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Mathematics and Computation in Imaging Science and Information Processing July-December, 2003

Organized by Institute of Mathematical Sciences and Center for Wavelet. Approximation, and Information Processing, National University of Singapore. Collaboration with the Wavelet Center for Ideal Data Representation. Co-chairmen of the organizing committee: Amos Ron (UW-Madison),

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Mathematics and Computation in Imaging Science and Information Processing July-December, 2003

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  1. Organized by Institute of Mathematical Sciences and Center for Wavelet. Approximation, and Information Processing, National University of Singapore. Collaboration with the Wavelet Center for Ideal Data Representation. Co-chairmen of the organizing committee: Amos Ron (UW-Madison), Zuowei Shen (NUS), Chi-Wang Shu (Brown University) Mathematics and Computation in Imaging Science and Information ProcessingJuly-December, 2003

  2. Wavelet Theory and Applications: New Directions and Challenges, 14 - 18 July 2003 Numerical Methods in Imaging Science and Information Processing, 15 -19 December 2003 Conferences

  3. Albert Cohen Wolfgang Dahmen Ingrid Daubechies Ronald DeVore David Donoho Rong-Qing Jia Yannis Kevrekidis Amos Ron Peter Schröder Gilbert Strang Martin Vetterli Confirmed Plenary Speakers for Wavelet Conference

  4. IMS-IDR-CWAIP Joint Workshop on Data Representation, Part I on 9 – 11, II on 22 - 24 July 2003 Functional and harmonic analyses of wavelets and frames, 28 July - 1 Aug 2003 Information processing for medical images, 8 - 10 September 2003 Time-frequency analysis and applications, 22- 26 September 2003 Mathematics in image processing, 8 - 9 December 2003 Industrial signal processing (TBA) Digital watermarking (TBA) Workshops

  5. A series of tutorial sessions covering various topics in approximation and wavelet theory, computational mathematics, and their applications in image, signal and information processing. Each tutorial session consists of four one-hour talks designed to suit a wide range of audience of different interests. The tutorial sessions are part of the activities of the conference or workshop associated with. Tutorials

  6. To stay in the program longer than two weeks Please visit http://www.ims.nus.edu.sg for more information Membership Applications

  7. Wavelet Algorithms for High-Resolution Image Reconstruction Zuowei Shen Department of Mathematics National University of Singapore http://www.math.nus.edu.sg/~matzuows Joint work with (accepted by SISC) T. Chan (UCLA), R.Chan (CUHK) and L.X. Shen (WVU)

  8. Outline of the talk Part I: Problem Setting Part II: Wavelet Algorithms

  9. pixel intensity = matrix entry What is an image? image = matrix Resolution = size of the matrix

  10. Resolution = 64  64 Resolution = 256 256 I. High-Resolution Image Reconstruction:

  11. Four low resolution images (64  64) of the same scene. Each shifted by sub-pixel length. Construct a high-resolution image (256 256) from them.

  12. #4 #2 #1 relay lenses partially silvered mirrors Boo and Bose (IJIST, 97): taking lens CCD sensor array

  13. a1 a2 b1 b2 By permutation a3 a4 b3 b4 a1 b1 a2 b2 c1 d1 c2 d2 a3 b3 a4 b4 c1 c2 d1 d2 c3 d3 c4 d4 c3 c4 d3 d4 Four low resolution images Four 2  2 images merged into one 4 4 image: Observed high- resolution image

  14. Four 64 64 images merged into one by permutation: Observed high-resolution image by permutation

  15. Low-resolution pixel Modeling Consider: High-resolution pixels Observed image: HR image passing through a low-pass filter a. LR image: the down samples of observed image at different sub-pixel position.

  16. L f= g , After modeling and adding boundary condition, it can be reduced to : Where L is blurring matrix, g is the observed image and f is the original image.

  17. Regularization is required: Here R can be I, . It is called Tikhonov method ( or the least square ) g The problem L f = g is ill-conditioned.

  18. Wavelet Method • Let â be the symbol of the low-pass filter. Assume: • can be found such that • One can use unitary extension principle to obtain a set of tight frame systems.

  19. We can express the true image as where v() are the pixel values of the high-resolution picture. Let  be the refinable function with refinement mask a, i.e. Let  d be the dual function of  :

  20. The pixel values of the observed image are given by The observed function is The problem is to find v( ) from (a *v)(). From 4 sets low resolution pixel values reconstruct f, lift 1 level up. Similarly, one can have 2 level up from 16 set...

  21. We have or Do it in the Fourier domain. Note that

  22. (i) Choose (ii) Iterate until convergence: PropositionSuppose that and nonzero almost everywhere. Then for arbitrary . Generic Wavelet Algorithm:

  23. (i) Choose (ii) Iterate until convergence: Regularization: Damp the high-frequency components in the current iterant. Wavelet Algorithm I:

  24. Different between Tikhonov and Wavelet Models: • Ld instead of L*. • Wavelet regularization operator. Both penalize high-frequency components uniformly by . Matrix Formulation: The Wavelet Algorithm I is the stationary iteration for

  25. Decompose the n-th iterate, i.e. , into different scales: ( This gives a wavelet packet decomposition of n-th iterate.) • Denoise these coefficients of the wavelet packet by thresholding method. Wavelet Thresholding Denoising Method: Before reconstruction,

  26. (i) Choose (ii) Iterate until convergence: Wavelet Algorithm II: Where T is a wavelet thresholding processing .

  27. Tikhonov Algorithm I Algorithm II 4  4 sensor array: Original LR Frame Observed HR

  28. Algorithm II Tikhonov 4  4 sensor array:

  29. Numerical Examples: 22 sensor array: 1 level up 44sensor array: 2 level up

  30. 1-D Example: Signal from Donoho’s Wavelet Toolbox.Blurred by 1-D filter. Original Signal Observed HR Signal Tikhonov Algorithm II

  31. Displacement errore x Displaced low-resolution pixel Calibration Error: High-resolution pixels Problem no longer spatially invariant. Ideal low-resolution pixel position

  32. The lower pass filter is perturbed The wavelet algorithms can be modified

  33. Reconstruction for 4  4 Sensors: (2 level up) Original LR Frame Observed HR Tikhonov Wavelets

  34. Tikhonov Wavelets Reconstruction for 4  4 Sensors: (2 level up)

  35. Numerical Results: 2  2 sensor array (1 level up) with calibration errors: 4  4 sensor array (2 level) with calibration errors:

  36. Super-resolution: not enough frames Example: 4  4 sensor with missing frames: (0,1) (0,3) (0,0) (0,2) (1,0) (1,2) (1,1) (1,3) (2,1) (2,3) (2,0) (2,2) (3,0) (3,2) (3,1) (3,3)

  37. Super-resolution: not enough frames Example: 4  4 sensor with missing frames: (0,1) (0,3) (1,0) (1,2) (2,1) (2,3) (3,0) (3,2)

  38. Super-Resolution: Not enough low-resolution frames. • Apply an interpolatory subdivision scheme to obtain the missing frames. • Generate the observed high-resolution image w. • Solve for the high-resolution image u. • From u, generate the missing low-resolution frames. • Then generate a new observed high-resolution image g. • Solve for the final high-resolution image f.

  39. Reconstructed Image: Observed LR Final Solution

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