Wavelet Based Image Denoising Technique

— This paper proposes different approaches of wavelet based image denoising methods. The search for efficient image denoising methods is still a valid challenge at the crossing of functional analysis and statistics. In spite of the sophistication of the recently proposed methods, most algorithms have not yet attained a desirable level of applicability. Wavelet algorithms are useful tool for signal processing such as image compression and denoising. Multi wavelets can be considered as an extension of scalar wavelets. The main aim is to modify the wavelet coefficients in the new basis, the noise can be removed from the data. In this paper, we extend the existing technique and providing a comprehensive evaluation of the proposed method. and Pepper, and Speckle performed in this paper. A signal to noise ratio as a measure of the quality of denoising was preferred.


I. INTRODUCTION
The image usually has noise which is not easily eliminated in image processing.According to actual image characteristic, noise statistical property and frequency spectrum distribution rule, people have developed many methods of eliminating noises, which approximately are divided into space and transformation fields The space field is data operation carried on the original image, and processes the image grey value, like neighborhood average method, wiener filter, center value filter and so on.The transformation field is management in the transformation field of images, and the coefficients after transformation are processed.Then the aim of eliminating noise is achieved by inverse transformation, like wavelet transform [1], [2].Successful exploitation of wavelet transform might lessen the noise effect or even overcome it completely [3].There are two main types of wavelet transform -continuous and discrete [2].Because of computers discrete nature, computer programs use the discrete wavelet transform.The discrete transform is very efficient from the computational point of view.In this paper, we will mostly deal with the modeling of the wavelet transform coefficients of natural images and its application to the image denoising problem.The denoising of a natural image corrupted by Gaussian noise is a classic problem in signal processing [4].The wavelet transform has become an important tool for this problem due to its energy compaction property [5].Indeed, wavelets provide a framework for signal decomposition in the form of a sequence of signals known as approximation signals with decreasing resolution supplemented by a sequence of additional touches called details [6] [7].Denoising or estimation of functions, involves reconstituting the signal as well as possible on the basis of the observations of a useful signal corrupted by noise [8] [9] [10] [11].The methods based on wavelet representations yield very simple algorithms that are often more powerful and easy to work with than traditional methods of function estimation [12].It consists of decomposing the observed signal into wavelets and using thresholds to select the coefficients, from which a signal is synthesized [5].Image denoising algorithm consists of few steps; consider an input signal ( )and noisy signal ( ).Add these components to get noisy data ( ) i.e.

( ) ( ) ( ) 
Here the noise can be Gaussian, Poisson"s, speckle and Salt and pepper, then apply wavelet transform to get ( ).

( ) → ( ) 
Modify the wavelet coefficient ( ) using different threshold algorithm and take inverse wavelet transform to get denoising image ̂( ).

( ) → ̂( ) 
The system is expressed in Fig. 1.Image quality was expressed using signal to noise ratio of denoised image.

II. WAVELET TRANSFORM
The wavelet expansion set is not unique.A wavelet system is a set of building blocks to construct or represents a signal or function.It is a two dimensional expansion set, usually a basis, for some class one or higher dimensional signals.
The subspaces of ( ) spanned by these functions is defined as

* ( )+
̅̅̅̅̅̅̅ for all integers k from minus infinity to infinity.A two dimensional function is generated from the basic scaling function by scaling and translation by Whose span over k is for all integer.The multiresolution analysis expressed in terms of the nesting of spanned spaces as .The spaces that contain high resolution signals will contain those of lower resolution also.The spaces should satisfy natural scaling condition ( ) ⇔ ( ) which ensures elements in space are simply scaled version of the next space.The nesting of the spans of ( )denoted by i.e. ( )is in , it is also in , the space spanned by ( ).This ( )can be expressed in weighted sum of shifted ( )as Where the h (n) is scaling function.The factor √ used for normalization of the scaling function.The important feature of signal expressed in terms of wavelet function ( )not in scaling function ( ).The orthogonal complement of in is defined as , we require, For all appropriate .The relationship of the various subspaces is .The wavelet spanned subspaces such that , which extends to .In general this where is in the space spanned by the scaling function ( ), at , equation becomes (9) eliminating the scaling space altogether .The wavelet can be represented by a weighted sum of shifted scaling function ( ) as, For some set of coefficient ( ), this function gives the prototype or mother wavelet ( ) for a class of expansion function of the form, Where the scaling of is, is the translation in t, and maintains the norms of the wavelet at different scales.The construction of wavelet using set of scaling function ( ) and ( ) that could span all of ( ), therefore function ( ) ( ) can be written as First summation in above equation gives a function that is low resolution of g (t), for each increasing index j in the second summation, a higher resolution function is added which gives increasing details.The function d (j, k) indicates the differences between the translation index k, and the scale parameter j.In wavelet analysis expand coefficient at a lower scale level to higher scale level, from equation (10), we scale and translate the time variable to given as After changing variables m=2k+n, above equation becomes If we denote as * ( ) ) is expressible at scale j+1, with a scaling function only not wavelets.At one scale lower resolution, wavelets are necessary for the detail not available at a scale of j.We have Where the terms maintain the unity norm of the basis functions at various scales.If ( ) and ( ) are orthonormal, the j level scaling coefficients are found by taking the inner product By using equation ( 14) and interchanging the sum and integral, can be written as But the integral is inner product with the scaling function at a scale j+1giving The corresponding wavelet coefficient is Fig. 2 shows the structure of two stages down sampling filter banks in terms of coefficients.
In terms of next scales which requires wavelet as Substituting equation (15) and equation (11) into equation ( 20), gives Because all of these function are orthonormal, multiplying equation (20) and equation ( 21) by ( ) and integrating evaluates the coefficients as Fig. 3 shows the structure of two stages up sampling filter banks in terms of coefficients i.e. synthesis from coarse scale to fine scale [5] [6] [7].

III. DENOISING TECHNIQUE WITH EXISTING THRESHOLD
Noise is present in an image either in an additive or multiplicative form.An additive noise follows the rule, ( ) ( ) ( ) While the multiplicative noise satisfies Where ( ) is the original signal, ( )denotes the noise.When noise introduced into the signal it produces the corrupted image ( ).14].Gaussian Noise is evenly distributed over the signal.This means that each pixel in the noisy image is the sum of the true pixel value and a random Gaussian distributed noise value.Salt and Pepper Noise is an impulse type of noise, which is also referred to as intensity spikes.This is caused generally due to errors in data transmission.The corrupted pixels are set alternatively to the minimum or to the maximum value, giving the image a "salt and pepper" like appearance.Unaffected pixels remain unchanged.The source of this noise is attributed to random interference between the coherent returns [7], [8], [9] [10].Fully developed speckle noise has the characteristic of multiplicative noise.

A. Universal Threshold
The universal threshold can be defined as, N being the signal length, σ being the noise variance is well known in wavelet literature as the Universal threshold.It is the optimal threshold in the asymptotic sense and minimizes the cost function of the difference between the function.One can surmise that the universal threshold may give a better estimate for the soft threshold if the number of samples is large [13] [14].

B. Visu Shrink
Visu Shrink was introduced by Donoho [13].It uses a threshold value t that is proportional to the standard deviation of the noise.It follows the hard threshold rule.An estimate of the noise level σ was defined based on the median absolute deviation given by Where corresponds to the detail coefficients in the wavelet transform.VisuShrink does not deal with minimizing the mean squared error.Another disadvantage is that it cannot remove speckle noise.It can only deal with an additive noise.VisuShrink follows the global threshold scheme, which is globally to all the wavelet coefficients [9].

C. Sure Shrink
A threshold chooser based on Stein"s Unbiased Risk Estimator (SURE) was proposed by Donoho and Johnstone and is called as Sure Shrink.It is a combination of the universal threshold and the SURE threshold [15] [16].This method specifies a threshold value for each resolution level in the wavelet transform which is referred to as level dependent threshold.The goal of Sure Shrink is to minimize the mean squared error [9], defined as, Where ( )is the estimate of the signal, ( ) is the original signal without noise and n is the size of the signal.Sure Shrink suppresses noise by threshold the empirical wavelet coefficients.The Sure Shrink threshold t* is defined as Where denotes the value that minimizes Stein"s Unbiased Risk Estimator, is the noise variance computed from http://ijacsa.thesai.org/

VI. CONCLUSION
This technique is computationally faster and gives better results.Some aspects that were analyzed in this paper may be useful for other denoising schemes, objective criteria for evaluating noise suppression performance of different significance measures.Our new threshold function is better as compare to other threshold function.Some function gives better edge perseverance, background information, contrast stretching, in spatial domain.In future we can use same threshold function for medical images as well as texture images to get denoised image with improved performance parameter.

Figure 1 :
Figure 1: Block diagram of Image denoising using wavelet transform.
The wavelet expansion gives a time frequency localization of the signal.Wavelet systems are generated from single scaling function by Input //ijacsa.thesai.org/scaling and translation.A set of scaling function in terms of integer translates of the basic scaling function by

Figure 2 :
Figure 2: Two stages down sampling filter bank A reconstruction of the original fine scale coefficient of the signal made from a combination of the scaling function and wavelet coefficient at a course resolution which is derived by

Figure 3 :
Figure 3: Two stages up sampling filter In filter structure analysis can be done by apply one step of the one dimensional transform to all rows, then repeat the same for all columns then proceed with the coefficients that result from a convolution with in both directions[6][7][8][9][10][12].The two level wavelet decomposition as shown in fig 4.

TABLE 1 :
RESULT OF DIFFERENT TECHNIQUE WITH LENA.