Structured Matrix Problems and Algorithms

from Variational Image Deblurring

by

Ke Chen

Thu, 26 Jan 2012LAA and FM @ Liverpool

Joint work with R H Chan (CUHK), Bryan Williams (UoL), Yalin

Zheng and Simon Harding (UoL Eye and Vision Science)

Centre for Mathematical Imaging

Techniques

www.liv.ac.uk/cmit

Department of Mathematical Sciences

Imaging Introduction 1/2

A Discrete Image vs Surface function Variational

Monday, July 12, 2010

IMSE 2010, Brighton -

www.liv.ac.uk/www/cmit

Introduction 2/2

Process features of images (noise/patch)

4

A. Image Denoising:

Savage-C.(05 IJCM); TChan-C(06 SIAM MMS; 06 NumerAlg) ;

TChan-C.-Carter(07 ETNA); RChan-C.(07 IJCM; 08 SISC);

C.-Tai (2007 JSC); C.-Savage(07 BIT);

Brito-C. (10 SIAM IS; 10 IEEE TIP); Yang-C.-Yu (12 JCM, 12 IJNAM);

Zhang-C.-Yu(20120 IJCM; 12 SIAM NA)

A. Image Deblurrring:

RChan-C.(2010 SISC)

B. Image Inpainting:

Brito-C.(2008 JCM); Brito-C.(10 IJMM)

C. Image Segmentation:

Badshah-C.(2008 CiCP; 09 IEEE TIP; 10 CiCP);

Rada-C. (CiCP 2012)

D. Image Registration:

Chumchob-C.(2009 IJNAM, 2011 JCAM and 2012 NMPDE);

Chumchob-C.-Brito (2011 SIAM MMS)

Liverpool publications: Google Ke Chen

Thu, 26 Jan 2012 LAA and FM @ Liverpool

OUTLINE of this TALK

I. Image denoising

II. Image denoising and deblurring

III. Image blind deconvolution

7

Inverse Problem

Rudin-Osher-Fatemi(1992)

I. Image denoising

8

To solve Euler-Lagrange Equation

9

Competing Uni-grid Methods

divpzu

10

Multi-grid Methods for the E-L Eqn

Chan/Chan/Wan (95)

11

Multi-grid Methods for the E-L Eqn

12

Nonlinear Full Approximation Scheme

13

To solve the ROF TV Optimization by

Multilevel Methods

14

Nash’s Approach

15

Nash’s Approach

This is the FAS (NMG)

16

An Optimisation Smoother

17

Drawbacks of Previous Approach

Both Uni-level and

Multilevel gets stuck

18

T. Chan and Chen’s Approach

19

T. Chan/Chen (SIMMS, 2006)

Monday, July 12, 2010

IMSE 2010, Brighton -

www.liv.ac.uk/www/cmit

TV Results – Sharp Restoration

II. Image denoising and deblurring

z Ku

Variational ROF model: Opt vs PDE

Differential (Sparse) + Integral (Dense) Operator: Hard

Fixed Point Methods: Vogel, Chan et al x 2, Chang et al

Feasible Method: M Ng et al / Yin et al – Approx/ 2-stages

Non-feasible Method: MG / Multilevel

Thu, 26 Jan 2012 LAA and FM @ Liverpool

2

2

Total variation

1min ( ) | |

2

* *·| |

( )

uJ u u dxdy u z

uu z

u

K

K K K

‖ ‖

The Huang-Ng-Wen (2008, MMS) method

Thu, 26 Jan 2012 LAA and FM @ Liverpool

2

2

Total variation

2 2

2 2,

Stage 1 for and Stage 2 for

alterating

1min ( ) | |

2

1min ( , ) | |

2

min

2

s

v

u

u

u

v

J u u dxdy u z

J u u dxdy u z

approximated by the follo

v

wi

v v

ng

K

K

‖ ‖

‖ ‖ ‖ ‖

20 alternating iterations 40 problems to be solved in all

II(a). Attempts to develop a

multigrid / multilevel method Multigrids for PDE: linear with FP

Vogel et al (95) – negative results shown

Chan-Chan-Wan (97) – MG for L2-norm (not TV)

Mutilevel for Optimization: nonlinear / dim reduction

Chan-Chen (06) method for denoising ?for denoising and

deblurring?

Thu, 26 Jan 2012 LAA and FM @ Liverpool

II(b). A new direct optimization

method

Thu, 26 Jan 2012 LAA and FM @ Liverpool

, ,...,11 12

, ,...,11 12

1 12 2

, , 1 , 1,

1 1

,

2

,

1 1

1 2min ( ) | |22

| |

( ) ( ) ( )

1m

in ( ) [(K ) ]2

mi

u

n

u u

(

u u unn

u u unn

ij

n n

i j i j i j i j

i j

n n

i j

j j

j

j

i j

i

J u u dxdy Ku zu

J u u u u

K

z

u

J

‖ ‖

2

; , , ,

1 1

)n n

m m i j

m

u z

Joint work with R H Chan

(2008)

Level 1 minimization

Thu, 26 Jan 2012 LAA and FM @ Liverpool

loc 2

, , ,

2

, ,

1( ) ( )

2

1 ( )

2

i j i j i j t

i j i j t

J c T c c w Ku z

T c c w z

‖ ‖

‖ ‖

Let , , , , , , , ,

0

e R, e , K K Ke =K w1

0

i j i j i j i ji j i j i j tu u u uc c cuc

2 2 4( ) per pixel pixels = ( )O n n O n

Level 2 minimization

Thu, 26 Jan 2012 LAA and FM @ Liverpool

loc 2

, , ,

2

, ,

1( ) ( )

2

1 ( )

2

i j i j i j t

i j i j t

J c T c c w Ku z

T c c w z

‖ ‖

‖ ‖

Let, , , , ,, , ,

0

1

1

0e K K Ke =K w , R, e

1

1

0

i j i j i j ii j i j t i jjcu u u u uc c c

22 4n

( ) per pixel pixels = ( )4

O n O n

Thu, 26 Jan 2012 LAA and FM @ Liverpool

, ,( ) T T

i j t t i j tT c w wc w z

Consider how to generatetw

Task I – work to work out

partial sums of columns

Thu, 26 Jan 2012 LAA and FM @ Liverpool

2( log )O n n

Task II – work to work out

all RHS of local problems

Thu, 26 Jan 2012 LAA and FM @ Liverpool

2( log )O n n

Task III – work to work out

irregular partial sums of columns

Thu, 26 Jan 2012 LAA and FM @ Liverpool

2( log )O n n

Overall Algorithm: Initial Stage (FMM like) + Iteration Stage (Multilevel like)

Thu, 26 Jan 2012 LAA and FM @ Liverpool

II(c). Test results (A) quality

Thu, 26 Jan 2012 LAA and FM @ Liverpoolpsnr=25.80

Thu, 26 Jan 2012 LAA and FM @ Liverpool

II(d). Test results (B) Speed comparison

Thu, 26 Jan 2012 LAA and FM @ Liverpool

T.F. Chan and W.C. Wong.

IEEE transactions on Image Processing,

Vol. 7, No. 3, March 1998.

III. Image blind deconvolution

PDEs to solve

Non-uniqueness of the solution

Constraints to

guarantee

uniqueness of the

solution

2 1

1 2

[ ( , ), ( , )],

[ ( , ), ( , )],

[ ( , ), ( , )]

u x y h x y

u x c y d h x x y d

h x y u x y

Alternating Minimization Algorithm

Fair Double TV solution

A more difficult problem

Medical images on the other hand are not so easily tractable.

They are highly noisy and blurred

Preliminary Results

Encouraging

Motion Blur

1 1 1 1 1

1 1 1 1 1is approximated1

1 1 1 1 1by iterations25

1 1 1 1 1

1 1 1 1 1

k

Gaussian Blur

2 2

,

(x +y ) is approximated=exp( )

by iterations

- i j

i jk

Preliminary Work on Colour Deblurring

Conclusions

In Imaging Deblurring framework, (nonlinear)

regularisers complicate integral operators.

Optimisation multilevel methods effective.

Generalization to blind deblurring is non-

trivial and no-going work

Thu, 26 Jan 2012 LAA and FM @ Liverpool