Point-to-line mappings as Hough transforms

In 1962 [US Patent 3069654], Hough used a linear point-to-line mapping (PTLM) to detect large sets of collinear points in an image, by mapping the points into concurrent lines and detecting peaks where many lines intersect. In 1972, Duda and Hart [Commun. ACM 15 (1972) 11] pointed out that Hough's method is not practical, because the peaks need not lie in a bounded region. They (and others after them) therefore developed methods of detecting sets of collinear points using nonlinear point-to-curve mappings that map collinear points into concurrent curves whose intersections do lie in a bounded range. In this paper we show that any PTLM that maps collinear points into concurrent lines must be linear, and that no such PTLM can map all the sets of collinear points in an image into peaks that lie in a bounded region; thus Duda and Hart's objection applies to any PTLM-based Hough transform.

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