mathematics and computation in imaging science and information processing july december 2003 l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Mathematics and Computation in Imaging Science and Information Processing July-December, 2003 PowerPoint Presentation
Download Presentation
Mathematics and Computation in Imaging Science and Information Processing July-December, 2003

Loading in 2 Seconds...

play fullscreen
1 / 39

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


  • 114 Views
  • Uploaded on

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),

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Mathematics and Computation in Imaging Science and Information Processing July-December, 2003' - miller


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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),

Zuowei Shen (NUS),

Chi-Wang Shu (Brown University)

Mathematics and Computation in Imaging Science and Information ProcessingJuly-December, 2003
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

Conferences
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

Confirmed Plenary Speakers for Wavelet Conference
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)

Workshops
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.

Tutorials
membership applications
To stay in the program longer than two weeks

Please visit http://www.ims.nus.edu.sg

for more information

Membership Applications
slide7

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)

slide8

Outline of the talk

Part I: Problem Setting

Part II: Wavelet Algorithms

slide9

pixel intensity

= matrix entry

What is an image?

image = matrix

Resolution = size of the matrix

slide10

Resolution = 64  64

Resolution = 256 256

I. High-Resolution Image Reconstruction:

slide11

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.

slide12

#4

#2

#1

relay lenses

partially silvered mirrors

Boo and Bose (IJIST, 97):

taking lens

CCD sensor

array

slide13

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

slide14

Four 64 64 images merged into one by permutation:

Observed high-resolution image by permutation

slide15

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.

slide16

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.

slide17

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.

slide18

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.
slide19

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  :

slide20

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...

slide21

We have

or

Do it in the Fourier domain. Note that

slide22

(i) Choose

(ii) Iterate until convergence:

PropositionSuppose that and nonzero almost everywhere. Then for arbitrary .

Generic Wavelet Algorithm:

slide23

(i) Choose

(ii) Iterate until convergence:

Regularization:

Damp the high-frequency components in the current iterant.

Wavelet Algorithm I:

slide24

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

slide25

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,

slide26

(i) Choose

(ii) Iterate until convergence:

Wavelet Algorithm II:

Where T is a wavelet thresholding processing .

slide27

Tikhonov Algorithm I Algorithm II

4  4 sensor array:

Original LR Frame Observed HR

slide28

Algorithm II

Tikhonov

4  4 sensor array:

slide29

Numerical Examples:

22 sensor array: 1 level up

44sensor array: 2 level up

slide30

1-D Example: Signal from Donoho’s Wavelet Toolbox.Blurred by 1-D filter.

Original Signal Observed HR Signal

Tikhonov Algorithm II

slide31

Displacement errore x

Displaced low-resolution pixel

Calibration Error:

High-resolution

pixels

Problem no longer spatially invariant.

Ideal low-resolution pixel position

slide32

The lower pass filter is perturbed

The wavelet algorithms can be modified

slide33

Reconstruction for 4  4 Sensors: (2 level up)

Original LR Frame Observed HR

Tikhonov Wavelets

slide34

Tikhonov

Wavelets

Reconstruction for 4  4 Sensors: (2 level up)

slide35

Numerical Results:

2  2 sensor array (1 level up) with calibration errors:

4  4 sensor array (2 level) with calibration errors:

slide36

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)

slide37

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)

slide38

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.
slide39

Reconstructed Image:

Observed LR Final Solution