SPATIALLY-ADAPTIVE WAVELET DENOISING SCHEME WITH LINEAR COLOR SHRINKAGE

Takahiro Saito, Nobihiro Fujii, Shingo Yoshioka, and Takashi Komatsu

Department of E&I Frontiers, High-Tech Research Center, Kanagawa University 3-27-1 Rokkakubashi, Kanagawa-ku, 221-8686, Yokohama, Japan

phone: +(81)45 481 5661, fax: +(81)45 491 7915, email: saitot01@kanagawa-u.ac.jp web: http://slat4.ee.kanagawa-u.ac.jp

ABSTRACT To remove signal-dependent noise of a digital color camera, this paper presents a new color-image denoising scheme in a wavelet transform domain, which is newly developed from the classic spa-tially-adaptive wavelet denoising scheme with the Wiener-type linear shrinkage function. The classic Wiener-type linear shrinkage function works well for monochrome images; to utilize linear inter-channel color cross-correlations, usually a noisy input color image undergoes the color-transformation from the RGB color space to the luminance-and-chrominance color space, and then the lumi-nance and the chrominance components are separately denoised. However, this color-denoising approach cannot take advantage of nonlinear inter-channel color cross-correlations caused by color edges, and cannot cope with actual signal-dependent noise. To utilize the noise’s signal-dependencies, this paper constructs a new Wiener-type linear color-shrinkage denoising scheme where the inter-channel color cross-correlations are directly utilized in the RGB color space, and its shrinkage parameters are spatially-adaptively controlled according to not only local statistics in a noisy input color image, but also the noise’s signal-dependencies. Experimental simulations demonstrate that our denoising scheme alleviates artifacts caused by denoising, and improves picture quality of denoised images, as well as we have expected.

1. INTRODUCTION Digital camera’s noise shows some signal-dependency [1], [2]. The variance of signal-dependent noise not only differs among the three primary color channels, but also depends on signal intensity. It is not easy to remove the signal-dependent noise selectively from output color images. Many types of denoising method have been proposed [2] ~ [12], and they can efficiently remove signal-independent noise, but many of them do not necessarily work well for actual signal-dependent noise.

To cope with signal-dependent noise of a digital color camera, this paper presents a new color-image denoising scheme in a wave-let transform domain, which is newly developed from the classic spatially-adaptive wavelet denoising scheme with the Wiener-type linear shrinkage function [4]. The classic Wiener-type linear shrink-age function works well for monochrome images; but its direct ap-plication to each primary color channel does not utilize inter-channel color cross-correlations, produces several artifacts as side effects, and does not achieve efficient noise removal. In denoised color images, remaining noise often appears to be artificial notice-able color spots and/or color smears, and specific artifacts referred to as the wavelet outliers, due to the cancellation of small wavelet transform coefficients, can be seen. For removing noise from noisy input color images more efficiently, usually the color images un-dergo the color-transformation from the original primary color space to the luminance-and-chrominance color space, and then the lumi-

nance and the chrominance components are separately denoised. This color-image denoising approach utilizes linear inter-channel color cross-correlations. However, this approach cannot take advan-tage of nonlinear inter-channel color cross-correlations caused by color edges, and cannot cope with the actual signal-dependent noise whose variance differs from one primary color channel to another and depends on an unknown true signal value. To utilize these noise’s properties, we construct a new Wiener-type linear color-shrinkage denoising scheme where the inter-channel color cross-correlations are directly utilized in the RGB color space, and its shrinkage parameters are spatially-adaptively controlled according to not only the local statistics in a noisy input color image, but also the noise’s signal-dependencies. For that purpose, this paper intro-duces the l2 norms of color differences and color sums into the clas-sic l2 variational model whose solution yields the classic Wiener-type linear shrinkage function, and then derives a new Wiener-type linear color-shrinkage scheme for color-image denoising in a wave-let transform domain.

2. SIGNAL-DEPENDENCY OF NOISE IN A DIGITAL COLOR CAMERA

In a digital color camera, several factors cause signal-dependency of additive noise [1], [2]. The factors are as follows: sensor’s shot noise, unevenness of sensor’s amplifiers, inverse gamma correction, color filters, white balancing, color interpolation, and so on. The variance of signal-dependent noise not only depends on signal intensity, but also differs among the three primary color channels.

Signal-dependent noise in a digital color camera is modeled as the following additive noise model [2]:

( ), , , , (1)

, : Noisy observation, : Signal, : Signal-dependent noise,: Gaussian noise with zero mean unit variance,

: Standard deviation of noise .

o C C C C C C g

o

g

F S N S w S N C R G B

F S NN

w N

= + = + ⋅ =

The function w(S) defines noise’s signal-dependency and deter-mines the standard deviation of noise. A typical example of w(S) of a certain digital color camera is shown in Fig. 1.

3. CLASSIC SHRINKAGE DENOISING IN A WAVELET TRANSFORM DOMAIN

3.1 Classic Wiener-type Linear Shrinkage The classic shrinkage functions used for wavelet denoising are de-rived from the lp-l2 type variational denoising problem [13]:

( )2inf , (2)

: Noisy input wavelet transform coefficient,: Denoised wavelet transform coefficient,

pof

o

f f f

ff

λ+ ⋅ −

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where the first lp-norm term is the regularization term that enables us to select a proper denoised wavelet transform coefficient, and the second l2-norm term is the data-fidelity term. In the lp-l2 variational denoising problem of (2), each wavelet transform coefficient is independently optimized irrespective of the other wavelet transform coefficients.

The optimization problem of (2) is easily solved in a closed form, and its optimization leads to the classic shrinkage denoising schemes. In the cases of p = 0, p = 1, and p = 2, the minimization of (2) yields the hard shrinkage, the soft shrinkage, and the Wiener-type linear shrinkage, respectively. The Wiener-type linear shrink-age, which is used in the spatially-adaptive wavelet denoising scheme [4], is defined by

. (3)1 of fλ

λ= ⋅

+

0

5

10

15

20

25

30

35

0 50 100 150 200 250

Signal Intensity S

w

ISO1600 R ISO1600 G ISO1600 BISO800 R ISO800 G ISO800 BISO400 R ISO400 G ISO400 BISO200 R ISO200 G ISO200 BISO100 R ISO100 G ISO100 B

Figure 1 - Measured noise’s signal-dependency functions, w’s, of a certain digital color camera for the three primary color channels and

the various ISO-sensitivity values

3.2 Spatially-Adaptive Denoising Scheme The Wiener-type linear shrinkage of (3) coincides with the well-known linear minimum mean squared error (MMSE) estimator of the true wavelet transform coefficient. We assume that an original clean image S is corrupted by additive white Gaussian noise N with known standard deviation σn, the wavelet transform coefficients of the noisy image Fo are given by fo = s + n where s and n are the wavelet transform coefficients of the clean image S and the noise N, respectively, and s is an independent Gaussian variable with known standard deviation σs. Under these assumptions, the linear MMSE estimator of the true wavelet transform coefficient s is given by

2 2

2 2 . (4)1

s no

s n

f fσ σσ σ

= ⋅+

The comparison of (3) and (4) shows that the parameter λ in (3) corresponds to the signal-to-noise ratio. In (4), it is assumed that the signal’s standard deviation σs is deterministic and known, but actu-ally σs is not known. Hence, the linear MMSE estimator is con-structed as the following approximate estimator:

2 22 2

2 2

ˆˆ, : Estimate for . (5)

ˆ1s n

o s ss n

f fσ σσ σ

σ σ= ⋅

+

The performance of the approximate linear MMSE estimator of (5) depends on the quality of the estimate of the signal variance. At each pixel k, an estimate of the signal variance is formed based on a local neighborhood N(k) centered at k. Assuming the correlation between variances of neighboring wavelet transform coefficients is high, the estimate of the signal variance at each pixel k is given by an approximate maximum likelihood (ML) estimator, as follows:

( ) ( )( )( )

( )

22 21ˆ max 0 , (6)

, : Number of wavelet transform coefficients in .

s o nj N k

k f jM

M N k

σ σ∈

⎛ ⎞= ⋅ −⎜ ⎟⎜ ⎟

⎝ ⎠∑

In [4], the two-stage spatially-adaptive denoising algorithm was proposed, in which a parameter of a priori marginal exponential distribution for each signal’s standard deviation σ(k) is estimated from the wavelet transform coefficients that are initially denoised by using the approximate ML estimate of (6), an approximate maxi-mum a posteriori (MAP) estimate for σ(k) using the exponential prior distribution is calculated, and the approximate MAP estimate is used for the second stage of the denoising with (5). 3.3 Application to Color-Image Denoising The direct application of the classic shrinkage denoising schemes to each primary color channel does not achieve efficient noise removal. To remove noise more efficiently, usually a noisy input color image undergoes the linear color-transformation from the RGB color space to the luminance-and-chrominance color space such as the YCrCb color space, and then the luminance and the chrominance compo-nents are separately denoised. This approach utilizes linear inter-channel color cross-correlations for denoising. However, this con-ventional approach cannot take advantage of nonlinear inter-channel color cross-correlations caused by color edges, and cannot cope with actual signal-dependent noise of a digital color camera.

4. NEW WIENER-TYPE LINEAR COLOR-SHRINKAGE DENOISING SCHEME

To utilize the noise’s signal-dependencies for denoising much more efficiently, we construct a new Wiener-type linear color-shrinkage denoising scheme in a wavelet transform domain, in which the inter-channel color cross-correlations are directly utilized in the RGB color space, and its shrinkage parameters are spatially-adaptively controlled according to not only the local statistics in a noisy input color image, but also the noise’s signal-dependencies. 4.1 Wiener-type Linear Color-Shrinkage For brief expression, in the following, noisy input wavelet transform coefficients of three primary color channels are simply denoted by ro, go, bo. In addition, their denoised ones are simply denoted by r, g, b. The classic Wiener-type linear shrinkage function is derived from the well-known l2-l2 variational problem [13]. By introducing the l2 norms of color differences and color sums into the variational prob-lem, we form a new l2-l2 color variational problem, in which for each pixel its colors are separately optimized. Given true values, b, r, of the B and the R channels, we formulate an energy function FG [g| b, r] of a denoised value g of the G channel as follows:

( )

2 2 2

2 2 2

F ,

. (7)

G

G o

g b r g g r g r

g b g b g g

α β

α β λ

⎡ ⎤ = + ⋅ − + ⋅ +⎣ ⎦

+ ⋅ − + ⋅ + + ⋅ −

The l2 norms of g - r and g - b operate to preserve identically-varying variations in which the two color signals change in phase with each other, whereas the l2 norms of g + r and g + b operate to preserve oppositely-varying variations in which one color signal increases while the other color signal decreases [12]. Since for most of natural color images the occurrence probability of the identically-varying variations is considered higher than that of the oppositely-varying variations, and hence the two non-negative parameters α, β are set so that the inequality α > β may be satisfied. The last energy term in (7) is a data-fidelity term, and the parameter λG is set for each pixel so that its values may be proportional to the signal-to-noise ratio, viz. the ratio of the signal variance to the noise variance, while taking into account the noise’s signal-dependencies measured in advance.

16th European Signal Processing Conference (EUSIPCO 2008), Lausanne, Switzerland, August 25-29, 2008, copyright by EURASIP

The minimization of (7) gives a Wiener-type linear color-shrinkage function Sh[go | b, r; α, β, λG] whose computational for-mula is expressed in a closed fashion, as follows:

1 2 2

1 2

Sh , ; , , (8)

, , ,1 2 2 1 2 2

0 , 0 .

o G o

G

G G

G

g g b r c g c b c r

c c

α β λ

λ α βα β λ α β λ

α β λ

⎡ ⎤= = × + × + ×⎣ ⎦−

= =+ + + + + +

> ≥ ≥

This Wiener-type linear color-shrinkage function is the straightfor-ward extension of the classic Wiener-type linear shrinkage function of (3) to the 3D primary color space.

The color-shrinkage denoising schemes for the other color channels are easily obtained by permuting r, g and b in the Wiener-type linear color-shrinkage function Sh[go | b, r; α, β, λG] for the G channel. Each of the three different Wiener-type linear color-shrinkage functions for the three primary color channels has the other two color channels as its parameters, and the three functions are dependent on each other. Hence, actual color-image denoising is achieved by repeating the three Wiener-type linear color-shrinkage functions for the three primary color signals alternately. The color-image denoising algorithm with the Wiener-type linear color-shrinkage functions is formed in an iterative fashion: [Iterative Wiener-type Linear Color-Shrinkage Denoising Scheme] 0) Initial setting:

( ) ( ) ( ){ }0 0 0; ; (9)o o or r g g b b← ← ←

1) Iterative update: ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1

1

1

Sh , ; , ,

Sh , ; , , (10)

Sh , ; , ,

n n no R

n n no G

n n no B

r r g b

g g b r

b b r g

α β λ

α β λ

α β λ

+

+

+

⎧ ⎫⎡ ⎤← ⎣ ⎦⎪ ⎪⎪ ⎪⎪ ⎪⎡ ⎤←⎨ ⎬⎣ ⎦⎪ ⎪⎪ ⎪⎡ ⎤←⎪ ⎪⎣ ⎦⎩ ⎭

2) Convergence test: If all the variables, r(n), g(n), b(n), are convergent, then stop the iteration; otherwise set n = n + 1 and return to 1). The algorithm converges mostly within not more than 5 iterations. 4.2 Convergence Analysis of the Iterative Scheme Defining the primary color vectors by

( ) ( ) ( ) ( ) ( )( )0 0 0 0, , , , , , (11)TT n n n nr g b r g b= =p p

the initial setting of (9) and the iterative update of (10) are expressed in a vector form, as follows:

( )00 , (12)=p p

( ) ( )10

1 2 2

1 2 2

1 2 2

1 2

, (13)0 0 0

, 0 0 , 0 ,0 0 0

, , , , .1 2 2 1 2 2

n n

R R R

G G G

B B B

CC C

C C

c c cc c c

c c c

c c C R G Bλ α βα β λ α β λ

+ = ⋅ + ⋅

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟= =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

−= = =

+ + + + + +

p D p T p

D T

Starting from the initial setting of (12), the iterative application of (13) yields the following solutions:

( )1

00

; 1, 2, . (14)n

n i n

i

n−

=

⎡ ⎤⎛ ⎞= ⋅ + ⋅ =⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∑p T D T p

By the spectral resolution, the transient matrix T is factorized:

( ) ( )1 2 3 1 2 3

1 2 3

(15), , ,

: Eigenvalues of , ,: Unit eigenvector corresponding to .

T

i

i i

κ κ κκ κ κ κ

κ

= ⋅ ⋅

= =

≥ ≥

T Q Λ QQ q q q Λ diag

Tq

If the absolute values of the eigenvalues, κ1, κ2, κ3, are less than 1.0, the iterative application of (13) will converge and will provide a stationary solution. Since the parameters, α, β, λR, λG, λB, are set so that the following inequalities can be satisfied,

0 , 0 , 0 , 0 , (16)R G Bα β λ λ λ> ≥ ≥ ≥ ≥ the following inequalities are always satisfied.

( ) ( )210 , , , . (17)

2 1 4 2CC

c C R G Bα βα β β λ

−< = < =

− + + +

In the parameter setting of (16), the necessary and sufficient condi-tion for the convergence is given by

2 2 2 2 2 2 2 2 22 1. (18)R G G B B R R G Bc c c c c c c c c⋅ + ⋅ + ⋅ + ⋅ ⋅ < The inequalities of (17) ensure that the necessary and sufficient condition of (18) always holds true, and the convergence solution is given by

( )S

1 0

1 2 3 1 01 2 3

1 0

1 1 1 2 2 2 3 3 3

lim

1 1 1 (19)1 1 1

, , , .

n

n

R

G

B

T T T

c rc gc b

κ κ κ

→∞=

⋅⎛ ⎞⎛ ⎞ ⎜ ⎟= ⋅ + ⋅ + ⋅ ⋅ ⋅⎜ ⎟ ⎜ ⎟− − −⎝ ⎠ ⎜ ⎟⋅⎝ ⎠= ⋅ = ⋅ = ⋅

p p

P P P

P q q P q q P q q

The matrix Pi is the projection matrix onto the 1-D linear subspace spanned by qi.

For ease of interpretation on (19), let us assume that the three parameters, λR, λG, λB, are equal to the constant value λ, as follows:

0 . (20)R G Bλ λ λ λ= = = ≥ In this case, the transient matrix T has the eigenvalues:

1 2 2 3 2 22 , , . (21)1 2 2

c c c α βκ κ κα β λ−

= = = − =+ + +

The unit eigenvector q1 corresponding to κ1 is given by

( )1 1,1,1 3 , (22)T=q and the projection of the primary color vector p=(r, g, b)T onto the eigenspace spanned by q1 means the calculation of its luminance component L, as follows:

( ) ( )1, 3 . (23)L r g b= = + +p q On the other hand, the eigenspace spanned by q2 and q3 is the 2-D linear subspace perpendicular to q1, and the projection of p onto this 2D eigenspace means the calculation of its two chrominance com-ponents, D1, D2, as follows:

( ) ( )1 2 2 3, , , . (24)D D= =p q p q In this case, the convergence solution is given by

( )S L L 0 CH CH 0

L 1 1 CH 2 2 3 3

L CH

CH L

lim (25)

, , ,

, ,1 4 1 3

0 1 .

n

nT T T

γ γ

λ λγ γβ λ α β λ

γ γ

→∞= = ⋅ ⋅ + ⋅ ⋅

= ⋅ = ⋅ + ⋅

= =+ + + + +

≤ ≤ <

p p P p P p

P q q P q q q q

The matrix PL is the projection matrix onto the luminance space, whereas the matrix PCH is the projection matrix onto the 2-D chrominance space. The linear transformation of (25) means the following denoising scheme: an input primary color vector po is first decomposed into its luminance vector PL·po and its chrominance vector·PCH·po, the decomposed vectors are shrunk by multiplying them by the two scalar constants, γL, γCH, respectively, and then a denoised primary color vector pS is synthesized by adding the two shrunken vectors.

Of course, the linear transformation of (19) in a general case means a more flexible denoising scheme whose color transforma-tion and shrinkage are automatically adaptable to signal variations

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and signal-dependencies of noise to be removed for each pixel; for that purpose, this paper proposes the two-stage denoising algorithm described in the next section. 4.3 Two-Stage Denoising Algorithm The variable S of the signal-dependency function w in (1) is the unknown true image, and hence in fact this paper adopts a two-stage denoising algorithm. At its first denoising stage, a noisy input color image Fo is denoised by the spatially-adaptive wavelet denoising method with the iterative Wiener-type linear color-shrinkage de-noising scheme, in which the three parameters, λR, λG, λB, are spa-tially-adaptively set at each pixel k, as follows:

( ) ( )

( )

2,

2,

2,

2,

ˆ; , , (26)

ˆ, : Estimate of a signal variance at a pixel for its color ,

: Noise-variance for the color over the entire image,

s CC

n C

s C

n C

kk C R G B

k k C

C

σλ

σ

σ

σ

= =

where the estimates of signal variances are given by the approxi-mate maximum likelihood (ML) estimator of (6). At the second denoising stage, instead of the unknown true image S, the image F denoised at the first stage is used, and the noisy input color image Fo is denoised again by the spatially-adaptive wavelet denoising method with the iterative Wiener-type linear color-shrinkage de-noising scheme, in which the three parameters, λR, λG, λB, are spa-tially-adaptively set at each pixel k, as follows:

( ) ( )( )( )

( )

2,

2

2,

ˆ; , , (27)

ˆ, : Estimate of a signal variance at a pixel for its color ,

:Signal-dependency function for the color ,: Color signal of the image denoised at the first stage.

s CC

C C

s C

C

C

kk C R G B

w F k

k k C

w CF C F

σλ

σ

= =

5. EXPERIMENTAL SIMULATIONS In the simulations, the bi-orthogonal shift-invariant wavelet trans-form is used [14]. The denoising simulations are conducted on the noisy test images, which are produced by adding artificial signal-dependent noise equivalent to ISO 1600 sensitivity, shown in Fig. 1, to the original noise-free images of Fig. 2.

Fig. 6 shows a portion of the image denoised at the second stage of our new color wavelet shrinkage approach. Fig. 5 shows a portion of the image denoised through the application of the classic spatially-adaptive wavelet denoising scheme (SAWD) [4] with the Wiener-type linear shrinkage function in the YCrCb color space. As shown in Fig. 5, the SAWD removes signal-dependent noise, but as side effects it produces visible artifacts, referred to as the wavelet outliers. The comparison between Fig. 5 and Fig. 6 demonstrates that utilizing the inter-channel color cross-correlations and the noise’s signal-dependencies directly in the RGB color space for color-image denoising alleviates the side effects, and improves pic-ture quality of denoised images, as well as we have expected. How-ever, even our new denoising method cannot perfectly suppress the wavelet outliers in smoothly-varying image regions.

Table 1 compares peak SNR’s [dB] of denoised images given by the first stage of our denoising approach, the second stage of our denoising approach, the SAWD [4] in the RGB color space, the SAWD [4] in the YCrCb color space, our previously proposed color total-variation denoising approach (Color-TV Denoising) [12], the bilateral filter [9], and the 3D transform domain collaborative filter-ing (Collaborative Filtering) [11], for the case of signal-dependent noise equivalent to ISO 1600 sensitivity and the four color images shown in Fig. 2. Our new color wavelet shrinkage approach shows the highest peak SNR’s, preserves fines image textures while sup-pressing the wavelet outliers; and restores color images giving the most natural visual impression.

6. CONCLUSIONS To remove signal-dependent noise of a digital color camera, this paper presents a new color-image denoising scheme in a wavelet transform domain, which is newly developed from the classic spa-tially-adaptive wavelet denoising scheme with the Wiener-type linear shrinkage function [4]. To utilize the noise’s signal-dependencies and the inter-channel color cross-correlations for de-noising, this paper constructs a new Wiener-type linear color-shrinkage denoising scheme where the inter-channel color cross-correlations are directly utilized in the RGB color space, and its shrinkage parameters are spatially-adaptively controlled according to not only local statistics in a noisy input color image, but also the noise’s signal-dependencies. Experimental simulations demonstrate that our proposed spatially-adaptive Wiener-type linear color-shrinkage denoising scheme alleviates artifacts caused by denoising, achieves a good balance between efficient removal of noise and preservation of fines image textures, restores images giving a natu-ral visual impression, and improves picture quality of denoised im-ages, as well as we have expected.

REFERENCES [1]G. Cortelazzo, G.A. Mian, and R. Parolari, “Statistical character-istics of granular camera noise,” IEEE Trans. Circuits Syst. Video Technol., vol.4, no.66, pp.536-543, 1994. [2] T. Saito, Y. Ishii, R. Nosaka, and T. Komatsu, “Removal of signal-dependent noise for a digital camera,” in Proc. SPIE-IS&T Electronic Imaging, vol.6502 (Digital Photography), San Jose, USA, 2007, pp.6502E:1-6502E:12. [3]D.L. Donoho, “Denoising by soft-thresholding,” IEEE Trans. Inform. Theory, vol.41, no.4, pp.613-627, 1995. [4]M. K. Mihcak, et al., “Low-complexity image denoising based on statistical modeling of wavelet coefficients”, IEEE Signal Proc-essing Letters, vol.6, no.12, pp.300-303, 1999. [5]S.G. Chang, et al., “Spatially adaptive wavelet thresholding with context modeling for image denoising,” IEEE Trans. Image Proc-essing, vol.9, no.9, pp.1522-1531, 2000. [6]J. Portilla, et al., “Image denoising using Gaussian scale mixtures in the wavelet domain,” IEEE Trans. Image Processing, vol.12, no.11, pp.1338-1351, 2003. [7]B. Goossens, et al., “Wavelet domain image denoising for non-stationary noise and signal-dependent noise,” in Proc. 2006 IEEE Int. Conf. Image Processing, Atlanta, USA, 2006, pp.1425-1428. [8]L. Rudin, S. Osher, and E. Fetami, “Nonlinear total variation based noise removal algorithm,” Physica D, vol.60, pp.259-268, 1992. [9]C. Tomasi and R. Manduchi, “Bilateral filtering for gray and color images,” in Proc. 1998 IEEE Int. Conf. Computer Vision, Bombay, India, 1998, pp.839-846. [10]A. Buades, B. Coll, and J.M. Morel,”A review of image denois-ing algorithms, with a new one,” Multiscale Model. Simul., vol.4, no.2, pp.490-530, 2005. [11]K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Processing, vol.16, no.8, pp.2080-2095, 2007. [12]T. Saito and T. Komatsu, “Total-variation approach and wavelet shrinkage for color-image denoising with inter-channel cross-correlations,” IEEE 3rd Int. Symposium on Communications, Con-trol and Signal Processing, St. Julian, Malta, 2008, pp.494-499. [13]M. Elad, “Why simple shrinkage is still relevant for redundancy representations?” IEEE Trans. Information Theory, vol.52, no.12, pp.5559-5569, 2005. [14]M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, “Im-age coding using wavelet transform,” IEEE Trans. Image Process-ing, vol.1, no.2, pp.205-220, 1992.

16th European Signal Processing Conference (EUSIPCO 2008), Lausanne, Switzerland, August 25-29, 2008, copyright by EURASIP

Figure 2 – Set of original noise-free color images used for denoising

simulations: No.1 ~ No.4 from the top left to the bottom right.

Table 1 – PSNR’s [dB] of denoised images given by our denoising approach and the existing denoising methods, in the noisy case

equivalent to the ISO 1600 sensitivity

Image No. 1 Image No. 2 Image No. 3 Image No. 4R 21.09 21.45 21.33 21.00G 19.66 19.88 19.63 19.61B 23.41 23.54 23.18 23.13

Total 21.12 21.37 21.14 21.01Image No. 1 Image No. 2 Image No. 3 Image No. 4

R 27.69 28.62 30.05 29.13G 27.31 28.24 29.68 28.63B 27.98 29.06 30.56 29.44

Total 27.65 28.63 30.08 29.05R 27.93 28.75 30.57 29.81G 27.39 28.25 29.92 29.17B 28.58 29.39 31.02 30.32

Total 27.94 28.77 30.48 29.74R 25.39 26.66 28.92 27.39G 24.45 25.78 28.09 26.48B 26.90 28.03 29.94 28.99

Total 25.47 26.73 28.92 27.50R 26.81 28.08 29.64 28.96G 26.47 27.61 29.47 28.42B 27.22 28.59 29.78 28.27

Total 26.82 28.07 29.63 28.54R 27.69 28.55 30.81 29.46G 27.21 27.91 29.35 28.53B 27.98 28.99 30.46 28.88

Total 27.61 28.46 30.16 28.94R 24.02 24.86 26.69 25.50G 22.81 23.50 25.69 24.20B 25.32 25.96 27.53 26.56

Total 23.93 24.66 26.57 25.31R 25.71 27.26 29.48 28.30G 24.75 25.98 28.60 27.05B 25.85 27.61 29.41 28.68

Total 25.41 26.89 29.14 27.95

Denoising Methods

Noisy TestColor Images

Color-TVDenoising [12]

Bilateral Filter[9]

CollaborativeFiltering [11]

Our Approach(First Stage)

Our approach(Second Stage)

SAWD [4]in RGB

SAWD [4]in YCrCb

Figure 3 – Portion of the original noise-free image No.3

Figure 4 – Portion of the noisy test image

Figure 5 – Portion of the image denoised by the spatially-adaptive wavelet denoising scheme (SAWD) [4] in the YCrCb color space

Figure 6 – Portion of the image denoised at the second stage of our

new color wavelet shrinkage approach

16th European Signal Processing Conference (EUSIPCO 2008), Lausanne, Switzerland, August 25-29, 2008, copyright by EURASIP