wavenumber k = 2π/λ. A
bucket (single-pixel) detector collects the light transmitted through or reflected from
the object for a transmissive or reflective [56, 101] ghost imaging setup, respectively.
For each controllable random phase φ(n)(ξ, η), where n = 1, . . . , N , impressed onto
2.2 Single-Pixel Imaging 32
the SLM, the corresponding intensity distribution I(n)(x, y) = |E(n)(x, y)|2 at the
object plane is calculated, where N denotes the total number of measurements. The
ghost image G(x, y) is determined from the correlation function
G(x, y) = 〈(IB − 〈IB〉)I(x, y)〉 (2.3)
where 〈· · · 〉 denotes ensemble averaging and
I(n)B =
∫ ∫dx dy I(n)(x, y) T (x, y) (2.4)
is the total signal collected by the bucket detector for the nth measurement and
T (x, y) is the intensity transmission function of the object. Note that the number
of measurements N required to reconstruct the object is governed by the number of
pixels needed to resolve the object.
The concepts of compressive sensing can be used to reduce the number of mea-
surements required to resolve the object (see section 1.3). In a similar fashion as
the single-pixel camera, the image of the object can be reconstructed following the
discussion used in section 1.3. This has recently been shown experimentally by Katz
et al. [85] using a pseudothermal ghost imaging setup (see Fig. 2.1(b)).
Next we show results of an experimental work on computational [82,83] and com-
pressive [85] thermal ghost imaging, for completeness. The same data collected using
the schematic shown in Fig. 2.1(b) was used for both experiments. We have used a
2.2 Single-Pixel Imaging 33
HeNe laser (λ = 632.8 nm) and a phase-only spatial light modulator (from Boulder
Nonlinear) to create speckle patterns. In both cases, the intensity distribution of the
reference beam (at the object plane) is determined computationally. A transparency
of the logo of the University of Rochester (UR), located a distance L = 152 cm from
the SLM surface, was used as our object and a large-area bucket detector (New-
port powermeter) collects light transmitted through the object for each realization n.
This process was repeated N times. The reconstruction of the computed ghost image
(object), using Eqn. 2.3, is shown in Fig. 2.2(a).
For the compressive ghost imaging experiment, we have used the discrete cosine
transform (DCT) as our sparsifying basis (see section 1.3.1). The reconstructed image,
with a better signal-to-noise ratio, is shown in Fig. 2.2(b).
(a) (b)
Figure 2.2: Experimental demonstration of single-pixel imaging. In (a), we show adirect reconstruction of the ghost image of the logo of the University of Rochester(UR), using computational ghost imaging techniques. In (b), we use a transformbasis (discrete cosine transform) and compressive sensing techniques to reconstructthe object using fewer realizations (measurements) than in (a).
2.3 Compressive Quantum Imaging: Theory 34
2.3 Compressive Quantum Imaging: Theory
In this section, we present a theoretical description for two compressive quantum
imaging setups. The first configuration is called entangled-photon compressive ghost
imaging. This configuration is also the focus of our experimental work, where bipho-
tons generated by a spontaneous parametric downconversion process (SPDC) are used
as light sources. We also describe a similar configuration we call single-photon single-
pixel compressive imaging. Here heralded single photons from an SPDC process are
used as a light source.
The coincidence rate at the two detector positions x1 and x2 for both configu-
rations (shown in Figures 2.3 and 2.4) is given by the normally ordered correlation
function [102, 103]
C(x1,x2) = 〈ψ| E(−)(x1)E(−)(x2)E
(+)(x2)E(+)(x1) |ψ〉 ,
=∣∣∣〈0| E(+)(x2)E
(+)(x1) |ψ〉∣∣∣2
(2.5)
where E(+)(x) and E(−)(x) is the positive- and negative- frequency part of the electric-
field operator at position x and |ψ〉 is the biphoton state. In most previous ghost
imaging configurations [23,25,38,39,42,43], a spatially resolving detector is used only
in the reference arm and a bucket detector is used in the object arm. The ghost image
is thus contained in
C(x2) =
∫dx1 C(x1,x2) (2.6)
2.3 Compressive Quantum Imaging: Theory 35
However, unlike ghost imaging setups that employ a spatially resolving detector such
as a charge coupled device (CCD) or raster scanning a single-pixel detector in the
reference arm, bucket detectors are used in both arms in our case. All the object
information is thus contained in the integrated coincidence signal
Cm =
∫dx2 dx1 C(x1,x2) (2.7)
where m denotes the mth measurement, as we will describe in more detail in Sec. 2.4.
2.3.1 Entangled-Photon Compressive Ghost Imaging
The schematic for entangled-photon compressive ghost imaging is given in Fig. 2.3.
The inset shows the unfolded Klyshko picture. Note that the object is imaged onto
the amplitude-only spatial light modulator (SLM). The two-photon amplitude for
setup is given by
〈0| E(+)(x2)E(+)(x1) |ψ〉
=
∫dxs dξ dxi dη dζ h(ζ,x2) Am(ζ) h(η, ζ) L(η)
×h(xi,η)ψ(xs,xi) h(ξ,xs) T (ξ) h(x1, ξ) (2.8)
where
L(x) = exp[−ik/(2f)x2] (2.9)
2.3 Compressive Quantum Imaging: Theory 36
bucket detector
BBO
SLM
f
object
bucket
detector
ζ
d1’
d2”
xs
ξ
η
x2
x1
xi
d2
d2’
d1
d2
d2’ d
1d
1’
objectSLM
BBOf
1
d2’
1
d1+d
2
1
f+ =
d2”
Figure 2.3: Schematics for compressive quantum ghost imaging. The object andspatial light modulator (SLM) planes are conjugate to each other. Inset: the corre-sponding unfolded Klyshko picture.
is the transfer function of the lens and h(x,x′) is the Fresnel free-space propagation
kernel, which under the paraxial approximation is h(x,x′) ∝ exp[ik/(2d)(x − x′)2].
Here f is the focal length of the lens, k = 2π/λ is the wavenumber and d is the longitu-
dinal separation between the x- and x′- planes. The quantity we wish to determine,
the transmission function of the object, is given by T (x) and Am(x) is the two-
dimensional random pattern imprinted onto the amplitude SLM with m = 1, . . . ,M ,
where M is the total number of realizations (measurements). The random pattern
Am(x) used in conjunction with the bucket detector map the spatial information con-
tained in the object transmission function T (x) into a sequence of coincidence signals
Cm
encoded by the different realizations of Am.
In the spontaneous parametric downconversion process (SPDC), the biphoton
2.3 Compressive Quantum Imaging: Theory 37
state can be approximated by ψ(xs,xi) ∝ δ(xs − xi) for a thin nonlinear crystal and
narrow bandpass filters before the detectors [24]. Under such conditions (which are
good approximations in our experiment and many experiments on quantum ghost
imaging) and after substituting the expressions for the Fresnel free-space propagation
kernel and the transfer function of the lens, the two-photon amplitude becomes
〈0| E(+)(x2)E(+)(x1) |ψ〉
=
∫dζ dξAm(ζ) T (ξ) e
ik
2d′′2(ζ−x2)2
eik2[ 1d′1(x1−ξ)2+ 1
d1ξ2]
×∫dη e
ik2[
d1d2(d1+d2)
η2+ 1d′2(ζ2−2η·ζ)]
∫dxs e
ik2[( 1
d1+ 1
d2)x2
s−2( ηd2
+ ξd1
)·xs]
(2.10)
After carrying out the integration over xs and when the thin lens equation 1/(d1 + d2)+
1/d′2 = 1/f is satisfied, the two-photon amplitude further simplifies to
〈0| E(+)(x2)E(+)(x1) |ψ〉
∝∫dξAm(−Mξ) T (ξ) e
ik2(M
2
d′′2+ 1
d1+ 1
d′1+M2
d′2)ξ2
eik
2d′′2(x2
2+2Mξ·x2)e
ik
2d′1(x2
1−2x1·ξ)
(2.11)
The integrated coincidence signal for the setup shown in Fig. 2.3, after integrating
2.3 Compressive Quantum Imaging: Theory 38
over x1 and x2 and using Eqn. 2.11 for the two-photon amplitude becomes
Cm =
∫dx1dx2
∣∣∣⟨0∣∣∣E(+)(x1) E
(+)(x2)∣∣∣ψ⟩∣∣∣
2
∝∫dξ |Am(−Mξ)|2 |T (ξ)|2
∝∑
n
|Am(−Mξn)|2 |T (ξn)|2 (2.12)
where M = d′2/(d1 + d2) is the magnification of the system and the finite size of
the SLM pixel are used, with n = 1, . . . , N , where N is the number of pixels in the
SLM. Note that the object and SLM planes are conjugate to each other and bucket
detectors are used in both arms. The last expression for the integrated coincidence
signal (Eqn. 2.12) can be rewritten in matrix form and will be used in the context of
compressed sensing later.
2.3.2 Single-Photon Single-Pixel Compressive Imaging
The schematic for the heralded single-photon compressive imaging is shown in Fig. 2.4.
In this configuration, similar to the entangled-photon compressive GI setup, biphotons
generated by SPDC are used. However, in this configuration, one of the biphotons, in
arm 2 of Fig. 2.4, is used to herald the presence of a single photon in arm 1. Similar
to the derivation given above in section 2.3.1, the two-photon probability amplitude
is given by
2.3 Compressive Quantum Imaging: Theory 39
〈0| E(+)(x2)E(+)(x1) |ψ〉
=
∫dxs dξ dη dζ dxi h(xi,x2) ψ(xs,xi) h(ξ,xs) T (ξ) h(η, ξ)
×L(η) h(ζ,η) Am(ζ) h(x1, ζ)
∝∫dζ dξAm(ζ) T (ξ) e
ik2( 1d2
+ 12f
)ξ2e
ik2( 12f
+ 1d′1
)ζ2
e− ik
d′1ζ·x2
1
×∫dxs e
ik2( 1d2
+ 1d1
)x2s e
− ikd2
xs·x2 e− ik
d1xs·ξ
∫dη e−
ik2f
(ξ+ζ)·η
(2.13)
bucket
detector
BBO
SLM
f2f
2fobject
bucket detector
ζ
d1
d2
d1’
xs
xi
ξ
η
x2
x1
d2
d1
2f 2f
object SLM
BBO fd
1’
Figure 2.4: Schematics for compressive single-photon single-pixel imaging. The objectis imaged onto the plane of the spatial light modulator (SLM). Inset: the correspond-ing unfolded Klyskho picture.
2.3 Compressive Quantum Imaging: Theory 40
After evaluating the ‘η’, ‘ζ ’, ‘xs’ integrals, the two-photon amplitude simplifies to,
〈0| E(+)(x2)E(+)(x1) |ψ〉
∝∫dξAm(−ξ) T (ξ) e
ik2( 1d1
+ 1f+ 1
d′1)ξ2
eik
d′1ξ·x2
1
× e− ik
2( 1d2
+ 1d1
)−1( 1d2
x2+1d1
ξ)2. (2.14)
The integrated coincidence signal for the setup shown in Fig. 2.4, after evaluating
the integrals over x1 and x2 and using the two-photon amplitude (Eqn. 2.14), is given
by
Cm =
∫dx1dx2
∣∣∣⟨0∣∣∣E(+)(x1) E
(+)(x2)∣∣∣ψ⟩∣∣∣
2
∝∫dξ |Am(−ξ)|2 |T (ξ)|2
∝∑
n
|Am(−ξn)|2 |T (ξn)|2 . (2.15)
Note that the object is imaged onto the SLM with unity magnification and the in-
tegrated coincidence signal for the heralded single-photon compressive imaging setup
becomes proportional to Eqn. 2.12 with M = 1.
The expressions for the integrated coincidence signal for both setups (see Eqns. 2.12
and 2.15) can be re-written in matrix form as C = AT, with Amn ≡ |Am(−xn)|2 and
Tn ≡ |T (xn)|2. Most natural images are sparse when expressed in the proper basis
2.4 Experimental Setup 41
such as that of the discrete cosine transform 1 or the wavelet transform used in JPEG
compression. Suppose the object intensity transmission function T is K-sparse in the
basis Φ, i.e., only K of its coefficients are non-zero. When the measurement matrix A
is taken to be a random matrix (such as a matrix whose entries are independent and
identically Gaussian or Bernoulli distributed), it has been shown that the restricted
isometry property (RIP) is satisfied [65, 69, 70]. Then according to the theory, the
vector T gives the desired result by minimizing ‖ΦTT‖1 subject to the condition
C = AT, in which ‖v‖1 =∑
i |vi| is the ℓ1 norm of v. The error in determining T
is bounded from above if M & O[Klog(N/K)] measurements are used. This number
can be much smaller than that of the Nyquist-Shannon criterion.
2.4 Experimental Setup
We demonstrate entangled photon compressive ghost imaging. The experimental
setup is depicted in Fig. 2.5. A continuous-wave Ar-ion laser, operating at a wave-
length of 363.8 nm, was used to pump a BBO (β-Barium Borate) nonlinear crystal.
1The two-dimensional discrete cosine transform (DCT) of a function f(x, y) is defined as
C(u, v) = α(u)α(v)
N−1∑
x=0
N−1∑
y=0
f(x, y) cos
[π(2x+ 1)u
2N
]cos
[π(2y + 1)v
2N
],
for u, v = 0, 1, . . . , N − 1 and α(u) and α(v) are defined as α(u) =√
1
N (√
2
N ) for u = 0 (u 6= 0)
respectively. The inverse DCT is defined as
f(x, y) =N−1∑
u=0
N−1∑
v=0
α(u)α(v)C(u, v) cos
[π(2x+ 1)u
2N
]cos
[π(2y + 1)v
2N
],
for x, y = 0, 1, . . . , N − 1
2.4 Experimental Setup 42
The BBO was cut for a type-II phase matching angle to generate a pair of orthog-
onally polarized signal and idler photons (degenerate at a wavelength of 727.6 nm)
that propagate collinearly. The pump was then spatially separated from the down-
converted degenerate photons using a UV grade fused silica dispersion prism. A
polarizing beam splitter (PBS1 in Fig. 2.5) was used to send the orthogonally po-
larized photons into the object and reference arms. A phase-only reflective spatial
light modulator (from Boulder Nonlinear: 512× 512 pixels, pixel pitch 15 µm), sand-
wiched between orthogonal polarizers (the two ports of PBS2 in Fig. 2.5), was used to
mimic an amplitude-only SLM. A half-wave plate was used to rotate the polarization
of the photons before impinging on the SLM. The face of the nonlinear crystal is
imaged using a lens (L in Fig. 2.5) of focal length f = 25 cm onto the object and the
amplitude-only SLM with a magnification of 3.
We group the native pixels of the SLM into cells with a size of 4 × 4 pixels, so
that we effectively have an array with N = 128× 128 pixels. We then impress known
but random binary patterns onto the SLM. An example of such two-dimensional
random binary pattern is shown in the inset of Fig. 2.5. We use identically dis-
tributed Bernoulli random variables (with values of 0 or 1) with equal probability.
The photons transmitted through the optical system are coupled into a multimode
fiber and registered by single-photon counting module (SPCM-AQR-14, from Perkin-
Elmer) detectors in both arms. A 10-nm FWHM bandwidth spectral filter (centered
at 727.6 nm) is placed in front of each detector. Coincidence circuitry (with a time
2.5 Image Reconstruction 43
window of 12 ns) was used to measure coincidence events between the avalanche
photodiodes (APDs) in the reference and object arms.
A
BBBO
Ar+ Laser
PBS2
PBS1
objectSLM
HWP
L
Figure 2.5: Setup for entangled photon compressive ghost imaging. PBS, polarizingbeam splitter; SLM, spatial light modulator; L, imaging lens; HWP, half-wave plate;BBO, β-Barium Borate crystal. A and B represent bucket detectors used for coin-cidence measurement. Inset: example of a two-dimensional random binary patternimpressed onto the SLM.
2.5 Image Reconstruction
It is important to point out the data acquisition techniques commonly used in ther-
mal and quantum ghost imaging (GI) setups here before we discuss the process for
the entangled photon compressive GI. In thermal GI, a random speckle pattern in the
reference arm is measured using a spatially resolving array of detectors and is corre-
lated with the bucket signal collected in the object arm. This procedure is repeated
M times and the ghost image is reconstructed using Eqns. 2.3 and 2.4. In quantum
2.5 Image Reconstruction 44
GI, two single-photon sensitive detectors (APDs, for example) are used in the signal
and idler arms due to the inherently weak nature of entangled photon sources. In the
reference arm, a spatially resolving APD is scanned in the transverse plane. A bucket
detector is used in the object arm. For each detector position in the transverse plane
of the reference arm, a coincidence measurement between the two detector signals
results in the ghost image of the object.
The image reconstruction for the entangled compressive ghost imaging is as fol-
lows. A two-dimensional random binary amplitude mask (Am) was sent to the spatial
light modulator (SLM) via a computer and coincidence measurements (Cm) between
the two bucket detectors were performed form = 1, . . . ,M whereM is the total num-
ber of measurements. For each random pattern impressed onto the SLM, coincidence
events were integrated for 9 seconds. In the experiment, we have used two objects
(the logo of the University of Rochester and the Greek letter Ψ) in the test arm of the
ghost imaging setup, as shown in the insets of Fig. 2.6(a) and (b). On average, the
singles counts in the object (reference) arms are: 19.5 k counts/s (25.6 k counts/s)
for the logo of the University of Rochester and 28.3 k counts/s (25.7 k counts/s) for
the Greek letter Ψ, respectively. The coincidence rate is about ∼ 1% of the singles
rate.
Compressive sensing works, i.e. we are able to fully recover a signal (image) from
undersampled measurements, because most signals (images) are sparse under certain
basis transformation (representation). To show the sparsity of the objects used in the
2.5 Image Reconstruction 45
experiment, we have used the two-dimensional discrete cosine transform (2D-DCT)
as a representational basis. As can be seen in Fig. 2.6(c) and (d), the objects are
sparse in the chosen basis (Φ).
(a) (b)
(c) (d)
0
1
2
3
4
Figure 2.6: Experimental image reconstruction using compressive sensing algorithms.Reconstructed ghost image of (a) the Greek letter Ψ and (b) the University ofRochester (UR) logo. The insets show the masks used in the test arm of the ghostimaging setup. (c, d) The absolute value of the calculated two-dimensional discretecosine transforms of the insets in (a) and (b), respectively.
The reconstruction of the object intensity transmission function (T), was accom-
plished by minimizing ‖ΦT‖ℓ1 subject to C = AT using the gradient projection
algorithm [104, the algorithm is described in Appendix A]. Here the ℓ1 norm of v
2.5 Image Reconstruction 46
gives the sum of the absolute value of the elements of v, i.e., ‖v‖1 =∑N
i=1 |vi|. The
results of the reconstruction, for the maximum number of measurements (M = 6300),
are shown in Fig. 2.6(a) and (b). Comparisons with the original masks of the objects
(insets in the same figure) show that we have a good reconstruction.
2000 3000 4000 5000 60000.02
0.04
0.06
0.08
0.1
M
MS
E
Figure 2.7: The calculated mean-squared error of the reconstructed ghost imagesof the logo of the University of Rochester (UR) (●) and the Greek letter Ψ (�) asfunctions of the number of measurements M .
To better quantitatively characterize the fidelity of the compressed sensing image
reconstruction algorithm, we have used the mean-squared error (MSE) as our metric.
The MSE is defined as
MSE =1
N||x− x||22 . (2.16)
Here x is the reconstructed image, x represents the original mask, ‖v‖2 is the Eu-
2.5 Image Reconstruction 47
clidean (ℓ2) norm of v and N is the number of resolution cells, in our case N =
128 × 128. Fig. 2.7 shows the calculated MSE as a function of the number of mea-
surements. As can be seen from the figure, the MSE flattens out for M > 4500 (27%
of the Nyquist limit of 128× 128), with values of 0.06 for the University of Rochester
(UR) logo and 0.03 for the Greek letter Ψ.
We have also characterized our CS results in terms of the signal-to-noise ratio
(SNR). The signal and noise are calculated as the mean intensity of the bright pixels
and the standard deviation of the dark background pixels, respectively [85]. The
maximum SNR we obtained, for the maximum number of measurements M = 6300,
is SNR=8 (SNR=10) for the object mask UR (Ψ) as shown in Fig. 2.8, respectively.
2000 3000 4000 5000 6000
1
3
5
7
9
M
SN
R
Figure 2.8: The calculated signal-to-noise ratio of the reconstructed ghost images ofthe University of Rochester (UR) logo (●) and the Greek letter Ψ (�) as functionsof the number of measurements M .
2.5 Image Reconstruction 48
Here we mention how our entangled-photon compressive ghost imaging recon-
struction results [74] compare with other compressed sensing based reconstructions.
It is important to note here that the conditions under which most of the experimen-
tal results [78, 81, 85–87, 105–111] described below are found are quite different. The
light source (coherent, pseudothermal, quantum), the image acquisition techniques,
the compressed sensing algorithms used, the dimensions (pixel size) of the object, the
number of measurements used for reconstruction, the sparsity level of the object, and
so on, could be different.
In [78, 81, 105], Takhar and co-workers describe the operation of the single-pixel
camera, the first application of the theory of compressed sensing. Duarte et al. [78]
have shown image reconstruction of the letter R (of size N = 256× 256 pixels) using
compressed sensing methods using only M = 1300 random measurements, i.e., using
only 2% of the measurements Nyquist limit (50× sub-Nyquist). They have calculated
values of MSE of 0.01 for image reconstruction of the letter R (of size N = 1282 pixels)
using M = N/10 measurements.
In [85], Katz et al. show image reconstruction using compressive sensing methods
for a pseudothermal ghost imaging setup (similar to Fig. 2.1(b)). For a double-slit
mask, the mean-square error (MSE) is calculated to be 0.05 (0.04) for the image
reconstructed using M = 256 (512) realizations, i.e., using 15% (30%) of the number
of measurements corresponding to the Nyquist limit, respectively. They have also
shown image reconstruction of a grayscale object with an MSE value of 0.005 using
2.5 Image Reconstruction 49
60% of the Nyquist limit.
One other application of compressed sensing (CS) is in magnetic resosance imaging
(MRI) [86, 87, 106–111]. For example, in [108], Ma et al. describe full body MR
image reconstruction using compressed sensing algorithms. For the full body image
of dimensions 924×208 pixels, they show image reconstruction with SNR values of
24.12 for a sampling ratio (M/N) of 38.38%, where M and N are the number of
measurements and the size of the image (number of pixels), respectively. In [107],
Ni et al. show image reconstruction of a 256×256 angiography image and a knee image
with a 320×320 pixel resolution using only 25% of the measurement Nyquist limit
with reconstruction errors of about -45 dB using chirp sensing matrices. In [111],
Huang et al. show results of brain MR image reconstruction using 20% sampling
using different CS algorithms [86, 108, 110, 111]. The maximum calculated value of
the signal-to-noise ratio was 20.35 dB.
We note here that in some cases, the result of our experimental work [74] is
comparable to others [85], where the MSE is about the same. Duarte et al. calculated
values of MSE of 0.01 for image reconstruction of the letter R (of size N = 128× 128
pixels) when using M = N/10 measurements. In our case, the results shown in
Fig. 2.6 (c) and (d) are taken using 38% of the measurements Nyquist limit. We
have used entangled photons for our experiment, while [78, 81, 105] have all used a
coherent light source. On the other hand, image reconstruction with much better
SNR are shown in [107–109].
2.6 Photon Efficiency Comparison 50
2.6 Photon Efficiency Comparison
We compare the performance of our CS procedure with other approaches to image
formation. For our demonstrations, there are 128 × 128 = 1.6 × 104 pixels in the
object. We obtain a very good image using 6300 measurements (see Fig. 2.6) and
a highly acceptable image using only 2000 measurements (see Fig. 2.8). Thus, we
are able to obtain good images while performing far fewer measurements than there
are pixels in the field to be imaged. We did not make any systematic attempt to
minimize the total number of photons used to form the image. It is nonetheless
interesting to examine the photon-efficiency of our CS process. Using the numbers
reported above, we estimate that approximately 1.4 × 107 detected biphotons were
used to obtain either of the images of Fig. 2.6. This number is considerably smaller
than the number required by conventional quantum ghost imaging, in which a point
detector is raster scanned in the reference arm. In this case, assuming a shot noise
limited system, we would need to collect approximately 100 photons per pixel to
achieve a SNR of 10 (the maximum SNR for the University of Rochester logo, see
Fig. 2.8). There are 128× 128 pixels in the image, but for raster scanning we utilize
only 1 part in 128×128 of the emitted photons. Thus, the required number of photons
is 100 × 1284 = 2.6 × 1010, which is three orders of magnitude higher than our CS
approach. Thus the CS method could be preferable in low-light-level, i.e., photon
starved, applications.
2.7 Summary 51
2.7 Summary
We have presented an experimental demonstration of image reconstruction at low
light levels using entangled photons from an SPDC source and using compressive
sensing (CS) algorithms. We have shown that CS can lead to high-resolution images
with a dramatically improved SNR. A relatively higher photon efficiency has also
been inferred as compared to ghost imaging using raster scanning. For the objects
used in the experiment, high-fidelity ghost image reconstruction was achieved using
only 27% of the number of measurements corresponding to the Nyquist limit. In
addition, unlike most ghost imaging (quantum or thermal) experiments where spa-
tially resolving detectors are a requirement, we have used only single-pixel (bucket)
detectors in both the reference and test arms. This could have an important impact
in quantum imaging where photon counting arrays are an expensive and cumbersome
resource and may have applications in secure image transmission [112] and optical
encryption [113].
Chapter 3
Speckle Averaging Effects in
Thermal Ghost Imaging
In chapter 2, we have presented a theoretical and experimental work on the impli-
cations of compressed sensing (CS) to quantum ghost imaging (GI). We have shown
experimentally that, compared to GI based on raster scanning, CS-based quantum
GI has a relatively high photon-efficiency. We have also shown that the need for pho-
ton counting arrays that are expensive and that require intensive electronics could
be mitigated by using a configuration that combines a spatial light modulator and
single-pixel (bucket) detector. The emphasis in the previous chapter has mainly been
on the detection part of a quantum GI setup.
In the present chapter, we study the effect of the field statistics of the illuminating
source on the quality of ghost images. We show theoretically and experimentally that
52
3.1 Introduction 53
a thermal GI setup can produce high quality images even when low-contrast speckle
patterns are used as an illuminating source. We show that as long as the collected
signal is mainly caused by the random fluctuation of the incident speckle field, as
opposed to other noise sources, the quality of the ghost image formed is not degraded
even when the detectors used are so slow that they respond only to the intensity-
averaged speckle patterns [114].
3.1 Introduction
Laser speckle patterns have attracted the attention of the scientific community and
have been studied after the first operation of the cw HeNe laser since the early
1960s [115–118, and references therein]. The term “speckle pattern” conventionally
refers to the intensity distribution produced by the mutual interference of a set of
randomly generated wave fronts, such as those obtained when scattering a coherent
laser beam off a rough surface or from a spatially disordered sample. The statistical
properties of such speckle fields have been studied by many authors [117, 118].
A diverse range of applications, based on the use of speckle phenomena, have
been developed in recent years. Some of these applications include 3D mapping
and range finding [119,120], metrology for biomedical applications [121–123], random
lasers [124], imaging of strongly interacting quantum systems [125], etc. The suc-
cess of many speckle-based technologies and techniques inevitably relies on the facts
that a speckle field can have high spatial and temporal randomness and that the de-
3.1 Introduction 54
tecting devices have enough spatial and temporal resolution to monitor the dynamic
variations of individual speckles [114].
In many circumstances, the contrast ratio is used to quantify the amount of vari-
ation within a speckle pattern. The contrast ratio can be defined as [126]
K =σ(I)
〈I〉 , (3.1)
where σ(I) ≡√
〈I2〉 − 〈I〉2 is the standard deviation of the intensity variation of the
speckles and 〈· · · 〉 denotes either temporal or spatial ensemble average.
If the response time of the detector is slow compared to the variation of the speckle
patterns, i.e., when the detector used in such a system is slow in the sense that its
refresh rate cannot keep up with the temporal variations of the illuminating speckle
field, the effective illuminating field that the system measures essentially becomes the
intensity average (or sum) of multiple mutually uncorrelated speckle patterns. The
contrast ratio of such an intensity-averaged speckle pattern obeys an inverse log-log
relation with the speckle averaging factorM , i.e., the contrast ratio K of such speckle
pattern scales with the speckle averaging factor M as M−1/2 [126]. The quantity M
can be calculated as the ratio between the integration time τd of the detector and the
coherence time τc of the random speckle field, i.e, as M = τd/τc. In other words, M
indicates the number of independent speckle patterns that are averaged together in
each measurement. Due to the reduction of the contrast ratio as M increases, the
performance of many speckle-based metrology and imaging techniques would quickly
3.1 Introduction 55
deteriorate as the integration (response) time of the detector increases [123].
Ghost imaging has recently been performed using speckle fields [23, 25, 39, 43, 55,
57, 127–132]. As it has been described before, ghost imaging is an indirect imaging
method that acquires the image of an object through spatial intensity correlation mea-
surements (see section 1.2). Unlike conventional imaging techniques, ghost imaging
uses a nonspatially resolving bucket detector to collect the optical signal directly from
the object either through reflection or transmission, and therefore it can be advanta-
geous in scenarios where using a detector array is restricted or difficult. We have also
seen, in chapter 2, how using compressive sensing in quantum ghost imaging setup
solves the need for expensive photon counting array of detectors. Ghost imaging also
offers great potential for imaging objects located in optically harsh environments [133]
or for imaging through turbulent and scattering media [134–136].
The use of slow detectors in many speckle-based imaging methods degrades the
image quality thus produced. One might naturally expect that this extends to ghost
imaging systems as well. However, in this chapter we show both theoretically and ex-
perimentally that this is actually not the case and that the image quality of a thermal
ghost imaging system can remain high even though the refresh rate of the detectors
is much slower than the coherence time of the illuminating speckle field, as long as
the fluctuations in the detected signal are due predominantly to the randomness of
the speckle pattern itself and not due to noise in the detection system [114, 130].
This chapter is organized as follows. In section 3.2, I present a detailed theoretical
3.2 Theory 56
study of the effect of the field statistics of the illuminating source on the quality of
ghost image formation. The effect of the speed of the detector on the quality of ghost
images is also studied. The experimental details for the thermal ghost imaging are
given in section 3.3. Analysis of the ghost image formation and the contrast-to-noise
ratio for different speckle averaging factors is carried out in section 3.4. A summary
is presented in section 3.5.
3.2 Theory
As we have described before in the introductory part to ghost imaging (see section 1.2)
and section 2.2, we can form the ghost image by calculating the correlation function
of the background-subtracted object and reference signals. Here a light source with
strong transverse spatial correlation, for example a speckle pattern split using a beam
splitter, propagates in the two arms of the ghost imaging configuration. In the object
arm, the speckle pattern is projected on a transmissive or reflective object, and all the
light that is transmitted or reflected is collected by a bucket detector. In the reference
arm, the speckle pattern is directly collected by a spatially resolving detector, e.g., a
CCD camera. The beam in this arm does not interact with the object. The object
is illuminated with N known random intensity patterns, and for each illuminating
pattern, the total energy that is transmitted through (or reflected from) the object
is recorded.
We now make the assumption that the detected signal is primarily given by the
3.2 Theory 57
random intensity variation of the speckle fields and that other noise sources, e.g.,
detector dark noise, can be neglected [137]. The image can then be acquired using
the following background-subtracted correlation formula:
G(x) =1
N
N∑
n=1
(I(n)B − 1
N
N∑
n=1
I(n)B
)(I(n)(x)− 1
N
N∑
n=1
I(n)(x))
=1
N
N∑
n=1
I(n)B I(n)(x)− 1
N2
N∑
n=1
I(n)B
N∑
n=1
I(n)(x), (3.2)
where x = (x, y) represents the two-dimensional Cartesian coordinates, N denotes
the total number of measurements, and
I(n)B =
∑
x
I(n)(x)T (x), (3.3)
is the total signal collected by the bucket detector for the nth measurement, I(n)(x) is
the intensity of the illuminating field collected by the camera for the nth measurement
at location x, and T (x) is the intensity transmission function of the object. For
simplicity, we here assume that the object has binary transmission; i.e., the value of
T (x) is either zero or unity. Note that the second term in Eqn. 3.2 is the product
of the background from two uncorrelated signals, and by subtracting this term, we
ensure that 〈G(x)〉 = 0 for any pixels where T (x) = 0.
The quality of the ghost image is characterized by the contrast-to-noise ratio
(CNR), i.e., by the ratio of the background-subtracted signal to its noise. For an
3.2 Theory 58
object with binary transmission, the CNR is defined by the following expression [137,
138]:
CNR =〈gin〉 − 〈gout〉√σ2in + σ2
out
, (3.4)
where 〈gin〉 ≡ 〈G(xin)〉 and 〈gout〉 ≡ 〈G(xout)〉 are the ensemble averages for the
ghost image signal at a point xin (inside the transmitting regions of the object) where
T (xin) = 1 and xout (outside the transmitting regions the object) where T (xout) = 0,
respectively. Similarly σ2in and σ2
out are the variances of the signal at xin and xout,
respectively.
We next derive an expression for the CNR of such a ghost imaging system analyt-
ically in terms of the statistical properties of the illuminating speckle field with three
simplified but reasonable assumptions: (1) the intensity of the illuminating patterns
are statistically independent of each other, (2) the intensity at each pixel is indepen-
dent from that at each other pixel, and (3) the detected signal fluctuation is primarily
given by the random intensity variation of the speckle fields, and other noise sources,
e.g., detector dark noise, can be neglected. Note that the second assumption implies
that we can replace the ensemble averages in Eqn. 3.4 with spatial average, and in
fact we use spatial average in our simulation and experiment to calculate the CNR of
the obtained ghost image.
The intensity transmission function of the object we wish to image, represented
by T (x), is assumed to be binary for simplicity, i.e. T (x) = 0 or 1. The object is
also assumed to be pixelated and to transmit at a total of T pixels. The number
3.2 Theory 59
of transmitting pixels T is generally calculated as the ratio of the transparent area
of the object to the speckle size of the illuminating field. The object is illuminated
with N known random intensity patterns given by I(n)(x) for n = 1, . . . , N . For each
illuminating pattern, the total energy transmitted through the object is recorded,
denoted by I(n)B .
Using the first assumption, i.e., the statistical independence of the intensity of the
illuminating patterns from each other, we calculate the expected imaging signal as
〈G(x)〉 = 〈 1N
N∑
n=1
I(n)B I(n)(x)− 1
N2
N∑
n=1
N∑
m=1
I(n)B I(m)(x)〉
= 〈 1N
N∑
n=1
I(n)B I(n)(x)− 1
N2
N∑
n=1
I(n)B I(n)(x)− 1
N2
N∑
n=1
N∑
m=1m6=n
I(n)B I(m)(x)〉
= 〈IBI(x)〉 −1
N〈IBI(x)〉 −
N(N − 1)
N2〈IB〉〈I(x)〉
〈G(x)〉 =N − 1
N
[〈IIB〉 − 〈I〉〈IB〉
], (3.5)
where we have introduced the shorthand I ≡ I(x) in the last line. Similarly, the
variance is given, after a lengthy calculation, by
σ2(x) =1
N
(N − 1
N
)2
〈I2I2B〉+(N − 1)(N − 2)
N3
[〈I2〉〈IB〉2 + 〈I〉2〈I2B〉 − 〈IIB〉2
]
− 21
N
(N − 1
N
)2 [〈I2IB〉〈IB〉+ 〈II2B〉〈I〉
]+ 2
(N − 1)(3N − 4)
N3〈IIB〉〈I〉〈IB〉
+N − 1
N3〈I2〉〈I2B〉 − 2
(2N − 3)(N − 1)
N3〈I〉2〈IB〉2. (3.6)
We can further simplify the equations for the expected imaging signal (Eqn. 3.5)
3.2 Theory 60
and the variance (Eqn. 3.6) by using the second assumption that the intensity at each
pixel is independent from one another by substituting in the following relations for
the various expectation values [114, 137]
〈I〉 = µ1, (3.7)
〈IB〉 = Tµ1, (3.8)
〈I2〉 = µ2, (3.9)
〈I2B〉 = T (T − 1)µ21 + Tµ2, (3.10)
〈IIB〉 = (T − T (x))µ21 + T (x)µ2, (3.11)
〈I2IB〉 = (µ3 − µ2µ1)T (x) + Tµ2µ1, (3.12)
〈II2B〉 =[µ3 − µ2µ1 + 2µ1(T − 1)(µ2 − µ2
1)]T (x) + Tµ2µ1
+ T (T − 1)µ31, (3.13)
〈I2I2B〉 =[µ4 − µ2
2 + 2µ1(T − 1)(µ3 − µ2µ1)]T (x)
+ Tµ22 + T (T − 1)µ2µ
21, (3.14)
where µr ≡ 〈Ir〉 is the r-th moment of the intensity probability distribution of each
illuminating pattern.
We see that, using Eqns. 3.7, 3.8 and 3.11, the expected imaging signal (Eqn. 3.5)
simplifies to
〈G(x)〉 =(N − 1
N
)(µ2 − µ2
1) T (x). (3.15)
3.2 Theory 61
The background-subtracted signal in the numerator of the CNR (Eqn. 3.4), after
using Eqns. 3.7– 3.14, is given by,
〈gin〉 − 〈gout〉 =(N − 1
N
)σ2I , (3.16)
where we have defined σ2I ≡ µ2 − µ2
1 as the variance of the intensities in each illumi-
nating pattern. Similarly we find the noise-squared (the denominator of the CNR) is
given by
σ2in + σ2
out =
(N − 1
N2
)[2Tσ4
I + (5− 6/N)(2µ2µ
21 − µ4
1
)
− (2− 3/N)µ22 − (1− 1/N) (4µ3µ1 − µ4)
](3.17)
=
(N − 1
N2
)[(2T − 2 + 3/N)σ4
I + (1− 1/N)γ4I
], (3.18)
where γ4I ≡ 〈(I−〈I〉)4〉 = µ4−4µ3µ1+6µ2µ21−3µ4
1 is the fourth-order moment about
the mean of each illuminating pattern.
Substituting these relations in Eqn. 3.4, the CNR is given by the remarkably
simple formula
CNR =
[N − 1
(2T − 2 + 3/N) + (1− 1/N) (γI/σI)4
]1/2. (3.19)
Here the total number of transmitting pixels T is generally given as the ratio of the
total transmitting area of the object and the average speckle size (spatial coherence
3.2 Theory 62
area) of the illuminating speckle pattern [137, 138]. Note that the second and fourth
moments about the mean of the intensity fluctuation for each illuminating speckle
field are given by σ2I ≡ 〈I2〉 − 〈I〉2 and γ4I ≡ 〈(I − 〈I〉)4〉, respectively, and we use
here the shorthand I ≡ I(x).
From the expression for the contrast-to-noise ratio (Eqn. 3.19), we see that the
CNR is determined by the number of measurements N , number of transmitting pixels
T , and the quantity (γI/σI)4, also known as the fourth standardized moment, or the
kurtosis, of the intensity distribution of the illuminating speckle fields. Note that
Eqn. 3.19 is valid for any illuminating field with arbitrary statistical properties, as long
as the pixel intensities are statistically independent. This expression for CNR also
indicates that the image quality of a thermal ghost imaging system is affected by the
fourth standardized moment (γI/σI)4 of the intensity fluctuation of the illuminating
field, rather than by the contrast ratio K = σI/〈I〉 of a speckle pattern as in many
conventional speckle-based methods.
In the usual limit of large N , this reduces further to
CNRN→∞ ∼[
N
2(T − 1) + (γI/σI)4
]1/2. (3.20)
3.2.1 Fast Detection Speed
When the detectors are fast enough to record individual speckle patterns (M = 1),
which obey negative exponential intensity statistics [126], we can easily show that
3.2 Theory 63
γ4I = 9〈I〉4 and σI = 〈I〉2. Consequently, the CNR of the ghost image with background
subtraction is given by [137]
CNR =
[N − 1
2T + 7− 6/N
]1/2. (3.21)
In the usual limit of large N and T ≫ 1, the CNR ∼ [N/(2T )]1/2.
3.2.2 Slow Detection Speed
The derivation given above for the contrast-to-noise ratio (CNR) is for the conven-
tional thermal ghost imaging (M = 1) method, whereby correlations are done between
the intensity distribution in the reference arm and the bucket signal for each real-
ization. Now we consider the case where we do collective frame averaging (M 6= 1)
before the correlations are carried out to form the ghost image, thereby simulating
slow detection speed.
When the detectors are slow, the ghost imaging system responds only to the inten-
sity average of M independent speckle patterns for each measurement. In such cases,
the ghost image can still be expressed using Eqn. 3.2, but with the expressions for the
bucket detector signal and the spatially resolved camera signal modified to take into
account the intensity sum of M independent speckle patterns for each measurement,
specifically,
I(n)B,M =
M∑
m=1
I(n,m)B , (3.22)
3.2 Theory 64
and
I(n)M (x) =
M∑
m=1
I(n,m)(x). (3.23)
Following Goodman [139–142], the sum of M speckle patterns follows a gamma
probability density function (PDF), i.e.,
p(x) =1
Γ(M)µMxM−1e−x/µ (3.24)
where µ is the mean intensity of a speckle pattern and Γ denotes the Gamma function.
The nth moment for such a distribution is
〈xn〉 =
∫xnp(x) dx
=1
Γ(M)µM
∫xn+M−1 e−x/µ dx
=µn
Γ(M)Γ(n+M)
where we have used the definition for the Gamma function.1
Using straightforward mathematics, one can show that the fourth-order moment
about the mean γ4IM and the variance σ2IM
of the intensity of the effective illuminating
speckle field are given by
γ4IM = 3(M + 2)M〈I〉4 = 3(M + 2)
M3〈IM〉4, (3.25)
1The Gamma function is defined as Γ(z) =∫∞
0e−t tz−1 dz
3.2 Theory 65
and
σ2IM
=M〈I〉2 = 1
M〈IM〉2 (3.26)
where 〈I〉 and 〈IM〉 are the expected values of the intensity for each independent
speckle pattern and the intensity-averaged speckle field at any pixel, respectively.
The ratio of the fourth moment (γIM ) to the variance (σIM ) is thus given by
(γIM/σIM )4 = 3(1 + 2/M). (3.27)
Expressions for the fourth and second order moments about the mean are given
explicitly in Eqns. 3.25 and 3.26, respectively. The dependence of speckle contrast
ratio K and kurtosis as a function of the speckle averaging factor M is shown graph-
ically in Fig. 3.1. Substituting Eqn. 3.27 into Eqn. 3.19 gives the general expression
for the CNR as a function of the collective frames (M), the number of transmitting
pixels (T ) and the number of measurements (N).
Substituting these relations into Eqn. 3.19, one can obtain the following remark-
ably simple expression for the CNR of a ghost imaging system that only responds to
intensity-averaged speckle fields:
CNR =
[N − 1
2T + 1 + 6/M − 6/(MN)
]1/2. (3.28)
We see that the CNR is very weakly dependent on M . The above expression for
the CNR shows that, even though the contrast ratio of the effective speckle fields
3.2 Theory 66
3
6
9
100 101 102 103
collective frame number M
100
100
10-4
101 102 103
collective frame number M
(a) (b)
Figure 3.1: Normalized second- and fourth-order moments about the mean (a) andkurtosis (γIM/σIM )4 (b) as functions of the speckle averaging factorM . Here the linesare the theory [cf. Eqns. 3.25 and 3.26], and symbols are the calculated results fromone typical numerical simulation realization [83, 118].
“seen” by a ghost imaging system decreases rapidly as the detectors become slow,
the quality of the ghost image actually remains approximately the same as long as
the transmitting area of the object is much larger than the spatial coherence area
of the individual speckle fields. This surprising result comes from the fact that the
quantity that affects the image quality of a ghost imaging system is the kurtosis of
the intensity fluctuation of the illuminating speckle field, which actually converges to
a constant value of 3 as M becomes larger than 10 (see Eqn. 3.27 and Fig. 3.1(b)).
Furthermore, in most practical situations, the transmitting area of the object is much
larger than the coherence area of individual speckle field, i.e., T ≫ 1. In such cases,
the image quality becomes essentially independent of the speckle averaging factor
M [cf. Eqns. 3.19 and 3.28]. Note that Eqn. 3.19 is a generalized result for thermal
ghost imaging systems using illuminating fields having arbitrary statistical properties,
3.3 Experimental Setup 67
however Eqn. 3.28 is a special case of this result for an illuminating field in the form
of the intensity sum of multiple speckle patterns.
3.3 Experimental Setup
Our thermal ghost imaging system is illustrated in Figure 3.2. A collimated HeNe
laser beam is used to illuminate a phase-only spatial light modulator (SLM, from
Boulder Nonlinear), which is programmed to impose a uniformly distributed random-
phase distribution onto the incident beam [83]. The first-order diffracted beam forms
speckle patterns with negative exponential intensity distributions [118] at the focal
plane of a Fourier lens and is used as the illuminating source. The generated illumi-
nating field is then projected, using an imaging lens and a beam splitter, onto the
object and reference planes. In the object plane, the speckle pattern is projected
onto a transmissive object, and all the transmitted light is collected by a large-area
bucket detector (Newport powermeter) placed behind the object. In the reference
arm, the intensity distribution of the illuminating speckle field is directly collected
by a detector array, a camera in our case. By using many uncorrelated speckle pat-
terns and correlating the signals collected by the bucket detector and the camera, one
can obtain a ghost image of the object using the background-subtracted correlation
function given by Eqn. 3.2.
3.4 Experimental Results 68
Laser
SLMFourier Lens
Aperture
Imaging
Lens
BS
CCD
Bucket
DetectorObject
beam-
expander
Figure 3.2: Schematics of our thermal ghost imaging setup. The spatial light modu-lator (SLM) is used to impress a sequence of random phase distribution on the laserfield. BS: beam splitter; CCD: charge coupled device
3.4 Experimental Results
In our experiment, our object is a double slit mask, whose transmitting area is ap-
proximately 200 times the average speckle size of each independent speckle field, i.e.,
T ≈ 200. The CCD and bucket signals are averaged for M uncorrelated speckle
patterns before the two signals are correlated using Eqn. 3.2 to mimic the use of slow
detectors that respond to the average of M independent speckle patterns. Note that,
for M = 1, our system reduces to a conventional ghost imaging system in which the
detectors respond to each independent speckle pattern.
We make the measurements for speckle averaging factor M equal to 1, 5, 15,
and 25, respectively, to study quantitatively the effect of the response time of the
detection system on the quality of the “ghost image” that we obtain. For each value
3.4 Experimental Results 69
(a) (b)
(c)
normalized intensity0 1
(f)(e)
500 µm
(d)
Figure 3.3: (a) Representative speckle pattern of the sort used in our experiments and(b) the intensity average of 25 patterns of the sort shown in (a). The statistics of thetwo patterns are very different, as described in the text. Nonetheless, ghost imagesobtained under the two conditions are essentially identical. (c) A ghost image of adouble slit mask (1.2 mm long, 100 µm wide, and with 40 µm gap in between) takenusing individual speckles and (d-f) a ghost image taken using the intensity average ofM = 5, 15 and 25 individual speckle patterns, respectively. In each case, N = 10 000measurements were used to obtain the ghost image.
3.4 Experimental Results 70
of M , we take 10 000 effective measurements. Figures 3.3(a) and 3.3(b) show two
typical illumination patterns recorded by the CCD camera for M = 1 and M = 25,
respectively. The speckle contrast ratio K for the two cases is 1 and 0.2, respectively.
The ghost images after 10 000 effective measurements for M = 1 is as shown in
Fig. 3.3(c). Similar ghost images for M = 5, 15 and 25 are shown in Figs. 3.3(d-f),
respectively. We can see that there is no obvious difference in image quality, which
is consistent with our theoretical prediction.
1000 3000 5000 7000 9000
1.5
2.5
3.5
CN
R
Number of Measurements N
M = 1
M = 5
M = 15
M = 25
Experiment Simulation
Figure 3.4: CNR as a function of the number of measurements N for the ghostimaging system that responds to different numbers M of averaged speckle patternsfor each measurement. Here the symbols are experimental results, and the lines aresimulation realizations.
To better demonstrate our theoretical predictions, we plot in Fig. 3.4 (as symbols)
3.5 Summary 71
our measured CNR of the ghost image as a function of the number of measurement
N for four different values of the speckle averaging factorM . It can be seen that even
though the response time of the detectors in the four cases is very different, there
is no obvious difference in the resulting image quality. Also shown in Fig. 3.4 (as
lines) are the results of numerical simulation. The agreement between simulation and
laboratory measurement is very good. The slight disagreement may be due to other
noise sources (such as camera dark noise) that are not considered in the simulation.
Note that in our experiment, the coherence area of the speckle field is approximately
100 pixels, whereas in our model we assumed that each pixel experienced independent
intensity fluctuations. However, we have performed extensive numerical simulations,
such as those reported in Fig. 3.4, which show that the predictions of our model are
not influenced by the average speckle size with respect the pixel size of the camera.
3.5 Summary
In this chapter, we have presented a theoretical analysis with experimental demon-
stration which shows that the image quality of a thermal ghost imaging system is
essentially independent of the response time of the detectors as compared to the co-
herence time of the illuminating speckle fields. This surprising result arises from the
fact that the image quality of a ghost imaging system is actually only weakly de-
pendent on the kurtosis of the intensity fluctuation of the illuminating speckle field
that the detectors respond to. As the detecting system becomes slow and sees only
3.5 Summary 72
an average of multiple speckle patterns, the contrast ratio K of the effective speckle
field decreases monotonically, but the kurtosis actually converges to a value of 3 for
thermal light (for M & 10). Consequently, the quality of the ghost image is almost
not affected by the detector speed as long as all nonspeckle noise, such as the detector
dark current noise, is small compared to the fluctuation of the averaged speckle fields.
While most thermal ghost imaging systems demonstrated to date have used pseu-
dothermal light whose coherence time can be controlled to match the speed of the
detectors, the possibility of performing ghost imaging with true thermal light has
always been considered intriguing and highly desirable [55,143]. The work presented
here shows that there need not be any blurring of the final image even when the
detection system is much slower than the coherence time of the thermal light source,
as long as the illumination is strong enough that shot noise and detector noise can
be neglected [114]. This result opens up the possibility of using slow detectors for
thermal ghost imaging with quickly varying thermal speckle fields and may shed light
on other applications using speckle fields as well.
Chapter 4
Discriminating Orthogonal
Single-Photon Images
4.1 Introduction
Quantum state discrimination (QSD) deals with finding optimal measurement schemes
(procedures) that determine the state of a quantum system [144–146]. In quantum in-
formation theory and quantum computing, the information is encoded in the state of
the quantum system. If, for example, a quantum system constitutes two nonorthogo-
nal states, distinguishing the states with certainty is not possible [147]. Determining
the exact state of a quantum system plays important roles in many quantum protocols
and QSD plays an important task in cryptography, in communication applications
and possibly in ultra-low light level image recognition [148].
Quantum state discrimination deals with the following problem. Suppose a quan-
73
4.1 Introduction 74
tum system is prepared in one of many nonorthogonal states given by {ρi}, with a
priori probabilities {pi}. Since exact determination of the actual state of the sys-
tem is generally not possible (unless the states are mutually orthogonal), what is the
best measurement scheme that leads to the determination of the initial state that
the quantum system was prepared in? Depending on the figure of merit that one
chooses to optimize, there are three different strategies (approaches) used for optimal
discrimination of quantum states.
The first scheme is called discrimination with minimum error [149,150]. Here the
probability of making an error in identifying the quantum state is minimized and
each possible outcome indicates some corresponding state. This was experimentally
demonstrated for two non-orthogonal polarization states at the Helstrom bound by
Barnett and Riis [151] and more states by [152–154].
Unlike in the first scheme, if we allow for the possibility of obtaining inconclusive
results, error-free discrimination is possible. This constitutes the second scheme and is
called unambiguous discrimination and is possible only when the allowed states are all
linearly independent [155–157]. Experimental verification of error-free measurement
of polarization for two non-orthogonal states was carried out by Huttner et al. [158,
159] and more states by [160].
The third scheme is a generalization of the unambiguous discrimination strategy
and allows for the possibility of linear dependence between the states. Due to this
possibility, there will always be errors associated when determining some states. The
4.1 Introduction 75
maximum confidence measurement strategy was introduced by Croke et al. [161]
and the scheme maximizes the confidence with which one identifies a given state
ρi. Here the confidence is defined as the the ratio between the number of correct
detection events and the total number of detection events when the outcome i is
detected [145,161]. Experimental verification of maximum confidence quantum state
discrimination has been shown by Mosley et al. [162] and Croke et al. [163].
Quantum state discrimination between two (or more) known orthogonal quantum
states has been performed experimentally for nearly all types of quantum states except
for images. Single photon image state discrimination is particularly difficult because
most single photon detector arrays are poorly suited to single photon image detection.
In this chapter we present results of an experiment demonstrating the quantum state
discrimination between two known single photon images using holographic methods.
Many protocols for encoding information onto an optical beam limit the infor-
mation content of an individual photon to one bit of information in classical optical
communication systems, or to one qubit of information for quantum information pro-
cessing protocols [164]. However, recent work has emphasized the vast Hilbert space
and thus the vast potential information content of a single photon. The use of the
orbital angular momentum states, such as Laguerre-Gauss (LG) states, of the photon
is one such example [165–167]. Since these states form an infinite basis, in principle
there is no limit to the information content that can be carried by a single photon.
Spatially encoded qudits [168], hyperentangled photon pairs [169] and multiphoton
4.1 Introduction 76
entanglement [170, 171] are such examples of the large information content of quan-
tum light field. Entanglement of a large number of photons, of the order of 100,
has been demonstrated recently [172]. Other examples include entanglement between
two photons generated by the process of parametric down conversion that can exist
in a high-dimensional Hilbert space; pixel entanglement between two optically en-
tangled d = 3 and d = 6 qudits was demonstrated experimentally using transverse
position-momentum entangled biphotons [173] and large-alphabet quantum key dis-
tribution using energy-time entangled biphotons with dimensionality d = 1024 was
experimentally demonstrated [174].
In the previous two chapters, the emphasis was on (ghost) image (GI) formation.
We have studied the use of compressive sensing in quantum GI in chapter 2 and the
effects of the statistics of the illuminating field on thermal GI quality in chapter 3.
The study in this chapter is concerned not with image formation, but with image
discrimination at low light levels. Here we describe an experimental procedure that
we have used to impress transverse image information onto an individual photon. If
the image is a member of a predefined basis set, we can determine which image is car-
ried by the photon by performing a single measure using holographic matched filtering
techniques. We should note here that Mair et al. [165] have used simplex holograms to
measure single-photon orbital angular momentum states. In this chapter we demon-
strate that arbitrary image states from a known basis set may be distinguished by
performing a single measure using a multiplexed hologram. This procedure should
4.2 Theory 77
be contrasted with the earlier work of [175], in which an image was impressed upon
a single photon, but the image was read out in a statistical fashion, one pixel at a
time, and thus required an ensemble of events to reproduce the image.
In this chapter, we present results from a proof-of-principle experiment which dis-
tinguishes between a basis set of two orthogonal images [164]. Quantum GI discrim-
ination for upto four spatially nonoverlapping objects have been recently carried out
by Malik et al. [176]. However, much larger sets of images (orthogonal or nonorthogo-
nal) can be used. In the classical field of matched filtering, as many as 10 000 images
have been fixed onto a single large-scale holographic memory [177].
This chapter is organized as follows. In section 4.2 we present the basic idea
of our single-photon image discrimination. The experimental details are given in
section 4.3 and the results on hologram characterization and single-photon image
discrimination are presented in sections 4.4 and 4.5, respectively. A summary is
presented in section 4.6.
4.2 Theory
Single photon image state discrimination was carried out using holographic methods.
The basic concept of our approach is illustrated in Fig. 4.1. A multiple-exposure
hologram was formed (parts a-d) using N different transmission objects (Ai) with
reference beams applied from N different directions (θi’s in the figure). We then
pass a single photon through one of the N objects as shown in part (e) and allow it
4.2 Theory 78
to fall onto the multiplexed-hologram constructed by the procedure shown in parts
(a)-(d). This photon will then diffract into one of N output directions depending
upon which image was impressed onto the photon. This procedure is well known in
optical information processing [72, 178] and is related to the more general method of
matched filtering [179] (see Sec. 1.4). The general methods used in classical image
discrimination apply equally well to quantum-mechanical light fields.
Let us first describe image reconstruction for a simplex hologram, i.e., a holo-
graphic material exposed by a single pair of object and reference beams. We assume
that the holographic recording material is illuminated simultaneously by an object
wave of field strength T (x) and a reference wave of field strength R(x) so that the
total field at the hologram is E(x) = T (x) + R(x), where x = (x, y) represents the
two-dimensional Cartesian coordinates. We assume that after development, the trans-
mission t(x) of the hologram is proportional to the local optical intensity [180, 181]
so that t(x) ∝ |T (x) +R(x)|2 or that
t(x) ∝ |T (x)|2 + |R(x)|2 + T (x)R∗(x) + T ∗(x)R(x). (4.1)
During the reconstruction process in the conventional holography [180, 181], the
hologram is illuminated with a wave identical to the reference wave R(x) used in
recording the hologram. The field leaving the hologram is thus given by Eout(x) ∝
4.2 Theory 79
object Ai hologram
unknown
image
single
photon
output wave
(if image is )
object A1
holographic
plate
reference
(plane wave)
reference
(plane wave)
holographic
plate
(a) exposure 1 (b) exposure 2
(e) single-photon readout
holographic
plate
reference
(plane wave)
reference
(plane wave)
holographic
plate
(c) exposure N-1 (d) exposure N
.
.
.
.
.
.
θ1
object A2
θ2
object AN-1 object A
N
θN-1
θN
A2
output wave
(if image is ) A1
output wave (if image is ) AN
Figure 4.1: Concept of the single photon image discrimination experiment. Amultiple-exposure hologram was formed using N transmission objects (Ai) and refer-ence beams applied from different directions (θi), as shown in parts (a)–(d). (e) Afterthe hologram was developed, an unknown image was impressed onto a single photonby passing a heralded single photon through an object (Ai) from the predefined set.The form of the image is determined by diffraction from the multiplexed hologramformed in parts (a)–(d). Depending on the image carried by the single photon, thehologram diffracts the image into the corresponding unique direction.
R(x)t(x) or by
R(x)|T (x)|2 +R(x)|R(x)|2 + T (x)|R(x)|2 + T ∗(x)R2(x). (4.2)
The third term in this expression is the one leading to standard holographic re-
4.2 Theory 80
construction, and if R(x) is nearly uniform across the aperture of the hologram we
see that this term just reproduces the amplitude distribution T (x) of the original
object. If, however, the hologram is illuminated (read) by a replica of the structured
object beam T (x), as in the case of holographic matched filtering, the situation is
more complicated. We find that Eout(x) ∝ T (x)t(x) or by
T (x)|T (x)|2 +R(x)|R(x)|2 + T (x)2R∗(x) + |T (x)|2R(x). (4.3)
In this case, the fourth term is the one leading to the diffracted output beam, and
we see that, due to the structured nature of the read beam, the transverse structure
of the object beam will be imprinted onto the diffracted beam.
We next consider a multiplexed hologram created by exposing N object-reference
field pairs sequentially. Let the ith reference field (assumed to be plane waves) be
represented by Ri(x) = R exp[iki ·x] and the object field as Ti(x) = Ai(x) exp[iφi(x)].
Here Ai(x) and φi(x) are the transverse amplitude and phase functions of the ith
object field at the hologram plane, respectively. Without loss of generality, we assume
that R, Ai, and φi are all real. Similar to Eqn. 4.1 for the simplex hologram, the
transmission function of the multiplexed hologram is approximated by
t(x) ∝N∑
i=1
|Ti(x) +Ri(x)|2
∝N∑
i=1
RAi(x) {exp[i(φi(x) + ki · x)] + c.c}+ Ai(x)2 +R2 (4.4)
4.3 Experimental Details 81
After the hologram was exposed and developed, if the jth object beam is incident
onto the hologram, the amplitude of the image diffracted into the ki-direction is given
by
Ej(x) ∝ Tj(x) t(x)
∝N∑
i=1
RAj(x)Ai(x) exp[i(φj(x) + φi(x) + ki · x)]. (4.5)
If we now assume zero local spatial overlap between the illuminating and the ith object
beams, i.e., Aj(x)Ai(x) = δij A2i (x), then the amplitude of the image diffracted into
the desired k-direction becomes
Ej(x) ∝ A2j(x) exp[i(φj(x) + kj · x)], (4.6)
and the multiplexed hologram projects the incident light from each image Aj(x) into
the corresponding kj-direction and each image can be uniquely detected.
4.3 Experimental Details
The experimental setup used for writing the multiplexed (biplexed, to be more spe-
cific) hologram is shown schematically in Fig. 4.2. We have used two spatially non-
overlapping transmission masks as our objects, i.e, we use stencils of yin and yang
symbol as objects A1 and A2, respectively (represented as A and B in the inset of
Fig. 4.2). Objects A and B have no spatial overlap and as such constitute orthogonal
4.3 Experimental Details 82
objects. The hologram is a thick angularly multiplexed phase transmission holo-
gram and is made using PFG-01, a fine-grained red-sensitive silver halide emulsion
on a glass plate substrate [182]. The emulsion has a peak light sensitivity of about
100 µJ /cm2 at 630 nm.
A B
HeNeLaser
Timer/Shutter
Hologram
Object
object
rotationstage
NPBS
Figure 4.2: Laboratory setup for writing the multiplexed hologram. Biplex hologramsare exposed using a HeNe laser and a pair of object-reference beam combinationssequentially. A shutter is used to electronically control the exposure time. For eachexposure, the reference-object pair is selected using a rotation stage and a translationstage. NPBS, nonpolarizing beamsplitter
Since the holographic material has a peak sensitivity in the red, we use a HeNe
laser (λ =632 nm) as the light source for recording the holograms. A nonpolarizing
beamsplitter (NPBS) is used to split the HeNe laser into object and reference beams.
Each beam has a power of ∼ 300µW after the NPBS. The object beam passes through
the object stencil (A or B), which are mounted on a translation stage and is imaged
onto the hologram recording medium with a 50 mm focal length lens, along with one
of the two reference beams. A precision translation stage allows us to reproducibly
4.3 Experimental Details 83
place either object A or B in the object plane. A mirror mounted on a rotation stage,
in the other port of the NPBS, sets the direction of the reference beam A (B) for each
exposure. The biplexed hologram is then formed by illuminating the hologram by the
object-reference pairs (A-A and B-B) sequentially. The exposure time for each pair
is set at 350 ms by an electronically controlled shutter.
TCSPC
A
Hologram
B
HeCdLaser
BiBO
Trigger
Object
A B
NPBS
Figure 4.3: Laboratory setup for the single-photon image readout. Heralded sin-gle photons are sent through either object A or B, during the image-discriminationphase of the experiment, and are then detected at either detector A or B. TCSPC,time-correlated single-photon counter; BiBO, Bismuth Borate crystal; NPBS, nonpo-larizing beamsplitter
The setup used for the single-photon readout (image-discrimination) phase of the
experiment is shown schematically in Fig. 4.3. A HeCd laser operating at λ=325 nm
is used to pump a 10-mm-long nonlinear crystal (BiBO) cut for collinear type-I phase
matching. The nonlinear cyrstal is angle-tuned to produce, through the process of
spontaneous parametric downconversion (SPDC), degenerate biphotons of the same
4.4 Hologram Characterization 84
polarization at 650 nm. A UV grade fused silica dispersion prism is used to separate
the biphotons from the pump beam and are sent to a nonpolarizing beamsplitter
(NPBS). The object beam is created by heralding single photons emitted by the
SPDC process. That is, one output port of the NPBS is coupled directly through a
multimode optical fiber to a single-photon sensitive detector which serves as a trigger.
The presence of a photon in the other output port of the NPBS, hereafter called the
image photon, is heralded by the trigger. The image photon passes through the
object transmission mask (A or B) and the biplexed hologram along the same path
as when the hologram was exposed. Depending on the image carried by the incoming
photons, the image photons are diffracted from the hologram and are coupled through
multimode fibers to detectors A or B. All three detectors used in the experiment are
Perkin-Elmer single-photon counting module (SPCM) detectors. Detection events
are counted with a PicoQuant PicoHarp 300, a time-correlated single-photon counter
(TCSPC).
4.4 Hologram Characterization
We have characterized the performance of the biplex holograms using a HeNe beam
used to write the hologram. The quality of the reconstructed images when the holo-
gram is read out by a plane-wave reference beams, as in conventional holography, is
shown in parts (a) and (b) of Fig. 4.4. Here we see that the reconstructed images are
accurate replicas of the stencil objects (yin and yang symbols). For the single-photon
4.5 Single-Photon Image Discrimination 85
readout phase of the experiment, parts (a) and (b) of Fig. 4.5 show the diffracted
(a)
(b)hologram
read beam
(plane wave)
read beam
(plane wave)reconstructed
image
reconstructed
image
Figure 4.4: Object reconstruction using a plane wave read beam.
beams when the hologram is illuminated by one of the image-bearing beams.
As we have described in Sec. 4.2, when the hologram is read out using the image-
bearing (object) beams, the transverse structure of the object beam will be imprinted
onto the diffracted beam. This behavior is apparent in the data shown in parts (a)
and (b) of Fig. 4.5. Quantitatively, we have measured a peak diffraction efficiency of
about 24% (19%) for objects A (B), and we find that the cross talk between them is
negligible.
4.5 Single-Photon Image Discrimination
As we have described in Sec. 4.2, when stencil A (B) is used as the object, the image
discrimination photons are diffracted by the hologram into the direction of reference
beam A (B), where single-photon sensitive detectors A (B) are located (see Fig. 4.3).
4.5 Single-Photon Image Discrimination 86
(a)
(b)hologram
input
output
input output
Figure 4.5: Hologram readout with an image carrying read beam.
Coincidence events between the heralding and the image-discrimination photons are
measured for the four object-detector combinations: (1) A-A, (2) A-B, (3) B-A, and
(4) B-B. The total number of coincidences for each object-detector combination for
54 min of integration are reported in Table 4.1. The singles rate (for the heralding
and image-discrimination photons) and the accidental coincidences are also included
in Table 4.1. For better visualization, these results (raw and accidental coincidences
and their ratio) are also shown graphically in Fig. 4.6.
The singles rate, s, for the ith channel, i.e., the heralding and image-discrimination
channels, is given by,
si = ǫiR + bi, (4.7)
where ǫi is the collection and detection efficiency of the ith channel, R is the pair
generation rate of the SPDC process and bi is the background counts of the ith
channel, which includes detector dark counts and counts from stray light measured
4.5 Single-Photon Image Discrimination 87
with the pump beam blocked.
Raw coincidences are the number of coincident events generated within the 500 ps
coincidence window. The raw coincidence C is the sum of real and accidental co-
incident events. Coincident events arising from coincidences between the herald
and image-discrimination photon pairs are real coincidences and the rate is given
by ǫhǫidR, where ǫid(ǫh) are the collection and detection efficiency of the image-
discrimination (heralding) channels, respectively. Accidental (or spurious) coincident
events arise between (a) a background count and a heralding photon (ǫhbidR∆t),
where ∆t is the coincidence window, (b) a background count and an image-discrimination
photon (ǫidbhR∆t), (c) two background counts (bhbid∆t), and (d) two uncorrelated
photons from a multipair event (ǫhǫidR2∆t). The accidental rate A is thus given by
the sum of the above four coincident events as
A = shsid∆t
= ǫhbidR∆t + ǫidbhR∆t + bhbid∆t + ǫhǫidR2∆t. (4.8)
The number of accidental coincidences arising from events (a–d) given above can
be measured by counting the number of image-discrimination photons that arrive
20±0.25 ns after the heralding photons. The total number of raw coincidences C is
given by
C = ǫhǫidR + A. (4.9)
4.5 Single-Photon Image Discrimination 88
The single-event count rates are sh ∼500 k counts/s for the trigger (in the herald-
ing arm) and sid ∼450 counts/s (sid ∼250 counts/s) for detector A (B), in the image
discrimination arm (see Table 4.1). In practice, the high degree of loss in the image
discrimination arm (sid ≪ sh) implies that the accidental coincidences are dominated
by coincidences between background counts and heralding photons. The low collec-
tion efficiency in the image discrimination arm is due to a combination of factors:
transmission losses at the image mask and the biplex hologram, reflection losses at
lenses, mirrors and beamsplitters, coupling losses from coupling a highly multimode
image into a multimode optical fiber, and alignment issues caused by using differ-
ent laser wavelengths for the hologram exposure (633 nm), single-photon generation
(centered at 650 nm), and single-photon alignment (using 660 nm laser).
Table 4.1: Image-discrimination results showing the total number of raw coincidences(C), accidental coincidences (A), and C/A ratio for each object-detector combination.Also shown are the heralding photon and image-discrimination photon singles rates,sh and sid. Background rates are bh ≃ 1000 ± 32 Hz, bid,A = 420 ± 20 Hz, andbid,B = 249± 16 Hz. 1/R∆t represents the maximum possible C/A ratio as discussedin the text.
Object-detector sh (kHz) sid (Hz) C A C/A 1/R∆tA-A 437 473 5738± 75 337± 18 16.99± 0.95 143± 3A-B 522 252 185± 14 201± 14 0.93± 0.09 N/AB-A 444 414 289± 17 287± 17 1.01± 0.08 N/AB-B 511 273 4401± 66 229± 15 19.24± 1.30 210± 5
We see from Table 4.1 (also Fig. 4.6) that when the readout beam is object A, the
total number of raw coincidence for the A-A combination is much higher than the
A-B combination, with accidental coincidences being about the same for both cases.
4.5 Single-Photon Image Discrimination 89
ABA
B
0
2000
4000
6000R
awC
oin
c.
ABA
B
0
2000
4000
6000
Acc
iden
tal
Co
inc.
ABA
B
0
5
10
15
20C
/AR
atio
DetectorObject
DetectorObject
Detector Object
(b)(a)
(c)
Figure 4.6: Single-photon image-discrimination results. Total number of raw coin-cidences (C), accidental coincidences (A), and C/A ratio for each object-detectorcombination.
Similar conclusion can be inferred when object B is used. We use the ratio of true
coincidences NAA (NBB) to false coincidences NAB (NBA) as a metric to quantify the
fidelity of our system. The ratio
fi = Nii/Nij, i, j = A,B and i 6= j, (4.10)
is found to be 31.2 (15.2) for A (B), respectively. This means that we can distinguish
object A from B with a confidence level of ∼96.8%. For object B, the confidence
level is ∼93.4%. Here the confidence level is calculated as 1 − 1/fi, where fi is as
defined above. The C/A ratios for the A-B and B-A object-detector combinations are
4.5 Single-Photon Image Discrimination 90
approximately unity as can be seen from the data in Table 4.1, i.e., nearly all of the
false events, NAB and NBA, can be attributed to accidental coincidences. One way
to increase the system fidelity is thus by improving the C/A ratios for the A-A and
B-B object-detector combinations. The C/A ratios can be improved by increasing
the total collection efficiency in the image-discrimination arm (the collection and
detection efficiency in the image-discrimination arm of the setup is extremely low for
image A and B as can be seen in Table 4.1) or by using detectors with reduced dark
counts for the image-discrimination photon, such as Perkin-Elmer SPCM-AQR-16
detectors that have only 25 dark counts/s.
Using Eqns. 4.8 and 4.9, the C/A ratio can be reduced to the following expres-
sion [164]:
C/A = 1 +1
R∆t
[(1 +
bhǫhR
)(1 +
bidǫidR
)]−1
, (4.11)
where, as before, ǫi is the collection and detection efficiency of the ith detector, bi
is the background count rate for the ith detector, R is the biphoton generation rate
of the crystal and ∆t is the duration of the coincidence window. The C/A ratio is
limited by (R∆t)−1 and the discrimination confidence level is limited by 1− R∆t in
the ideal setting where bi ≪ ǫiR. In the present experiment (R∆t)−1 ∼ 143 so that
the discrimination confidence level is bounded by 99.30%. To date, the best C/A ratio
in entangled biphoton sources is ∼1000 from a Raman-scattering process in optical
fiber [183].
4.6 Summary 91
4.6 Summary
In summary, we have studied image discrimination at the single photon level and
have shown that it is possible to impress an image onto an optical field comprised
of a single photon and subsequently sort these photons into classes determined by
the image that the photon carries [164]. We have used basis sets containing only two
locally spatially orthogonal (spatially separated) images, for which very good discrim-
ination was obtained with a multiplexed image hologram. For many applications, a
much larger basis set, possibly including nonspatially separated, or more generally,
nonorthogonal images, would be desirable. Image discrimination of nonorthogonal
images using numerical correlation methods has been investigated by [34] where Mor-
ris showed that distinguishing images with a confidence level of 97% requires about
250 photons. For a basis set involving nonorthogonal images the principles of unam-
biguous state discrimination in large Hilbert spaces [184] may be applied to design a
hologram which optimally discriminates images in the basis set. Limits to the number
of images that can be discriminated in a hologram are set by issues such as cross talk,
which tends to increase with the number of stored images, and diffraction efficiency,
which tends to decrease with the number of stored images. However, it is reassuring
to note that as many as 10 000 images have been stored in a holographic memory
under appropriate conditions [177].
Chapter 5
Conclusions and Discussion
In this thesis, I have presented both theoretical and experimental studies on ghost
image formation using compressive sensing methods and transverse image discrimina-
tion at low light levels. I have also presented a study of how the statistics of the light
field used in thermal ghost imaging affects the quality of ghost images thus formed.
The first two chapters deal with different aspects of ghost imaging. Ghost imaging
(GI) is a novel transverse imaging modality that exploits the spatial correlation be-
tween the reference and test beams to retrieve information about an unknown object.
The two spatially correlated beams travel through two separate optical paths and can
have the same (degenerate) or different (nondegenerate) wavelength [185]. The test
beam, which interacts with the object, is not spatially resolved while the reference
beam does not interact with the object but is spatially resolved by a detector.
In quantum ghost imaging, a spatially resolving single-pixel detector is raster
scanned in the reference arm (Dr) of the GI setup. Due to the inherently weak
92
5 Conclusions and Discussion 93
sources of entangled photons used for experiments in quantum GI, the signal-to-noise
(SNR) is improved by increasing the integration time (T ) for each single-pixel detector
position in the reference arm. If the number of pixels used to resolve the unknown
object N ≫ 1, as is usually the case, the total integration time (NT ) needed to
acquire the image of the unknown object could be prohibitively large for practical
applications.
In chapter 2, a new quantum GI setup was introduced. The configuration of
the test arm is the same as in other quantum GI setups (see, for example, [23]).
However a spatial light modulator (SLM), at the location of the spatially resolving
single-pixel detector Dr, is used instead in the reference arm. All the light that is
reflected from the SLM is collected by a bucket detector. Here the new and highly
evolving field of compressive sensing is used in reconstructing the ghost image. The
number of measurements required to resolve the object is not N , unlike in raster-
scanned quantum GI, but is dictated by the the sparsity level (K) of the object in
the transforming basis and is given by O[Klog(N/K)]. One clear benefit here is the
improvement in image acquisition time.
The ghost image reconstruction using the gradient projection method was carried
our for two object: the logo of the University of Rochester and the Greek letter Ψ.
High quality images were reconstructed using only 27% of the number of measure-
ments corresponding to the Nyquist limit. The fidelity of the compressed sensing
image reconstruction algorithm was also characterized by using the mean-squared
5 Conclusions and Discussion 94
error (MSE) as a metric. We have found values of 0.06 and 0.03 for the logo of the
University of Rochester (UR) and the Greek letter Ψ, respectively, at 27% of the
measurement Nyquist limit.
We have also shown that the total number of photons needed for image recon-
struction using the compressed sensing based approach is much smaller than raster
scanned quantum GI. We note here that no systematic effort was made to mini-
mize the total number of photons needed for image reconstruction. We have used
the discrete cosine transform (DCT) as our image sparsifying basis. We should note
that since the image reconstruction was done offline, we could also search for other
transforming basis where the object could be represented more compactly than in
DCT.
In chapter 3, a systematic study on how the field statistics of the illuminating
source affects the quality of the ghost image formation was carried out, both theo-
retically and experimentally. In thermal ghost imaging experiments bright and high
contrast speckle patterns illuminate the object. One would expect that the quality of
the ghost image would be degraded as the contrast of the speckle patterns illuminating
the object decreases.
We have experimentally studied ghost image formation for varying contrast of the
illuminating speckle patterns. One of the metric used to quantify the quality of the
ghost image is the contrast-to-noise ratio (CNR). We have shown that the quality of
the ghost image is not affected even when low-contrast illuminating speckle patterns
5 Conclusions and Discussion 95
are used as long as the fluctuations in the detected signal are due predominantly to the
randomness of the speckle pattern itself and not due to noise in the detection system.
We have shown, both theoretically and experimentally, that the CNR depends on
the kurtosis and not on the contrast of the illuminating field. We derived how the
CNR depends on the number of measurements (N), the total number of transmitting
pixels of the object (T ) and the fourth standardized moment or the kurtosis given
by (γI/σI)4, where σ2
I and γ4I are the second- and fourth- order moments about the
mean, respectively, of the intensity fluctuation for each illuminating speckle field.
We have carried out an experiment where as many as 25 speckle patterns are
averaged together for each measurement. We have found good agreement between
the experimental results and the theoretical predictions. These findings could have
important practical implications in ghost imaging experiments where thermal source
with short coherence time and slow detectors are employed.
In chapters 2 and 3, we considered different aspects of ghost image formation and
reconstruction. In chapter 4, we studied image discrimination, using holographic-
matched filtering techniques, at low light levels. Here we considered locally spatially
orthogonal set of two objects as our basis. Heralded single photons from a sponta-
neous parametric downconversion process are used for encoding and discriminating
images from our predefined orthogonal basis set. We show experimentally that we
can discrimination two objects with a confidence level of >93.4%. Using better single-
photon detectors (that have lower dark counts than used in our experiment), we have
5 Conclusions and Discussion 96
shown that the confidence level can be as high as 99.34%. Despite the fact that our
predefined basis set consists of only two locally spatially orthogonal masks (objects),
holographic memory [177] could be used to extend the number of the predefined set.
An extension of these work using a quantum ghost imaging setup (see for exam-
ple, [23]) for upto four spatially nonoverlapping objects have been recently carried
out [176].
5 Conclusions and Discussion 97
Bibliography
[1] M. Kolobov, Quantum Imaging (Springer, New York, 2006).
[2] C. Fabre, J. B. Fouet, and A. Maıtre, “Quantum limits in the measurement of
very small displacements in optical images,” Opt. Lett. 25, 76 (2000).
[3] N. Treps et al., “Surpassing the standard quantum limit for optical imaging
using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[4] V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of super-resolution in
reconstruction of optical objects,” Phys. Rev. A 71, 043802 (2005).
[5] M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys.
71, 1539 (1999).
[6] L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” Journal of
Optics B: Quantum and Semiclassical Optics 4, S176 (2002).
[7] A. Gatti, E. Brambilla, and L. Lugiato, in Quantum imaging, Vol. 51 of Progress
in Optics, edited by E. Wolf (Elsevier, Amsterdam, 2008), pp. 251 – 348.
98
BIBLIOGRAPHY 99
[8] C. Thiel, T. Bastin, J. von Zanthier, and G. S. Agarwal, “Sub-rayleigh quantum
imaging using single-photon sources,” Phys. Rev. A 80, 013820 (2009).
[9] P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger,
“De broglie wavelength of a non-local four-photon state,” Nature 429, 158
(2004).
[10] M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase
measurements with a multiphoton entangled state,” Nature 429, 161 (2004).
[11] A. Mosset, F. Devaux, and E. Lantz, “Spatially noiseless optical amplification
of images,” Phys. Rev. Lett. 94, 223603 (2005).
[12] A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P.
Dowling, “Quantum interferometric optical lithography: Exploiting entangle-
ment to beat the diffraction limit,” Phys. Rev. Lett. 85, 27332736 (2000).
[13] M. D’Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quan-
tum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
[14] M. Tsang, “Quantum imaging beyond the diffraction limit by optical centroid
measurements,” Phys. Rev. Lett. 102, 253601 (2009).
[15] V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-rayleigh-
diffraction-bound quantum imaging,” Phys. Rev. A 79, 013827 (2009).
BIBLIOGRAPHY 100
[16] G. Scarcelli, A. Valencia, S. Gompers, and Y. Shih, “Remote spectral measure-
ment using entangled photons,” Appl. Phys. Lett. 83, 5560 (2003).
[17] M. C. Teich and B. E. A. Saleh, “Entangled-photon microscopy,”
Ceskoslovensky casopis pro fyziku 47, 3 (1997).
[18] E. Brambilla, L. Caspani, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “High-
sensitivity imaging with multi-mode twin beams,” Phys. Rev. A 77, 053807
(2008).
[19] I. Sokolov, M. Kolobov, A. Gatti, and L. Lugiato, “Quantum holographic tele-
portation,” Opt. Commun. 193, 175 (2001).
[20] A. Gatti, I. V. Sokolov, M. I. Kolobov, and L. A. Lugiato, “Quantum fluctu-
ations in holographic teleportation of optical images,” The European Physical
Journal D - Atomic, Molecular, Optical and Plasma Physics 30, 123 (2004).
[21] L. Magdenko, I. Sokolov, and M. Kolobov, “Quantum teleportation of optical
images with frequency conversion,” Opt. Spectrosc. 103, 62 (2007).
[22] A. Abouraddy, B. Saleh, A. Sergienko, and M. Teich, “Quantum holography,”
Opt. Express 9, 498 (2001).
[23] T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical
imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52,
R3429 (1995).
BIBLIOGRAPHY 101
[24] A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of
entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[25] R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-photon” coincidence
imaging with a classical source,” Phys. Rev. Lett. 89, 113601 (2002).
[26] A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging,
quantum and classical,” Phys. Rev. A 70, 013802 (2004).
[27] J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in
x-ray diffraction,” Phys. Rev. Lett. 92, 093903 (2004).
[28] Y. Shih, “Quantum imaging,” Selected Topics in Quantum Electronics, IEEE
Journal of 13, 1016 (2007).
[29] J. Shapiro and R. Boyd, “The physics of ghost imaging,” Quantum Information
Processing 11, 949 (2012).
[30] D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric
production of optical photon pairs,” Phys. Rev. Lett. 25, 8487 (1970).
[31] C. K. Hong and L. Mandel, “Theory of parametric frequency down conversion
of light,” Phys. Rev. A 31, 24092418 (1985).
[32] M. H. Rubin, “Transverse correlation in optical spontaneous parametric down-
conversion,” Phys. Rev. A 54, 53495360 (1996).
BIBLIOGRAPHY 102
[33] M. N. O’Sullivan, K. W. C. Chan, and R. W. Boyd, “Comparison of the signal-
to-noise characteristics of quantum versus thermal ghost imaging,” Phys. Rev.
A. 82, (2010).
[34] G. M. Morris, “Image correlation at low light levels: a computer simulation,”
Appl. Opt. 23, 3152 (1984).
[35] G. M. Morris, M. N. Werick, and T. A. Isberg, “Image correlation at low light
levels,” Opt. Lett. 10, 315 (1985).
[36] M. N. Wernick and G. M. Morris, “Image classification at low light levels,”
Journal of the Optical Society of America A 3, 2179 (1986).
[37] M. Morris, in Optical processing and computing, edited by H. Arsenault, T.
Szoplik, and B. Macukow (Academic Press, New York, 1989).
[38] R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and
classical coincidence imaging,” Phys. Rev. Lett. 92, 033601 (2004).
[39] A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with
thermal light,” Phys. Rev. Lett. 94, 063601 (2005).
[40] M. D’Angelo, A. Valencia, M. H. Rubin, and Y. Shih, “Resolution of quantum
and classical ghost imaging,” Phys. Rev. A 72, 013810 (2005).
BIBLIOGRAPHY 103
[41] D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation
of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74, 3600
(1995).
[42] A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato,
“Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53,
739 (2006).
[43] F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato,
“High-resolution ghost image and ghost diffraction experiments with thermal
light,” Phys. Rev. Lett. 94, 183602 (2005).
[44] G. Scarcelli, V. Berardi, and Y. Shih, “Can two-photon correlation of chaotic
light be considered as correlation of intensity fluctuations?,” Phys. Rev. Lett.
96, 063602 (2006).
[45] B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with gaussian-
state light,” Phys. Rev. A 77, 043809 (2008).
[46] P. H. S. Ribeiro, S. Padua, J. C. Machado da Silva, and G. A. Barbosa, “Con-
trolling the degree of visibility of Young’s fringes with photon coincidence mea-
surements,” Phys. Rev. A 49, 4176 (1994).
[47] A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-
photon Fourier optics,” J. Opt. Soc. Am. B 19, 1174 (2002).
BIBLIOGRAPHY 104
[48] B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality
between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816
(2000).
[49] D. Klyshko, Photons and Nonlinear Optics (Gordon and Breach, New York,
1992).
[50] A. Belinsky and D. Klyshko, “2-photon optics-diffraction, holography and trans-
formation of 2-dimensional signals,” Sov. Phys. JETP 78, 259 (1994).
[51] K. Wang and D.-Z. Cao, “Subwavelength coincidence interference with classical
thermal light,” Phys. Rev. A 70, 041801 (2004).
[52] Y. Cai and S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent
light radiation,” Phys. Rev. E 71, 056607 (2005).
[53] D.-Z. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging
systems,” Phys. Rev. A 71, 013801 (2005).
[54] G. Scarcelli, A. Valencia, and Y. Shih, “Experimental study of the momentum
correlation of a pseudothermal field in the photon-counting regime,” Phys. Rev.
A 70, 051802 (2004).
[55] X.-H. Chen, Q. Liu, K.-H. Luo, and L.-A. Wu, “Lensless ghost imaging with
true thermal light,” Opt. Lett. 34, 695 (2009).
BIBLIOGRAPHY 105
[56] R. Meyers, K. S. Deacon, and Y. Shih, “Ghost-imaging experiment by measur-
ing reflected photons,” Phys. Rev. A 77, 041801 (2008).
[57] D. Zhang, Y.-H. Zhai, L.-A. Wu, and X.-H. Chen, “Correlated two-photon
imaging with true thermal light,” Opt. Lett. 30, 2354 (2005).
[58] L. Basano and P. Ottonello, “Experiment in lensless ghost imaging with thermal
light,” Appl. Phys. Lett. 89, (2006).
[59] A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-
particle duality: From the microscopic to the macroscopic realm,” Phys. Rev.
Lett. 90, 133603 (2003).
[60] E. Brambilla, A. Gatti, M. Bache, and L. A. Lugiato, “Simultaneous near-field
and far-field spatial quantum correlations in the high-gain regime of parametric
down-conversion,” Phys. Rev. A 69, 023802 (2004).
[61] M. Bache, E. Brambilla, A. Gatti, and L. A. Lugiato, “Ghost imaging using
homodyne detection,” Phys. Rev. A 70, 023823 (2004).
[62] B. I. Erkmen and J. H. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost
imaging,” Phys. Rev. A 79, 023833 (2009).
[63] E. J. Candes and T. Tao, “Near-optimal signal recovery from random projec-
tions: Universal encoding strategies?,” IEEE Trans. on Info. Theory 52, 5406
(2006).
BIBLIOGRAPHY 106
[64] E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incom-
plete and inaccurate measurements,” Comm. on Pure and Appl. Math. 59, 1207
(2006).
[65] D. Donoho, “Compressed sensing,” IEEE Trans. on Info. Theory 52, 1289
(2006).
[66] J. Romberg, “Imaging via compressive sampling,” IEEE Sig. Proc. Mag. 25, 14
(2008).
[67] E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,”
Sig. Proc. Mag., IEEE 25, 21 (2008).
[68] E. J. Candes and J. Romberg, “Sparsity and incoherence in compressive sam-
pling,” Inverse Problems 23, 969 (2007).
[69] E. J. Candes and T. Tao, “Decoding by linear programming,” Info. Theory,
IEEE Trans. on 51, 4203 (2005).
[70] R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simple proof of the
restricted isometry property for randommatrices,” Constructive Approximation
28, 253 (2008).
[71] A. V. Lugt, “Signal detection by complex spatial filtering,” Information Theory,
IEEE Transactions on 10, 139 (1964).
[72] A. V. Lugt, “Coherent optical processing,” Proc. of the IEEE 62, 1300 (1974).
BIBLIOGRAPHY 107
[73] J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company
Publishers, Englewood, CO, 2004).
[74] P. Zerom, K. W. C. Chan, J. C. Howell, and R. W. Boyd, “Entangled-photon
compressive ghost imaging,” Phys. Rev. A 84, 061804 (2011).
[75] K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “Two-color ghost imaging,”
Phys. Rev. A. 79, (2009).
[76] B. F. Aull et al., “Three-dimensional imaging with arrays of geiger-mode
avalanche photodiodes,” Lincoln Laboratory Journal 105 (2004).
[77] M. A. Albota et al., “Three-dimensional imaging laser radar with a photon-
counting avalanche photodiode array and microchip laser,” Appl. Opt. 41, 7671
(2002).
[78] M. F. Duarte et al., “Single-pixel imaging via compressive sampling,” IEEE
Sig. Proc. Mag. 25, 83 (2008).
[79] D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state
tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
[80] A. Shabani et al., “Efficient measurement of quantum dynamics via compressive
sensing,” Phys. Rev. Lett. 106, 100401 (2011).
BIBLIOGRAPHY 108
[81] D. Takhar et al., “A new compressive imaging camera architecture using optical-
domain compression,” in Proc. of Computational Imaging IV at SPIE Elec-
tronic Imaging 6065, 43 (2006).
[82] J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008).
[83] Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detec-
tor,” Phys. Rev. A 79, 053840 (2009).
[84] R. E. Meyers and K. S. Deacon, “Lens-less quantum ghost imaging: New two-
photon experiments,” Vacuum 83, 244 (2008).
[85] O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl.
Phys. Lett. 95, 131110 (2009).
[86] M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of com-
pressed sensing for rapid MR imaging,” Magnetic Resonance in Medicine 58,
1182 (2007).
[87] U. Gamper, P. Boesiger, and S. Kozerke, “Compressed sensing in dynamic
MRI,” Magnetic Resonance in Medicine 59, 365 (2008).
[88] N. Shental, A. Amir, and O. Zuk, “Identification of rare alleles and their carriers
using compressed se(que)nsing,” Nucleic Acids Research 38, e179 (2010).
[89] M. Herman and T. Strohmer, “High-resolution radar via compressed sensing,”
Signal Processing, IEEE Transactions on 57, 2275 (2009).
BIBLIOGRAPHY 109
[90] M. Stuff, B. Thelen, N. Subotic, J. Parker, and J. Browning, “Optimization and
waveforms for compressive sensing applications in the presence of interference,”
2009 International Waveform Diversity and Design Conference, p. 213 (2009).
[91] C. Berger, S. Zhou, and P. Willett, “Signal extraction using compressed sensing
for passive radar with ofdm signals,” 11th International Conference on Infor-
mation Fusion, p. 1 (2008).
[92] X.-C. Xie and Y.-H. Zhang, “High-resolution imaging of moving train by
ground-based radar with compressive sensing,” Elec. Lett. 46, 529 (2010).
[93] G. Howland, P. Zerom, R. W. Boyd, and J. C. Howell, “Compressive sensing
LIDAR for 3D imaging,” CLEO:2011 - Laser Applications to Photonic Appli-
cations CMG3 (2011).
[94] G. A. Howland, P. B. Dixon, and J. C. Howell, “Photon-counting compressive
sensing laser radar for 3D imaging,” Appl. Opt. 50, 59175920 (2011).
[95] J. Ma, “Compressed sensing for surface characterization and metrology,” IEEE
T. Instrumentation and Measurement 59, 1600 (2010).
[96] R. Marcia and R. Willett, “Compressive coded aperture superresolution im-
age reconstruction,” IEEE International Conference on Acoustics, Speech and
Signal Processing, p. 833 (2008).
BIBLIOGRAPHY 110
[97] J. Yang, J. Wright, T. Huang, and Y. Ma, “Image super-resolution as sparse
representation of raw image patches,” IEEE Conference on Computer Vision
and Pattern Recognition, p. 1 (2008).
[98] S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and recon-
struction of sparse sub-wavelength images,” Opt. Exp. 17, 23920 (2009).
[99] W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M.
Mittleman, “A single-pixel terahertz imaging system based on compressed sens-
ing,” Appl. Phys. Lett. 93, (2008).
[100] See http://dsp.rice.edu/cs for many more applications.
[101] N. D. Hardy and J. H. Shapiro, “Ghost imaging in reflection: resolution, con-
trast, and signal-to-noise ratio,” Proc. of SPIE 7815, 78150L (2010).
[102] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge
University Press, Cambridge, England, 1995).
[103] M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of
two-photon entanglement in type-II optical parametric down-conversion,” Phys.
Rev. A 50, 5122 (1994).
[104] M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection
for sparse reconstruction: Application to compressed sensing and other inverse
problems,” IEEE J. of Sel. Top. in Sig. Proc. 1, 586 (2007).
BIBLIOGRAPHY 111
[105] M. Wakin et al., “An architecture for compressive imaging,” in IEEE Interna-
tional Conference on Image Processing (2006), pp. 1273–1276.
[106] M. Lustig, D. Donoho, J. Santos, and J. Pauly, “Compressed sensing MRI,”
Signal Processing Magazine, IEEE 25, 72 (2008).
[107] K. Ni, S. Datta, P. Mahanti, S. Roudenko, and D. Cochran, “Efficient determin-
istic compressed sensing for images with chirps and reed-muller codes,” SIAM
J. Img. Sci. 4, 931 (2011).
[108] S. Ma, W. Yin, Y. Zhang, and A. Chakraborty, “An efficient algorithm for com-
pressed MR imaging using total variation and wavelets,” in IEEE Conference
on Computer Vision and Pattern Recognition, 2008. CVPR 2008. (2008), pp.
1–8.
[109] B. Han, F. Wu, and D. Wu, “Image representation by compressed sensing,”
in IEEE International Conference on Image Processing, 2008. ICIP 2008. 15th
(2008), pp. 1344–1347.
[110] J. Yang, Y. Zhang, and W. Yin, “A fast alternating direction method for TVL1-
L2 signal reconstruction from partial Fourier data,” Selected Topics in Signal
Processing, IEEE Journal of 4, 288 (2010).
[111] J. Huang, S. Zhang, and D. Metaxas, “Efficient MR image reconstruction for
compressed MR imaging,” in Proceedings of the 13th international conference
BIBLIOGRAPHY 112
on Medical image computing and computer-assisted intervention: Part I, MIC-
CAI’10 (Springer-Verlag, Berlin, Heidelberg, 2010), pp. 135–142.
[112] M. Bondani, E. Puddu, I. P. Degiovanni, and A. Andreoni, “Chaotically seeded
parametric downconversion for ghost imaging,” JOSA B 25, 1203 (2008).
[113] P. Clemente, V. Duran, V. Torres-Company, E. Tajahuerce, and J. Lancis,
“Optical encryption based on computational ghost imaging,” Opt. Lett. 35,
2391 (2010).
[114] P. Zerom et al., “Thermal ghost imaging with averaged speckle patterns,” Phys.
Rev. A 86, 063817 (2012).
[115] J. D. Rigden and E. I. Gordon, “The granularity of scattered optical maser
light,” Proc. IRE 50, 2367 (1962).
[116] B. Oliver, “Sparkling spots and random diffraction,” Proceedings of the IEEE
51, 220 (1963).
[117] J. Dainty, in The Statistics of Speckle Patterns, Vol. 14 of Progress in Optics,
edited by E. Wolf (Elsevier, North-Holland, Amsterdam, 1977), pp. 1 – 46.
[118] J. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts
& Company, Publishers, Englewood, CO, 2006).
BIBLIOGRAPHY 113
[119] J. Garcıa, Z. Zalevsky, P. Garcıa-Martınez, C. Ferreira, M. Teicher, and Y.
Beiderman, “Three-dimensional mapping and range measurement by means of
projected speckle patterns,” Appl. Opt. 47, 3032 (2008).
[120] J. Garcıa, Z. Zalevsky, P. Garcıa-Martınez, C. Ferreira, M. Teicher, and Y.
Beiderman, “Projection of speckle patterns for 3D sensing,” Journal of Physics:
Conference Series 139, 012026 (2008).
[121] D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave
spectroscopy,” Phys. Rev. Lett. 60, 1134 (1988).
[122] G. Montaldo, M. Tanter, and M. Fink, “Time reversal of speckle noise,” Phys.
Rev. Lett. 106, 054301 (2011).
[123] D. A. Boas and A. K. Dunn, “Laser speckle contrast imaging in biomedical
optics,” Journal of Biomedical Optics 15, 011109 (2010).
[124] U. Bortolozzo, S. Residori, and P. Sebbah, “Experimental observation of speckle
instability in Kerr random media,” Phys. Rev. Lett. 106, 103903 (2011).
[125] C. Sanner et al., “Speckle imaging of spin fluctuations in a strongly interacting
Fermi gas,” Phys. Rev. Lett. 106, 010402 (2011).
[126] J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am.
66, 1145 (1976).
BIBLIOGRAPHY 114
[127] A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with
thermal light: Comparing entanglement and classical correlation,” Phys. Rev.
Lett. 93, 093602 (2004).
[128] Y.-H. Zhai, X.-H. Chen, D. Zhang, and L.-A. Wu, “Two-photon interference
with true thermal light,” Phys. Rev. A 72, 043805 (2005).
[129] R. Meyers, K. S. Deacon, and Y. Shih, “Ghost-imaging experiment by measur-
ing reflected photons,” Phys. Rev. A 77, (2008).
[130] G. Brida, M. V. Chekhova, G. A. Fornaro, M. Genovese, E. D. Lopaeva, and
I. R. Berchera, “Systematic analysis of signal-to-noise ratio in bipartite ghost
imaging with classical and quantum light,” Phys. Rev. A 83, 063807 (2011).
[131] G. Scarcelli, V. Berardi, and Y. Shih, “Can two-photon correlation of chaotic
light be considered as correlation of intensity fluctuations?,” Phys. Rev. Lett.
96, (2006).
[132] B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-
state light,” Phys. Rev. A 77, 043809 (2008).
[133] F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,”
Phys. Rev. Lett. 104, (2010).
[134] J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Exp. 17, 7916
(2009).
BIBLIOGRAPHY 115
[135] C. Li, T. Wang, J. Pu, W. Zhu, and R. Rao, “Ghost imaging with partially
coherent light radiation through turbulent atmosphere,” Appl. Phys. B. 99,
599 (2010).
[136] W. Gong and S. Han, “Correlated imaging in scattering media,” Opt. Lett. 36,
394 (2011).
[137] K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “Optimization of thermal
ghost imaging: high-order correlations vs. background subtraction,” Opt. Exp.
18, 5562 (2010).
[138] K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-order thermal ghost
imaging,” Opt. Lett. 34, 3343 (2009).
[139] J. Goodman, “Some effects of target-induced scintillation on optical radar per-
formance,” Proceedings of the IEEE 53, 1688 (1965).
[140] J. Dainty, “Some statistical properties of random speckle patterns in coherent
and partially coherent illumination,” Optica Acta: International Journal of
Optics 17, 761 (1970).
[141] J. Dainty, “Detection of images immersed in speckle noise,” Optica Acta: In-
ternational Journal of Optics 18, 327 (1971).
[142] J. Ohtsubo and T. Asakura, “Statistical properties of the sum of partially de-
veloped speckle patterns,” Opt. Lett. 1, 98 (1977).
BIBLIOGRAPHY 116
[143] H. C. S. Karmakar, Y. Zhai and Y. Shih, (unpublished).
[144] J. A. Bergou, “Discrimination of quantum states,” Journal of Modern Optics
57, 160 (2010).
[145] S. M. Barnett and S. Croke, “Quantum state discrimination,” Adv. Opt. Pho-
ton. 1, 238 (2009).
[146] A. Chefles, “Quantum state discrimination,” Contemporary Physics 41, 401
(2000).
[147] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Infor-
mation, 1st. ed. (Cambridge University Press, Cambridge, 2004).
[148] S. Franke-Arnold and J. Jeffers, “Unambiguous state discrimination in high
dimensions,” The European Physical Journal D 66, 1 (2012).
[149] C. W. Helstrom, Quantum Detection and Estimation (Academic Press, New
York, 1976).
[150] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-
Holland, Amsterdam, 1979).
[151] S. M. Barnett and E. Riis, “Experimental demonstration of polarization dis-
crimination at the helstrom bound,” Journal of Modern Optics 44, 1061 (1997).
BIBLIOGRAPHY 117
[152] R. B. M. Clarke, V. M. Kendon, A. Chefles, S. M. Barnett, E. Riis, and M.
Sasaki, “Experimental realization of optimal detection strategies for overcom-
plete states,” Phys. Rev. A 64, 012303 (2001).
[153] S. M. Barnett, “Minimum-error discrimination between multiply symmetric
states,” Phys. Rev. A 64, 030303 (2001).
[154] C.-L. Chou and L. Y. Hsu, “Minimum-error discrimination between symmetric
mixed quantum states,” Phys. Rev. A 68, 042305 (2003).
[155] I. Ivanovic, “How to differentiate between non-orthogonal states,” Physics Let-
ters A 123, 257 (1987).
[156] D. Dieks, “Overlap and distinguishability of quantum states,” Physics Letters
A 126, 303 (1988).
[157] A. Peres, “How to differentiate between non-orthogonal states,” Physics Letters
A 128, 19 (1988).
[158] B. Huttner, A. Muller, J. D. Gautier, H. Zbinden, and N. Gisin, “Unambiguous
quantum measurement of nonorthogonal states,” Phys. Rev. A 54, 3783 (1996).
[159] R. B. M. Clarke, A. Chefles, S. M. Barnett, and E. Riis, “Experimental demon-
stration of optimal unambiguous state discrimination,” Phys. Rev. A 63, 040305
(2001).
BIBLIOGRAPHY 118
[160] M. Mohseni, A. M. Steinberg, and J. A. Bergou, “Optical realization of optimal
unambiguous discrimination for pure and mixed quantum states,” Phys. Rev.
Lett. 93, 200403 (2004).
[161] S. Croke, E. Andersson, S. M. Barnett, C. R. Gilson, and J. Jeffers, “Maximum
confidence quantum measurements,” Phys. Rev. Lett. 96, 070401 (2006).
[162] P. J. Mosley, S. Croke, I. A. Walmsley, and S. M. Barnett, “Experimental real-
ization of maximum confidence quantum state discrimination for the extraction
of quantum information,” Phys. Rev. Lett. 97, 193601 (2006).
[163] S. Croke, P. J. Mosley, S. M. Barnett, and I. A. Walmsley, “Maximum confi-
dence measurements and their optical implementation,” The European Physical
Journal D 41, 589 (2007).
[164] C. J. Broadbent, P. Zerom, H. Shin, J. C. Howell, and R. W. Boyd, “Discrimi-
nating orthogonal single-photon images,” Phys. Rev. A 79, 033802 (2009).
[165] A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital
angular momentum states of photons,” Nature 412, 313 (2001).
[166] G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular
momentum of light: Preparation of photons in multidimensional vector states
of angular momentum,” Phys. Rev. Lett. 88, 013601 (2001).
BIBLIOGRAPHY 119
[167] J. Leach, J. Courtial, K. Skeldon, S. Barnett, S. Franke-Arnold, and M. Padgett,
“Interferometric methods to measure orbital and spin, or the total angular
momentum of a single photon,” Phys. Rev. Lett. 92, (2004).
[168] S. P. Walborn, D. S. Lemelle, M. P. Almeida, and P. H. S. Ribeiro, “Quantum
key distribution with higher-order alphabets using spatially encoded qudits,”
Phys. Rev. Lett. 96, (2006).
[169] J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of
hyperentangled photon pairs,” Phys. Rev. Lett. 95, (2005).
[170] P. Walther, M. Aspelmeyer, and A. Zeilinger, “Heralded generation of multi-
photon entanglement,” Phys. Rev. A 75, (2007).
[171] H. de Riedmatten, I. Marcikic, V. Scarani, W. Tittel, H. Zbinden, and N.
Gisin, “Tailoring photonic entanglement in high-dimensional Hilbert spaces,”
Phys. Rev. A 69, (2004).
[172] H. S. Eisenberg, G. Khoury, G. A. Durkin, C. Simon, and D. Bouwmeester,
“Quantum entanglement of a large number of photons,” Phys. Rev. Lett. 93,
(2004).
[173] M. N. O’Sullivan-Hale, I. Ali-Khan, R. W. Boyd, and J. C. Howell, “Pixel
entanglement: Experimental realization of optically entangled d=3 and d=6
qudits,” Phys. Rev. Lett. 94, (2005).
BIBLIOGRAPHY 120
[174] I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key
distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98,
(2007).
[175] R. M. Camacho, C. J. Broadbent, I. Ali-Khan, and J. C. Howell, “All-optical
delay of images using slow light,” Phys. Rev. Lett. 98, (2007).
[176] M. Malik, H. Shin, M. O’Sullivan, P. Zerom, and R. W. Boyd, “Quantum ghost
image identification with correlated photon pairs,” Phys. Rev. Lett. 104, 163602
(2010).
[177] X. An, D. Psaltis, and G. W. Burr, “Thermal fixing of 10,000 holograms in
LiNbO3:Fe,” Appl. Opt. 38, 386 (1999).
[178] G. M. Morris and N. George, “Matched filtering using band-limited illumina-
tion,” Opt. Lett. 5, 202 (1980).
[179] G. G. Turin, “An introduction to matched filters,” IRE Trans. on Inf. Theory
6, 311 (1960).
[180] P. Hariharan, Optical Holography: Principles, Techniques and Applications
(Cambridge University Press, Cambridge, 1984).
[181] R. Collier, C. Burckhardt, and L. Lin, Optical holography (Academic Press, New
York, 1971).
BIBLIOGRAPHY 121
[182] Our holograms were made using well-established laboratory procedures. (See,
for instance: [180] and [181]). The holographic recording material was purchased
from Slavich, Vilnius 2006, Lithuania.
[183] J. Fan and A. Migdall, “A broadband high spectral brightness fiber-based two-
photon source,” Opt. Exp. 15, 2915 (2007).
[184] A. Peres and D. R. Terno, “Optimal distinction between non-orthogonal quan-
tum states,” Journal of Phys. A 31, 7105 (1998).
[185] K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “Two-color ghost imaging,”
Phys. Rev. A. 79, (2009).
[186] J. Barzilai and J. Borwein, “Two-point step size gradient methods,” IMA Jour-
nal Of Numerical Analysis 8, 141 (1988).
[187] D. P. Bertsekas, Nonlinear Programming, 2nd ed. (Athena Scientific, Boston,
1999).
[188] J. Nocedal and S. J. Wright, Numerical Optimization (Springer-Verlag, Berlin,
2000).
[189] C. Paige and M. Saunders, “LSQR - an algorithm for sparse linear-equations
and sparse least-squares,” ACM Transactions On Mathematical Software 8, 43
(1982).
BIBLIOGRAPHY 122
[190] S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pur-
suit,” SIAM Journal On Scientific Computing 20, 33 (1998).
[191] R. Tibshirani, “Regression shrinkage and selection via the lasso,” Journal Of
The Royal Statistical Society Series B- Methodological 58, 267 (1996).
[192] B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani, “Least angle regression,”
Annals Of Statistics 32, 407 (2004).
[193] D. Malioutov, M. Cetin, and A. Willsky, “Homotopy continuation for sparse
signal representation,” Proceedings of the IEEE International Conference on
Acoustics, Speech, and Signal Processing 5, 733 (2005).
[194] M. Osborne, B. Presnell, and B. Turlach, “A new approach to variable selection
in least squares problems,” IMA Journal Of Numerical Analysis 20, 389 (2000).
[195] S. Wright, Primal-Dual Interior-Point Methods (SIAM Publications, Philadel-
phia, PA, 1997).
[196] B. A. Turlach, “On algorithms for solving least squares problems under an ℓ1
penalty or an ℓ1 constraint,” Proc. American Statistical Association; Statistical
Computing Section, pp. 2572 (2005).
[197] M. Figueiredo and R. Nowak, “An EM algorithm for wavelet-based image
restoration,” IEEE Transactions On Image Processing 12, 906 (2003).
BIBLIOGRAPHY 123
[198] R. Nowak and M. Figueiredo, “Fast wavelet-based image deconvolution using
the EM algorithm,” Proceedings of the 35th Asilomar Conference on Signals,
Systems and Computers 1, 371 (2001).
[199] G. Davis, S. Mallat, and M. Avellaneda, “Greedy adaptive approximation,” J.
Construct. Approx. 12, 57 (1997).
[200] D. Donoho, M. Elad, and V. Temlyakov, “Stable recovery of sparse overcomplete
representations in the presence of noise,” IEEE Transactions On Information
Theory 52, 6 (2006).
[201] J. Tropp, “Greed is good: Algorithmic results for sparse approximation,” IEEE
Transactions On Information Theory 50, 2231 (2004).
[202] A. Chambolle, “An algorithm for total variation minimization and applica-
tions,” Journal Of Mathematical Imaging And Vision 20, 89 (2004).
[203] See http://sparselab.stanford.edu/.
[204] See http://users.ece.gatech.edu/˜justin/l1magic/.
[205] S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point
method for large-scale ℓ1-regularized least squares,” Selected Topics in Signal
Processing, IEEE Journal of 1, 606 (2007).
[206] See http://www.lx.it.pt/˜mtf/GPSR/ for the gradient projection for the sparse
reconstruction (GPSR) [104] code.
BIBLIOGRAPHY 124
[207] The CVX code is freely available at http://cvxr.com/cvx/.
Appendix A
Algorithms for Compressed
Sensing
In compressed sensing [63–66] we deal with finding unique solutions to an underdeter-
mined set of linear equations. Before we introduce the algorithm used to reconstruct
the ghost images in chapter 2 (see Fig. 2.6), consider the following set of linear equa-
tions:
Ax = y (A.1)
where A is an invertible N × N matrix, y is a vector in RN . One way of finding a
solution to the linear system of equations in Eqn. A.1 is to minimize the objective
125
A Algorithms for Compressed Sensing 126
(cost) function F (x), explicitly given by,
F (x) ≡ 1
2〈x,Ax〉 − 〈y,x〉
=1
2xTAx− yTx. (A.2)
subject to x ∈ RN . Here (·)T represents the transpose of a vector and 〈a,b〉 denotes
the dot product of the vectors a and b. We see the equivalence of Eqns. A.1 and A.2
by setting the gradient of F (x) to zero.
Although there are many methods, such as conjugate gradient, gradient projec-
tion [186–188], of solving the unconstrained minimization of the objective function
F (x) in Eqn. A.2, we use the steepest-descent method to show the general approach.
In this method, we start with an arbitrary initial guess for x, i.e. x(0). The objective
function decreases most quickly if we choose our search direction dk ∈ RN as the
negative gradient of F at xk, i.e.,
dk = −∇F (xk), (A.3)
and the next step (iterate) is given by
xk+1 = xk + αkdk, (A.4)
where the step size αk is determined by finding the minimizer of the objective function
A Algorithms for Compressed Sensing 127
at xk and dk. That is a line search procedure is carried out to find αk that minimize
the objective function along a line through
αk = argminαF (xk + αdk). (A.5)
The above steps ( A.3– A.5) are repeated until we get an approximate solution.
There are many methods and algorithms that have been recently used for com-
pressed sensing problems [189–202]. There are also a number of toolboxes and codes
developed by many groups (see for example [203–205]).
The gradient projection method by Figueiredo et al. [104,206], that we have used
for image reconstruction in chapter 2, can be used to solve the following unconstrained
optimization problem
minx
1
2||y −Ax||22 + τ ||x||1 , (A.6)
which includes a quadratic (ℓ2) error term combined with a sparseness-inducing (ℓ1)
regularization term. Here x ∈ Rn, y ∈ R
k, andA is an k×nmatrix, τ is a nonnegative
parameter, and ||v||p = (∑
i |vi|p)1/p denotes the ℓp norm.
If we make the following substitution for x:
x = u− v, u ≥ 0, v ≥ 0, (A.7)
the optimization problem (Eqn. A.6) can be reformulated as a bound-constrained
A Algorithms for Compressed Sensing 128
quadratic program:
minu,v
1
2||y −A(u− v)||22 + τ1T
nu+ τ1Tnv, (A.8)
s.t. u ≥ 0
v ≥ 0.
Here ui = (xi)+ and vi = (−xi)+ for all i = 1, 2, ..., n, where (.)+ denotes the positive-
part operator defined as (x)+ = max{0, x} and 1n = [1, 1, ..., 1]T is the vector consist-
ing of n ones. In more standard form (see Eqn. A.2), the bound-constrained quadratic
program (Eqn. A.8) becomes
minz
cTz+1
2zTBz ≡ F (z), (A.9)
s.t. z ≥ 0,
where z =
[u
v
], c = τ12n +
[−ATy
ATy
], and B =
[ATA −ATA
−ATA ATA
].
The gradient projection method is then applied to Eqn. A.9. After an initial guess
(z(0)), the next iterate z(k+1) is determined through the following steps. We set, for
some scalar parameter α(k) > 0, which is determined using a line search similar to
Eqn. A.5,
w(k) = (z(k) − α(k)∇F (z(k)))+. (A.10)
A Algorithms for Compressed Sensing 129
After picking a second scalar λ(k) ∈ [0, 1], we determine the next iterate through
z(k+1) = z(k) + λ(k)(w(k) − z(k)). (A.11)
Explicit expressions for α(k) and λ(k) are given in [104]. A convergence test is per-
formed and the above steps are repeated until an approximate solution is found.
Another program that we have used is called CVX [207]. CVX is a freely available
Matlab-based program for various optimization problems. In chapter 1, we have
described how different ℓ1 and ℓ2 minimization are using a numerical example. Here
we have the optimization problem:
minx
||x||1 subject to y = Ax, (A.12)
The following snippet is used to do ℓ1 minimization to generate Fig. 1.3(a) in
chapter 1.
cvx_begin
variable xp(n);
minimize(norm(xp, 1));
subject to
A*xp==y;
cvx_end