Minimal surfaces: a geometric three dimensional segmentation approach

Abstract

Summary.A novel geometric approach for three dimensional object segmentation is presented. The scheme is based on geometric deformable surfaces moving towards the objects to be detected. We show that this model is related to the computation of surfaces of minimal area (local minimal surfaces). The space where these surfaces are computed is induced from the three dimensional image in which the objects are to be detected. The general approach also shows the relation between classical deformable surfaces obtained via energy minimization and geometric ones derived from curvature flows in the surface evolution framework. The scheme is stable, robust, and automatically handles changes in the surface topology during the deformation. Results related to existence, uniqueness, stability, and correctness of the solution to this geometric deformable model are presented as well. Based on an efficient numerical algorithm for surface evolution, we present a number of examples of object detection in real and synthetic images.

Computing geodesics and minimal surfaces via graph cuts

“…Numerical optimization is performed via level-sets. The same approach can be used for 3D segmentation via minimal surfaces (see [7] for details). Further generalizations of geodesic active contours and some anisotropic metrics are discussed in [17].…”

“…The results in (e) are for a simple 6-neighborhood system. The results in Figure 8(f) use 26-neighborhoods with geometrically justifies weights in (7).…”

From high energy physics to low level vision

“…(2), andÎ is the unit vector in the I direction. 5 5 The mean curvature ow can be written as @ @t x I = g x I : Fixing the xy coordinates amounts to moving the surface via its mean curvature feature components I, thereby preserving the edges at which these component are small: Along the edges (cli s), the surface normal is almost parallel to the x-y plane. Thus, I(x y) hardly evolves along the edges while the ow drives other regions of the image towards a minimal surface at a more rapid rate.…”

Representing Diffusion MRI in 5-D Simplifies Regularization and Segmentation of White Matter Tracts

“…One of the main advantages of the level set theory is its natural extension to -dimensional images, which is usually difficult to achieve with standard segmentation models. Thus, the extension to three-dimensional images is commonly used, e.g., [27], and even though some of the properties of the 2-D curves, such as the property of shrinking to a point under curvature flow, do not hold in the 3-D case, the main part of the theory remains valid and works well for segmentation of 3-D objects. The extension to even higher dimensions is straightforward, which will allow us to use the level set method in the 5-D POS.…”