The Ecological Statistics of Good Continuation: Multiscale Markov Models for Contours

Background Previous models of good continuation [e.g., Williams & Jacobs 95] make the firstorder Markov assumption, i.e., if we parametrize a curve by arc length t, the tangent direction of the contour at t+1 only depends on the tangent at t. The goal of this study is to use humanmarked boundary contours in a large database of natural images to empirically determine the validity of this model. Methods
Experiment 1: We measure the distribution of lengths of contours segmented at local curvature maxima. If the firstorder Markov assumption holds, the lengths of the segments would have an exponential distribution.
Results
Experiment 1: We observe a power law, instead of an exponential law, in the distribution of the contour segment length. The probability is inversely proportional to the square of segment length. The power law justifies the intuition that contours are multiscale in nature; the firstorder Markov assumption is shown to be empirically invalid.
Conclusion Any algorithm for contour processing has to be intrinsically multiscale. Higherorder Markov models exploit information across scales and lead to an efficient algorithm for multiscale contour completion. 