Biorthogonal filterbanks and energy preservation property in image compression

The energy preservation property is among the most widely used properties of orthogonal transforms in image compression because the reconstruction error can be computed as the sum of the subband distortions. Thus, this is a key point in the use of efficient bit allocation techniques such as rate-distortion algorithms. Therefore, we study the nonorthogonality of biorthogonal filterbanks with reference to energy preservation from both theoretical and applicative points of view. We calculate the Riesz bounds as energy preservation bounds for filterbanks and discrete wavelet transforms, and then connect these results with the Riesz bounds of the related continuous wavelet transform. The simultaneous use of biorthogonal filterbanks and rate-distortion algorithms is then discussed as the issue of estimating the reconstruction error as an additive function of the subband distortion. We propose a weighted sum of the subband distortions as an estimate, whose accuracy is calculated by a wide range of experiments. This accuracy is shown to be correlated to the Riesz bounds of the filterbanks. We conclude that from this point of view, most of the usual biorthogonal filterbanks may be considered as nearly orthogonal.