Kruger (1998),
Geisler et al. (2001), and
Sigman et al. (2001) examined the joint statistics of oriented filter responses in natural images, and found statistical correlations suggestive of proximity, colinearity, parallelism, and cocircularity principles. However, no attempt was made to restrict the analysis to filters responding to contours, let alone common contours. These statistics therefore arise from within and between a mixture of structures: contours, texture, and shading flows, and do not lead directly to a model for the perceptual organization of contours.
Geisler et al. (2001) have taken this work one step further. Using human participants to trace perceived contours, the second-order spatial statistics of oriented elements lying on a common contour are estimated. However, the contour tracing method employed by Geisler et al. does not recover the ordering of the elements along a contour. Thus a contour is represented as an unordered set of oriented elements, and the statistics relate arbitrary pairs of tangents on a contour.
We have argued that a statistical encoding of the ordering of elements along a contour is critical to understanding contour grouping. A defining property of contours is their one-dimensionality. Without this, basic properties of contours, such as curvature, closure, concavity, and convexity, cannot be defined. In continuous space, contours may be parameterized by a real variable, e.g., arc length. To maintain this property in a discrete encoding, a contour must be represented not as a set but as a sequence of local elements.
For these reasons, we define the problem of contour grouping as the recovery of sequences of tangents projecting from the contours of a scene. We model these sequences as Markov chains. The Markov approximation captures the local nature of the physical processes that give rise to these contours, and is consistent with the monotonically decreasing nature of the autocorrelation function of natural images, with the spatiotopic structure of early visual cortex and with the well studied psychophysical principle of proximity. Application of the Markov approximation allows us to understand the problem of contour grouping by characterizing the statistics of local grouping between successive elements comprising a contour.
Our model reflects the fact that the statistical dependencies between neighbouring tangents on a contour are much stronger than those between distant tangents on a contour. In the approach of
Geisler et al. (2001), the power of the strong statistics relating neighboring tangents is diluted with the weak statistics relating distant tangents. This leads to substantial differences between our statistics and their statistics.