COMPUTER VISION, GRAPHICS, AND IMAGE PROCESSING 21, 383-394 (1983) Approximating Point-Set Images by Line Segments Using a Variation of the Hough Transform1 PHILIP R. THRIFT AND STANLEY M. DUNN Computer Vision Laboratory, Computer Science Center; Universityof Maryland, CollegePark, Maryland 20742 Received February 24, 1982 A transform method is presented for the detection of curves in noisy point-set images and is used to detect line segments in such images. The results of this transform are compared with results from the Hough transform. 1, INTRODUCTION Statistical geometry is the study of how to restore pure geometric objects when we can only observe deformed versions of them. The case with which this paper is concerned is that when the pure geometric objects are finite subsets of curve segments from a given parametric family. If we call the pure image I, then D(I), the deformed version, consists of a (random) point subset of I with additive noise superimposed, with perhaps an additional point process as background noise. In this paper we shall present an approximation method, similar to the Hough transform as described in [1] and to the chamfer matching in [2]. The generalized Hough transform as described by O'Gorman and Clowes [3] differs in the function by which each counter is incremented. O'Gorman and Clowes chose to increment the cell by the value of the gradient at the location (x, y). This was done to ensure that strong edges in the image are found. Shapiro and Iannino [4] model the effects of noise by assuming that the actual location of the point is within some fixed region of the given estimate. Two characterizations are presented in [4], one that bounds the total maximum error, and another that bounds the maximum error in one direction. Sklansky [5] treats the problem of additive noise by modification of the point-spread function of the matched filter. The product of the image point (signal and noise) with the shifted signal value is added to the accumulator for all possible shifts. Additional Hough transform references are given in [6]. Distributional considera- tions will be examined later. The general problem, which covers a wide range of distributional assumptions, appears below. We suppose f~ c R a is some parameter space with ~ = ({(x, y): f(~, x, y) = 0, (x, y)~ Ra):~ ~ a) a family of curves in R2 parameterized by ~. Let (xl, Yl) ..... (x u, y~r) be a set of N points of R2 that arise as a deformed version of a set of segments of M curves parameterized by (1..... ~M of 0~. We shall assume that (Xl, Yl) ..... (xN, Yu) arises as a deformation of a point-process realization of the segments, plus additive noise. IThe support of the U.S. Air Force Office of Scientific Research under Grant AFOSR-77-3271 is gratefully acknowledged, as is the help of Janet Salzman in preparing this paper. 383 0734-189X/83 $3.00 Copyright © 1983by Academic Press,Inc. All fights of reproduction in any formreserved.