Approximation of Signals by Predict Wavelet Transform

Abstract

The article presents a general outline of the signal theory using predict wavelet transform. The predict wavelet transform (1) is a new attitude to a multiresolution signal analysis by discrete wavelet transform [8]. It is implemented in accordance with a lifting scheme, which allows realizing a signal filtration by biorthogonal wavelets [10,9]. Thanks to it, there exists a possibility to receive an optimal, biorthogonal filter for analysis of chosen signal characteristic. The article describes the signal approximation method by lifting scheme (9). This method is a generalization of the predict interpolation proposed by Wim Sweldens [6,10]. It allows to predict odd signal samples using a polynomial degree much lower than an interpolation polynomial degree. This solution not only enables approximating by algebraic polynomial but also by some base functions. Thus, this method is more flexible and optimal. Using an orthogonal Gram’s polynomial (23) or a trigonometric polynomial for approximation eliminates a problem of ill conditioning of matrix coefficients (19). Such conditioning can cause large rounding errors during performing computer operation, so it leads to incorrect results. The article enables to get acquainted with the relation between classical and predict wavelet transform (6). The method of obtaining a biorthogonal highpass filter and lowpass filter (analysis and synthesis signal) is shown on the base of a second rank predictor example. The corresponding filters are received by summation of flow-ways between Predict and Update coefficients. The calculated filter coefficients are described by wavelet and scaling functions in a graphical form (7, 8). The summary chapter presents the use of lifting scheme during multiresolution image analysis (9) and irregular meshes analysis 3D (10).