 
Summary: In Energy Minimization Methods in Computer Vision and Pattern Recognition at Nice, September,
2001. Figueiredo, Zerubia, Jain, Eds., Lecture Notes in Computer Science, Springer.
A Markov Process Using Curvature
for Filtering Curve Images
Jonas August and Steven W. Zucker
Center for Computational Vision and Control
Departments of Electrical Engineering and Computer Science
Yale University
51 Prospect Street, New Haven, CT 06520
fjonas.august,steven.zuckerg@yale.edu
Abstract. A Markov process model for contour curvature is introduced via a stochastic dif
ferential equation. We analyze the distribution of such curves, and show that its mode is the
Euler spiral, a curve minimizing changes in curvature. To probabilistically enhance noisy and
low contrast curve images (e.g., edge and line operator responses), we combine this curvature
process with the curve indicator random eld, which is a prior for ideal curve images. In partic
ular, we provide an expression for a nonlinear, minimum mean square error lter that requires
the solution of two elliptic partial dierential equations. Initial computations are reported,
highlighting how the lter is curvatureselective, even when curvature is absent in the input.
1 Introduction
Images are ambiguous. One unpleasant consequence of this singular fact is that we cannot compute
