Wavelet functions to estimate velocity in spatiotemporal signals

This paper establishes a framework to estimate velocity from spatiotemporal signals using the wavelet transform and multiresolution techniques. Initial theory is derived with the assumption that the spatiotemporal signal can be represented by a polynomial of order M. Wavelet functions are derived for polynomials of different degree from which the velocity can be estimated as the ratio of two of the four components of a two dimensional (2-D) wavelet transform of the signal. We have characterized two classes of wavelet and scaling functions: one with nonuniform support and another with symmetry and uniform support. For a wavelet function of order M, the velocity estimates are exact if the signal can be represented by a polynomial of the same order or less. In many cases, the velocity error is very low, even when there is no match. We also present the error estimates for three different signals: a polynomial of degree four, a sinusoid (polynomial of degree infinity), and a function with analytical value for the velocity. The paper also demonstrates how error in the velocity estimates can be reduced by using multiresolution techniques. Even though results are presented using one-dimensional (1-D) signals, the extension to higher dimensions (images) is straightforward and uses the same wavelet functions derived in this paper