2003 Wavelet Analysis and Its Applications  
A pre-processing design using neural networks is proposed for multiwavelet filters. Various numerical experiments are presented and a comparison is given between neural network pre-processing and a preprocessing for solving linear systems. Neural network pre-processing produces a good approximation for a large number of terms and converges repidly. In memoriam Michihiro Nagase To appear in J. of Wavelets, Multiresolution and Information Processing. Résumé On propose un préprocesseur
more » ... cesseur neuromimétique pour le filtrage des multi-ondelettes. On présente de nombreux résultats numériques que l'on compare avec avec des préprocesseurs pour la solution de systèmes linéaires. Le préprocesseur neuromimétique aproxime bien les grands systèmes et converge rapidement. Introduction Since wavelets are solutions of multiscale equations, they cannot easily be studied and applied as mathematical objects without the use of computers. This paper is no exception. Multiwavelets consist in several scaling functions and wavelets. It is believed that multiwavelets are ideally suited to multichannel signals like color images which are two-dimensional three-channel signals and stereo audio signals which are one-dimensional two-channel signals. For instance, for a two-channel signal, which consists of a two-vector sequence of bits, {x k }, the lowpass and highpass filters are 2 × 2 matrix functions corresponding to two scaling functions and two wavelets, respectively. Multiscaling functions and multiwavelets can simultaneously have orthogonality, linear phase, symmetry and compact support. This situation cannot occur in the scalar case with real scaling functions and real wavelets. The simplest scalar wavelet in L 2 (R) is the Haar system, see Meyer[1], Section 3.2, with the indicator function of the interval [0, 1] as scaling function. Alpert[2] generalized the Haar system to one-dimensional discontinuous multiwavelets with vanishing moments in L 2 (R). Using fractal interpolation, Geronimo, Hardin, and Massopust[3] constructed a pair of real-valued one-dimensional symmetric scaling functions with short support, and Donavan, Geronimo, Hardin, and Massopust[4] constructed a corresponding pair of real-valued one-dimensional wavelets (DGHM) with short support. Strang and Strela[5, 6] used matrix methods in the time domain to construct the DGHM wavelets and also a nonsymmetric pair. Assuming that the scaling functions have sufficiently many vanishing moments, Ashino and Kametani[7] introduced r-regular multiwavelets in L 2 (R n ) and proved a general existence theorem, following Meyer's general existence theorem (see Meyer[1], Theorem 2 of Section 3.6 and Proposition 4 of Section 3.7). Jia and Shen[8] investigated multiresolution on the basis of shift-invariant spaces, proved a general existence theorem and gave examples to illustrate the general theory. Using Lawton's results[9] on complex-valued filters, Cooklev[10] and Cooklev et al.[11] obtained one-dimensional perfect-reconstruction filter banks given by a pair of analyzing and synthesizing orthogonal linear-phase two-channel multiwavelet filters. Plonka[12], Cohen, Daubechies and Plonka[13], Plonka and Strela[14], Shen[15], Strela[16], and many others, have obtained important results on the existence, regularity, orthogonality and symmetry of multiwavelets. Definitions and properties of multiwavelets, filters and filter banks can be found, for instance, in Ashino, Nagase, and Vaillancourt[18] and Zheng[19] and in the monograph by Keinert[20]. To start with multiwavelet filtering, we need to get scaling coefficients at high resolution. In the case of multiwavelets constructed by means of d multiscaling functions, there are d input channels for each sample of data, because frequently used multiwavelets have multiscaling functions with similar support widths. For fast multiwavelet algorithms, a given data needs to be pre-processed into d inputs to reduce their sizes. In the case of scalar wavelets, samples of a given function are used as coefficients in the expansion of the function in terms of the scaled and shifted scaling function, because, at very fine resolution, the scaling function is close to a constant multiple of a translated delta function. But in the multiwavelet case, simply using nearby samples as the scaling coefficients is a bad choice, because each of the d scaling functions may not be close to a constant multiple of a translated delta function even at very high resolution. For these reasons, data samples need to be pre-processed, or prefiltered, to produce reasonable coefficient values of the expansion in terms of the multiscaling functions at the finest scale. The design of prefilters have been based on interpolation (Xia, Geronimo, Hardin and Suter[21]), quadrature rules (Johnson[22]), approximation (Hardin and Roach[23]) and orthogonal projection (Vrhel and Aldroubi[24]). The field of neural networks started some fifty years ago but has found solid application only in the past twenty years and it is developing rapidly. Neural networks described in Rumelhart and McClelland[26] are composed of simple elements operating in parallel. These elements are inspired by biological nervous systems. As in nature, the network function is determined largely by the connections between elements. A neural network described in Demuthl and Beale [27] can be trained to perform a particular function by properly choosing the values of the connections (weights) between elements. Commonly, neural networks are adjusted, or trained, so that a particular input leads to a specific target output. The network is adjusted by comparing the output and the target, until the network output matches the target. Typically, many such input/target pairs are used, in this supervised learning, to train a network. Neural networks have been trained to perform complex functions in various fields of application including pattern recognition, identification, classification, speech, vision and control systems. Today, neural networks have been trained to solve problems that are difficult for conventional computers or human beings. The supervised training methods are commonly used, but other networks can be obtained from unsupervised training techniques or from direct design methods. Unsupervised networks can be used, for instance, to identify groups of data. Certain kinds of linear networks and Hopfield networks are designed directly. Several kinds of design and learning techniques can enrich the users' choices. Neural networks allow data-adaptive pre-processing designs. Adaptability greatly reduces the computation cost if we abandon the goal of perfect reconstruction. In this paper, we propose a variable pre-processing using neural networks which can be adapted to each data. To obtain an approximate solution to a structural problem for certain types of multiscaling functions we propose
doi:10.1142/9789812796769_0092 fatcat:p5bwqftuznfb7lei2gqkfh6cea