A Multiscale Data Representation for Distributed Sensor Networks

Abstract

Characterization of the scaling and wavelet functionsThe sensors are irregularly spaced on a 2D grid. We assume that the sensors sample a slowly varying field. To reconstruct the original field, we interpolate the sensor values over R 2 in the following manner. The approximate value of the field at any arbitrary location is equal to the value recorded by the sensor closest to it. Thus, the function space F spanned by the reconstructed field is the space of piecewise constants whose discontinuities lie on voronoi cell boundaries.Our goal is to first find a suitable set of functions that span the function space F . Motivated by the Haar wavelet transform, we pick one function that captures the average value of the sensor field, and N funtions that capture the deviation of each sensor value from the average. Since we use (N + 1) functions that span the N dimensional function space F , we refer to the functions as frame functions. Example of such frame functions in 1 − D are shown in Figure 1. The scaling function is constant over the entire region. The wavelet functions have discontinuities only at the boundaries of one voronoi cell. The wavelet functions have zero mean. Thus, the dot product of a wavelet function and the scaling function is zero.We need to pick the parameters of the frame functions (namely the heights of the wavelet functions) appropriately. We choose these parameters such that the set of frame functions form a Parseval tight frame [1]. For Parseval tight frames, we can establish bounds on the energy of the error when wavelet coefficients are thresholded.Let the support of the voronoi regions be ∆ i , for i = 1, 2, ..., N . LetDefine the scaling function as The i th wavelet function, W i , takes the value k i for all the voronoi cells except at the i th cell, where it takes value k i . Since the average value of the wavelet functions are zero, we haveTo satisfy the Parseval tight frame conditions, we look at the energy of the signal in the signal and wavelet domains. The energy of the signal is given byThe scaling coefficient s and the wavelet coefficient w are given by s =