IMAGE DENOISING USING WAVELET TRANSFORMS WITH ENHANCED DIVERSITY Alexandru Isar 1 , Sorin Moga 2 , Corina Naforniţă 1 , Marius Oltean 1 and Ioana Adam 1 1 Electronics and Telecommunications Faculty/Communications Univ “Politehnica” 2 LUSSI ENST-Bretagne 2 Bd. V. Parvan, Timisoara, Romania Technopôle de Brest Iroise, CS 83818, 29238 BREST Cedex, France alexandru.isar@etc.utt.ro ABSTRACT The performance of image-denoising algorithms using wavelet transforms can be improved significantly by matching the parameters of those transforms with the input image. For every noisy image, there is a best pair of parameters formed by a mother wavelets and a best primary resolution, which maximizes the output Peak Signal to Noise Ratio, PSNR. Unfortunately this best pair is not known in advance. The denoising algorithms are sensitive to the parameters of the wavelet transform used. This sensitivity can be reduced using a diversity enhanced wavelet transform, obtained by computing few wavelet transforms with different parameters. After the filtering of each detail coefficient, the corresponding wavelet transforms are inverted and the estimated image, having a higher PSNR, is extracted. This paper presents a denoising algorithm based on diversity enhanced wavelet transforms. Some comparisons with the best available results will be given in order to illustrate its effectiveness. 1. INTRODUCTION In 1992 David Donoho has introduced a method of signal denoising based on adaptive nonlinear filtering in the wavelets transform domain, [1]. Let us suppose that the useful signal x is additively perturbed by a noise n i . To estimate the signal x, from an additive mixture of useful signal and noise, x i = x + n i , Donoho, [1], proposed the following method: 1. The Wavelet Transform, WT, of the signal x i is computed. 2. A filtering procedure is applied in the wavelet transform domain, obtaining the signal y 0 . 3. Taking the inverse wavelet transform, IWT, of the signal y 0 , the denoised version of the signal x i , named x 0 , is obtained. There are a lot of papers dealing with denoising methods, due to the important number of parameters involved. In this paper the construction of two Diversity Enhanced Wavelet Transforms, DE WTs, will be described. Some simulation results will be given in order to illustrate the effectiveness of the new transforms. Other potentially applications of those new WTs also exist. For example these WTs can be used for watermarking. They can substitute in this respect the Discrete Cosine Transform, DCT or the Discrete Wavelet Transform, DWT, taking into account their advantages: the translations invariance, the enhanced directional selectivity and the low sensitivity with the transforms parameters. 2. SOME WTS USED IN DENOISING There are two kinds of WTs: redundant and non redundant. The first wavelet transform used in denoising applications was the Discrete Wavelet Transform, DWT. This transform is most commonly used in its maximally decimated form (Mallat’s dyadic filter tree [2], [3]). The DWT is an orthogonal (non redundant) WT. It has two main disadvantages, [4-7]: the lack of shift invariance and the poor directional selectivity. It is also dependent on the selection of its parameters. Some WTs, used in denoising algorithms are presented in table 1. The following denominations were used: Orthogonal Real DWT - OR DWT, Biorthogonal Real DWT - BR DWT, Complex DWT - C DWT, Dual Tree Complex WT - DT CWT, Cycle Spinning OR DWT - CS OR DWT, Undecimated DWT - U DWT and Shiftable Multiscale Transform - SMT. In [8] is proposed a very redundant shift invariant OR DWT, associated to a denoising algorithm named Cycle Spinning, CS. This represents a first example of enhanced diversity in the wavelet domain. Other redundant WTs are proposed in [9] and [10]. All the WTs presented in table 1 have two parameters: the mother wavelets, MW and the primary resolution, PR, (number of iterations). The importance of their selection is highlighted in [11]. For a given noisy image, there is only one pair of such parameters that maximizes the denoising algorithm output PSNR. The selection of the parameters pair that maximizes the output PSNR for a given noisy image is very difficult due to the noise that perturbs the useful image.