OPTICAL TRANSFORMS IN DIGITAL

HOLOGRAPHY

Holo-05,Varna, May 21-25, 2005

L. Yaroslavsky

The problem of mutual correspondence between optical transformations and their computer representations is addressed and different computer representations of basic optical transforms such as convolution, Fourier and Fresnel integral transforms are briefly reviewed.

“Direct” imaging: convolution integral

ddhyxayxa ,,,~

dxdy

F

yfxfiyxaffa yx

yx 2exp,,

Transform imaging: Integral Fourier Transform

xa(x)

Z

f

Input plane Output plane

Transform imaging: Integral Fresnel Transform

dxdyZ

yfxfiyxa

ffa

yx

yx

22

2exp,

,

f

F F

a(x)

x

The conformity and mutual correspondence principles

between analogue and digital signal transformations

Discretizationand

quantization

Input digital signal

Continuous input signal (image or hologram)

Continuous output signal (output image or computer generated hologram)

Digital signaltransformations

in computer

Reconstructionof the continuous

signal

Equivalent continuous transformation

Output digital signal

The conformity principle requires that digital representation of signal transformations should parallel that of signals. Mutual correspondence between continuous and digital transformations is said to hold if both act to transform identical input signals into identical output signals. According to these principles, digital processors incorporated into optical information systems should be regarded and treated along with signal digitization and signal reconstruction devices as integrated analogous units

Let be a continuous signal as a function of spatial co-ordinates given by a vector

xa x

xkxx ddddk xkxx rrrr

k and

,

(1)

are discretization and reconstruction basis functions defined in the discretization and reconstruction devices coordinates dx and

rx , respectively, x

Mathematical formulation of signal discretization and reconstruction

is a vector of the sampling intervals, k is a vector of signal sample indices.

At the signal discretization, signal samples ka are computed as

dxxkxxaa ddk (2)

assuming certain relationship between signal and sampling device coordinate systems x and dx

Signal reconstruction from the set of their samples .

ka is described as

k

rrk

r xkxaxa ~ (3)

It is understood that although result xa~ of the signal reconstruction from its discrete representationobtained according to Eq. 3 is not, in general, identical to the initial signal xa

substitute for the initial signal in the given application.

, it can serve as a

According to the conformity principle, Eqs. 2 and 3 form the base for adequate discrete representation of signal transformations.

DISCRETE REPRESENTATION OF THE CONVOLUTION INTEGRAL

xukxx dddk

xukxx rrrk

dxahdxhaxb

Discretization basis functions

Signal reconstruction basis functions

Signals belong to the class:

1

0

N

n

rrn xunxxa

X

ddn dxxunxxaa

For such signals,

dxhxukadxhaxbk

rrk

Samples of xb

X

ddk dxxukxxab ~

dxunxukxxha rrdd

nn

nnknha

where

dxxuuxmxhh rdrdm

are:

duwhere and du are shifts, in fractions of the discretization interval, of sample positions with

respect to input and output signal coordinate systems, correspondingly.

Equation (4) represents is the canonical equation of signal domain digital filtering. Equation (5) defines how discrete point spread function of a digital filter can be found that corresponds to a given convolution point spread function .

mh .h

For digital filtering, this equation is replaced by

1

0

hN

nnknk hb (5)

(6)

(4)

DISCRETE REPRESENTATION OF CONVOLUTION INTEGRAL: continuous PSF of a digital filter

1

0

1

0

1

0

~~~ b hb N

k

rN

nnkn

N

k

rk xkxahxkxbxb

1

0

1

0

~~~N

k

rN

n

dn xkxdxnkah

h

dxnkxkxhaN

k

N

n

drn

h1

0

1

0

~~~where rukk ~ dunn ~

and

According to the mutual correspondence principle, given point spread function of a digital filter , point spread function of an equivalent continuous filter can be found as following.

nh

xb~

reconstructed from its N samples kb

Signal discretization

Input digital signal

Continuous input signal

Continuous output signal

Digital filter Reconstructionof the continuous

signal

Equivalent continuous filter

Output digital signal

1

0

1

0

~~~,

N

k

N

n

drneq

h

xnkxkxhxh

.d .r

nh

DISCRETE REPRESENTATION OF CONVOLUTION INTEGRAL: continuous MTF of the digital filter

Frequency response (MTF) of the digital filter

dxdpfxixhpfH eqeq 2exp,,

1

0

1

0

2exp~~~b hN

k

N

n

drn dxdpfxixnkxkxh

1

0

1

0

~2exp2exp2exp~2exp

N

k

drN

nn xkpfidpidxfxixxnpih

h

pfSVpfpDFRpfH rreq ,,

1

0

1

0

2exp~2exphh N

n

dn

N

nn xunpihxnpihpDFR

where

dxfxixf rr 2exp

dxpxixp dd 2exp

xNpfNN

xpf

xNpfpfSV

,sincdsin

sin,

Discrete frequency response (DFR) of the digital filter

Frequency response of the signal reconstruction device

Frequency response of the signal sampling device

Term responsible for filter space variance

DISCRETE REPRESENTATION OF CONVOLUTION INTEGRAL: continuous MTF of the digital filter

f

SV(f-p)=

p

DFR(p)=

Base band

DFR(p) in the base band [-1/2Δx , 1/2Δx ]

1

0

2exphN

n

dn xunpih

xNpfNN ,sincd

dxfxixf rr 2exp

dxpxixp dd 2exp

DISCRETE REPRESENTATION OF CONVOLUTION INTEGRAL: continuous MTF of the digital filter (ctnd)

Filter space variance is associated with finiteness of the number of signal samples:

pfxpfNNpfSVNN

,sincdlim,lim

Theorem 1.

DFT coefficients of the digital filter impulse response are samples of its Discrete Frequency Response

Theorem 2

DFR of the digital filter is a discrete sinc-interpolated function of its samples

1

0

1

0

;sincd2exphh N

n hhr

N

nn N

rxpNxpnihpDFR

where

h

N

nnr N

nrih

h

2exp1

0

Continuous frequency response of the digital filter

with PSF that computes signal local mean in the window of 5/64 of signal size

r

Filter base band x/1

Discrete sinc-function xNxN

xxxxN

sinsin

,sincd

Discrete sinc-function is a discrete analog of the continuous sampling sinc-function, which is a point spread function of the ideal low-pass filter. In distinction to the sinc-function, discrete sinc-function is a periodical function with period NΔx or 2NΔx depending on whether N is odd or even number. Its Fourier spectrum is a sampled version of the frequency response of the ideal low pass filter

N is an odd number

N is an even number

Continuous (red dots) and discrete (blue line) sinc-functions for odd and even number of samples N

NΔx 2NΔx

Frequency response of the ideal low pass filter (red) and Fourier transform of the discrete sinc-function (blue)

Discrete Representation of Integral Fourier Transform:

xukxaxaN

krk

1

0

x

a(x)

xu

Continuos sampled signal Continuous signal spectrum

dxfxiexpxaf

2

dxfxixukxafN

krk 2exp

1

0

fxukfia r

N

kk

2exp1

0

dxfxixf r 2exp

Signal spectrum samples

f

f

f

v

is frequency response of signal reconstruction device

dffvrff dr

fxvrukfiaN

kk 2exp

1

0

Second term is disregarded

fxvrukfiaN

kkr

2exp1

0

(7)

dffvrffxukfia dr

N

kk 2exp

1

0

dfxukfiffvrf dr 2exp

Cardinal sampling: , no sampling grid shifts ( )fNx 1

1

0

2exp1 N

kkr N

kria

NDiscrete Fourier Transform (DFT)

Discrete Representation of Integral Fourier Transform:DFT, Shifted DFT, DCT and DcST

Cardinal sampling: , sampling grid shiftsfNx 1 0, vu

Shifted Discrete Fourier Transform (SDFT(u,v)) Refs. 1:

N

uri

N

kri

N

kvia

N

N

kk

vur 2exp2exp2exp

1 1

0

,

DFT plays a fundamental role in digital holography thanks to the availability of Fast Fourier Transform (FFT) algorithm.

1

0

21cos

N

kkr r

N

ka

1

0

21sin

N

kkr r

N

ka

Important special cases of Shifted DFTs are Discrete Cosine (DCT) and Discrete cosine-Sine (DcST) Transforms

DCT DcST

DCT and DcST are SDFT(1/2,0) of signals that exhibit even and, correspondingly, odd symmetry ( ). They have fast computational algorithms that belong to the family of fast Fourier Transform algorithms. Using fast DCT and DCsT algorithms, one can efficiently implement fast boundary effect free digital convolution.

kNk aa 12

0, vu

(8)

(9)

(10)

Using SDFTs, one can carry out continuous spectrum analysis with sub-pixel resolution and arbitrary signal re-sampling with ideal discrete-sinc-interpolationRefs.1

1. L.P. Yaroslavsky, Shifted Discrete Fourier Transforms, In: Digital Signal Processing, Ed. by V. Cappellini, and A. G. Constantinides, Avademic Press, London, 1980, p. 69- 74.

Sampling in -scaled coordinates: , no sampling grid shifts ( ): fNx 1

Scaled Discrete Fourier Transform (ScDFT; it is also known under names “chirp-transform” and “Fractional Fourier TransformRef.2-4”):

1

0

;, 2exp1 N

kk

vur N

kria

N

N

uri

N

kri

N

kvia

N

N

kk

vur

2exp2exp2exp1 1

0

,

Sampling in -scaled coordinates: , sampling grid shifts ( ) fNx 1

Shifted Scaled Discrete Fourier Transform (ShScDFT,):

0, vu

Discrete Representation of Integral Fourier Transform:Shifted and Scaled DFT

0, vu

N

ri

N

kiDFT

N

kiaDFTIDFT

N

kria

N k

N

kkr

2221

0

expexpexp2exp1

For computational purposes, it is convenient to express ScDFT via canonical DFT that can be computed using FFT algorithms ( denotes element-wise, or Hadamard product of vectors).

(11)

(12)

This algorithm enables signal re-sampling, in arbitrary scale (sub-sampling and up-sampling), with ideal discrete sinc-interpolation

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2. Rabiner L.R., Schafer R.W., Rader C.M., The chirp z-transform algorithm and its applications, Bell System Tech. J., 1969, v. 48, 1249-12923. Rabiner L. R., Gold B., Theory and applications of digital signal processing, Prentice Hall, Englewood Cliffs, N.J., 19754. Bailey D. H., Swatztrauber P.N., The fractional Fourier Transform and applications, SIAM Rev., 1991, v. 33, 297-301

Point spread function PSFd (r,f) of the discrete Fourier analysis

N

rui

N

kri

N

kvia

NT

N

k

Tk

uvr

TT

2exp2exp2exp

1 1

0

,,

N

rui

N

kri

N

kvidxxukxxa

NT

N

k

Tdd

2exp2exp2exp1 1

0

N

rui

N

vrkidxxukxdffxif

NT

N

k

Tdd

2exp2exp2exp1 1

0

df

N

rui

N

vrkidxxukxfxif

N

N

k

TTdd

1

0

2exp2exp2exp1

1

0

,, 2exp2exp2exp,N

k

TTdd

vuDFA N

rui

N

vrkidxxukxfxifrPSF TT

dffrPSFfN

vuDFA

vur ,

1 ,,,,

PSFd (r,f) links signal spectrum and its samples obtained by signal DFTs of its samples:

f vu

r,

It can be found as:

(14)

(15)

1

0

2exp2exp2exp,N

k

TTddd N

rui

N

vrkidxxukxfxifrPSF

N

vNxf

Nu

N

rNuifxfN

vrNN T

dTdT

2

1

2

1

2

12exp;sincd

Point spread function of discrete Fourier analysis (ctnd)

fffNr

fNrfPSF d

21

;sincd,

21 Nvuu TdT xNf 1

Signal discretization

Input digital signal

Continuous input signal

Discrete Fourier Transforms(ScShDFTs)

Discrete Fourier Transformer

Input signal spectrum samples TT vu

r,,

dxxfxif dd 2exp

115 120 125 130 135 140

0

0.5

15-times zoomed spectrum of a sinusoidal signal with frequency f=129

115 120 125 130 135 140-0.2

0

0.2

0.4

0.6

0.8

5-times zoomed spectrum of a sinusoidal signal with frequency f=130.5

115 120 125 130 135 140-0.2

0

0.2

0.4

0.6

5-times zoomed spectrum of two above sinusoidal signals

115 120 125 130 135 140

0

0.5

15-times zoomed spectrum of a sinusoidal signal with frequency f=129

115 120 125 130 135 140

0

0.5

15-times zoomed spectrum of a sinusoidal signal with frequency f=130

115 120 125 130 135 140

0

0.5

1

5-times zoomed spectrum of two above sinusoidal signals

Resolving power of discrete spectrum analysis

Resolving spectra of two sinusoidal signals with close frequencies (129 and 130 , (left) and 129 and 130.5 (right) units)

xxkxNZ

xkxPSF d

FZ

,sincd)(,

PSF OF NUMERICAL RECONSTRUCTION OF HOLOGRAMS RECORDED IN FAR DIFFRACTION ZONE

For a hologram sampling device with frequency response Φ(.), point spread function of numerical reconstruction of Fourier holograms is obtained as:  

The point spread function is a periodical function of k:

;1 1 kPSFNgkPSF FZNgr

FZ r

(g is integer). It generates σN samples of object wavefront masked by the frequency response of the hologram recording and sampling device, the samples being taken with discretization interval

Δx/σ = λZ/ σSH =λZ/ σNΔf

within the object size So= λZ/ Δf.

The case σ =1 corresponds to a “cardinal” reconstructed object wavefront sampled with discretization interval Δx= λZ/ SH =λZ/ NΔf . When σ >1 , reconstructed discrete wavefront is σ -times over-sampled, or σ -times zoomed-in. One can show that in this case the reconstructed object wavefront is a discrete sinc-interpolated version of the “cardinal” one.

fNZSZx H where - wave length, - object-to-hologram distance; - number of hologram samples,

Z N, - hologram sampling interval f

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Discrete Representation of 2-D Integral Fourier Transform: 2-D Separable, Rotated and Affine DFTs

1

0

1

0 2,

121

,

1 2

2exp2exp1 N

k

N

llksr N

lsia

N

kri

NN

y

x

DC

BA

y

x~

~

xA fxAN ~/1 1 xB fyBN ~/1 2 yC fxCN ~/1 1 yD fyDN ~/1 2

1

0

1

0 2211,,

1 2

2expN

k

N

l DBCAlksr N

sl

N

rl

N

sk

N

rkia

y

x

y

x~

~

cossin

sincos

1

0

1

0,, sincos2exp

N

k

N

llksr N

rlsk

N

slrkia

x~ y~ xf yf

Separable cardinal sampling: , , with no shifts in coordinate systems that coinside with those of signal and its 2-D spectrum

xfNx 11

Separable 2-D Discrete Fourier Transform (2-D DFT)

xfNx 11

Sampling in a coordinate system affine transformed with respect to that of the signal

Affine Discrete Fourier Transform (AffDFT)

where ; ; ; ; ; - signal sampling intervals; , - signal spectrum sampling intervals

Sampling, with equal sampling intervals in coordinate systemrotated with respect to that of signal through angle θ

Rotated Discrete Fourier Transform (RotDFT)

(18)

(19)

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Original imageAvailability of shift and scale and rotation angle parameters in SDFT, ScDFT and RotDFT enables fast algorithms for image scaling, rotation and general re-sampling with ideal discrete sinc-interpolationRef. 5

Comparison of image rotation using bicubic spline (top) and discrete sinc-interpolation

72x5o-rotated image with sincd-interpolation (left), rotation error(middle) rotation error spectrum right)

72x5o-rotated image with “bicubic” interpolation (left), rotation error(middle) rotation error spectrum right)

Base band

Base band

5. L. Yaroslavsky, Digital Holography and Digital Image Processing, Kluwer Academic Publ., Boston, 2004

xukxaxaN

krk

1

0

Discrete Representation of Integral Fresnel Transform

dxfxixaf 2exp

Input signal sampled representation with sampling grid shift

1

0

2expN

kkr fvxufrxkia

dxfvxufrxkxixxi r 2expexp 2

dffvxufrxkxfiffi d

2expexp 2

For discrete representation of Fresnel integral, only this term is used:

1

0

2expN

kkr fvxufrxkia

The last two terms describe contribution of signal and transform sampling devices. In the assumption that PSFs of sampling and reconstruction devices are delta-functions they can be ignored

dffvrff dr Sampling signal transform with sampling grid shift

dx

Z

fxixaf

2~~exp~~

xufv

1

0

2/exp

N

kkr N

wrkia

Shifted Scaled Discrete Fresnel Transform (ShScDFrT):

22 fNZ

fNZx

Discrete Representation of Integral Fresnel Transform: Discrete Fresnel Transforms

Cardinal sampling: , with no shifts, in coordinate systems collinear with those of signal and its transform

fNZx

Canonical Discrete Fresnel Transform (DFrT):

1

0

2/exp

N

kkr N

rkia

22 fNZ

Sampling in -scaled coordinates : , with shifts , in coordinate systems collinear with those of signal and its transform

xu fv

/vuw

N

ri

N

kri

N

kia

N

N

kkr

221

02

2

exp2expexp1

DFrT can be expressed via DFT and computed using FFT:

fNZx Sampling in -scaled coordinates : , with shifts , in coordinate systems collinear with those of signal and its transform; chirp-function in the transform is ignored

xu fv

1

02

2

2expexp1 N

kkr N

wrki

N

kia

N

Shifted Scaled Partial Discrete Fresnel

Transform (ShScPDFrT):

(21)

(22)

(23)

(24)

22 fZ

Cardinal sampling: , with shifts , in coordinate systems collinear with those of signal and its transform

fNZx

Focal plane invariant Discrete Fresnel Transform (FPIDFrT):

1

0

2/2exp

N

kkr N

Nrkia

2/ Nvuw

Discrete Fresnel Transforms, ctnd:

Images are restored from a hologram copied from PDF file of the paper: E.Ciche, P. Marquet, Chr. Depeursinge, Spatial filtering of zero order and twin-image elimination in digital off-axis holography, Appl. Opt., v. 30, No. 23, Aug. 2000

Numerical reconstruction of images on different distances from a hologram using canonic DFrT

Numerical reconstruction of images on different distances from a hologram using Focal plane invariant DFrT

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Invertibilityof Discrete Fresnel Transforms and frincd-function

If one computes, for a sampled signal , , , direct Shifted DFrT with depth and shift parameters ( ) and then inverts it with inverse Shifted DFrT with depth and shift parameters ( ), one obtains

ka 1,...,1,0 Nk w,

w,

1

0

22, 2;;frincdexpexp

1 N

nn

wk qNwknqN

N

wnia

N

wki

Na

Where , and 22 11 q www

N

xri

N

qri

NxqN

N

r

2expexp1

;;frincd1

0

2

is a frincd-function, an analog of sincd-function of the DFT, identical to it when .

In numerical reconstruction of holograms, frincd-function is a convolution kernel that links object and its “out of focus” reconstruction.

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Discrete frinc-function and its focal plane invariant version: dependence on focusing parameter q

50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1abs(frincd)

q=0

q=0.02

q=0.08

q=0.18q=0.32

q=0.5

1

0

2 2expexp1

;;frincdN

k N

kriqki

NrqN

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k

abs(Frinc(256,q))

q=0

q=0.0001

q=0.0002

q=0.0004

q=0.0008

q=0.0016

N

kri

N

Nkqki

NxqN

N

k

2expexp1

;;frincd1

0

x

10

0*q

Frincd(256,q,x) for q=0:0.01:2.56

50 100 150 200 250

50

100

150

200

250

q=1

Frincd-function: approximations

The value of the focusing parameter q=1

is the threshold after which aliasing begins

In numerical reconstruction of

holograms, q=λZ/NΔf2

As it was shown in Ref.YaroChina,

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0.2

0.4

0.6

0.8

1

z

Frincd(2048;0.5;s-Vq):Magnitude

0 200 400 600 800 1000 1200 1400 1600 1800 2000-3

-2

-1

0

1

2

3

z

Frincd(2048;0.5;s-Vq):PhasePhase

Ratio of the left and right parts of the equation

N

ri

N

irN

2

exp;1;frincd

1rectexp;;frincd

2

Nq

r

qN

ri

Nq

irqN

For integer r,

Magnitude

Discrete representation of integral Fresnel Transform: Convolutional Discrete Fresnel Transform

y 12

dppfipidxpxixadxfxixaf 2expexp2expexp 22

Integral Fresnel Transform can also be regarded as a convolution and represented through two Fourier Transforms:

1

0

21

0

221

0

;;frincdexp2exp1 N

kk

N

s

N

kkr kwrNa

N

sis

N

wrkia

N

Assuming that sampling intervals of signal and its transform are identical:x

Similarly to the above discrete Fourier and discrete Fresnel transforms ConvDFrT is an orthogonal transform with inverse ConvDFrT defined as

1

0

21

0

221

0

;;frincdexp2exp1 N

rr

N

s

N

rrk kwrN

N

sis

N

wrki

Na

When , ConvDFrT degenerates into an identical transform. When , it is identical to the canonical DFrT. Although ConvDFrT can be inverted for any , in numerical reconstruction of holograms it can be applied only for . If , aliasing may appear in form of overlapping periodical copies of the reconstruction result.

02 12

12 12

f

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(27)

PSF of reconstruction of holograms recorded in near diffraction zone: Fourier reconstruction algorithm

In digital holography, DFrT is used, under a name of Fourier reconstruction algorithm, for numerical reconstruction of optical holograms recorded in near diffraction zone. This process can be characterized by its point spread function (PSF) that links object wave front and object samples obtained from samples of its hologram in the numerical reconstruction.

Similarly to the above discussed case of DFT, PSF of numerical reconstruction of holograms by DFrT depends on parameters of the algorithm and of PSF of the hologram sampling device. General formulas are presented in Ref. 5. Here we, as an illustration, provide PSF of the reconstruction process for the case when point spread function of the hologram sampling device can be regarded as a delta-functions. In this case for “in focus” reconstruction

xxkxNN

kxxikxPSF o

NZFour

;sincd

/exp;, 2

22

PSF, “out of focus” reconstruction:

PSF, “in focus” reconstruction

xxkxN

N

NkNxxikxPSF oNZ

;

11;frincd

2/12/1/exp;, 22

0

2222

where is focusing parameter of the reconstruction algorithm and is its value that corresponds to the “in focus” reconstruction.

2 20

5. L. Yaroslavsky, F. Zhang, I. Yamaguchi, Point spread functions of digital reconstruction of digitally recorded holograms, In: Proceedings of SPIE Vol. 5642 , Information Optics and Photonics Technology , Guoguang Mu, Francis T. Yu, Suganda Jutamulia, Editors Jan 2005;

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As one can see aliasing free object size is equal to the period . Given size of the hologram , the period is . Therefore aliasing free object reconstruction using Fourier reconstruction algorithm is possible if

Otherwise the algorithm works as a “magnifying glass” capable of reconstructing of small -th fraction of the of object from -th fraction of the hologram provided the rest of the hologram is zeroed.

xNSo fNSh Ho SS 2

122 fNZ 2

2

Fourier reconstruction

Fourier reconstruction of the central part of

the hologram free of aliasing

Convolution reconstruction

Hologram reconstruction: Fourier algorithm vs Convolution algorithm

Z=33mm; μ2=0.2439

Z=83mm; μ2 =0.6618

Z=136mm; μ2 =1

Hologram courtesy Dr. J. Campos, UAB, Barcelona, Spain

Aliasing artifacts

All restorations are identical

Image is destroyed due to the aliazing

L. Yaroslavsky, Ph.D., Dr. Sc. Phys&Math,

ProfessorDept. of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv

University, Tel Aviv, Israelwww.eng.tau.ac.il/~yaro