<title>Analytic Hough transform</title>

Abstract

An analytic extension of the Hough Transform is introduced and analyzed, and an implementation is demonstrated. The Hough Transform in its usual implementation has proven to be a useful tool for image segmentation and feature extraction through identification of approximately coffinear point sets in images. The Analytic Hough Transform (AliT) algorithm significantly improves upon these results by operating specifically with the information in spatially quantized images to yield those pixel sets that exactly define digital lines in the image. The resulting pixel sets, while being subsets of a digital line set, need not be contiguous. Thus the AHT also represents an alternative to digital line tests that depend upon contiguity. An Inverse Analytic Hough Transform (IAHT) is also introduced. For a given quantized image the AliT segments its Hough parameter space into convex polygons that represent all real line sets that pass entirely through certain digital line pixel sets in the image. The IAHT converts these parameter space polygons into a pair of convex hulls in image space. A real line passes between these hulls if and only if it passes through every pixel connected with the parameter space polygon. Thus the IAHT generates a pair of simple geometric boundaries in image space that associate pixels with polygonal AliT solution regions. An implementation of the AliT is discussed and demonstrated. It is found that the AliT, with its exact results, can be a computationally attractive alternative to the usual implementation of a high resolution Hough Transform. Furthermore, the AliT and the IAHT effectively couple and efficiently find exact solutions to the problems of digital line detection and determination of associated real line parameters.