Transform Coding of Signals with Bounded Finite Differences: from Fourier to Walsh, to Wavelets
The digital spectral transform method is an important compression tool in signal and image processing applications. The fast Fourier transform algorithms, developed in the 60-s, facilitated the use of transform coding methods for redundancy elimination and efficient data representation. In order to determine the optimal zonal sampling method for a given transform, it is necessary to derive estimates of the transform spectra on a class of input signals. We present a unified approach for deriving upper bounds of spectra of orthogonal transforms on classes of input signals with bounded first and second order finite differences. Based on this approach we obtain estimates of spectra for classical discrete Fourier, Hartley, cosine, sine, as well as Walsh, Haar, and other wavelet transforms. These estimates allow one not only to select the significant transform coefficient packets a priori, but also to compute the maxima of mean-square errors of reconstruction for a given compression ratio and to compare the efficacy of different transforms based on that criterion.
KeywordsCompression Ratio Discrete Fourier Transform Spectral Component Binary Representation Orthogonal Wavelet
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