DECORRELATION TECHNIQUES IN IMAGE RESTORATION

Catalina Cocianu

Dept. of Computer Science, Academy of Economic Studies, Bucharest

Calea Dorobantilor #15-17, Bucuresti –1, Romania

Luminita State

Dept. of Computer Science, University of Pitesti, Pitesti

Caderea Bastliei #45, Bucuresti – 1, Romania

Panayiotis Vlamos

Ionian University, Corfu, Greece

Doru Constantin

Department of Computer Science, University of Pitesti, Pitesti, Romania

Keywords: De-correlation methods, principal component analysis, noise removal, shrinkage technique, learning from

data.

Abstract: The restoration can be viewed as a process that attempts to reconstruct or recover an image that has been

degraded by using some a priori knowledge about the degradation phenomenon. The multiresolution

support provides a suitable framework for noise filtering and image restoration by noise suppression. We

present the algorithms GMNR, a generalization of the MNR algorithm based on the multiresolution support

set for noise removal in case of arbitrary mean, and NFPCA. A comparative analysis of the performance of

the algorithms GNMR and NFPCA is experimentally performed against the standard AMVR and MMSE.

1 INTRODUCTION

The effectiveness of restoration techniques mainly

depends on the accuracy of the image modeling. The

image restoration tasks correspond to the process of

finding an approximation to the overall degradation

process and finding the appropriate inverse process

to estimate the original unknown image (Gonzales,

2002).

Noise is any undesired information that

contaminates an image and appears in images from a

variety of sources. The digital image acquisition

process is the primary process by which noise

appears in digital images. Typically, the noise can be

modeled with either a Gaussian, uniform or salt and

pepper distribution.

The image restoration tasks mainly correspond to

the process of finding an approximation to the

overall degradation process and finding the

appropriate inverse process to estimate the original

unknown image. The most successful denoising

algorithms fulfill at least the two following features.

They use translation invariant overcomplete

representations with local kernels selected to scale

and orientation and apply a multidimensional

shrinkage function based on joint observations of the

coefficients in the neighborhoods. Some of these

methods can be viewed as extensions of the classical

Wiener estimate which assumes a global Gaussian

behavior of both signal and noise. (Portilla, 2005)

In (Balster, Zheng, Ewing, 2003) a selective

wavelet shrinkage algorithm for digital image

denoising aiming to improve the performance and

computation scheme of a wavelet shrinkage

algorithm is proposed, the denoising methodology

incorporated in this algorithm involving two-

193

Cocianu C., State L., Vlamos P. and Constantin D. (2008).

DECORRELATION TECHNIQUES IN IMAGE RESTORATION.

In Proceedings of the International Conference on Signal Processing and Multimedia Applications, pages 193-196

DOI: 10.5220/0001933901930196

Copyright

c

SciTePress

threshold validation process for real time selection

of wavelet coefficients.

The MNR technique is essentially based on the

statistical significance of the wavelet coefficients

specifying the support. The statistical significance is

established, somehow heuristically in terms of

second order statistics. (Stark, 1995)

We present the algorithms GMNR, a

generalization of the MNR algorithm based on the

multiresolution support set for noise removal in case

of arbitrary mean, and NFPCA. A comparative

analysis of the performance of the algorithms

GNMR and NFPCA is experimentally performed

against the standard AMVR and MMSE.

The paper reports the conclusions experimentally

derived on the convergence rates and their

corresponding efficiency for specific image

processing tasks.

2 NOISE DECORRELATION

TECHNIQUE

The multiresolution support provides a suitable

framework for noise filtering and image restoration

by noise suppression. The procedure used is to

determine statistically significant wavelet

coefficients and from this to specify the

multiresolution support, therefore a statistical image

model is used as an integral part of the image

processing. The support is used subsequently to

hand-craft the filtering processing.

The MNR algorithm is (Stark, 1995),

Input: The image

0

X , the number of the

resolution levels p and the heuristic thresold k (the

value of k should be taken close to 3).

Step 1. Compute the sequence of image variants

{

}

pj

j

X

,1=

and the wavelet coefficients using the “À

Trous” algorithm (Stark, 1995)

() ()

()

∑∑

−−

−

++=

lk

jj

jj

kclrXklhcrX

11

1

2,2,,

() () ()

crXcrXcr

jjj

,,,

1

−

=ω

−

, (1)

where h is a discrete low-pass filter.

Step 2. Apply the significance test,

()

cr

j

,ω is significant

if and only if

()

jj

kcr σ≥ω , , for pj ,...,1=

Step

3. Compute the restored image,

() () ()

()

()

∑

=

ωωσ+=

p

j

jjjp

crcrgcrXcrX

1

,,,,,

~

, (2)

where g is defined by,

()

()

(

)

()

⎪

⎩

⎪

⎨

⎧

σ<ω

σ≥ω

=ωσ

jj

jj

jj

kcr

kcr

crg

,,0

,,1

,,

Output The restored image

X

~

.

In the following, we present the algorithm

GMNR, a generalization of the MNR algorithm

based on the multiresolution support set for noise

removal in case of arbitrary mean (Cocianu, 2003).

Let g be the original “clean” image,

η~

(

)

2

,σmN

and the analyzed image

η+= gf . The sampled

variants of f, g and

η

obtained using the two-

dimensional filter

ϕ

are given by,

(

)

(

)

(

)

cylxclfyxc −−ϕ= ,,,,

0

,

(

)

(

)

(

)

cylxclgyxI −−ϕ= ,,,,

0

,

(

)

(

)

(

)

cylxclyxE −−ϕη= ,,,,

0

,

000

EIc

+

=

. (3)

Consequently, the wavelet coefficients of

0

c

computed by the algorithm “À Trous” are,

() ()

=

⎟

⎠

⎞

⎜

⎝

⎛

−−

ψ=ω

jjj

c

j

ycxl

clfyx

2

,

2

,,

2

1

,

0

(

)

(

)

yxyx

E

j

I

j

,,

00

ω+ω= , (4)

where

()

⎟

⎠

⎞

⎜

⎝

⎛

φ−φ=

⎟

⎠

⎞

⎜

⎝

⎛

ψ

22

1

22

1 x

x

x

.

For any pixel

(

)

yx, , we get

() ()

=

⎟

⎠

⎞

⎜

⎝

⎛

−−

ϕ=

ppp

p

ycxl

clfyxc

2

,

2

,,

2

1

,

(

)

(

)

yxEI

pp

,yx,

+

=

. (5)

The representation of the image

0

c is given by,

() () ()

=ω+=

∑

=

p

j

c

jp

yxyxcyxc

1

0

,,,

0

() () () ()

∑∑

==

ω+ω++=

p

j

E

j

p

j

I

jpp

yxyxyxEyxI

11

,,,,

00

. ( 6)

Note that only

(

)

yxE

p

, and

()

∑

=

ω

p

j

E

j

yx

1

,

0

include

noise component. The mean of the noise can be

decreased using the following algorithm.

Step1. Determine the images

()

i

E , ni ≤≤1 , by

superimposing noise sampled from

()

2

,σmN on the

“white wall” image.

Step2. For all j, pj ≤

≤

1 , compute

j

c ,

(

)

i

j

E ,

ni

≤

≤

1 and the coefficients

()

i

E

j

c

j

ωω ,

0

using the “À

SIGMAP 2008 - International Conference on Signal Processing and Multimedia Applications

194

Trous” algorithm, where h is given by the filtering

mask

⎟

⎟

⎟

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎛

16

1

8

1

16

1

8

1

4

1

8

1

16

1

8

1

16

1

according to,

() ()

(

)

∑∑

−−

−

++=

lc

jj

jj

cylxcclhyxc

11

1

2,2,,

()

() ()

()

(

)

∑∑

−−

−

++=

lc

jji

j

i

j

cylxEclhyxE

11

1

2,2,,

() ()

(

)

yxcyxcyx

jj

c

j

,,,

1

0

−=ω

−

()

()

()

()

()

()

yxEyxEyx

i

j

i

j

E

j

i

,,,

1

−=ω

−

(7)

Step 3. Compute the image

I

~

by,

() ()

()

()

[

()

()

()

()

⎥

⎦

⎤

ω−ω+

+−=

∑

∑

=

=

p

j

E

j

c

j

n

i

i

pp

yxyx

yxEyxc

n

yxI

i

1

1

,,

,,

1

,

~

0

. (8)

Step 4. Compute a variant of the original image

0

I using the multiresolution filtering based on the

statistically significant wavelet coefficients.

Note that

I

~

computed at Step 3 is

,'

~

0

EII +=

where E’~

()

2

',' σmN

,

0'≈m and

()

22

' σ≈σE

.

An alternative approach in solving image

restoration task can be performed by PCA neural

network. The idea is to use features extracted from

the noise in order to compensate the lost information

and improve the quality of images.

The NFPCA algorithm is presented in the

following. We consider the additive normal

distributed degradation model. Let

0

I

be a RxC

matrix, where

CnnCC <

≤

= 2,

1

representing the

initial image of

L gray levels and let I be the

distorted variant resulted from

0

I

by superimposing

random noise

()

Σ,0N ,

Ri ,...,1=∀ ,

()

njjnk ,...,1−= ,

1

,...,1 Cj = ,

() () ()

kkIkI

jiji

η+=

0

,,

, where

•

ji

I

,

is the sequence of n pixels of the i-th row

from the

()

1−jn -th pixel to the nj-th of the

image

I

•

0

, ji

I

is the sequence of n pixels of the i-th row

from the

()

1−jn -th pixel to the nj-th of the

image

0

I

•

η

is a n-dimensional random vector

distributed

()

Σ,0N .

The algorithm for removing the noise component

proceeds in two stages:

• in the first stage the noise features

Φ

are

computed. The columns of

Φ

are the eigen

vectors of

Σ

, taken according to the decreasing

order of their corresponding eigen- values;

• in the second stage, using Φ , we apply a

noise removal method

M for cleaning each pixel

(

)

ji, of the de-correlated transformed image.

The restoration process of the image

I using the

learned features is performed as follows:

Step 1. Compute the image I’ by de-correlating

the noise component,

Ri ,...,1

=

∀

,

1

,...,1 Cj

=

,

''

0

,,,

η+Φ=Φ=

ji

T

ji

T

ji

III

, where

ηΦ=η

T

'~

(

)

',0

Σ

N ,

Λ

=

Σ

Φ

Φ

=

Σ

T

' ,

{

}

n

diag

λ

λ

λ

=

Λ

,...,,

21

.

Step 2. The noise component 'η is removed for

each pixel

P of the image I’ using the

multirezolution support of

I’ by the labeling method

of each wavelet coefficient of P, resulting

I”.

(

)

0

,,,

'"

ji

T

jiji

IIMNRI Φ≅=

,

Ri ,...,1

=

∀

,

1

,...,1 Cj

=

,

where

(

)

ji

IMNR

,

'

is produced by applying the

above mentioned method to

ji

I

,

' .

Step 3. An approximation

0

~

I

I

≅ of the initial

image

0

I

is produced by applying the inverse

transform of

T

T

Φ

to I” ,

0

,

0

,,,

"

~

jiji

T

jiji

IIII =ΦΦ≅Φ= , (9)

Ri ,...,1

=

∀

,

1

,...,1 Cj

=

Note that the decorrelation of the noise

component is performed by the computation carried

out at

Step 1 because the resulted image is

(

)

(

)

(

)

kkIkI

ji

T

ji

''

0

,,

η+Φ= , (10)

(

)

njjnk ,...,1

−

=

,

1

,...,1 Cj

=

,

where for each

(

)

njjnk ,...,1−

=

,

1

,...,1 Cj = ,

(

)

k'

η

~

(

)

kkkiki

N

,

2

,

2

,

,,0 λ=σσ .

When the assumption of zero mean noise is not

acceptable, the method GMNR to remove the noise

resulted by the decorrelation process instead.

In order to evaluate the performance of the

proposed noise removal algorithms, a series of

experiments were performed on different 256 gray

level images. We compared the performance of our

DECORRELATION TECHNIQUES IN IMAGE RESTORATION

195

algorithm NFPCA against MMSE (Umbaugh,

1998), AMVR (Umbaugh, 1998), and GMNR.

The values of the variances to model the noise in

images processed by NFPCA represent the

maximum of the variances per pixel resulted from

the decorrelation process. The implementation of the

GMNR algorithm used the masks

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎛

=

256

1

64

1

128

3

64

1

256

1

64

1

16

1

32

3

16

1

64

1

128

3

32

3

64

9

32

3

128

3

64

1

16

1

32

3

16

1

64

1

256

1

64

1

128

3

64

1

256

1

1

h and

⎟

⎟

⎟

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎛

=

20

1

10

1

20

1

10

1

5

2

10

1

20

1

10

1

20

1

2

h

A synthesis of the comparative analysis on the

quality and efficiency corresponding to the

restoration algorithms presented in the paper is

supplied in Table 1.

Table 1.

Restoration

algorithm

Type of

noise

Mean

error/pixel

MMSE 52.08

AMVR

U(30,80)

10.94

MMSE 50.58

AMVR

U(40,70)

8,07

MMSE 37.51

AMVR 11.54

GMNR 14.65

NFPCA

N(40,200)

12.65

MMSE 46.58

AMVR 9.39

GMNR 12.23

NFPCA

N(50,100)

10.67

REFERENCES

Bacchelli, S., Papi S. 2006, Image denoising using

principal component analysis in the wavelet domain.

In Journal of Computational and Applied

Mathematics, Volume 189, Issues 1-2, 1 May 2006,

Pages 606-621

Balster, E. J., Zheng, Y. F., Ewing, R. L. 2003, Fast,

Feature-Based Wavelet Shrinkage Algorithm for

Image Denoising. In International Conference on

Integration of Knowledge IntensiveMulti-Agent

Systems. KIMAS '03: Modeling, Exploration, and

Engineering Held in Cambridge, MA on 30

September-October 4, 2003

Cocianu, C., State, L., Stefanescu, V., Vlamos, P., 2006,

PCA-Based Data Mining Probabilistic and Fuzzy

Approaches with Applications in Pattern Recognition,

Proceedings of ICSOFT 2006, Portugal, pp. 55-60,

2006

Cocianu, C., State, L., Vlamos, P.,2002, On a Certain

Class of Algorithms for Noise Removal in Image

Processing:A Comparative Study, In Third IEEE

Conference on Information Technology ITCC-2002,

Las Vegas, Nevada, USA, April 8-10, 2002

Cocianu, C., State, L., Stefanescu, V., Vlamos, P., 2004,

On the Efficiency of a Certain Class of Noise Removal

Algorithms in Solving Image Processing Tasks, In:

Proceedings of the ICINCO 2004, Setubal, Portugal

Diamantaras, K.I., Kung, S.Y., Principal Component

Neural Networks: theory and applications, John

Wiley &Sons, 1996

Gonzales, R., Woods, R., Digital Image Processing,

Prentice Hall, 2002

Haykin, S., Neural Networks A Comprehensive

Foundation, Prentice Hall,Inc. 1999

Hyvarinen, A., Karhunen, J., Oja,E., Independent

Component Analysis, John Wiley &Sons, 2001

Hyvarinen, A., Hoyer, P., Oja, P., 1999. Image Denoising

by Sparse Code Shrinkage,

www.cis.hut.fi/projects/ica,

Pitas, I., 1993, Digital Image Processing Algorithms,

Prentice Hall

Portilla, J. 2005, Image Restoration using Gaussian Scale

Mixtures in Overcomplete Oriented Pyramids. In

SPIE's International Symposium on Optical Science

and Technology, SPIE's 50th Annual Meeting, Proc. of

the SPIE, vol. 5914, pp. 468-82, San Diego, CA, Aug

2005

Sonka, M., Hlavac, V., 1997, Image Processing, Analyses

and Machine Vision, Chapman & Hall Computing

Stark, J.L., Murtagh, F., Bijaoui, A., 1995, Multiresolution

Support Applied to Image Filtering and Restoration,

Technical Report

State, L, Cocianu, C, Vlamos, P.., 2001, Attempts in

Using Statistical Tools for Image Restoration

Purposes, In Proceedings of SCI2001, Orlando, USA,

July 22-25, 2001

Umbaugh, S., 1998, Computer Vision and Image

Processing, Prentice Hall

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