An Anisotropic A Contrario Framework for the Detection of Convergences in Images

Abstract

The a contrario framework for the detection of convergences in an image consists in counting, for each tested point, the number of elementary linear structures that converge to it (up to a given precision), and when this number is high enough, the point is declared to be a meaningful point of convergence. This is so far analogous to a Hough transform, and the main contribution of the a contrario framework is to provide a statistical definition of what "high enough" means: it means large enough to ensure that in an image where all elementary structures are distributed according to a background noise model, there is, in expectation, less than 1 detection. Our aim in this paper is to discuss, from a methodological viewpoint, the choice and the influence of the background noise model. This model is generally taken as the uniform independent distribution on elementary linear structures, and here, we discuss the case of images that have a natural anisotropic distribution of structures. Our motivating example is the one of mammograms in which we would like to detect stellate patterns (that appear as local convergences of spicules), and in which the linear structures are naturally oriented towards the nipple. In this paper, we show how to tackle the two problems of (a) defining and estimating an anisotropic "normal" distribution from an image, and of (b) computing the probability that a random structure, following an anisotropic distribution, converges to any given convex region. We illustrate the whole approach with several examples.