Subdivision-Based Active Contours --- Statistical optimality of Hermite splines for the reconstruction of self-similar signals --- The Role of Discretisation in X-Ray CT Reconstruction
Anaïs Badoual, Virginie Uhlmann, Michael McCann, EPFL STI LIB

Test Run • 29 May 2018

Abstract
In this talk we present a new family of active contours by exploiting subdivision schemes. Depending on the choice of the mask, such models have the ability to reproduce trigonometric or polynomial curves. They can also be designed to be interpolating, a property that is useful in user-interactive applications. Such active contours are robust in the presence of noise and to the initialization. We illustrate their use for the segmentation of bioimages. ------- Hermite splines are commonly used for interpolating data when samples of the derivative are available, in a scheme called Hermite interpolation. Assuming a suitable statistical model, we demonstrate that this method is optimal for reconstructing random signals in Papoulis’ generalized sampling framework. More precisely, we show the equivalence between cubic Hermite interpolation and the linear minimum mean-square error (LMMSE) estimation of a second-order Lévy process. ------- Discretization—representing a continuous-time function or operation with a discrete-time one—is unavoidable in solving inverse problems. In X-ray computed tomography (CT) reconstruction, the classical algorithm handles discretization "at the end". Modern approaches discretize "in the middle" or "at the beginning". In this talk, I will show how the latter provides algorithms that are mathematically rigorous and implementable. I will also discuss the choice of the basis function among pixels, B-splines, box splines, Kaiser-Bessel windows, and sincs.