Analytic approach to finding lines in images and estimating their uncertainty

Abstract

This paper presents a new approach to maximum likelihood fitting of lines in images. Standard approaches such as the Hough Transform are heuristic and dogged by the subtleties of quantization error. As an alternative, we take an analytic approach to the Hough Transform coupled with principles of robust maximum likelihood line fitting. We derive two line-parameter likelihood functions based respectively on the heteroscedastic Gaussian distribution and the symmetric bivariate Cauchy distribution. For these functions we establish a Nyquist rate and threshold values such that all local maxima (lines containing a minimum number of points) are found; the analytic form of the likelihood function is used to obtain precise estimates of the line parameters. Finally we obtain analytically the first order covariance of the estimated line parameters, and hence a confidence envelope about the fitted line. Contrary to popular belief this is not obtained by analyzing the shape of peaks in the Hough space. The motivation for this paper is to establish a rigorous framework for finding curves in images in a maximum likelihood sense; only armed this foundation can the more difficult problems in pattern recognition be addressed in a rigorous manner.