HYBRID IMAGE DENOISING USING WIENER FILTER WITH DISCRETE WAVELET TRANSFORM AND FRAMELET TRANSFORM

Removal of noise from an image is an essential part of image processing systems. In this paper a hybrid denoising algorithm which combines spatial domain Wiener filter and thresholding function in the wavelet and framelet domain is done. In this work three algorithms are proposed. The first hybrid denoising algorithm using Wiener filter with 2-level discrete wavelet transform (DWT), the second algorithm its using Wiener filter with 2-level framelet transform (FLT) and the third hybrid denoising algorithm its combines wiener filter with 1-level wavelet transform then apply framelet transform on LL of wavelet transform. The Wiener filter is applied on the low frequency subband of the decomposed noisy image. This stage will tend to cancel or at least attenuate any residual low frequency noise component. Then thresholding detail high frequency subbands using thresholding function. This approach can be used for grayscale and color images. The simulation results show that the performance of the first proposed hybrid denoising algorithm with discrete wavelet transform (db5 type) is superior to that of the second and third proposed algorithms and to that of the conventional denoising approach at most of the test noisy image with Gaussian noise and Slat & pepper noise while the third proposed denoising algorithm with hybrid wavelet & framelet transform is superior to that of the other proposed algorithms at noisy images with speckle noise.


INTRODUCTION
Image denoising restores the details of an image by removing unwanted noise. Digital images become noisy when these are acquired by a defective sensor or when these are transmitted through a noisy channel [1]. Noise may be classified as substitutive noise (impulsive noise: e.g., salt and pepper noise, random valued impulse noise, etc.), additive noise (e.g., additive white Gaussian noise) and multiplicative noise (e.g. speckle noise) [2]. However, in this paper the investigation has been done in salt & pepper noise, Gaussian noise and Speckle noise. In general, the goal of any noise removal scheme is to suppress noise as well as to preserve details and edges of image as much as possible. Many denoising methods have been proposed over the years, such as the Wiener filter, wavelet thresholding, anisotropic filtering, bilateral filtering, total variation method, and non-local methods. Among these, wavelet thresholding has been reported to be a highly successful method [3]. In wavelet thresholding a signal is decomposed into approximation (low-frequency) and detail (high-frequency) subbands, and the coefficients in the detail subbands are processed via hard or soft thresholding. The hard thresholding eliminates (sets to zero) coefficients that are smaller than a threshold; the soft thresholding shrinks the coefficients that are larger than the threshold as well. The main task of the wavelet thresholding is the selection of threshold value and the effect of denoising depends on the selected threshold: a bigger threshold will throw off the useful information and the noise components at the same time while a smaller threshold cannot eliminate the noise effectively. A major strength of the wavelet thresholding is the ability to treat different frequency components of an image separately; this is important, because noise in real scenarios may be frequency dependent. But, in wavelet thresholding the problem experienced is generally smoothening of edges [3]. In this work a hybrid denoising method is proposed to find the best possible solution, so that PSNR of the image after denoising is optimal. The proposed model is based on wavelet transform or/and framelet transform which has been successfully used in noise removal [4]and hybrid with Wiener filtering, which exploits the potential features of both wavelet transform and Wiener filter at the same time their limitations are overcome [3].

DISCRETE WAVELET TRANSFORM (DWT)
When DWT is applied to noisy image, image is divided into four sub bands as shown in Fig.  1(a).These sub bands are formed by separable applications of horizontal and vertical filters. Coefficients that are represented as sub bands LH1, HL1 and HH1 are detail images while coefficients are represented as sub band LL1 is approximation image. The LL1 sub band is further decomposed to obtain the next level of wavelet coefficients as shown in Fig. 1

FRAMELET TRANSFORM (FLT)
The three-channel filter bank, which is used to develop the FLT corresponding to a wavelet frame, Fig. 3 (a and b) show that the filter bank structure for 2D analysis FT and 2D synthesis FT, respectively.

WIENER FILTER
Wiener theory, formulated by Norbert Wiener, forms the foundation of data-dependent linear least square error filters. Wiener filters play a central role in a wide range of applications such as linear prediction, echo cancellation, signal restoration, channel equalization, time-delay estimation and additive noise reduction [6]. In this paper, the purpose of the Wiener filter is to filter out the noise that has corrupted a signal. This filter is based on a statistical approach.

a. Analysis b. Synthesis
Mostly all the filters are designed for a desired frequency response. Wiener filter deals with the filtering of an image from a different view. The goal of wiener filter is reduced the mean square error as much as possible [2]. Consider a signal x (m) observed in a broadband additive noise n (m) and model as [6]: Assuming that the signal and the noise are uncorrelated, it follows that the autocorrelation matrix of the noisy signal is the sum of the autocorrelation matrix of the signal x (m) and the noise n(m): = + (2) And we can also write Where , and are the autocorrelation matrices of the noisy signal, the noise-free signal and the noise respectively, and is the cross correlation vector of the noisy signal and the noise-free signal. Substitution of eq. 2 & 3 in the Wiener filter Equation ( = −1 ), yields [6]: Eq. 4 is the optimal linear filter for the removal of additive noise. In the following, a study of the frequency response of the Wiener filter provides useful insight into the operation of the Wiener filter. In the frequency domain, the noisy signal Y(f) is given by Where X(f) and N(f) are the signal and noise spectra respectively. For a signal observed in additive random noise, the frequency-domain Wiener filter is obtained as Where ( )and ( )are the signal and noise power spectra respectively. Dividing the numerator and the denominator of eq. 6 by the noise power spectra ( )and substituting the variable Where SNR is a signal-to-noise ratio.

PROPOSED ALGORITHM
In this work state three algorithms. Fig. 6 shows the flow chart of these algorithms: In the first proposal, 2-Level wavelet decomposition is applied on the noisy image, in the second proposal, 2-Level framelet transform is applied on the noisy image and in the third proposal, 1-Level wavelet transform then 1-Level framelet transform is applied on the noisy image. Then apply wiener filter for low frequency domain and soft thresholding for high frequency domains. The filtered decomposed image is reconstructed by applying inverse wavelet transform and inverse framelet transform to get the denoised image.      Fig. 9 shows denoising some color images with noisy image by Gaussian noise and the results of PSNR shown in Table 6.    Table 7.

CONCLUTION
Based on the experiments performed in this work, it can be concluded that the denoising methods depending on noise types, where the results illustrates that the proposed algorithm 1 is the best amongst three in terms of PSNR and MSE, in addition to be more uniform and consistent in most the types of images tested with Gaussian noise and slat & pepper noise while the proposed algorithm 3 illustrates that the Hybrid filter is able to recover much more detail of the original image and provides a successful way of image denoising with speckle noise