Vector quantization of image subbands: a survey <s> x(n) <s> We construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity. The order of regularity increases linearly with the support width. We start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction. The construction then follows from a synthesis of these different approaches. <s> BIB001 </s> Vector quantization of image subbands: a survey <s> x(n) <s> We consider coding gain and spatial localisation properties of DWT filters for still image compression. A high coding gain, relative to a highly correlated image source model, ensures that slowly varying image areas are coded efficiently. A low spatial width ensures that anomalies such as edges can also be coded efficiently. By considering an image model we design filters that simultaneously exhibit a high coding gain and low spatial width. Several existing filters are also examined. Several filter sets are then compared for image compression using a DWT and a JPEG type quantisation method. The filters with the best spatial width and coding gain properties outperform the other filters at low bit rates, especially with regard to ringing distortion.<<ETX>> <s> BIB002 </s> Vector quantization of image subbands: a survey <s> x(n) <s> The selection is considered of scaling functions for optimal signal representation by multidimensional biorthogonal wavelet frames. Criterion for optimality is the minimization of the mean-square approximation error at each level of the decomposition. Conditions are given under which the approximation error of the decomposition approaches zero as the level increases. Optimal and suboptimal families of filters for the realization of the scaling functions are explicitly defined. > <s> BIB003
If all four of the subbands are subjected to another stage of ltering and downsampling, this leads to a uniform decomposition, as depicted in Figure 4 (a). This decomposition is a balanced tree. If only the low band is further decomposed, this is referred to as an octave-band decomposition (also often called a logarithmic-tree or pyramid 1 decomposition). This situation, which is represented by a highly unbalanced tree, is shown in Figure 4 (b). In fact, for the xed maximum depth of the tree, this octave-band decomposition is as unbalanced a tree as one can have. For the xed maximum depth of the tree, the balanced tree produced by a uniform decomposition and the maximally unbalanced tree produced by a pyramidal decomposition represent the two extremes of possible choices. Anything between these extremes is called a wavelet packet 38] . An example of a wavelet packet is shown in Figure 4 imposed on the lowpass lters. The wavelet transform can be interpreted as a subband decomposition of the signal into a set of overlapping frequency channels having the same bandwidth on a logarithmic scale. In this paper, we will refer to this type of octave-band decomposition as a wavelet decomposition. The more general term \subband decomposition" will be used for the uniform-band case, and as the general term. The various subband/VQ systems surveyed used a wide variety of lters. One of the goals of signal decomposition theory is to nd conditions such that in the absence of noise or quantization, the overall reconstruction is perfect or nearly so. The simplest and shortest lter, the Haar wavelet, is important for historical and theoretical reasons, but is rarely used as part of a wavelet/SQ or wavelet/VQ scheme. By far the most common choice is one of the quadrature mirror lters (QMFs) that are listed in 81]. Many authors refer to these as \Johnston's near-perfect reconstruction lters," or near-PR lters. The second most popular choice is one of the family of Daubechies orthogonal lters BIB001  . Also common are the biorthogonal lters   , which possess the advantage of linear phase. More rarely used are the Gabor functions, which are Gaussians modulated by complex exponentials. Use of the latter for wavelet transforms is motivated by the fact that they have joint localization in the spatial/spatial-frequency domains that is optimal in a certain sense, and that, according to recent experiments, the majority of receptive eld pro les of the human visual system t quite well this type of function. The choice of lter re ects a trade-o among many factors, such as spatial localization, frequency selectivity, regularity, and coding gain. A number of papers have examined the question of selecting lter coe cients   BIB002 BIB003  , but as this survey is concerned with quantization, no more will be said on lters.