Construction of smooth orthogonal wavelets with compact support in R[superscript d]

Ph.D.Yang Wan

Arbitrary smooth orthogonal nonseparable wavelets in

Abstract. For each r ∈ N, we construct a family of bivariate orthogonal wavelets with compact support that are nonseparable and have vanishing moments of order r or less. The starting point of the construction is a scaling function that satisfies a dilation equation with special coefficients and a special dilation matrix M: the coefficients are aligned along two adjacent rows, and |det(M) | =2. We prove that if M 2 = ±2I, e. g., M = ( 02 1)orM=(1), then the smoothness of the wavelets 10 1 −1 improves asymptotically by 1 − 1 2 log2 3 ≈ 0.2075 when r is incremented by 1. Hence they can be made arbitrarily smooth by choosing r large enough

Compactly Supported Orthogonal Symmetric Scaling Functions

. Daubechies [5] showed that, except for the Haar function, there exist no compactly supported orthogonal symmetric scaling functions for the dilation q = 2. Nevertheless, such scaling functions do exist for dilations q ? 2 (as evidenced by Chui and Lian's construction [3] for q = 3); these functions are the main object of this paper. We construct new symmetric scaling functions and introduce the "Batman" family of continuous symmetric scaling functions with very small supports. We establish the exact smoothness of the "Batman" scaling functions using the joint spectral radius technique. 1. Introduction Compactly supported wavelets are typically constructed from a compactly supported single scaling function that generates a multiresolution analysis [5, 14]. It is important (and nontrivial) to construct scaling functions (and hence wavelets) with desirable properties, such as orthogonality, high regularity, symmetry and small support. Recall that a multiresolution analysis (MRA) with d..

Construction Of Compactly Supported Symmetric Scaling Functions

. In this paper we study scaling functions of a given regularity for arbitrary dilation factor q. We classify symmetric scaling functions and study the smoothness of some of them. We also introduce a new class of continuous symmetric scaling functions, the "Batman" functions, that have very small support. Their smoothness is established. 1. Introduction Compactly supported wavelet functions are typically constructed from multiresolution analyses whose scaling functions are compactly supported, see [6] or [15]. It is an important problem to construct scaling functions (and hence wavelets) that possess desirable properties. These properties usually include high regularity, symmetry and small supports. Recall that a multiresolution analysis with dilation factor q, where q 2 Z and jqj ? 1, is a sequence of nested subspaces of L 2 (R) \Delta \Delta \Delta ae V \Gamma2 ae V \Gamma1 ae V 0 ae V 1 ae V 2 ae \Delta \Delta \Delta (1.1) such that V j = span ae f(q j x \Gamma k) : k 2 Z ..

Arbitrarily Smooth Orthogonal Nonseparable Wavelets in R²

. For each r 2 N, we construct a family of bivariate orthogonal wavelets with compact support that are nonseparable and have vanishing moments of order r or less. The starting point of the construction is a scaling function that satisfies a dilation equation with special coefficients and a special dilation matrix M : the coefficients are aligned along two adjacent rows, and jdet(M)j = 2. We prove that if M 2 = \Sigma2I , e. g., M = \Gamma 0 2 1 0 \Delta or M = \Gamma 1 1 1 \Gamma1 \Delta , then the smoothness of the wavelets improves asymptotically by 1 \Gamma 1 2 log 2 3 ß 0:2075 when r is incremented by 1. Hence they can be made arbitrarily smooth by choosing r large enough. Key words. nonseparable wavelets, smooth orthogonal scaling function, regularity AMS subject classifications. Primary 42C15; Secondary 26B35, 41A25, 41A63, 65D20 1. Introduction. Since the introduction by Daubechies [7] of compactly supported orthogonal wavelet bases in R 1 with arbitrarily high ..

Fast Fréchet queries

Inspired by video analysis of team sports, we study the following problem. Let P be a polygonal path in the plane with n vertices. We want to preprocess P into a data structure that can quickly count the number of inclusion-minimal subpaths of P whose Fréchet Distance to a given query segment Q is at most some threshold value e. We present a data structure that solves an approximate version of this problem: it counts all subpaths whose Fréchet Distance is at most e, but this count may also include subpaths whose Fréchet Distance is up to (2+3 \sqrt 2) . For any parameter n¿=¿s¿=¿n 2, our data structure can be tuned such that it uses O(s polylog n) storage and has O((n/\sqrt) polylog n) query time. For the special case where we wish to count all subpaths whose Fréchet Distance to Q is at most e·length(Q), we present a structure with O(n polylog n) storage and O(polylog n) query time

An Online Adaptive Model for Location Prediction

Context-awareness is viewed as one of the most important aspects in the emerging pervasive computing paradigm. Mobile context-aware applications are required to sense and react to changing environment conditions. Such applications, usually, need to recognize, classify and predict context in order to act efficiently, beforehand, for the benefit of the user. In this paper, we propose a mobility prediction model, which deals with context representation and location prediction of moving users. Machine Learning (ML) techniques are used for trajectory classification. Spatial and temporal on-line clustering is adopted. We rely on Adaptive Resonance Theory (ART) for location prediction. Location prediction is treated as a context classification problem. We introduce a novel classifier that applies a Hausdorff-like distance over the extracted trajectories handling location prediction. Since our approach is time-sensitive, the Hausdorff distance is considered more advantageous than a simple Euclidean norm. A learning method is presented and evaluated. We compare ART with Offline kMeans and Online kMeans algorithms. Our findings are very promising for the use of the proposed model in mobile context aware applications