Image Denoising Using WEAD
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The document proposes a hybrid method called Wavelet Embedded Anisotropic Diffusion (WEAD) for image denoising. WEAD is a two-stage filter that first applies anisotropic diffusion to reduce noise, followed by wavelet-based Bayesian shrinkage. This reduces the convergence time of anisotropic diffusion, allowing the image to be denoised with less blurring compared to anisotropic diffusion or wavelet methods alone. Experimental results on various images demonstrate that WEAD achieves better denoising performance than anisotropic diffusion or Bayesian shrinkage methods, as measured by higher PSNR and SSIM scores and fewer required iterations.
![IMAGE DENOISING USING WAVELET EMBEDDED
ANISOTROPIC DIFFUSION (WEAD)
Jeny Rajan*, M.R Kaimal †
*Network Systems & Technologies Ltd (NeST), Technopark Campus, Trivandrum INDIA, Email :jenyrajan@rediffmail.com
†
Dept. Of Computer Science, University of Kerala, Trivandrum, INDIA , Email :mrkaimal@yahoo.com
Keywords: Anisotropic Diffusion, Bayesian Shrinkage,
Denoising, Wavelets.
Abstract
In this paper a PDE based hybrid method for image denoising
is introduced. The method is a bi-stage filter with anisotropic
diffusion followed by wavelet based bayesian shrinkage. Here
efficient denoising is achieved by reducing the convergence
time of anisotropic diffusion. As the convergence time
decreases, image blurring can be restricted and will produce a
better denoised image than anisotropic or wavelet based
methods. Experimental results based on PSNR, SSIM and
edge analysis shows excellent performance of the proposed
method.
1 Introduction
Image denoising has a significant role in image pre
processing. As the application areas of image denoising are
more, there is a big demand for efficient denoising
algorithms. In this work, we developed a new method;
Wavelet Embedded Anisotropic Diffusion (WEAD), and
applied it to denoise images corrupted with additive Gaussian
noise. The intention behind this method is to reduce the
convergence time of anisotropic diffusion and thereby
increase its performance. The proposed method produces
excellent results when compared with various wavelet
shrinkage [6], [7], [8] and non-linear diffusion methods [1],
[2].
Second order partial differential equations have been used as
efficient methods for removing noise from images. One of the
most commonly used PDE based denoising technique since
its introduction is the Perona-Malik method [1]. The Perona –
Malik equation for an image u is given by
[ ] ),(),(,)( 00
yxuyxuuucdiv
t
u
t
=∇∇=
∂
∂
=
(1)
where ∇u is the gradient of the image u, div is the divergence
operator and c is the diffusion coefficient. The diffusion
coefficient c is a non-decreasing function and diffuses more
on plateaus and less on edges and thus edges are preserved.
Another objective for the selection of c(.) is to incur
backward diffusion around intensity transitions so that edges
are sharpened, and to assure forward diffusion in smooth
areas for noise removal [2]. Some of the commonly employed
diffusivity functions are given in [12]. The equation (1) is
studied as an efficient tool for noise removal and scale space
analysis of images.
Wavelet based methods are always a good choice for image
denoising and has been discussed widely in literatures for the
past two decades [6]-[8],[11],[12].Wavelet shrinkage permits
a more efficient noise removal while preserving high
frequencies based on the disbalancing of the energy of such
representations [4]. The technique denoises image in the
orthogonal wavelet domain, where each coefficient is
thresholded by comparing against a threshold; if the
coefficient is smaller than the threshold, it is set to zero,
otherwise it is kept or modified.
Wavelet shrinkage depends heavily on the choice of a
thresholding parameter and the choice of this threshold
determines, to a great extent the efficacy of denoising. The
denoising process is based on the fact that the wavelet
transform compresses most of the L2
energy of the signal in a
restricted number of large coefficients. The procedure can be
summarized in three steps
(Z)WY'
T(Y,λZ
W(x)Y
1
)
−=
=
=
(2)
where x is the affected signal, W(.) and W-1
is the forward and
inverse wavelet transform operators. T(Y,λ ) denotes the
denoising operator with soft or hard threshold λ. Of the
various methods based on wavelet thresholding,
VisuShrink[6], SureShrink[7], BayesShrink[8] and its
variants are the most popular. VisuShrink uses one of the well
known thresholding rules: the universal threshold. In addition,
subband adaptive systems have superior performance, such as
SureShrink , which is a data driven system. Recently,
BayesShrink [8], which is also a data driven subband adaptive
technique, is proposed and outperforms VisuShrink and
SureShrink. In the proposed method BayesShrink is used
along with anisotropic diffusion to get a better performance
than stand alone Anisotropic diffusion or BayesShrink.
This work does not attempt to investigate in deep the
theoretical properties of the proposed model in general
settings. Our primary goal is to demonstrate that how the
performance of PDE based denoising methods can be
improved by using the proposed hybrid method. The paper is
Appeared in the Proceedings of IEE International Conference on Visual Information Engineering (VIE) 2006, pp 589 - 593](https://image.slidesharecdn.com/imagedenoisingusingwaveletembedded-150115042534-conversion-gate02/85/Image-Denoising-Using-WEAD-1-320.jpg)
![organized as follows. In section II Bayesian denoising
technique is discussed. Section III and IV explains the
proposed method, experimental results and comparison of the
proposed method with other popular models. Finally
conclusion and remarks are included in section V.
2 Denoising Using Bayesian Shrinkage
The Bayesian Shrinkage estimates a soft-threshold that
minimizes the Bayesian risk. The Bayesian risk estimation is
subband dependent. The threshold is mathematically derived
in [8]. The generalized Gaussian distribution (GCD),
following [9] is
{ }|]|),([exp),()(,
β
βσ βσαβσ xXXCGG xX
−= (5)
-∞ <x< ∞, σX>0,β>0, where
2/1
1
)/1(
)/3(
),(
Γ
Γ
= −
β
β
σβσα XX (6)
and
Γ
=
β
βσαβ
βσ
1
2
),(.
),(
X
XC (7)
and
duuet tu 1
0
)( −
∞
−
∫=Γ (8)
is the gamma function. The parameter σX is the standard
deviation and β is the shape parameter. Here the objective is
to find a soft threshold T that minimizes the Bayes risk,
2
^
|
2
^
)(
−=
−= XXEEXXETr XYX
(9)
where )2,(|),(
^
ση xNXYYTX ≈= and
βσ ,X
GGX ≈ .
Denote the optimal threshold by T*
)(minarg),(* Tr
T
XT =βσ (10)
which is a function of the parameters σX and β. In [8] it is
shown that for general σ, the threshold TB can be written as
( )
X
X
B
T
σ
σ
σ
2
= (11)
where σ2
is the noise variance and σX2
the signal variance.
By using the threshold in (11), we can restore the image much
better than by using VisuShrink or Sure Shrink, where
VisuShrink uses a threshold choice, Mlog2σ . This can be
unwarrantedly large due to its dependence on the number of
samples. SureShrink uses a hybrid of the universal threshold
and the SURE threshold, derived from Stein’s unbiased risk
estimator.
3 Proposed Model
In the proposed model the Bayesian Shrinkage of the non-
linearly diffused signal is taken. The equation can be written
as
)( '
1−
= nsn
IBI (12)
where Bs is the Bayesian shrink and '
1−n
I is anisotropic
diffusion as shown in (1) at (n-1)th
time. Numerically (12) can
be written as
( )nnsn
tdIBI ∆+= −1
(13)
where Bs can be calculated by finding Tb as mentioned in eqn
(11) after taking wavelet transform of '
1−n
I .
The intention to develop this method is to decrease the
convergence time of the anisotropic diffusion. It is understood
that the convergence time for denoising is directionally
proportional to the image noise level. In the case of
anisotropic diffusion, as iteration continues, the noise level in
image decreases (till it reaches the convergence point), but in
a slow manner. But in the case of Bayesian shrinkage, it just
cut the frequencies above the threshold and that in a single
step. An iterative Bayesian Shrinkage will not incur any
change in the detail coefficients from the first one. Now
consider the proposed algorithm, here the threshold for
Bayesian shrinkage is recalculated each time after anisotropic
diffusion, and as a result of two successive noise reduction
step, it approaches the convergence point much faster than
anisotropic diffusion.
As the convergence time decreases, image blurring can be
restricted, and as a result image quality increases. The whole
process is illustrated in Fig.2. Fig.2(a) shows the convergence
of the image processed by Perona-Malik anisotropic
diffusion. The convergence point is at P. ie. at P we will get
the better image, with the assumption that the input image is a
noisy one. If this convergence point P can be shifted towards
y-axis, its movement will be as shown in Fig 2 (b).ie. if we
pull the point P towards y-axis, it will move in a left-top
fashion. Here the Bayesian shrinkage is the catalyst, which
pulls the convergence point P of the anisotropic diffusion
towards a better place. The method can be extended to other
PDE based methods like fourth order PDEs, Total-variation
minimization, Complex diffusion etc.
Yy
Scale space images
Anisotropic
Diffusion
BayesShrink
Fig 1: Block diagram of the proposed denoising algorithm.
The iteration process will continue till the input signal y is
converged to Y.](https://image.slidesharecdn.com/imagedenoisingusingwaveletembedded-150115042534-conversion-gate02/85/Image-Denoising-Using-WEAD-2-320.jpg)
![4 Experimental Results & Comparative Analysis
Experiments were carried out on various types of images.
Comparisons and analysis were done on the basis of MSSIM
(Mean Structural Similarity Index Matrix) and PSNR (Peak
Signal to Noise Ratio)
The MSSIM[10] is used to evaluate the overall image quality
and is defined as
),(
1
),(
1
j
M
j
j
yxSSIM
M
YXMSSIM ∑=
= (14)
where X and Y are the reference and the distorted images
respectively, M is the number of local windows in the image,
SSIM is the Structural Similarity Index Matrix, xj and yj are
the image contents at the jth
local window. The Structural
Similarity Index Matrix (SSIM) is defined as
( )( )
( )( )2
22
1
22
21
22
),(
CC
CC
yxSSIM
yxyx
xyyx
++++
++
=
σσµµ
σµµ
(15)
(a) (b) (c)
(d) (e)
Fig 3: Noise removal with Anisotropic Diffusion, BayesShrink and Propose method
a) Original Image b) Noisy Image
(PSNR 20.24 dB, SSIM 0.3788)
c) Denoised with Anisotropic
diffusion( PSNR 25.87 dB, SSIM
0.6744, iterations :315 d)
Denoised with BayesShrink (PSNR
26.71 dB, SSIM 0.7267, iterations
: 1) e)Denoised with proposed
method (PSNR 27.67 dB, SSIM
0.7836, iterations : 51)
(a) (b) (c)
Fig 2: Working of WEAD (a) Shows the convergence of a noisy image (convergence at P). If this P can be shifted towards left,
image quality can be increased and time complexity can be reduced. Illustrated in (b). (c) shows the signal processed by WEAD. It
can be seen that the convergence point is shifted to left and moved upwards.](https://image.slidesharecdn.com/imagedenoisingusingwaveletembedded-150115042534-conversion-gate02/85/Image-Denoising-Using-WEAD-3-320.jpg)
![Fig 3 shows the image denoised with anisotropic diffusion,
Bayesshrink and proposed method. It is observed that the
proposed method reduces the number of iterations of
anisotropic diffusion from 315 to 51 and improves the image
quality. It can be seen that there is 10% improvement over
anisotropic diffusion and around 5% over Bayesshrink in
preserving image structure. Based on PSNR also it can be
seen that the proposed method performs better than the other
two. The graphs in Fig.4 shows comparative analysis of
anisotropic diffusion, bayesshrink and proposed method. It is
clear that the performance of the methods depends on image
type and noise levels. But in both cases, whether anisotropic
diffusion or bayesshrink gives better results, the performance
of the proposed method seems to be much better than the
other two. It can be seen that the proposed method preserves
image structures much better than anisotropic diffusion and
bayesshrink. Also the number of iterations required for the
proposed method to produce the better image is much less
than that of anisotropic diffusion. The experiment is repeated
for various types of images with varying noise levels and
seems that the method proposed is giving better results than
anisotropic diffusion and bayesshrink.
Conclusion
A method to improve the performance of nonlinear
anisotropic diffusion is proposed. The method produces a
converged image with less number of iterations preserving
image edges better than anisotropic diffusion.
Acknowledgements
The authors would like to thank the Scientists of ADRIN
(Hyderabad, India) for helpful discussions. They would also
like to thank the authorities of NeST (Trivandrum, India) for
providing the necessary facilities for doing the experiments.
References
[1] P. Perona and J. Malik, “Scale-space and edge detection
using anisotropic diffusion”, IEEE Trans. Pattern
Analysis and Machine Intelligence., vol 12. No. 12, pp
629-639, July 1990.
[2] Yu-Li You, Wenguan Xu, Allen Tannenbaum and
Mostafa Kaveh, “Behavioral Analysis of Anisotropic
Diffusion in Image Processing”, IEEE Trans. Image
Processing, vol. 5, no. 11, pp 1539-1553, November
1996.
[3] W. Rudin, “Real and Complex Analysis”, New York :
McGraw-Hill, 1996.
[4] Gabriel Cristobal, Monica Chagoyen, Boris Escalante-
Ramirez, Juan R Lopez, “Wavelet-based denoising
methods, A comparative study with applications in
microscopy (Unpublished work style),” Proc. SPIE’s
International Symposium on Optical Science,
Engineering and Instrumentation, Wavelet Applications
in Signal and Image Processing IV, Vol. 2825, Denver,
CO. 1996.
[5] Raghuram Rangarajan, Ramji Venkataramanan,
Siddharth Shah, “Image Denoising Using Wavelets”, ----
---,2002.
[6] David L Donoho, “Ideal spatial adaptation by wavelet
shrinkage”, Biometrika, 81(3) : 425-455, August 1994.
[7] David L. Donoho and Iain M. Johnstone, “Adapting to
Unknown Smoothness via Wavelet Shrinkage,” Journal
of American StatisticalAssociation, 90(432):1200-1224,
December 1995
[8] S. Grace Chang, Bin Yu and Martin Vetterli, “Adaptive
Wavelet Thresholding for Image Denoising and
Compression,” IEEE Trans. Image Processing, Vol 9,
No. 9, Sept 2000, pg 1532-1546.
[9] R.L. Joshi, V.J. Crump and T.R. Fisher, “Image subband
coding using arithmetic and trellis coded quantization”,
IEEE Trans. Circuits Syst. Video Technol., vol. 5, pp
515-523, Dec. 1995.
[10] Zhou Wang, Alan Conard Bovik, Hamid Rahim Sheik
and Erno P Simoncelli, “ Image Quality Assessment :
From error visibility to structural similarity”, IEEE
Trans. Image Processing, Vol. 13, No. 4, 2004.
[11] R.R Cofiman and David L. Donoho, “Translation
invariant denoising”, Wavelets and Statistics, Lecture
Notes in Ststistics, Springer Verlag, 1995.
[12] Pavel Mrazek, Joachim Weickert and Gabriele Steidl,
“Correspondence between Wavelet Shrinkage and
Nonkinear Diffusion”, Scale-Space 2003, LNCS 2695,
pp. 101-116, 2003](https://image.slidesharecdn.com/imagedenoisingusingwaveletembedded-150115042534-conversion-gate02/85/Image-Denoising-Using-WEAD-5-320.jpg)
Image Denoising Using WEAD
- 1. IMAGE DENOISING USING WAVELET EMBEDDED ANISOTROPIC DIFFUSION (WEAD) Jeny Rajan*, M.R Kaimal † *Network Systems & Technologies Ltd (NeST), Technopark Campus, Trivandrum INDIA, Email :jenyrajan@rediffmail.com † Dept. Of Computer Science, University of Kerala, Trivandrum, INDIA , Email :mrkaimal@yahoo.com Keywords: Anisotropic Diffusion, Bayesian Shrinkage, Denoising, Wavelets. Abstract In this paper a PDE based hybrid method for image denoising is introduced. The method is a bi-stage filter with anisotropic diffusion followed by wavelet based bayesian shrinkage. Here efficient denoising is achieved by reducing the convergence time of anisotropic diffusion. As the convergence time decreases, image blurring can be restricted and will produce a better denoised image than anisotropic or wavelet based methods. Experimental results based on PSNR, SSIM and edge analysis shows excellent performance of the proposed method. 1 Introduction Image denoising has a significant role in image pre processing. As the application areas of image denoising are more, there is a big demand for efficient denoising algorithms. In this work, we developed a new method; Wavelet Embedded Anisotropic Diffusion (WEAD), and applied it to denoise images corrupted with additive Gaussian noise. The intention behind this method is to reduce the convergence time of anisotropic diffusion and thereby increase its performance. The proposed method produces excellent results when compared with various wavelet shrinkage [6], [7], [8] and non-linear diffusion methods [1], [2]. Second order partial differential equations have been used as efficient methods for removing noise from images. One of the most commonly used PDE based denoising technique since its introduction is the Perona-Malik method [1]. The Perona – Malik equation for an image u is given by [ ] ),(),(,)( 00 yxuyxuuucdiv t u t =∇∇= ∂ ∂ = (1) where ∇u is the gradient of the image u, div is the divergence operator and c is the diffusion coefficient. The diffusion coefficient c is a non-decreasing function and diffuses more on plateaus and less on edges and thus edges are preserved. Another objective for the selection of c(.) is to incur backward diffusion around intensity transitions so that edges are sharpened, and to assure forward diffusion in smooth areas for noise removal [2]. Some of the commonly employed diffusivity functions are given in [12]. The equation (1) is studied as an efficient tool for noise removal and scale space analysis of images. Wavelet based methods are always a good choice for image denoising and has been discussed widely in literatures for the past two decades [6]-[8],[11],[12].Wavelet shrinkage permits a more efficient noise removal while preserving high frequencies based on the disbalancing of the energy of such representations [4]. The technique denoises image in the orthogonal wavelet domain, where each coefficient is thresholded by comparing against a threshold; if the coefficient is smaller than the threshold, it is set to zero, otherwise it is kept or modified. Wavelet shrinkage depends heavily on the choice of a thresholding parameter and the choice of this threshold determines, to a great extent the efficacy of denoising. The denoising process is based on the fact that the wavelet transform compresses most of the L2 energy of the signal in a restricted number of large coefficients. The procedure can be summarized in three steps (Z)WY' T(Y,λZ W(x)Y 1 ) −= = = (2) where x is the affected signal, W(.) and W-1 is the forward and inverse wavelet transform operators. T(Y,λ ) denotes the denoising operator with soft or hard threshold λ. Of the various methods based on wavelet thresholding, VisuShrink[6], SureShrink[7], BayesShrink[8] and its variants are the most popular. VisuShrink uses one of the well known thresholding rules: the universal threshold. In addition, subband adaptive systems have superior performance, such as SureShrink , which is a data driven system. Recently, BayesShrink [8], which is also a data driven subband adaptive technique, is proposed and outperforms VisuShrink and SureShrink. In the proposed method BayesShrink is used along with anisotropic diffusion to get a better performance than stand alone Anisotropic diffusion or BayesShrink. This work does not attempt to investigate in deep the theoretical properties of the proposed model in general settings. Our primary goal is to demonstrate that how the performance of PDE based denoising methods can be improved by using the proposed hybrid method. The paper is Appeared in the Proceedings of IEE International Conference on Visual Information Engineering (VIE) 2006, pp 589 - 593
- 2. organized as follows. In section II Bayesian denoising technique is discussed. Section III and IV explains the proposed method, experimental results and comparison of the proposed method with other popular models. Finally conclusion and remarks are included in section V. 2 Denoising Using Bayesian Shrinkage The Bayesian Shrinkage estimates a soft-threshold that minimizes the Bayesian risk. The Bayesian risk estimation is subband dependent. The threshold is mathematically derived in [8]. The generalized Gaussian distribution (GCD), following [9] is { }|]|),([exp),()(, β βσ βσαβσ xXXCGG xX −= (5) -∞ <x< ∞, σX>0,β>0, where 2/1 1 )/1( )/3( ),( Γ Γ = − β β σβσα XX (6) and Γ = β βσαβ βσ 1 2 ),(. ),( X XC (7) and duuet tu 1 0 )( − ∞ − ∫=Γ (8) is the gamma function. The parameter σX is the standard deviation and β is the shape parameter. Here the objective is to find a soft threshold T that minimizes the Bayes risk, 2 ^ | 2 ^ )( −= −= XXEEXXETr XYX (9) where )2,(|),( ^ ση xNXYYTX ≈= and βσ ,X GGX ≈ . Denote the optimal threshold by T* )(minarg),(* Tr T XT =βσ (10) which is a function of the parameters σX and β. In [8] it is shown that for general σ, the threshold TB can be written as ( ) X X B T σ σ σ 2 = (11) where σ2 is the noise variance and σX2 the signal variance. By using the threshold in (11), we can restore the image much better than by using VisuShrink or Sure Shrink, where VisuShrink uses a threshold choice, Mlog2σ . This can be unwarrantedly large due to its dependence on the number of samples. SureShrink uses a hybrid of the universal threshold and the SURE threshold, derived from Stein’s unbiased risk estimator. 3 Proposed Model In the proposed model the Bayesian Shrinkage of the non- linearly diffused signal is taken. The equation can be written as )( ' 1− = nsn IBI (12) where Bs is the Bayesian shrink and ' 1−n I is anisotropic diffusion as shown in (1) at (n-1)th time. Numerically (12) can be written as ( )nnsn tdIBI ∆+= −1 (13) where Bs can be calculated by finding Tb as mentioned in eqn (11) after taking wavelet transform of ' 1−n I . The intention to develop this method is to decrease the convergence time of the anisotropic diffusion. It is understood that the convergence time for denoising is directionally proportional to the image noise level. In the case of anisotropic diffusion, as iteration continues, the noise level in image decreases (till it reaches the convergence point), but in a slow manner. But in the case of Bayesian shrinkage, it just cut the frequencies above the threshold and that in a single step. An iterative Bayesian Shrinkage will not incur any change in the detail coefficients from the first one. Now consider the proposed algorithm, here the threshold for Bayesian shrinkage is recalculated each time after anisotropic diffusion, and as a result of two successive noise reduction step, it approaches the convergence point much faster than anisotropic diffusion. As the convergence time decreases, image blurring can be restricted, and as a result image quality increases. The whole process is illustrated in Fig.2. Fig.2(a) shows the convergence of the image processed by Perona-Malik anisotropic diffusion. The convergence point is at P. ie. at P we will get the better image, with the assumption that the input image is a noisy one. If this convergence point P can be shifted towards y-axis, its movement will be as shown in Fig 2 (b).ie. if we pull the point P towards y-axis, it will move in a left-top fashion. Here the Bayesian shrinkage is the catalyst, which pulls the convergence point P of the anisotropic diffusion towards a better place. The method can be extended to other PDE based methods like fourth order PDEs, Total-variation minimization, Complex diffusion etc. Yy Scale space images Anisotropic Diffusion BayesShrink Fig 1: Block diagram of the proposed denoising algorithm. The iteration process will continue till the input signal y is converged to Y.
- 3. 4 Experimental Results & Comparative Analysis Experiments were carried out on various types of images. Comparisons and analysis were done on the basis of MSSIM (Mean Structural Similarity Index Matrix) and PSNR (Peak Signal to Noise Ratio) The MSSIM[10] is used to evaluate the overall image quality and is defined as ),( 1 ),( 1 j M j j yxSSIM M YXMSSIM ∑= = (14) where X and Y are the reference and the distorted images respectively, M is the number of local windows in the image, SSIM is the Structural Similarity Index Matrix, xj and yj are the image contents at the jth local window. The Structural Similarity Index Matrix (SSIM) is defined as ( )( ) ( )( )2 22 1 22 21 22 ),( CC CC yxSSIM yxyx xyyx ++++ ++ = σσµµ σµµ (15) (a) (b) (c) (d) (e) Fig 3: Noise removal with Anisotropic Diffusion, BayesShrink and Propose method a) Original Image b) Noisy Image (PSNR 20.24 dB, SSIM 0.3788) c) Denoised with Anisotropic diffusion( PSNR 25.87 dB, SSIM 0.6744, iterations :315 d) Denoised with BayesShrink (PSNR 26.71 dB, SSIM 0.7267, iterations : 1) e)Denoised with proposed method (PSNR 27.67 dB, SSIM 0.7836, iterations : 51) (a) (b) (c) Fig 2: Working of WEAD (a) Shows the convergence of a noisy image (convergence at P). If this P can be shifted towards left, image quality can be increased and time complexity can be reduced. Illustrated in (b). (c) shows the signal processed by WEAD. It can be seen that the convergence point is shifted to left and moved upwards.
- 4. where µx and µy are the estimated mean intensity along x and y directions and σx and σy are the standard deviation respectively. σxy can be estimated as −− − = ∑= yix N i ixy yx N µµσ )(( 1 1 1 (16) where K1,K2«1 is a small constant and L is the dynamic range of the pixel values (255 for 8 bit grayscale images). The second parameter used for evaluation is the PSNR which is defined as C1 and C2 in (15) are constants and the values are given as ( )2 11 LKC = (17) and ( )2 22 LKC = (18) = rms b PSNR 10 log20 (19) where b is the largest possible value of the signal (typically 255 or 1), and rms is the root mean square difference between two images. (a) (b) (c) (d) (e) (f) Fig 4. Comparative Analysis of Anisotropic Diffusion, BayesShrink and Proposed Method. (a) and (c) based on PSNR and (b) and (d) based on Mean SSIM. For (a) and (b) the image used is gray (shown in (e) ) and for (c) and (d) the image used is Lena (shown in (f)). It can be seen that in both cases the proposed method performs better than the other two.
- 5. Fig 3 shows the image denoised with anisotropic diffusion, Bayesshrink and proposed method. It is observed that the proposed method reduces the number of iterations of anisotropic diffusion from 315 to 51 and improves the image quality. It can be seen that there is 10% improvement over anisotropic diffusion and around 5% over Bayesshrink in preserving image structure. Based on PSNR also it can be seen that the proposed method performs better than the other two. The graphs in Fig.4 shows comparative analysis of anisotropic diffusion, bayesshrink and proposed method. It is clear that the performance of the methods depends on image type and noise levels. But in both cases, whether anisotropic diffusion or bayesshrink gives better results, the performance of the proposed method seems to be much better than the other two. It can be seen that the proposed method preserves image structures much better than anisotropic diffusion and bayesshrink. Also the number of iterations required for the proposed method to produce the better image is much less than that of anisotropic diffusion. The experiment is repeated for various types of images with varying noise levels and seems that the method proposed is giving better results than anisotropic diffusion and bayesshrink. Conclusion A method to improve the performance of nonlinear anisotropic diffusion is proposed. The method produces a converged image with less number of iterations preserving image edges better than anisotropic diffusion. Acknowledgements The authors would like to thank the Scientists of ADRIN (Hyderabad, India) for helpful discussions. They would also like to thank the authorities of NeST (Trivandrum, India) for providing the necessary facilities for doing the experiments. References [1] P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion”, IEEE Trans. Pattern Analysis and Machine Intelligence., vol 12. No. 12, pp 629-639, July 1990. [2] Yu-Li You, Wenguan Xu, Allen Tannenbaum and Mostafa Kaveh, “Behavioral Analysis of Anisotropic Diffusion in Image Processing”, IEEE Trans. Image Processing, vol. 5, no. 11, pp 1539-1553, November 1996. [3] W. Rudin, “Real and Complex Analysis”, New York : McGraw-Hill, 1996. [4] Gabriel Cristobal, Monica Chagoyen, Boris Escalante- Ramirez, Juan R Lopez, “Wavelet-based denoising methods, A comparative study with applications in microscopy (Unpublished work style),” Proc. SPIE’s International Symposium on Optical Science, Engineering and Instrumentation, Wavelet Applications in Signal and Image Processing IV, Vol. 2825, Denver, CO. 1996. [5] Raghuram Rangarajan, Ramji Venkataramanan, Siddharth Shah, “Image Denoising Using Wavelets”, ---- ---,2002. [6] David L Donoho, “Ideal spatial adaptation by wavelet shrinkage”, Biometrika, 81(3) : 425-455, August 1994. [7] David L. Donoho and Iain M. Johnstone, “Adapting to Unknown Smoothness via Wavelet Shrinkage,” Journal of American StatisticalAssociation, 90(432):1200-1224, December 1995 [8] S. Grace Chang, Bin Yu and Martin Vetterli, “Adaptive Wavelet Thresholding for Image Denoising and Compression,” IEEE Trans. Image Processing, Vol 9, No. 9, Sept 2000, pg 1532-1546. [9] R.L. Joshi, V.J. Crump and T.R. Fisher, “Image subband coding using arithmetic and trellis coded quantization”, IEEE Trans. Circuits Syst. Video Technol., vol. 5, pp 515-523, Dec. 1995. [10] Zhou Wang, Alan Conard Bovik, Hamid Rahim Sheik and Erno P Simoncelli, “ Image Quality Assessment : From error visibility to structural similarity”, IEEE Trans. Image Processing, Vol. 13, No. 4, 2004. [11] R.R Cofiman and David L. Donoho, “Translation invariant denoising”, Wavelets and Statistics, Lecture Notes in Ststistics, Springer Verlag, 1995. [12] Pavel Mrazek, Joachim Weickert and Gabriele Steidl, “Correspondence between Wavelet Shrinkage and Nonkinear Diffusion”, Scale-Space 2003, LNCS 2695, pp. 101-116, 2003