Unitary Extension Principle: Ten Years After

Zuowei ShenZuowei ShenDepartment of MathematicsDepartment of Mathematics

National University of SingaporeNational University of Singapore

Outline

Unitary Extension Principle (UEP) Applications in Image Processing New Development

Wavelet Tight Frame

Let be countable. is a tight frame if

It is equivalent to

A wavelet system is the collection of the dilations and the shifts of a finite set

Unitary Extension Principle

Function is refinable with mask if

Let , where , with wavelet masks

Define .

Unitary Extension Principle: (Ron and Shen, J. Funct. Anal., 1997),

is a tight frame provided

Why the Unitary Extension Principle?

Constructions of waveletsbecome painless

Symmetric spline wavelets with short support;

Wavelets for practical problems.

Lead to Pseudo-splines

Provide a better approximation order for truncated wavelet series.

Splines, orthonormal and interpolatory refinable functions

are special cases of pseudo-splines.

First Introduced in: I. Daubechies, B, Han, A. Ron and Z. Shen, Framelets: MRA-based

constructions of wavelet frames, Applied and Computation Harmonic Analysis, 14, 1—46, 2003.

Regularity analysis and … B. Dong and Z. Shen Pseudo-splines, wavelets and framelets, Applied

and Computation Harmonic Analysis, 22 (1), 78—104, 2007.

I. Daubechies, B, Han, A. Ron and Z. Shen, Framelets: MRA-based constructions of wavelet frames, Applied and Computation Harmonic Analysis, 14, 1—46, 2003

C. K. Chui, W. He, J. Stöckler, Compactly supported tight and sibling frames with maximum vanishing moments, Applied and Computation Harmonic Analysis, 13, 224—262, 2002

Lead to Oblique Extension Principle

Nonstationary tight frames

Nonstationary tight frames have been studied extensively by

C. Chui, W. He and J. Stockler.

Compactly supported, symmetric tight frames with infinite order of smoothness and vanishing moment by using (nonstationary pseudo-splines).

B. Han, Z. Shen, Compactly Supported Symmetric Wavelets With Spectral Approximation Order, (2006).

Characterization of spaces

Characterization of various space norms by wavelet frame coefficients has been studied by L. Borup, R. Grinbonval, and M. Nieslsen; Y. Hur and A. Ron

Link the characterizations to frames in Sobolev spaces with their duals in dual Sobolev spaces.

B. Han and Z. Shen, Dual Wavelet Frames and Riesz Bases in Sobolev Spaces, preprint (2007)

Application I:

Filling missing data

Inpainting

Given 64 x 64 image First approximated 128 x 128image

Given 64 X 64 image First approximated 128 X 128 image Result 128 X 128 image

Given 64 X 64 image

Result 128 X 128 image Given 128 X 128 image

Matrix Representations

Let rows of be frame, i.e. Decomposition: Reconstruction: can be generated by tight frame filters obtained

via UEP

Algorithm

Decomposition Threshold

ReconstructionReplace the data on by the known data g

B-Spline tight frame derived by UEP is used.

Convergence and minimization

the sequence converges to a solution of the minimization problem

Convergence and minimization the sequence converges to a solution of

the minimization

J. Cai, R. Chan, Z. Shen, A Framelet-based Image Inpainting Algorithm, Preprint (2006)

Numerical Experiments

Observed Image Framelet-Based Method

PSNR=33.83dB

PDE Method

PSNR=32.91dB

Numerical Results

Observed Image Framelet-Based Method

PSNR=33.10dB

Minimizing the functional without penalty term

PSNR=30.70dB

Application II:

Deconvolution

SettingQuestion: Given

How to find ?How to find ?

Regularization Methods: Solving a system of linear Regularization Methods: Solving a system of linear equations;equations;

•Designing a tight (or bi) frame system with being one of the Designing a tight (or bi) frame system with being one of the masks using UEP;masks using UEP;

•Reducing to the ``problem of recovering wavelet coefficients’’;Reducing to the ``problem of recovering wavelet coefficients’’;

•Deriving an algorithm from the tight frame system designed;Deriving an algorithm from the tight frame system designed;

•Proving convergence of the algorithm;Proving convergence of the algorithm;

•Analyzing the minimization properties of the solution.Analyzing the minimization properties of the solution.

A. Chai and Z. Shen, Deconvolution by tight framelets, A. Chai and Z. Shen, Deconvolution by tight framelets, Numerische MathematikNumerische Mathematik to appear . to appear .

Ideas Ideas

Algorithm

Decomposition Replace the known data g

ThresholdReconstruction

Project onto the set of non-negative

vectors

• R. Chan, T. Chan, L. Shen, Z. Shen: Wavelet algorithms for highR. Chan, T. Chan, L. Shen, Z. Shen: Wavelet algorithms for high resolution image reconstruction, SIAM Journal on Scientific resolution image reconstruction, SIAM Journal on Scientific Computing, 24 (2003) 1408-1432Computing, 24 (2003) 1408-1432..

Ideas started inIdeas started in

Using bi-frames derived from biorthogonal wavelets, it performs Using bi-frames derived from biorthogonal wavelets, it performs better than the regularization method.better than the regularization method.

Other’s work

Wavelet-Vaguelette decomposition by Donoho

Mirror wavelet method by Mallat et al.

Wavelet Galerkin method, inverse truncated operator under wavelet basis by Cohen et al.

Iterative threshold method, sparse representation of solution under wavelet basis given by Daubechies et al.

High-Resolution Image Reconstruction

Resolution = 64 64 Resolution = 256 256

Four low resolution images (64 64) of the same scene.

Each shifted by sub-pixel length.

Construct a high-resolution image (128 128) from them.

#2

#4

#1

taking lens

CCD sensorarray

relay lenses

partially silvered mirrors

Modeling:

High-resolution

pixels

4

1

2

1

4

1

2

11

2

1

4

1

2

1

4

1

LR image: the down samples of observed image at different sub-pixel position. Observed image: HR image passing through a low-pass filter a. Reducing to a deconvolution problem

Reconstruction high resolution image

Original LR Frame Observed HR

Regularization Wavelets

Infrared Astronomy Imaging:

Chopped and Nodded Process

Numerical Results: 1D Signals

K=37, N=128 Original Projected Landweber Framelet Method

Ex 1

Ex 2

Numerical Results: Real Images

Observed Image from United

Kingdom Infra-Red Telescope

Projected Landweber’s

Iteration

Framelet-Based Method

J. Cai, R. Chan, L. Shen, Z. Shen, Restoration of Chopped and Nodded Images by Framelet, preprint (2006).

Restoring chopped and nodded images by tight frames, Proc. SPIE Symposium on Advanced Signal Processing: Algorithms, Architectures, and Implementations, Vol. 5205, 310-319, San Diego CA, August, 2003.

Another Example

One of the frame in a video

Before enhancement

After enhancement

R. Chan, Z. Shen, T. Xia A framelet algorithm for enchancing video stills, R. Chan, Z. Shen, T. Xia A framelet algorithm for enchancing video stills, Applied and Computational Harmonic AnalysisApplied and Computational Harmonic Analysis

Reference frame

t

Displacement error

Improving resolution

of reference

frame

704-by-578 image of f100 by bilinear interpolation

704-by-578 image of f100 by tight frame method using 20 frames from the movie

Bilinear method Tight frame method

Video Enhancement

New Development Two systems: one represents piecewise

smooth function sparsely, the other represents the texture sparsely

Two systems: one is a frame (or Riesz basis) in one space and the other in its dual space.

It is better to have such two systems

satisfying some `dual’ relations

New Development

Let

be a compactly supported refinable function with some

smoothness. Define

Can the corresponding wavelet system be a Riesz

basis for some space?

New Development

Let

be a compactly supported refinable function

with some smoothness. Will

form a frame in some space?

New Development

B. Han and Z. Shen, Dual Wavelet Frames and Riesz Bases in Sobolev Spaces, preprint (2007)

This paper takes a new approach to handle all the questions raised before. Most of questions are solved, many new interesting directions are opened.

Thanks!