CAPlets : wavelet representations without wavelets

MultiResolution (MR) is among the most effective and the most popular approaches for data representation. In that approach, the given data are organized into a sequence of resolution layers, and then the “difference” between each two consecutive layers is recorded in terms of detail coefficients. Wavelet decomposition is the best known representation methodology in the MR category. The major reason for the popularity of wavelet decompositions is their implementation and inversion by a fast algorithm, the so-called fast wavelet transform (FWT). Another central reason for the success of wavelets is that the wavelet coefficients capture very accurately the smoothness class of the function hidden behind the data. This is essential for the understanding of the performance of key wavelet-based algorithms in compression, in denoising, and in other applications. On the downside, constructing wavelets with good space-frequency localization properties becomes involved as the spatial dimension grows. An alternative to the sometime-hard-to-construct wavelet representations is the always-easy-to-construct (and slightly older) non-orthogonal pyramidal algorithms. Similar to wavelets, the (linear, regular, isotropic) pyramidal representations are based on some method for linear coarsening (by a decomposition filter) of their data, and a complementary method for linear prediction (by a prediction filter) of the original data from the coarsened one. The first step creates the resolution layers and the second allows for trivial extractions of suitable detail coefficients. The decomposition and reconstruction algorithms in the pyramidal approach are as fast as those of wavelets. In contrast with orthonormal wavelets, the representation is redundant, viz. the total number of detail coefficients exceeds the original size of the data: denoting by s the ratio between the size of the data at two consecutive resolution layers, the “redundancy ratio” in the pyramidal representation is s s − 1 . In this paper, we introduce and study a general class of pyramidal representations that we refer to as CompressionAlignment-Prediction (CAP) representations. The CAP representation is based on the selection of three filters: the low-pass decomposition filter, the low-pass prediction filter, and the full-pass alignment filter. Like previous pyramidal algorithms, CAP are implemented by a simple, fast, wavelet-like decomposition and a trivial reconstruction. The primary goal of this paper is to establish the precise way in which the CAP representations encode the smoothness class of the underlying function. Remarkably, the CAP coefficients provide the same characterizations of TriebelLizorkin spaces and Besov spaces as the wavelet coefficients do, provided that the three CAP filters satisfy certain requirements. This means, at least in principle, that the performance of CAP-based algorithms should be similar to their wavelet counterparts, despite of the fact that, when compared with wavelets, it is much easier to develop CAP representations with “customized” or “optimal” properties. Moreover, upon assuming the prediction filter to be interpolatory, we extract from the CAP representation a sister CAMP representation (“M” for “modified”). Those CAMP representations strike a phenomenal balance between performance (viz., smoothness characterization) and space localization. Our analysis of the CAP representations is based on the existing theory of framelet (redundant wavelet) representations. AMS (MOS) Subject Classifications: Primary 42C15, Secondary 42C30