International Journal of Computer Applications (0975 – 8887) Volume 97– No.15, July 2014 27 Reconstruction of Wavelet Coefficients K. Mathew Karpagam University, Coimbatore, Tamilnadu-641021, India S. Shibu, Ph.D K.R.Gouri Amma College Of Engineering For Women, Cherthala, Kerala, India ABSTRACT A comparative study of different methods of reconstruction of wavelet coefficients is presented. The following are the different techniques for the reconstruction of wavelet coefficients. To start with, we show how to design and construct Daubechies four coefficient wavelet system which are orthogonal and compactly supported wavelets. Then we outline the multi resolution analysis of wavelets using Mallat transform. Multi resolution analysis can be illustrated by the decomposition and reconstruction of wavelet system using Laplacian Pyramid. To construct wavelet systems with finite support and regularity using orthonormal and interpolariting units, only multicomponent wavelets are possible. When image function is expressed in terms of scaling functions and wavelet functions of higher resolution, we need to consider only few wavelet coefficients and wavelet coefficients are dominant only near edges. The wavelet coefficient near edges can be estimated using wavelet transform maxima and statistical inference of coefficients using Markov tree model. An alternate method is the reconstruction of wavelet coefficients using total variation minimisation. If we use thresholding, so that when we neglect wavelet coefficients having values less than a threshold value there is ringing across edges. Hence we propose an improved algorithm of reconstruction of wavelet coefficients using zero padding and cycle spinning. PSNR of images with wavelet based interpolation and denoising by cycle spinning is moderately high. Keywords Wavelets of finite support, multiresolution analysis, multicomponent wavelets, cycle spinning, Markov tree model, wavelet transform maxima, total variation minimisation. 1. INTRODUCTION Wavelets are a powerful tool for digital image analysis or time frequency analysis of image. A wavelet is a small wave, which is oscillatory and it contains both the analysing shape and the window (a finite support). One can construct wavelets ψ such that the dilated and translated family ? ?2,,221z n jjjn jntt?????????????? ? ? ? is an orthonormal basis of L 2 (R). Orthogonal wavelets dilated by carry signal variation at resolution .The construction of these bases is related to multiresolution signal approximation.n-refers to the translation of signal and j refers to the scale and n refers to time location. The wavelet function ? ? tn j,? has a time spread ? ?nj?? related as ? ? ? ? dt t t n jt222, ? ??? and energy (frequency)spread ??????^,n j w? ? is given by ? ?? ?? ? ? ??? ?? ?dn j2^0^,221?? ??????? is the dilation parameter which changes the support of ψ in time and rescales ψ and changing the translation parameter n makes changes its location. It is observed that small scales corresponds to high frequency. Wavelet functions are located both in time and frequency, but it cannot be exact localisation due to uncertainty principle so instead of exact localisation, the function is restricted to wavelet Heisenberg box. The localisation measures ? ?jn x ?? and ??????^jn? ? ? are often represented by rectangles in time frequency plane. These rectangles have same, but their sides are stretched and contracted by the factor and . All information about the transformed signal is preserved when the wavelet transform is sampled on certain discrete subset of time frequency plane. The values of continuous transform in those points are coefficients of a corresponding wavelet basis series expansion. The image function is decomposed in terms of orthonormal basis function n jn jn jC f,,, ? ?? where ? ? dt t f t f Cjn jn n j ?? ? ) ( ,*,? ? This is a doubly infinite sum over both time index n and scale index j and however sum can be made finite with no error because in expansion, only few coefficients are dominant. Multiresolution analysis allows us to decompose a signal or image function into approximation and details. These coefficients can be computed using various bank filters or Daubechies filters or Laplacian pyramid structure. Consider one dimensional image function ? ?? ? ?? ?? ?J j j kjk jk k jkk jd C t f00 0, ,? ? where ? called scaling function and ψ is is called wavelet. k j ,0? and jk? are generated by translation and dilations of ψ. The expansion functions which are composed of integer translations and binary scaling of the real square integral functions, then ? ? k xjjk j? ?0002 22,? ? where Z k j ? ,0 ? ?2L n ? ?, represent width of functionk j ,0? , 202j control amplitude of the function and shape of
