Reconciling Distance Functions and Level Sets

Abstract

This paper is concerned with the simulation of the partial differential equation driven evolution of a closed surface by means of an implicit representation. In most applications, the natural choice for the implicit representation is the signed distance function to the closed surface. Osher and Sethian have proposed to evolve the distance function with a Hamilton-Jacobi equation. Unfortunately the solution to this equation is not a distance function. As a consequence, the practical application of the level set method is plagued with such questions as When do we have to reinitialize the distance function? How do we reinitialize the distance function?, which reveal a disagreement between the theory and its implementation. This paper proposes an alternative to the use of Hamilton-Jacobi equations which eliminates this contradiction: in our method the implicit representation always remains a distance function by construction, and the implementation does not differ from the theory anymore. This is achieved through the introduction of a new equation. Besides its theoretical advantages, the proposed method also has several practical advantages which we demonstrate in three applications: (i) the segmentation of the human cortex surfaces from MRI images using two coupled surfaces (X. Zeng, et al., in

Fast PDE approach to surface reconstruction from large cloud of points

“…One popular solution consists of the re-initialization of the volumetric function [33,34], which is used for locally re-distancing the level sets without affecting the motion of the zero level set. An interesting alternative was proposed by Gomes and Faugeras [35], who defined a different implementation of the Hamilton-Jacobi PDE that inherently preserves the distance function.…”

On advances in differential-geometric approaches for 2D and 3D shape analyses and activity recognition

“…The set of valid signed-distance functions forms a nonlinear space but its geometry is seldom utilized to keep representations valid [45]. Since this method is not based on a Riemannian metric, it does not naturally provide geodesic paths on shape spaces of curves.…”

Novel Approach to Segment the Inner and Outer Boundaries of the Bladder Wall in T2-Weighted Magnetic Resonance Images

“…(16) is extended to the ROI using the method proposed by Gomes and Faugeras. 8 The extension of the speed function makes the algorithm focus on the movement of the initial contours, and therefore it avoids the possible inner boundaries when the appearance of the bladder wall is affected. The mean intensity value of the internal region is fixed in order to make the segmentation more reliable and less dynamic; as sometimes, the perivesical fats that are near the bladder wall have similar appearances to the bladder wall itself due to the partial volume effect, they become included in the outer boundary based on their large intensity contrasts to the high-signal intensity background.…”