Convex Hulls, Occluding Contours, Aspect Graphs and the Hough Transform

The Hough transform is a standard technique for finding features such as lines in images. Typically edgels or other features are mapped into a partitioned parameter or Hough space as individual votes. The target image features are detected as peaks in the Hough space. In this paper we consider not just the peaks but the mapping of the entire shape boundary from image space to the Hough parameter space. We analyse this mapping and illustrate correspondences between features in Hough space and image space. Using this knowledge we present an algorithm to construct convex hulls of arbitrary 2D shapes with smooth and polygonal boundaries as well as isolated point sets. We also demonstrate its extension to the 3D case. We then show how this mapping changes as we move the origin in image space. The origin can be considered as a vantage point from which to view the object and the occluding contour can be extracted easily from Hough space as those points where R = 0. We demonstrate the potential for tracking of transitions in the mapping to be used to construct an aspect graph of arbitrary 2D and 3D shapes.