 Open Access
 Total Downloads : 882
 Authors : Shamaila Khan, Anurag Jain , Ashish Khare
 Paper ID : IJERTV2IS1149
 Volume & Issue : Volume 02, Issue 01 (January 2013)
 Published (First Online): 07022013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Denoising of Images Based on Different Wavelet Thresholding by Using Various Shrinkage Methods using Basic Noise Conditions
Shamaila Khan 
Prof. Anurag Jain 
Prof. Ashish Khare 
M.Tech.(CSE) 
HOD (CSE) 
HOD (IT) 
RITS, Bhopal 
RITS, Bhopal 
RITS, Bhopal 
ABSTRACT
Wavelet transforms enable us to represent signals with a high degree of scarcity. Wavelet thresholding is a signal estimation technique that exploits the capabilities of wavelet transform for signal denoising. The aim of this paper is to study various thresholding techniques such as Sure Shrink, Visu Shrink and Bayes Shrink and determine the best one for image denoising. This paper presents an overview of various threshold methods for image denoising. Wavelet transform based denoising techniques are of greater interest because of their performance over Fourier and other spatial domain techniques. Selection of optimal threshold is crucial since threshold value governs the performance of denoising algorithms. Hence it is required to tune the threshold parameter for better PSNR values. In this paper, we present various wavelet based shrinkage methods for optimal threshold selection for noise removal.
General Terms
Image denoising, Wavelet based methods.
Keywords
Denoising, Spatial domain methods, Wavelet shrinkage, optimal threshold selection

INTRODUCTION
In general, an image may be contaminated by noise during acquiring and transmission. The noise present in the images may appear as additive or multiplicative components which have been modelled in a number of ways in the literature [1],[17] such as Gaussian noise, Speckle noise, Salt & Pepper noise, Impulse noise etc… As the occurrence of noisy pixels in the image is random in nature, their distributions are modelled using probabilistic methods [20] [24]. In most of the real time applications such as medical imaging, satellite image data analysis, remote applications etc.., the noisy components have to be removed to ensure faithful information retrieval from the images. A common defect in the imaging system is unwanted non linearity in the sensor and display system. Post processing correction
of sensor signals and preprocessing correction of display signals can reduce degradations substantially [1]. Hence preprocessing is essential in any information analysis and retrieval system. Denoising is one of the preprocessing techniques which have drawn much attention of the researchers over a few decades. In this paper, present a detailed survey of various noise removal techniques, with a focus on threshold computing methods is presented since choosing the threshold is crucial in the process of denoising. This paper is organized as follows. Section 2 presents denoising procedure and classification of denoising methods. Section 3 discusses about the wavelet based denoising techniques. Various threshold methods and the tradeoffs involved in selecting an optimal threshold are presented in Section 4. Finally, discussions on observations and conclusion are presented in Section 5.

METHODS OF DENOISING
If f(x,y) be the uncorrupted image of size NXN and n(x,y) be the noise function, then the noisy image observation g(x,y) with additive noise.
The process of denoising is nothing but the estimation of the information from noisy observation. With this background, the state of art denoising methods can be categorized as follows.

Spatial Filtering Techniques
Spatial filtering is the method of choice in situations when only additive noise is present. This category consists of mean filter and the order statistics filter such as Median filter, Maximum and Minimum filter, Midpoint and Alpha trimmed median filter. Arithmetic and Geometric mean filters are well suited for random noise like Gaussian or uniform noise. The Contraharmonic filter is well suited for impulse noise, but it requires the prior knowledge about the noise (light or dark). As found in the literature [1],[17], median filter can perform well in removing impulse noise while the number of passes of the median filter has to be
kept as low as possible, since larger number of passes may result in blurred images. The process of spatial filtering consists of moving the filter mask (Fig: 1) from point to point in an image. At each point (x,y) the response of filter at that point is calculated. The mask may be of any size of interest (3X3, 5X5, 7X7 etc…). Also, it has to be noted that size of the filter mask affects the performance of the filter [15].
Another class of filters which fall under spatial filters is adaptive filter, which method changes behavior based on the statistical characteristics of the image inside the filter region defined by m x n rectangular window. These filters can offer superior denoising performance with the cost of increased complexity [17] [24].
Adaptive median filter is the prime variant of adaptive filter. Filter mask size is altered according to the parameters calculated
in the mask area considered originally. It performs well for the impulse noise with low spatial density and seeks to preserve details while smoothing nonimpulse noise too. Researchers have shown interest to evolve adaptive iterative median filter which outperforms even for high density noises [26].

Frequency domain filtering
Frequency domain filtering can be used for periodic noise reduction and removal. This category of filters include band pass filter, band stop filter, Notch (Reject/Pass) filters. The appropriate filter can be chosen with the prior knowledge of noise distribution. The various Fourier domain filtering techniques such as Inverse filter, Wiener filter and least square filter are found in literature. A simple method of removing multiplicative noise like speckle noise too has been proposed namely homomorphic filtering [1] [17]. Fourier transform has been found to be an important image processing tool for image processing and analysis. The major advantage of Fourier domain analysis is that, it can explore the geometric characteristics of a spatial domain image [2]. It has been used for the removal of additive noises from the images. Unlike Fourier transform, Wavelet transform shows localization in both time and frequency and hence it has proved itself to be an efficient tool for a number of image processing applications including noise removal [19]. Fourier transform based methods are less useful because, they cannot work on non stationary signals and they can capture only global features. But in the real scenario, as the images are only piecewise smooth and the noise distributions are random in nature, Fourier transform cannot perform well for the stochastic noise, but wavelets can do. Hence wavelet based noise removal has attracted much attention of the researchers for several years [4], [6]. A detailed study on wavelet based Denoising techniques is presented in the next section


WAVELET DENOISING
Wavelet transform is the mathematical tool used for various image processing applications such as noise removal, feature extraction, compression and image analysis. The general method of wavelet based denoising is that, the noisy image may first be transformed to wavelet domain [2] [6].
The transformed image appears as four subbands (A, V, H, and D) as shown in Fig 1 based on the level of decomposition j. 2D discrete wavelet transform leads to decompsition of approximate coefficients at level j into four components i.e. the approximation at level j+1 and details in three orientations (Horizontally, Vertically and diagonally) [25]. Since the noisy components are of high frequency, the three higher bands may contain the noisy components [25], and proper threshold may be applied to smooth the noisy wavelet coefficients followed by the inverse 2DDWT may be applied to reconstruct the denoised image. Selection of optimal threshold is crucial for the performance of denoising algorithm. Threshold is selected based on the image and noise priors such as mean and variance [10] [23]. Selection of optimal threshold along with various types of wavelet threshold methods is presented in the next section.
Fig.1 One DWT decomposition step

WAVELET BASED THRESHOLD METHODS

Sure Shrink
Sure Shrink is more explicitly adaptive to unknown smoothness and has better largesample MSE properties. This method is a subband adaptive threshold scheme, based on Steins unbiased estimator for risk (SURE) (quadratic loss function) [68]. One gets an estimate of the risk for a particular threshold value t. minimizing the risks in t gives a selection of the threshold value. Sure Shrink is a thresholding by applying subband adaptive threshold, a separate threshold is computed for each detail subband based upon SURE (Steins unbiased estimator for risk), a method for estimating the loss 2 in an unbiased fashion. In our case let wavelet coefficients in the jth subband be { Xi : i =1,,d }, is the soft threshold. applied to the image data, resulting in an estimate of the mean vector. This estimate is sparse and much less noisy than the raw image data [14]. The SURE principle just described has a serious drawback in situations of extreme sparsity of the wavelet coefficients. In such cases the noise contributed o the SURE profile by the many coordinates at which the signal is zero, swamps the information contributed to the SURE profile by the few coordinates where the signal is nonzero. Consequently, Sure Shrink uses a Hybrid scheme [16].

Bayes Shrink
Bayes Shrink has attracted much attention since it sets different thresholds for every subband. Here subbands are frequency bands that differ from each other in level and direction. The relationship between the wavelet transforms of the degraded image, uncorrupted image and generalized Gaussian noise with distribution:
N (0, 2) (Y, X and V respectively), can be modeled as:
Y = X+V (1)
Since huge information about the noise is available at the diagonal coefficients of first level wavelet decomposition (HH1) the noise variance is calculated using the robust estimator. Wm are the wavelet coefficients in each scale and M is the total number of wavelet coefficients. With this background, the threshold using Bayes shrink is calculated.
The Bayes shrink method is effective for images corrupted by Gaussian noise. Bayes shrink is less sensitive to the presence of noise in the areas around the edges [9] [11]. However, the presence of noise in flat regions of the image is perceptually more noticeable by the human visual system. Bayes shrink performs little Denoising in high
4.4 Neigh Shrink
In the spatial domain, it is well known that an adaptive Wiener method based on estimation from local information is very efficient for digital image enhancement. In the wavelet domain, despite the decorrelating properties of the wavelet transform, as pointed out in the introduction, there still exist significant residual statistical dependencies between neighbor wavelet coefficients. Our goal is to exploit this dependency to improve the estimation of a coefficient given its noisy observation and a context (spatial and scale neighbors).
One of the simplest wavelet shrinkage rules for an N x N image is the universal threshold
activity subregions to preserve the sharpness of edges but completely denoised the flat subparts of the image.
= 2 Â²logNÂ²
(7)
The risk function values are equal to the risk in coefficient values. A mere least square estimate does not denoise the original image [21].Hence to estimate wavelet coefficients.
4.3 Bivariate Shrink
New shrinkage function which depends on both coefficient and its parent yield improved results for wavelet based image denoising. Here, we modify the Bayesian estimation problem as to take into account the statistical dependency between a coefficient and its parent. Then,
The universal threshold grows asymptotically and removes more noise coefficients as N tends to infinity. The universal threshold is designed for smoothness rather than for minimizing the errors. So is more meaningful when the signal is sufficiently smooth or the length of the signal is close to infinity. Natural image, however, is usually neither sufficiently smooth nor composed of infinite number of pixels. In fact, if we suppose that an optimal threshold which minimize MSE (or maximized PSNR), is , is always much less than 1.0 for natural image. Especially we got very similar value for different kinds and size of
y =w +n
(2)
images when we applied soft thresholding rule.
1 1 1
y2=w2+n2 (3)
4.5 Trade off between Threshold, PSNR
Where y and y are noisy observations of w and w and n
and Complexity
1 2 1 2 1
and n2 are noise samples.
Then, mathematically it can be written as
y=w + n (4)
w= (w1, w2) (5)
n= (n1, n2) (6)
Fig.2 New Bivariate shrinkage function
Selection of optimal threshold determines the efficiency of the Denoising algorithm [10]. The common measure of quality in images in peak signal to noise ratio are defined as PSNR=y=10(ylo1g, y2)(255) 2/MSE (db) (8)
10
10
Here MSE is the mean square error whose magnitude quantifies the presence of noise and the performance of Denoising algorithm. As discussed in section – IV wavelet based shrinkage algorithms give better estimate of the noise priors and hence the threshold with the expense of high computational complexity. It is very crucial to select the threshold value with less computational complexity and with significant improvements in PSNR.
Fig. 2 Salt & Pepper Noise with Sure Shrink Method


Evaluation Criteria
The above said methods are evaluated using the quality measure Peak Signal to Noise ratio which is calculated using the formula:
PSNR= 10log 10 (255) 2/MSE (db) (9)
Where, MSE is the mean squared error between the original image and the reconstructed denoised image. It is used to evaluate the different denoising scheme like Neigh shrink and Modified Neigh shrink.

Experiments
Quantitatively assessing the performance in practical application is complicated issue because the ideal image is normally unknown at the receiver end. So this paper uses the following method for experiments. One original image is applied with Gaussian noise with different variance. The methods proposed for implementing image denoising using wavelet transform take the following form in general. The image is transformed into the orthogonal domain by taking the wavelet transform. The detail wavelet coefficients are modified according to the shrinkage algorithm. Finally, inverse wavelet is taken to reconstruct the denoised image. In this paper, different wavelet bases are used in all methods. For taking the wavelet transform of the image, readily available MATLAB routines are taken. In each subband, individual pixels of the image are shrinked based on the threshold selection. A denoised wavelet transform is created by shrinking pixels. The inverse wavelet transform is the denoised image.

Results and Discussions
For the above mentioned three methods, image denoising is performed using wavelets from the second level to fourth level decomposition and the results are shown in figure (3) and able if formulated for second level decomposition for different noise variance as follows. It was found that three level decomposition and fourth level decomposition gave optimum results. However, third and fourth level decomposition resulted in more blurring. The experiments were done using a window size of 3X3, 5X5 and 7X7. The neighborhood window of 3X3 and 5X5 are good choices.
Fig. 1 Gaussian Noise with Sure Shrink Method
Original Image Noisy Image Denoised Image
Original Image Noisy Image Denoised Image
Fig. 3 Speckle Noise with Sure Shrink Method
Original Image Noisy Image Denoised Image
Results based on PSNR values obtained by applying on different methods.
Table 1. Type of noise: Gaussian Noise
Lena
Barbara
Mandrill
My_Image
Sure Shrink
24.9186
24.8543
26.1505
26.444
Bayes Shrink
23.9419
23.4202
24.369
24.7382
Neigh Shrink
26.5673
27.8746
24.8382
31.0067
Bivariate Shrink
71.9643
71.4069
96.2018
77.7025
Table 2. Type of noise: Salt and Pepper Noise
Lena
Barbara
Mandrill
My_Image
Sure Shrink
18.5611
19.2412
18.7047
20.9524
Bayes Shrink
18.567
19.1534
18.5042
20.8096
Neigh Shrink
22.6607
22.6526
23.4103
22.625
Bivariate Shrink
75.7063
71.4103
69.1896
72.0309
Lena
Barbara
Mandrill
My_Image
Sure Shrink
19.4768
21.2934
20.7625
22.9878
Bayes Shrink
16.9754
18.5228
17.3399
20.754
Neigh Shrink
24.5477
23.3882
23.3804
23.7186
Bivariate Shrink
75.6929
71.4178
69.1993
71.9989
Lena
Barbara
Mandrill
My_Image
Sure Shrink
19.4768
21.2934
20.7625
22.9878
Bayes Shrink
16.9754
18.5228
17.3399
20.754
Neigh Shrink
24.5477
23.3882
23.3804
23.7186
Bivariate Shrink
75.6929
71.4178
69.1993
71.9989
Table 3. Type of noise: Speckle Noise

Conclusion
In this paper, the image denoising using discrete wavelet transform is analyzed. The experiments were conducted to study the suitability of different wavelet bases and also different window sizes. Among all discrete wavelet bases, coiflet performs well in image denoising. Experimental results show that Bivariate Shrink Method gives better result than Sure Shrink, Bayes Shrink and Neigh Shrink methods when applied on series of images.

Future Scope
In this paper three types of noise are involved and applied to number of images after introducing a particular type of shrink method. This work can be further elaborated in other types of noise like Brownian noise, Poisson noise etc. to produce a wide range of denoising methods.

References

Anil K.Jain 2003, Fundamentals of Digital Image Processing PHI

Boggess & Narcowich, 2002, A First Course in Wavelets with Fourier Analysis, Prentice Hall

David L. Donoho 1993 DeNoising by Soft Thresholding, IEEE Trans. Info. Theory 43, pp. 933936.

David L. Donoho, Iain M. Johnstone, GÃ©rard Kerkyacharian, Dominique Picard 1993 "Wavelet Shrinkage: Asymptopia".

D. L. Donoho and I. M. Johnstone 1994, Ideal spatial adaptation by wavelet shrinkage, Biometrika, vol. 81, no. 3, pp. 425455.

David L. Donoho and Iain M. Johnstone 1995,Adapting to Unknown Smoothness via Wavelet Shrinkage, Journal of the American Statistical Association Vol. 90, No. 432 pp. 12001224.

Fengxia Yan, Lizhi Cheng, and Silong Peng 2008, A New Interscale and Intrascale Orthonormal Wavelet Thresholding for SUREBased Image Denoising, IEEE Signal Processing Letters, Vol. 15.

Florian Luisier, Thierry Blu, and Michael Unser (June 2010), SURELET for Orthonormal WaveletDomain Video Denoising, IEEE Transactions On Circuits And Systems For Video Technology, Vol. 20, No. 6.

S.Grace Chang, Bin Yu, Martin Vetterli 2000, Adaptive wavelet thresholding for denoising and compression, IEEE Trans. On Image processing, Vol.9, No.9, PP. 15321546.

Hamed Pirsiavash, Shohreh Kasaei, and Farrokh Marvasti, 2005, An Efficient Parameter Selection Criterion for Image Denoising, IEEE International Symposium on Signal Processing and Information Technology.

Iman Elyasi, and Sadegh Zarmehi 2009, Elimination Noise by Adaptive Wavelet Threshold World Academy of Science, Engineering and Technology.

Lakhwinder Kaur, Savita Gupta and R.C. Chauhan, Image Denoising using Wavelet Thresholding.

Levent Sendur, Ivan W. Selesnick 2002,Bivariate Shrinkage Functions for WaveletBased Denoising Exploiting Interscale Dependency IEEE Transactions On Signal Processing, Vol. 50, No. 11.

Leavline, E.J.; Sutha, S 2011, Gaussian noise removal in gray scale images using fast Multiscale Directional Filter Banks, IEEE International Conference on Recent Trends in Information Technology (ICRTIT 2011) pp 884 – 889

Loupas, T.; McDicken, W.N.; Allan, P.L.( Jan 1989), An adaptive weighted median filter for speckle suppression in medical ultrasonic images, IEEE Transactions on Circuits and SystemsVolume: 36 Issue:1 pp 129 135

Martin Raphan, and Eero P. Simoncelli (August 2008), Optimal Denoising in Redundant Representations IEEE Transactions On Image Processing, Vol. 17, No. 8.

Rafael C.Gonzalez, Richard E.Woods 2002, Digital Image Processing, Second Edition, Pearson Education Asia.

Ramin Eslami and Hayder Radha 2003,The Contourlet Transform for Image Denoising Using Cycle Spinning, IEEE Trans. Image Processing pp. 19821986.

Rao R M and A S Bopardikar 2000, Wavelet Transforms Introduction to theory and Applications, Pearson Education, Asia.

Saeed V. Vaseghi 2000, Advanced Digital Signal Processing and Noise Reduction,, John Wiley & Sons Ltd.

Sardy, S 2000, Minimax Threshold for Denoising Complex Signals with Waveshrink, IEEE Transactions On Signal Processing, , VOL 48; PART 4, pages 10231028

Shan GAI, Peng LIU, Jiafeng LIU, Xianglong TANG, (2010) A New Image Denoising Algorithm via Bivariate Shrinkage Based on Quaternion Wavelet Transform , Journal of Computational Information Systems 6:11 3751 3760

S.Sudha, G.R.Suresh, R.Sukanesh 2007, Wavelet based Image denoising using adaptive thresholding, International conference on Computational Intelligence and Mltimedia Applications, PP. 296300.

Tony F. Chan, Jianhong[Jackie] shen 2005 , Image Processing and Analysis,, Society for Industrial and Applied Mathematics, Philadelphia.

Vetterli M Kovacevic J 1995, Wavelets and Sub band Coding, Prentice Hall.

Wei Li; Yanxia Sun; Shengjian Chen 2009, A New Algorithm for Removal of HighDensity Salt and Pepper Noises, IEEE 2nd International Congress on Image and Signal Processing.pp.14.