Denoising Techniques Based on the Multiresolution Representation

Denoising Techniques Based on the Multiresolution Representation

Cătălina COCIANU

Dept. of Computer Science, Academy of Economic Studies, Bucharest, România Luminiţa STATE

Dept. of Computer Science, University of Piteşti, Piteşti, România Viorica ŞTEFANESCU

Dept. of Mathematics, Academy of Economic Studies, Bucharest, România Panayiotis VLAMOS

Hellenic Open University, Athens, Greece

So far, considerable research efforts have been invested in the are of using statistical methods for image processing purposes yielding to a significant amount of models that aim to improve as much as possible the still existing and currently used processing techniques, some of them being based on using wavelet representation of images. Among them the simplest and the most attractive one use the Gaussian assumption about the distribution of the wavelet coefficients. This model has been successfully used in image denoising and restoration. The limitation comes from the fact that only the first-order statistics of wavelet coefficients are taking into account and the higher-order ones are ignored. The dependencies between wave-let coefficients can be formulated explicitly, or implicitly. The multiresolution representation is used to develop a class of algorithms for noise removal in case of normal models. The mul-tiresolution algorithms perform the restoration tasks by combining, at each resolution level, according to a certain rule, the pixels of a binary support image. The values of the support image pixels are either 1 or 0 depending on their significance degree. At each resolution lev-el, the contiguous areas of the support image corresponding to 1-value pixels are taken as possible objects of the image. Our work reports two attempts in using the multiresolution based algorithms for restoration purposes in case of normally distributed noise. Several re-sults obtained using our new restoration algorithm are presented in the final sections of the paper.

Keywords: multiresolution support, wavelet transform, filtering techniques, statistically sig-nificant wavelet coefficients.

Introduction

The effectiveness of restoration tech-niques mainly depends on the accuracy of the image modeling. A long series of image de-gradation models have been proposed under various working assumptions. One of the most popular degradation models is the linear continuous image-degradation where it is as-sumed that the image blur can be modeled as a superposition with an impulse response that may be space variant and its output is subject to an additive noise.

The restoration can be viewed as a process that attempts to reconstruct or recover a de-graded image using some a priori knowledge about the degradation mechanism.

Thus restoration techniques are oriented to-ward modeling the degradation and applying

the inverse process in order to recover the original image. This approach usually in-volves formulating a criterion of goodness that will yield some optimal estimate of the desired result.

Generally speaking, the multiresolution algo-rithms perform the restoration tasks by com-bining, at each resolution level, according to a certain rule, the pixels of a binary support image. The values of the support image pix-els are either 1 or 0 depending on their signi-ficance degree. At each resolution level, the contiguous areas of the support image cor-responding to 1-value pixels are taken as possible objects of the image. The multireso-lution support set is a data structure suitable for developing noise removal algorithms that perform the restoration tasks by combining,

at each resolution level, according to a cer-tain rule, the pixels of a binary support im-age. The values of the support image pixels are either 1 or 0 depending on their signific-ance degree. At each resolution level, the contiguous areas of the support image cor-responding to 1-value pixels are taken as possible objects of the image.

So far, considerable research efforts have been invested in the are of using statistical methods for image processing purposes yielding to a significant amount of models that aim to improve as much as possible the still existing and currently used processing techniques, some of them being based on us-ing wavelet representation of images. Among them the simplest and the most attractive one use the Gaussian assumption about the distri-bution of the wavelet coefficients. This mod-el has been successfully used in image de-noising and restoration. The limitation comes from the fact that only the first-order statis-tics of wavelet coefficients are taking into account and the higher-order ones are ig-nored. The dependencies between wavelet coefficients can be formulated explicitly, or implicitly. Moreover, most wavelet models can be loosely classified into two categories: those exploiting inter-scale dependencies and those exploiting intra-scale dependencies [7]. Typically, on each resolution level, the mag-nitudes of wavelet coefficients corresponding to images are strongly correlated [8]. The wavelet coefficients can be thought on a quad-tree-like structure such that when a pa-rental node is of small magnitude, those of its descendents are very likely to be small as well. A working assumption is that the wave-let coefficient distributions are Gaussian mix-ture for all subbands.

The spatial clustering trend of wavelet coef-ficients are extensively exploited by different compression algorithms as, for instance, the EQ coder and the morphological coder [6]. For example, in case of the EQ coder, the re-sulted wavelet coefficients are independent of zero mean and slow varying variance. This technique proved useful in developing de-noising applications in signal processing, where local statistics are estimated from the

data [9]. Also, methods were spatially vary-ing variances were assumed have been pro-posed, these models being able to take into account the inter-scale dependencies.

2. Image Denoising Using a Scale-Space Mixture Model

A model explicitly combining the inter-scale and intra-scale dependencies of image wave-let coefficients was proposed by Liu and Moulin [9]. The model uses a simple classifi-cation technique and is based on the follow-ing empirical observations: wavelet coeffi-cients of large magnitude are typically repre-sentative for edges, textures as well as noisy areas, while those of small magnitude rather correspond to smooth regions.

The design of this model is motivated by the zero tree coding technique. Let T be a signi-ficance threshold. In each subband except the first fine scale, the wavelet coefficients are partitioned into two classes based on the magnitude of their parents: W_sig is the set of coefficients that have significant parents (>T) and W_insig is the set of coefficients that have insignificant parents. Hence, the size of the each of the two classes is controlled by the significance threshold T. However, the two classes have quite different statistics. Since the histogram of the coefficients in W_insig is highly concentrated around zero, while the histogram of W_sig is more spread out.

The described statistical model can be ap-plied to image denoising as follows. Assume that the initial clean image I is disturbed by additive white noise of zero mean and va-rianceσ2, producing the imageI'=I +η. The goal is to determine a good estimation of I given the image I’.

The estimation problem can be formulated in the wavelet domain: the image coefficients

I ~

have to be estimated using the empirical coefficientsI~'=I~+η. The algorithm can be described as follows.

For each wavelet coefficient, depending on the subband it belongs to, execute:

Step 1. For each of the first three fine sub-bands (with horizontal, vertical and diagonal

orientations), wavelet coefficients within the subband are modeled as identically distri-buted Laplacian with mean zero and variance

2 ~_j I

σ , where j stands for the index of the se-lected subband. The variance estimate is computed as,

{

}

{

2

}

2 ~ max0, ~,' σ σ = Var I I∈subbandj − j I (1)

Consequently the maximum a posteriori (MAP) estimates of I~ result by applying a soft threshold ₂ ~ 2 2 j I σ σ

λ = to each noisy coef-ficient.

Step 2. For each of the higher subbands, wavelet coefficients are clustered into one of the classes W_sig and W_insig according to the magnitude of their estimated parent with re-spect to T. Coefficients in W_sig are modeled as identically distributed Laplacian of zero mean. Their estimated variances are,

{

}

{

2

}

2 ~ max0, ~,' σ σ = ∈ _sig − j I Var I I W (2)

The wavelet coefficients belonging to W_insig have small magnitudes and they represent smooth image areas. Assuming that σˆ_i2 is the true variance of I~_i , the MAP estimation is, i i i i I I ~' ˆ ˆ ˆ ˆ 2 2 2 σ σ σ + = (3)

Note that the course band coefficients are not processed because of their very high SNR. The approach can be described as a top-down denoising process. Initially, the coarse scale coefficients are identified, then the algorithm propagates from parental nodes to their des-cendent subbands until the highest subband is reached. The coefficient estimates and the parental node significance information are used to process the next subband.

3. The Proposed Denoising Wavelet Based Model

Our attempt aims the development of a de-noise intra-scale algorithm based on the mul-tiresolution support set of the images.

The multiresolution representation is used to develop a class of noise removal algorithms in case of normal models. The multiresolu-tion algorithms perform the restoramultiresolu-tion tasks by combining, at each resolution level, ac-cording to a certain rule, the pixels of a bi-nary support image. The values of the sup-port image pixels are either 1 or 0 depending on their significance degree. At each resolu-tion level, the contiguous areas of the support image corresponding to 1-value pixels are taken as possible objects of the image. Our work reports two attempts in using the multi-resolution based algorithms for restoration purposes in case of normally distributed noise. Several results obtained using our new restoration algorithm are presented in the fi-nal sections of the paper.

The multiresolution support set is a data structure suitable for developing noise re-moval algorithms. The multiresolution algo-rithms perform the restoration tasks by com-bining, according to a certain rule and at each resolution level, the pixels of a binary sup-port image. The values of the supsup-port image pixels are either 1 or 0 depending on their significance degree. At each resolution level, the contiguous areas of the support image corresponding to 1-value pixels are taken as possible objects of the image. The multireso-lution support is the set of all support images. The multiresolution support can be computed using the statistically significant wavelet coefficients.

Let j be a certain multiresolution level. Then, for each pixel (x, y) of the input image I, the

multiresolution support at the level j is,

(

I; j,x,y

)

=1⇔

M I contains significant

information at the level j and the pixel (x,y).

If we denote by ψ be the mother wavelet function, then the generic computing scheme of the multiresolution support set is described as follows.

Step 1. Compute the wavelet transform of the input image usingψ.

Step 2. Compute M

(

I;j,x,y

)

using the

statis-tically significant wavelet coefficients for each resolution level j and for each pixel

(x,y).

The computation of the wavelet transform of a one dimensional signal is performed by the algorithm “À Trous”. The algorithm can be extended to perform this computation in case of two-dimensional signals as, for instance, image signals. Let f be the continuous

sig-nal function and let φ be a low pass filter having the dilatation property

( ) (

)

∑

− = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ l l x l h x _φ φ 2 2 1 (4)

where h is a discrete valued low-pass filter. If we denote by

{

c0

( )

k

}

the sampled signal f

computed via ϕ, then

( )

k f

( ) (

x x k

)

c₀ = ,φ − . (5)

The set

{ }

cj

( )

k at the resolution level j is giv-en by,

( )

( )

_⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = j j j k f x x k c 2 , 2 1 _φ , (6) therefore, we get

( )

∑

_⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − l j j l k x l h k x 1 2 2 2 1_{ϕ ϕ} (7) and

( )

( )

∑

( )

∑

( )

( )

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ₋ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ₋ = _{− − − −} l j j l j j j k f x h l x k l hl f x x k l c _{1 1 1 1} 2 , 2 1 2 , 2 1 _{ϕ ϕ}

( )

(

)

∑

− − + = l j j k l c l h ₁ 2 1 .

The wavelet coefficients ω_jare computed us-ing the terms cj−1 and cjas

( )

k c_j

( )

k c_j

( )

k

j = −1 −

ω . (8)

The wavelet coefficients can be also ex-pressed as [12],

( )

( )

_⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = j j j k f x x k 2 , 2 1 _ψ ω (9) where

( )

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ 2 2 1 2 2 1 x x x _{φ φ} ψ .

Using the resolution levels1,2,...,p, where p is

a given parameter, the “À Trous” algorithm computes the wavelet coefficients according to the following scheme [12].

Input: The sampled signal

{

c₀

( )

k

}

For j=0,1,…,p do Step 1. j=j+1; computecj−1

( )

k ,

( )

∑

( )

(

−

)

− + = l j j j k hlc k l c 1 1 2 . Step 2. Compute ωj

( )

k =cj−1

( )

k −cj

( )

k End-for

Output: The set

{

ωj

( )

k,cp

}

_j=1,...,p.

Note that the computation of cj

( )

k carried out

in Step 1 imposes that either the periodicity condition cj

(

k+N

)

=cj

( )

k or the continuity

property cj

(

k+N

)

=cj

( )

N holds.

Since the representation of the original sam-pled signal is

( )

( )

∑

( )

= + = p j j p k k c k c 1 0 ω (10)

in case of images, the values of c₀ are com-puted for each pixel (x,y) as follows,

( )

( )

∑

( )

= + = p j j p x y x y c y x c 1 0 , , ω , . (11)

If the input image I encodes a noise compo-nent η, then the wavelet coefficients also en-code some information about η. A label pro-cedure is applied to each ω_j

( )

x,y in order to remove the noise component from the wave-let coefficients computed for I. In case for each pixel (x,y) of I, the distribution of the coefficients is available, the significance lev-el corresponding to each component

( )

x y

j ,

ω can be established using a statistic-al test. We say that I is locstatistic-al constant at the resolution level j in case the amount of noise in I at this resolution level can be neglected. Let Η0 be the hypothesisΗ0: I is local con-stant at the resolution level j. In case there is significant amount of noise in I at the resolu-tion level j, we get that the alternative

hypo-thesis⎬Η₀:ω_j

( )

x, y ~N

( )

0,σ2_j . In order to define the critical region W of the statistical test we proceed as follows. Let 0<ε <1 be

the a priori selected significance level and let ε

z be such that when ⎬Η₀ is true,

( )

(

)

∫

− ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ − = < = − ε ε ε σ σ π ω ε z z j j j x y z t ₂ dt 2 2 exp 2 1 , 1 cÜÉu . (12)

In other words, the probability of rejecting ⎬Η₀ (hence acceptΗ₀) when ⎬Η₀ is true is

ε and consequently, the critical region is

[

z_ε z_ε

]

W = − , . Accordingly, the significance level of the wavelet coefficients is given by

the rule: ω_j

( )

x,y is a significant coefficient if and only ifω_j

( )

x, y ∉W.

Usually, zε is taken askσj, where k is a

se-lected constantk≈3, because

( )

(

_j x y >k _j

)

= P

(

_j

( )

x y >k _j

)

+P

(

_j

( )

x y <−k _j

)

= P ω , σ ω , σ ω , σ

( )

(

_j x y k _j

)

(

P

(

_j

( )

x y k _j

)

)

Pω > σ = − ω ≤ σ =2 , 21 ,

( )

(

ω σ

)

(

ε

)

ε σ_j < ⇒ j , > j ≥21− k P x y k z

Using the significance level, we set to 1 the statistically significant coefficient and

re-spectively we set to 0 the non-significant ones. The restored image I~ is computed as

( )

( )

∑

(

( )

)

( )

= + = p j j j j p x y g x y x y c y x I 1 , , , , , ~ _{σ ω ω} (13) where g is defined by

(

( )

)

( )

( )

⎪⎩ ⎪ ⎨ ⎧ < ≥ = j j j j j j k y x k y x y x g σ ω σ ω ω σ , , 0 , , 1 , , .

4. The Filtering Technique of the Images Distorted by General Normal Distributed Noise

Let g be the original “clean” image,

η~_N

(

_m,σ2

)

_{and the analyzed image f =g+η.} The sampled variants of f, g and η obtained using the two-dimensional filter ϕ are given by,

( )

x y f

( ) (

l c x l y c

)

c₀ , = , ,ϕ − , − ,

( )

x y g

( ) (

l c x l y c

)

I₀ , = , ,ϕ − , − ,

( )

x y

( ) (

l c x l y c

)

E₀ , = η , ,ϕ − , − , 0 0 0 I E c = + .

Consequently, the wavelet coefficients of c0 computed by the algorithm “À Trous” are

( )

( )

⎟ = ⎠ ⎞ ⎜ ⎝ ⎛ − − = _{j j j} c j x y f l c l x c y 2 , 2 , , 2 1 , 0 _ψ ω

( ) ( )

⎟ = ⎠ ⎞ ⎜ ⎝ ⎛ − − ψ η + _{j j} j y c x l c l c l g 2 , 2 , , , 2 1

( )

( )

_⎟⎟ = ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − = j j j j j j y c x l c l y c x l c l g 2 , 2 , , 2 1 2 , 2 , , 2 1 _{ψ η ψ} -=ωI_j0

( )

x,y +ωE_j0

( )

x,y , where

( )

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ 2 2 1 2 2 1 x x x _{φ φ}

ψ . For any pixel

( )

x,y , we get

( )

( )

_⎟⎟ = ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − = p p p p x y f l c l x c y c 2 , 2 , , 2 1 , ϕ I_p

( )

å? ç +E_p

( )

x,y . The representation of the image c₀ is given by

( )

=

( )

+

∑

( )

=

= p j c j p x y x y c y x c 1 0 0 , , ω ,

=

( )

( )

∑

( )

∑

( )

= = + + + p j E j p j I j p p x y E x y x y x y I 1 0 1 0 _{, ,} , , ω ω . (14)

Note that onlyE_p

( )

x,y and

∑

( )

= p j E j x y 1 0 _, ω

in-clude noise component in (14). The mean of the noise can be decreased using the follow-ing algorithm.

Step1. Determine the images_E( )i _{, 1≤i≤n, by}

superimposing noise sampled from _N

(

_m,σ2

)

on the “white wall” image.

Step2. For all j, 1≤ j≤ p, compute c_j, ( )i

j

E , n

i≤ ≤

1 and the coefficients _E( )i

j c

j ω

ω0, using the “À Trous” algorithm, according to,

( )

∑∑

( )

(

− −

)

− + + = l c j j j j x y hl cc x l y c c , , ₁ 2 1 , 2 1 ( )

( )

∑∑

( )

( )

(

− −

)

− + + = l c j j i j i j x y hl c E x l y c E , , ₁ 2 1 , 2 1

( )

x y c_j

( )

x y c_j

( )

x y c j0 , = −1 , − , ω and ( )

( )

x y E( )_ji

( )

x y E( )_ji

( )

x y i E j , = −1 , − , ω

Step 3. Compute the image I~ by

( )

∑

( )

( )

( )

∑

( )

( )

( )

= = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ₋ + − = n i p j i E j c j i p p x y E x y x y x y c n y x I 1 1 0 _{, ,} , , 1 , ~ _{ω ω} . Step 4. Compute a variant of the original

im-age I₀ using the multiresolution filtering based on the statistically significant wavelet coefficients.

Note that I~ computed at Step 3 is ,' ~ 0 E I I = + where E’~N

(

m,'σ'2

)

, m'≈0 and E

( )

σ'2 ≈σ2.

4. Concluding Remarks and Suggestions for Further Work

A series of experiments were performed, dif-ferent 256 gray level images being

prepro-cessed aiming the contrast enhancement, in-creasing enlighting and noise removing by filtering them. Our experiments use the aver-aging and respectively binomial filtering techniques. The parameters involved in the mentioned algorithm were tuned taking into account the following factors: the distortion degree of the inputs, the particular smoothing filter, the volume of the resulting accepted data.

The implementation of the proposed algo-rithm 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 .

Some of the results produced by the proposed algorithm are depicted in Table 1. Our tests were performed on distorted images processed by the masks h₁ and h₂and the Gaussian distributions N(40,100) and

respec-tively N(50,200), assumed to model the addi-tive noise.

In our opinion, the performance of efficiency of the proposed algorithm can be significant-ly improved by narrowing the class of processed nodes, namely by taking into

ac-count and process only the coefficients be-longing to the setW_sig. Also, improvements are expected to come from different refine-ments of the proposed algorithm.

Table 1. Restoration algo-rithm Type of noise Mean er-ror/pixel

Using the mask h₁ N(40,100) 12.6

Using the mask h₂ 10.53

Using the mask h₁ N(50,200) 16.16

Using the mask h₂ 12.74

Extensions of this methodology in case of more general assumptions concerning the sta-tistical properties, as for instance modeling in case of generalized Gaussian mixtures are aimed. These developments are still in progress and the results are going to be re-ported elsewhere.

6. References

[1] A. Chambolle, R.A. DeVore, N-Y. Lee, J. Lucier, “Nonlinear Wavelet Image Processing: Variational Problems, Compres-sion and Noise Removal through Wavelet Shrinkage”, IEEE Trans. On Image Proc., Vol. 7, No. 1, Jan. 1998

[2] C.K. Chui, Wavelets, A Mathematical Tool for Signal Analysis, SIAM, Philadel-phia, PA, 1997

[3] C. Cocianu, L. State, V. Stefanescu, P. Vlamos, “Noise Removal Techniques Using the Multiresolution Representation”, Proc. of 32nd International Conference on Comput-er&Industrial Engineering (ICC&IE), Irel-and, August 11th-13th, 2003

[4] C. Cocianu, L. State, V. Stefanescu, P. Vlamos, “On the Efficiency of a Certain Class of Noise Removal Algorithms in Solv-ing Image ProcessSolv-ing Tasks”, Proceedings of the ICINCO 2004, Setubal, Portugal

[5] R. Gonzales, R. Woods, Digital Image Processing, Prentice Hall, 2002

[6] S. LoPresto, K. Ramchandran, M.T. Orc-hard, “Image coding based on mixture mod-eling of wavelet coefficients and a fast esti-mation-quantization framework”, Data Com-pression Conference 97, Snowbird, Utah, 1997

[7] P. Moulin, J. Liu, “Analysis of multireso-lution image denoising schemes using gene-ralized-gaussian and complexity priors”, IEEE Trans. On Info. Theory, vol 45, Apr. 1999

[8] P. Moulin, J. Liu, “Image denoising based on scale-space mixture modeling of wavelet coefficients”,

http://citeseer.nj.nec.com/liu99image.html [9] M. K. Mihcak, L. Kozintsev, K. Ram-chandran, “Local statistical modeling of wavelet image coefficients and its application to denoising”, Proc. IEEE Int. Conf. Acoust., Speech and Signal Proc., Detroit, Mar. 1999 [10] L. State, C. Cocianu, V. Stefanescu, P. Vlamos, “PCA-Based Data Mining Probabil-istic and Fuzzy Approaches with Applica-tions in Pattern Recognition”, Proceedings of ICSOFT 2006, Portugal, pp. 55-60, 2006 [11] L. State, C. Cocianu, V. Stefanescu, P. Vlamos, “A New Approach of Image Resto-ration”, Proceedings of the 27th Conference on Computer&Industrial Engineering, 11-13 oct 2000, Beijing

[12] J.L. Stark, F. Murtagh, A. Bijaoui, “Multiresolution Support Applied to Image Filtering and Restoration”, Technical Report, 1995

[13] M.Vetterli, J.Kovacevic, Wavelets and Subband Coding, Prentice Hall, Englewood Cliffs:N.J.,1995