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

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Mathematics and Computation in Imaging Science
and Information ProcessingJuly-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),
• Zuowei Shen (NUS),
• Chi-Wang Shu (Brown University)

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Conferences

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

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Confirmed Plenary Speakers for Wavelet Conference

• Albert Cohen
• Wolfgang Dahmen
• Ingrid Daubechies
• Ronald DeVore
• David Donoho
• Rong-Qing Jia

• Yannis Kevrekidis
• Amos Ron
• Peter Schröder
• Gilbert Strang
• Martin Vetterli

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Workshops

• 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)

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Tutorials

• 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.

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Membership Applications

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

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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)
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Outline of the talk
Part I Problem Setting Part II
Wavelet Algorithms
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What is an image?
image matrix
Resolution size of the matrix
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I. High-Resolution Image Reconstruction
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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.
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Boo and Bose (IJIST, 97)
taking lens
CCD sensor array
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Four 2 ? 2 images merged into one 4 ? 4 image
Observed high- resolution image
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Four 64? 64 images merged into one by permutation
Observed high-resolution image by permutation
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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.
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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.
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The problem L f g is ill-conditioned.
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• One can use unitary extension principle to
  obtain a set of tight frame systems.

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Let ? be the refinable function with refinement
mask a, i.e.
Let ? d be the dual function of ?
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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...
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Do it in the Fourier domain. Note that
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Generic Wavelet Algorithm
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Regularization
Damp the high-frequency components in the current
iterant.
Wavelet Algorithm I
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Matrix Formulation
The Wavelet Algorithm I is the stationary
iteration for
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Wavelet Thresholding Denoising Method
Before reconstruction,
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Wavelet Algorithm II
Where T is a wavelet thresholding processing .
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4 ? 4 sensor array
Original LR Frame Observed HR
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4 ? 4 sensor array
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Numerical Examples
2?2 sensor array 1 level up
4?4 sensor array 2 level up
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1-D Example Signal from Donohos Wavelet
Toolbox.Blurred by 1-D filter.
Original Signal
Observed HR Signal
Tikhonov
Algorithm II
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Calibration Error
High-resolution pixels
Problem no longer spatially invariant.
Ideal low-resolution pixel position
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The lower pass filter is perturbed The wavelet
algorithms can be modified
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Reconstruction for 4 ? 4 Sensors (2 level up)
Original LR Frame Observed HR
Tikhonov
Wavelets
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Reconstruction for 4 ? 4 Sensors (2 level up)
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Numerical Results
2 ? 2 sensor array (1 level up) with calibration
errors
4 ? 4 sensor array (2 level) with calibration
errors
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Super-Resolution Not enough low-resolution
frames.

1. Apply an interpolatory subdivision scheme to
   obtain the missing frames.
2. Generate the observed high-resolution image w.
3. Solve for the high-resolution image u.
4. From u, generate the missing low-resolution
   frames.
5. Then generate a new observed high-resolution
   image g.
6. Solve for the final high-resolution image f.

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Reconstructed Image
Observed LR
Final Solution