Mathematics and Computation in Imaging

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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),



Zuowei Shen (NUS),



Chi
-
Wang Shu (Brown University)

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

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

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)

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.


Membership Applications


To stay in the program longer than two
weeks


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


for more information

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)



Part I: Problem Setting



Part II: Wavelet Algorithms



Outline of the talk

What is an image?


image = matrix


pixel intensity

= matrix entry

Resolution = size of the matrix

I. 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 (256


256) from them.

#2

#4

Boo and Bose (IJIST, 97):

#1

taking lens

CCD sensor

array

relay
lenses

partially silvered
mirrors

Four 2


2
images merged into one 4


4
image:

a
1

a
2

a
3

a
4

b
1

b
2

b
3

b
4

c
1

c
2

c
3

c
4

d
1

d
2

d
3

d
4

Four low resolution images

Observed high
-

resolution image

a
1

b
1

a
2

b
2

c
1

d
1

c
2

d
2

a
3

b
3

a
4

b
4

c
3

d
3

c
4

d
4

By
permutation

Four 64


64

images merged into one by permutation:

Observed high
-
resolution image by
permutation

Modeling

Consider
:

Low
-
resolution
pixel

High
-
resolution

pixels

4
1
2
1
4
1
2
1
1
2
1
4
1
2
1
4
1
Observed image: HR image passing through a

low
-
pass filter
a.

LR image: the down samples of observed image

at different sub
-
pixel position
.


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.



The problem
L
f = g
is ill
-
conditioned.

g
*
1
*
)

(
L
R
L
L




g

g
*
1
*
)
(
L
L
L

.
)

(
*
*
g
f
L
R
L
L



Here
R
can be
I,

. It is called
Tikhonov

method
( or the least square )

Regularization is required:

Wavelet Method



Let

â

be the symbol of the low
-
pass filter. Assume:



can be found such that

d
d
b
,
b
,
a
ˆ
ˆ

ˆ
1
ˆ
ˆ
ˆ
ˆ
}
0
\{
2
2







Z
b
b
a
a
d
d



One can use unitary extension principle to obtain a
set of tight frame systems.

Let


be the refinable function with refinement mask
a
, i.e.


Let


d

be the dual function of


:



.


,

0








d
We can express the true image as

where
v
(

)

are the pixel values of the high
-
resolution picture.





,

2

2
2









d
v
f
Z
.

)
2
(

)
(
4


2





Z





a
The pixel values of the observed image are given by





2
*

,


Z



v
a
The
observed function

is



.

)
2
/
(



)
(


2






Z





d
a
g
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...


Do it in the Fourier domain. Note that

(1)


.

1
ˆ
ˆ

ˆ
ˆ
}
0
\{
2
2




Z



b
b
a
a
d
d
We have



.

ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
0
\
2
2
v
v
b
b
v
a
a
d
d












Z



or



.

ˆ
ˆ
ˆ
ˆ
ˆ
0
\
*
2
2
v
v
b
b
v
a
a
d
d












Z



Generic Wavelet Algorithm:

(
i) Choose



;
ˆ
2
2
0


,
L
v




(ii) Iterate until convergence:



.

ˆ
ˆ
ˆ
ˆ
ˆ
0
\
*
1
2
2
n
d
d
n
v
b
b
v
a
a
v













Z



Proposition

Suppose that and nonzero
almost everywhere. Then for
arbitrary .


1
ˆ
ˆ
0


a
a
d
0
||
ˆ
ˆ
||
2


v
v
n
0
ˆ
v
Regularization:

Damp the high
-
frequency components in the current iterant.

Wavelet Algorithm I:

(
i) Choose



;
ˆ
2
2
0


,
L
v




(ii) Iterate until convergence:



.

ˆ

ˆ
ˆ
)
1
(
ˆ
ˆ
0
\
*
1
2
2
n
d
d
n
v
b
b
v
a
a
v














Z




Matrix Formulation:

The Wavelet Algorithm I is the stationary iteration for




.

)
(
g
f
d
d
d
L
H
H
L
L







Different between Tikhonov and Wavelet Models:


L
d

instead of
L
*
.


Wavelet regularization operator
.

Both penalize high
-
frequency components uniformly by

.

Wavelet Thresholding Denoising Method:



Decompose the
n
-
th iterate, i.e. , into

different
scales: ( This gives a wavelet packet decomposition of n
-
th
iterate.)

n
v
b
ˆ
ˆ













,

ˆ
ˆ
ˆ
ˆ
ˆ

ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
0
,
0
\
1
0
2
2
n
j
d
J
j
d
n
J
d
n
v
b
a
b
b
a
v
b
a
a
v
b













Z


Denoise these coefficients



of the wavelet

packet by thresholding method.



n
j
v
b
a
b
ˆ
ˆ
ˆ
ˆ


Before reconstruction,

Wavelet Algorithm II:

(
i) Choose



;
ˆ
2
2
0


,
L
v




(ii) Iterate until convergence:







n
d
d
n
v
b
b
v
a
a
v
ˆ
ˆ
T
ˆ
ˆ
ˆ
,
\
*








0
0
1
2
2

Z
Where T is a wavelet thresholding processing .

4


4
sensor array:


Original



LR Frame


Observed HR

Tikhonov Algorithm I Algorithm II

4


4
sensor array:

Tikhonov

Algorithm II

SNR
Tikhonov
Algorithm I
Algorithm II
(dB)
PSNR
RE
PSNR
RE
PSNR
RE
Iter.
30
32.55
0.0437
33.82
0.0377
34.48
0.0350
9
40
33.88
0.0375
34.80
0.0337
35.23
0.0321
12
SNR
Tikhonov
Algorithm I
Algorithm II
(dB)
PSNR
RE
PSNR
RE
PSNR
RE
Iter.
30
29.49
0.0621
29.70
0.0601
30.11
0.0579
30
40
30.17
0.0573
30.30
0.0566
30.56
0.0549
45
2

2 sensor array: 1 level up

4

4

sensor array
: 2 level up

Numerical Examples:

1
-
D

Example:

Signal from Donoho’s Wavelet Toolbox.

Blurred by
1
-
D

filter.

Original Signal Observed HR Signal

Tikhonov Algorithm II

Ideal low
-
resolution
pixel position

High
-
resolution

pixels

Calibration Error:

Problem no
longer spatially
invariant.

Displaced low
-
resolution pixel

Displacement error

e
x

The lower pass filter is perturbed


The wavelet algorithms can be modified





Reconstruction for 4


4 Sensors: (2 level up)


Original



LR Frame


Observed HR

Tikhonov Wavelets

Reconstruction for 4


4 Sensors: (2 level up)

Tikhonov

Wavelets

Numerical Results:

2


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


Least Squares Model

Our Algorithm

SNR(dB)

PSNR

RE


*

PSNR

RE

Iterations

30

28.00

0.0734

0.0367

30.94

0.0524

2

40

28.24

0.0715

0.0353

31.16

0.0511

2



4


4 sensor array (2 level) with calibration errors:


Least Squares Model

Our Algorithm

SNR(dB)

PSNR

RE


*

PSNR

RE

Iterations

30

24.63

0.1084

0.0492

27.80

0.0752

5

40

24.67

0.1078

0.0505

26.81

0.0751

6



(0,0)

(1,1)

(0,2)

(1,3)

(2,0)

(3,1)

(2,2)

(3,3)

(0,1)

(0,3)

(1,0)

(2,1)

(1,2)

(2,3)

(3,0)

(3,2)

Example:

4


4 sensor with missing frames:

Super
-
resolution
: not enough frames

(0,1)

(0,3)

(1,0)

(2,1)

(1,2)

(2,3)

(3,0)

(3,2)

Example:

4


4 sensor with missing frames:

Super
-
resolution
: not enough frames

i.
Apply an interpolatory subdivision scheme to obtain
the missing frames.

ii.
Generate the observed high
-
resolution image
w
.

iii.
Solve for the high
-
resolution image
u
.

iv.
From
u
, generate the missing low
-
resolution frames.

v.
Then generate a new observed high
-
resolution image
g
.

vi.
Solve for the final high
-
resolution image
f
.

Super
-
Resolution:


Not enough low
-
resolution frames.

Tikhonov
Algorithm I
Algorithm II
PSNR
RE
PSNR
RE
PSNR
RE
27.44
0.0787
27.82
0.0753
27.76
0.0758
Reconstructed Image:

Observed LR Final Solution