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OPTICAL TRANSFORMS IN DIGITAL HOLOGRAPHY

L. Yaroslavsky. OPTICAL TRANSFORMS IN DIGITAL HOLOGRAPHY. Holo-05, Varna, May 21-25, 2005. a ( x ). x. f. F. F.

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OPTICAL TRANSFORMS IN DIGITAL HOLOGRAPHY

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  1. L. Yaroslavsky OPTICAL TRANSFORMS IN DIGITAL HOLOGRAPHY Holo-05,Varna, May 21-25, 2005

  2. a(x) x f F F 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 Transform imaging: Integral Fourier Transform Transform imaging: Integral Fresnel Transform Input plane Output plane a(x) x f Z

  3. Input digital signal Output digital signal Discretization and quantization Digital signal transformations in computer Reconstruction of the continuous signal Equivalent continuous transformation Continuous output signal (output image or computer generated hologram) Continuous input signal (image or hologram) The conformity and mutual correspondence principles between analogue and digital signal transformations 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

  4. Mathematical formulation of signal discretization and reconstruction Let be a continuous signal as a function of spatial co-ordinates given by a vector , (1) and are discretization and reconstruction basis functions defined in the discretization and reconstruction devices coordinates and , respectively, is a vector of the sampling intervals, is a vector of signal sample indices. At the signal discretization, signal samples are computed as (2) assuming certain relationship between signal and sampling device coordinate systems and . Signal reconstruction from the set of their samples is described as (3) It is understood that although result of the signal reconstruction from its discrete representation obtained according to Eq. 3 is not, in general, identical to the initial signal , it can serve as a substitute for the initial signal in the given application. According to the conformity principle, Eqs. 2 and 3 form the base for adequate discrete representation of signal transformations.

  5. DISCRETE REPRESENTATION OF THE CONVOLUTION INTEGRAL (4) Discretization basis functions Signal reconstruction basis functions where and are shifts, in fractions of the discretization interval, of sample positions with respect to input and output signal coordinate systems, correspondingly. Signals belong to the class: For such signals, Samples of are: (5) For digital filtering, this equation is replaced by where (6) 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 .

  6. Input digital signal Output digital signal Continuous input signal Continuous output signal Digital filter Signal discretization Reconstruction of the continuous signal Equivalent continuous filter reconstructed from its N samples DISCRETE REPRESENTATION OF CONVOLUTION INTEGRAL: continuous PSF of a digital filter 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. where and

  7. DISCRETE REPRESENTATION OF CONVOLUTION INTEGRAL: continuous MTF of the digital filter Frequency response (MTF) of the digital filter where 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

  8. p Base band f DFR(p) in the base band [-1/2Δx , 1/2Δx] SV(f-p)= DFR(p)= DISCRETE REPRESENTATION OF CONVOLUTION INTEGRAL: continuous MTF of the digital filter

  9. Continuous frequency response of the digital filter with PSF that computes signal local mean in the window of 5/64 of signal size Filter base band 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: 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 where

  10. Discrete sinc-function 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 or2NΔx 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 2NΔx NΔx Continuous (red dots) and discrete (blue line) sinc-functions for odd and even number of samples N Frequency response of the ideal low pass filter (red) and Fourier transform of the discrete sinc-function (blue)

  11. a(x) u x v f Discrete Representation of Integral Fourier Transform: Continuos sampled signal Continuous signal spectrum is frequency response of signal reconstruction device Signal spectrum samples Second term is disregarded (7)

  12. Discrete Representation of Integral Fourier Transform: DFT, Shifted DFT, DCT and DcST • Cardinal sampling: , no sampling grid shifts ( ) (8) Discrete Fourier Transform (DFT) DFT plays a fundamental role in digital holography thanks to the availability of Fast Fourier Transform (FFT) algorithm. • Cardinal sampling: , sampling grid shifts Shifted Discrete Fourier Transform (SDFT(u,v)) Refs. 1: (9) Using SDFTs, one can carry out continuous spectrum analysis with sub-pixel resolution and arbitrary signal re-sampling with ideal discrete-sinc-interpolationRefs.1 • Important special cases of Shifted DFTs are Discrete Cosine (DCT) and Discrete cosine-Sine (DcST) Transforms (10) 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. 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.

  13. Discrete Representation of Integral Fourier Transform: Shifted and Scaled DFT • Sampling in -scaled coordinates: , no sampling grid shifts ( ): Scaled Discrete Fourier Transform (ScDFT; it is also known under names “chirp-transform” and “Fractional Fourier TransformRef.2-4”): (11) 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). (12) This algorithm enables signal re-sampling, in arbitrary scale (sub-sampling and up-sampling), with ideal discrete sinc-interpolation • Sampling in -scaled coordinates: , sampling grid shifts ( ) Shifted Scaled Discrete Fourier Transform (ShScDFT,): (13) 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, 1249-1292 3. Rabiner L. R., Gold B., Theory and applications of digital signal processing, Prentice Hall, Englewood Cliffs, N.J., 1975 4. Bailey D. H., Swatztrauber P.N., The fractional Fourier Transform and applications, SIAM Rev., 1991, v. 33, 297-301

  14. Point spread function PSFd (r,f) of the discrete Fourier analysis PSFd (r,f) links signal spectrum and its samples obtained by signal DFTs of its samples: (14) It can be found as: (15)

  15. Point spread function of discrete Fourier analysis (ctnd) Input digital signal Signal discretization Discrete Fourier Transforms (ScShDFTs) Continuous input signal Input signal spectrum samples Discrete Fourier Transformer

  16. 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)

  17. 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: (16) where - wave length, - object-to-hologram distance; - number of hologram samples, , - hologram sampling interval The point spread function is a periodical function of k: (17) (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.

  18. Discrete Representation of 2-D Integral Fourier Transform:2-D Separable, Rotated and Affine DFTs • Separable cardinal sampling: , , with no shifts in coordinate systems that coinside with those of signal and its 2-D spectrum Separable 2-D Discrete Fourier Transform (2-D DFT) (18) • Sampling in a coordinate system affine transformed with respect to that of the signal Affine Discrete Fourier Transform (AffDFT) (19) where ; ; ; ; ; - signal sampling intervals; , - signal spectrum sampling intervals • Sampling, with equal sampling intervals in coordinate system rotated with respect to that of signal through angle θ Rotated Discrete Fourier Transform (RotDFT) (20)

  19. Original image Availability 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 72x5o-rotated image with “bicubic” interpolation (left), rotation error(middle) rotation error spectrum right) Base band 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) Base band 5. L. Yaroslavsky, Digital Holography and Digital Image Processing, Kluwer Academic Publ., Boston, 2004

  20. Discrete Representation of Integral Fresnel Transform Sampling signal transform with sampling grid shift Input signal sampled representation with sampling grid shift 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 For discrete representation of Fresnel integral, only this term is used:

  21. 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 (21) Canonical Discrete Fresnel Transform (DFrT): DFrT can be expressed via DFT and computed using FFT: (22) • Sampling in -scaled coordinates : , with shifts , in coordinate systems collinear with those of signal and its transform (23) Shifted Scaled Discrete Fresnel Transform (ShScDFrT): • Sampling in -scaled coordinates : , with shifts , in coordinate systems collinear with those of signal and its transform; chirp-function in the transform is ignored Shifted Scaled Partial Discrete Fresnel Transform (ShScPDFrT): (24)

  22. Discrete Fresnel Transforms, ctnd: • Cardinal sampling: , with shifts , in coordinate systems collinear with those of signal and its transform (24) Focal plane invariant Discrete Fresnel Transform (FPIDFrT): 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 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

  23. 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 Where , and (25) 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.

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

  25. As it was shown in Ref.YaroChina, Frincd(256,q,x) for q=0:0.01:2.56 Magnitude 50 100 100*q Phase 150 200 Ratio of the left and right parts of the equation 250 For integer r, 50 100 150 200 250 x Frincd-function: approximations q=1 The value of the focusing parameter q=1 is the threshold after which aliasing begins In numerical reconstruction of holograms, q=λZ/NΔf2

  26. Discrete representation of integral Fresnel Transform: Convolutional Discrete Fresnel Transform Integral Fresnel Transform can also be regarded as a convolution and represented through two Fourier Transforms: Assuming that sampling intervals of signal and its transform are identical: (26) Similarly to the above discrete Fourier and discrete Fresnel transforms ConvDFrT is an orthogonal transform with inverse ConvDFrT defined as (27) 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.

  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 PSF, “in focus” reconstruction (28) PSF, “out of focus” reconstruction: where is focusing parameter of the reconstruction algorithm and is its value that corresponds to the “in focus” reconstruction. (29) 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. 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;

  28. Hologram reconstruction: Fourier algorithm vs Convolution algorithm Fourier reconstruction of the central part of the hologram free of aliasing Fourier reconstruction Convolution reconstruction Image is destroyed due to the aliazing Z=33mm; μ2=0.2439 Aliasing artifacts Z=83mm; μ2 =0.6618 All restorations are identical Z=136mm; μ2 =1 Hologram courtesy Dr. J. Campos, UAB, Barcelona, Spain

  29. 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

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