In this paper, the Fourier analysis is used to derive the properties o
f an evolving curve. An arbitrary-kernel-repeatedly-smoothing (AKRS) e
volution of a curve is then introduced. It is shown that when the repe
ated number is large, the AKRS evolution of a curve is an approximatel
y discrete implementation of the scale-based evolution of this curve i
n the Euclidean space. As a special case, an exponential repeatedly sm
oothing is proposed to implement the scale-based evolution. It is show
n that in addition to its simple implementation and its desired approx
imation to a Gaussian kernel, an exponential function is a function th
at when it is selected as a repeatedly smoothing kernel, the motion (b
oth 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 propos
ed to extract dominant points. It is shown experimentally that the pro
posed method is robust to noise, object rotation and object changes in
sizes. (C) 1996 Pattern Recognition Society. Published by Elsevier Sc
ience Ltd.