Guaranteed Convergence of the Hough Transform

The straight-line Hough Transform using normal parameterization with a continuous voting kernel is considered. It transforms the collinearity detection problem to a problem of nding the global maximum of a two dimensional function above a domain in the parameter space. The principle is similar to robust regression using xed scale M-estimation. Unlike standard M-estimation procedures the Hough Transform does not rely on a good initial estimate of the line parameters: The global optimization problem is approached by exhaustive search on a grid that is usually as ne as computationally feasible. The global maximum of a general function above a bounded domain cannot be found by a nite number of function evaluations. Only if suucient a-priori knowledge about the smoothness of the objective function is available, convergence to the global maximum can be guaranteed. The extraction of a-priori information and its eecient use are the main challenges in real global optimization problems. Convergence in the Hough Transform is the ability to ensure that the global maximum is in the immediate neighborhood of the maximal grid point. More than thirty years after Hough patented the basic algorithm, it is still not clear how ne should the parameter space quantization be in order not to miss the true maximum. In this paper conditions for the convergence of the Hough Transform to the global maximum are derived. The necessary constraints on the variability of the objective (Hough) function are obtained by using the saturated parabolic voting kernel and by deening an image model with several application dependent parameters. Random errors in the location of edge points and background noise are allowed in the model 1 and lead to statistical convergence guarantees. Signiicant intermediate results are obtained on the structure of the peak region and on the spatial statistics of noise voting in the continuous kernel Hough Transform. Convergence strategies are studied and the necessary parameter space quantization intervals are derived. Guaranteed focusing policies for multi-resolution Hough algorithms are developed. The application of the theoretic results to images that deviate from the image model is considered and exempliied.