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In this paper we seek to exploit information geometry in order to define the Riemannian structure of the statistical manifold associated with the Chan-Vese active contour model. This Riemannian structure is obtained through a relationship between the contour's Mumford-Shah energy functional and the likelihood of the categorical latent variables of a Gaussian mixture model. Accordingly the natural metric of the statistical manifold formed by the contours is determined by their 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 natural gradient algorithm for the Chan-Vese active contour, with application to image segmentation. Because the proposed method performs optimization on the parameters natural manifold it attains dramatically faster convergence rates than the Euclidean gradient descent algorithm commonly used in the literature. Experiments performed on standard test images from the active contour literature are presented and confirm that the proposed natural gradient algorithm delivers accurate segmentation results in few iterations. Comparisons with methods from the state of the art show that the proposed method converges extremely fast, and could improve significantly the speed of many existing image segmentation methods based on the Chan-Vese active contour as well as enable its application to new problems. A MATLAB implementation of the proposed method is available at http://www.stats.bris.ac.uk/~p12320/code/SmoothNaturalGradient4ChanVese.rar.