Non-separable two-dimensional multiwavelet transform for image coding and compression

In most cases of two-dimensional (2D) wavelet applications in image processing, and in coding and compression, separable filters constructed by a tensor product a two one-dimensional (1D) filters are used. This approach imposes limitations on the design oi such filters. We therefore present a method of designing non-separable, orthogonal 2D wavelet functions and filter-banks. We also show how to obtain orthogonal filter-banks which have linear phase and any number of vanishing moments or approximation order. After; applying the 2D wavelet transform to an image, we use vector quantization (VQ) and zerotree for coding the wavelet coefficients in an efficient way. For VQ, we use the entropy-constrained vector quantizer (ECVQ). The bit allocation for each resolution level is determined automatically during the training process by a Lagrange multiplier method. Pt is shown that it is better Co combine coefficients, that relate to different wavelet functions and the same location, into the same vector, rather than combining neighbouring coefficients of the same wavelet function. We also present a different method for the coefficients coding, employing the fact that there exists similarity between the subbands, to increase the efficiency of data compression, This method is based on the embedded zerotree wavelet (EZW) algorithm, but introduces some modifications: We relate to vectors instead of dealing with each coefficient individually, and the method is adjusted to the specific structure of the transform we use. Quincunx multiwavelets with up to third order polynomial approximation, corresponding to filters with very small support, are considered in detail.