Repeatedly smoothing, discrete scale-space evolution and dominant point detection

In this paper, the Fourier analysis is used to derive the properties of an evolving curve. An arbitrary-kernel-repeatedly-smoothing (AKRS) evolution of a curve is then introduced. It is shown that when the repeated number is large, the AKRS evolution of a curve is an approximately discrete implementation of the scale-based evolution of this curve in the Euclidean space. As a special case, an exponential repeatedly smoothing is proposed to implement the scale-based evolution. It is shown that in addition to its simple implementation and its desired approximation to a Gaussian kernel, an exponential function is a function that when it is selected as a repeatedly smoothing kernel, the motion (both magnitude and direction) of a point on a curve from the (i - 1)th to ith instant is equal to the curvature of the ith smoothed curve at this point. Finally, a perimeter-controlled-evolution method is proposed to extract dominant points. It is shown experimentally that the proposed method is robust to noise, object rotation and object changes in sizes. (C) 1996 Pattern Recognition Society. Published by Elsevier Science Ltd.