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Terahertz Imaging with Compressed Sensing
Department of Electrical and Computer EngineeringRice University, Houston, Texas, USA
Wai Lam Chan
December 17, 2007
2
Mittleman Group (http://www.ece.rice.edu/~daniel)
THz Near-field microscopy (Zhan, Astley)
THz Imaging (Chan, Pearce)
THz Photonic Crystal structures (Prasad, Jian)
THz waveguides (Mendis, Mbonye, Diebel, Wang)
THz emission spectroscopy (Laib, Zhan)
Terahertz (THz) Research Group at Rice
T-rays and Imaging
What Are T-Rays?
100 103 106 109 101
2
101
5
101
8
102
1
T-Rays
Radio Waves
Microwaves
X-Rays
Gamma Rays
Visible Light
Hz
Imaging Throughout History
Daguerreotype (1839)
http://inventors.about.com/library/inventors/bldaguerreotype.htm
X-rays (1895)
http://inventors.about.com/library/inventors/blxray.htm
T-rays (1995)
B. B. Hu and M. C. Nuss, Opt. Lett., 20, 1716, 1995
Why Can T-Rays Help?
0 20 40 60 80 100
Time (ps)
0.2 0.4 0.6 0.8 1.0
Frequency (THz)
0.2 0.4 0.6 0.8 1.0
Frequency (THz)
E(t) E(f) |E(f)|
•Measurement of E(t)
•Subpicosecond pulses
•Submillimeter Wavelengths
T-Rays Provide
•Travel-time / Depth Information
•High depth resolution
•High spatial resolution
Benefits to Imaging
Subpicosecond pulses Linear Phase Over 1 THz in Bandwidth
Material Responses to T-rays
Water
Metal
Plastics
Strongly Absorbing
Highly Reflective
Transparent
8
Promising Applications of T-Rays
(Karpowicz, et al., Appl. Phys. Lett. vol. 86, 054105 (2005))
Zandonella, C. Nature 424, 721–722 (2003).
Space Shuttle Foam
Wallace, V. P., et. al. Faraday Discuss. 126, 255 - 263 (2004).
Diseased Tissue
Medical Imaging
Safety
SecurityConcealed Weapon
(Kawase, Optics & Photonics News, October 2004)
THz Time-domain Imaging
Object
THz TransmitterTHz Receiver
THz Time-domain Imaging
Object
THz TransmitterTHz Receiver
• Pixel-by-pixel scanning
• Limitations: acquisition time vs. resolution
• Faster imaging method
Just take fewer samples!
Compressed Sensing (CS)[Candes et al, Donoho]
Why CS works: Sparsity
• Many signals can be compressed in some representation/basis (Fourier, wavelets, …)
pixels largewaveletcoefficients
widebandsignalsamples
largeGaborcoefficients
• Reconstruct via nonlinear processing (optimization)
• Take fewer ( ) measurements
High-speed THz Imaging with Compressed Sensing (CS)
Measurements(projections)
(Donoho, IEEE Trans. on Information Theory, 52(4), pp. 1289 - 1306, April 2006)
“sparse” signal / object(K-sparse)
MeasurementMatrix
M << N
• Signal is -sparse• Few linear projections
Compressed Sensing (CS) Theory
1 2 3 4
5 6 7 8
9 10
11
12
13
14
15
16
sparsesignal (image)
informationrate
measurements
Measurement matrix
• Signal is -sparse• Few linear projections
• Random measurements will work!
Compressed Sensing (CS) Theory
1 2 3 4
5 6 7 8
9 10
11
12
13
14
15
16
sparsesignal (image)
informationrate
measurements
Measurement matrix(e.g., random)
Random can be …
…
1 2 M
…
1 2 M
Random 0/1
(Bernoulli)
Random
2-D Fourier
and many others …
• Reconstruction/decoding: given(ill-posed inverse problem) find
CS Signal Recovery
measurementssparsesignal
nonzeroentries
• Reconstruction/decoding: given(ill-posed inverse problem) find
• L2 fast, wrong
CS Signal Recovery
• Reconstruction/decoding: given(ill-posed inverse problem) find
• L2 fast, wrong
• L0 correct, slowonly M=K+1 measurements required to perfectly reconstruct K-sparse signal[Bresler; Rice]
CS Signal Recovery
number ofnonzeroentries
• Reconstruction/decoding: given(ill-posed inverse problem) find
• L2 fast, wrong
• L0 correct, slow
• L1 correct, mild oversampling [Candes et al, Donoho]
CS Signal Recovery
linear program
CS in Action Part I: CS-THz Fourier
Imaging
THz Fourier Imaging Setup
6cm 6cm 6cm
objectmask
THz transmitter (fiber-coupled PC antenna)
THz receiver
6cm
metal aperture
automated translation stage
N Fourier samples
THz Fourier Imaging Setup
6cm6cm
objectmask
THz transmitter
6cm
Fourier plane
pick only random measurements for
Compressed Sensing
Random 2-D Fourier
…
Measurement matrix…
THz Fourier Imaging Setup
automated translation
stage
polyethlene lens
object mask “R”(3.5cm x 3.5cm)
THz receiver
Fourier Imaging Results
Fourier Transform of object (Magnitude)
Inverse Fourier Transform Reconstruction (zoomed-in)
6.4 cm 4.5 cm
6.4
cm
4.5
cm
Resolution: 1.125 mm
Imaging Results with CS
Inverse FT Reconstruction
(4096 measurements)
CS Reconstruction (500 measurements)
4.5 cm
4.5
cm
CS Reconstruction (1000 measurements)
Imaging Using the Fourier Magnitude
6cm
objectmask
THz transmitterTHz receiver
6cm
metalaperture
translationstage
variable objectposition
Reconstruction with Phase Retrieval (PR)
• Reconstruct signal from only the magnitude of its Fourier transform
• Iterative algorithm based on prior knowledge of signal:– real-valued– positivity– finite support
• Hybrid Input-Output (HIO) algorithm
• Compressive Phase Retrieval (CPR)
(Fienup, Appl. Optics., 21(15), pp. 2758 - 2769, August 1982)
(Moravec et al.)
Imaging Results with Compressive Phase Retrieval (CPR)
6 cm
6 cm
Resolution: 1.875 mm
Fourier Transform of object (Magnitude-only)
CPR Reconstruction(4096 measurements)
6.4 cm
6.4
cm
Compressed Sensing Phase Retrieval (CSPR) Results
• Modified CPR algorithm with CS
Fourier Transform of object
(Magnitude-only)
CPR Reconstruction (4096 measurements)
CSPR Reconstruction (1000 measurements)
6.4 cm
6.4
cm
6 cm
6 c
m
CS in Action Part I: CSPR Imaging System
• THz Fourier imaging with compressed sensing (CS) and phase retrieval (PR)
• Improved acquisition speed
• Processing time
• Potential for:– Flaw or impurity detection– Imaging with CW source (e.g., QCL)
CS in ActionPart II: Single-Pixel THz
Camera
Imaging with a Single-Pixel detector?
(Lee A W M, et al., Appl. Phys. Lett. vol. 89, 141125 (2006))
• Continuous-Wave (CW) THz imaging with a detector array
• Real-time imaging
Single-Pixel Camera (Visible Region)
DMD
Random pattern onDMD array
(Baraniuk, Kelly, et al. Proc. of Computational Imaging IV at SPIE Electronic Imaging, Jan 2006)
imagereconstruction
DSP
DMD
Random 0/1 Bernoulli
…
Measurement matrix
…
….001010….
Random patterns for CS-THz imaging
• Random patterns on printed-circuit boards (PCBs)
THz Single-Pixel Camera Setup
THz receiver
Random pattern on
PCBsTHz transmitter (fiber-coupled PC antenna)
object mask
7cm6cm 42cm
THz Single-Pixel Camera Imaging Result
Object maskCS resconstruction
(200 measurements)CS resconstruction
(400 measurements)
THz Single-Pixel Camera Imaging Result
CS resconstruction (400 measurements)
CS resconstruction (200 measurements)
• image phase?
CS in ActionPart II: Single-Pixel THz camera
• First single-pixel THz imaging system with no raster scanning
• Potential for: – Low cost (simple hardware)– near video-rate acquisition
• Faster acquisition:– film negatives (wheels/sprockets)– more advanced THz modulation
techniques
Conclusions
• Terahertz imaging with Compressed Sensing– Acquire fewer samples high-speed image
acquisition– THz Fourier imaging with CSPR– Single-pixel THz camera
• Ongoing research– THz camera with higher speed and resolution– Imaging phase with CS– CS-THz tomography– Imaging with multiple THz sensors
43dsp.rice.edu/cs
Mittleman Group (http://www.ece.rice.edu/~daniel)
Contact info: William Chan
(wailam@rice.edu) Acknowledgement
Dr. Daniel MittlemanDr. Richard BaraniukDr. Kevin Kelly
Matthew MoravecDharmpal TakharKriti Charan
44
+ -
T-Ray System
THz Transmitter
Substrate LensFemtosecond Pulse
GaAs Substrate
DC Bias
Picometrix T-Ray Instrumentation System
Picometrix T-Ray Transmitter Module
Femtosecond Pulse
45
T-Ray System
T-Ray Control Box with Scanning Delay Line
Fiber Coupled Femtosecond Laser System
Sample
THz Transmitter THz Receiver
Optical Fiber
46
Summary of T-Rays
• Broad fractional bandwidth
• Direct measurement of E(t)
• Short wavelengths (good depth resolution)
• Unique material responses
47
• Signal is -sparse
• Samples
sparsesignal
nonzeroentries
measurements
Sampling1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
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