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
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