Exploiting information geometry to improve the convergence of nonparametric active contours

In this paper we seek to exploit information geometry in order to define the Riemannian metric of the manifold associated with nonparametric active contour models from the exponential family. This Riemannian metric is obtained through a relationship between the contour´s energy functional and the likelihood of the categorical latent variables of a mixture model. Accordingly contours form a statistical manifold equipped with a natural metric which is determined by the model´s Fisher information matrix. Mathematical developments show that this matrix has a closed-form analytic expression and is diagonal. Based on this, we subsequently develop a Riemannian steepest descent algorithm for the active contour, with application to image segmentation. Because the proposed method performs optimisation on the parameter´s natural manifold it attains dramatically faster convergence rates than the Euclidean gradient descent algorithm commonly used in the literature. A segmentation experiment on an ultrasound image is presented and confirms that the proposed natural gradient algorithm converges extremely fast and delivers accurate segmentation results in few iterations.