Diffeomorphic Active Contours

Abstract

In this study we present a geometric flow approach to the segmentation of three-dimensional medical images obtained from magnetic resonance imaging (MRI) or computed tomography (CT) scan methods, by minimizing a cost function. This energy term is based on the intensity of the original image and its minimum is found following a gradient descent curve in an infinitedimensional space of diffeomorphisms (Diff) to preserve topology. The general framework is reminiscent of variational shape optimization methods, but remains closer to general developments on deformable template theory of geometric flows. In our case, the metric that provides the gradient is defined as a right invariant inner product on the tangent space ( at the identity of the group of diffeomorphisms, following the general Lie group approach suggested by Arnold [2]. To avoid local solutions of the optimization problem and to mitigate the influence of several sources of noise, a finite set of control points is defined on the boundary of the template binary images, yielding a projected gradient descent on Diff.

Constrained Diffeomorphic Shape Evolution

“…The only change would be in the nature of the data attachment term, which must be designed appropriately to handle unlabeled curves and surfaces, or images. However, note that the algorithm is based on a Lagrangian formulation, which requires the evaluation of kernel sums over nonregular grids, as implied by the space of constrained velocities in (5). The approach can proceed with a standard implementation for small-to medium-size problems (no more than a couple of thousand diffeons in our experiments), but requires adapted numerical procedures for large sizes.…”

“…The approach can proceed with a standard implementation for small-to medium-size problems (no more than a couple of thousand diffeons in our experiments), but requires adapted numerical procedures for large sizes. In such contexts, one can use kernels that are computationally friendly over nonregular grids, such as the class studied in [21] (see also [5,12]). …”

“…Returning to the general finite-dimensional model, (5), the inexact matching formulation can be rewritten as a finite-dimensional optimal control problem, which requires one to minimize…”

“…This leads in turn to a new class of variational problems, for which the existence of solutions is proved in Sect. 5. After a brief reminder of the kind of solutions that are obtained in the unconstrained case (Sect.…”

Spaces and manifolds of shapes in computer vision: An overview

“…Because Sobolev metrics make moving along low frequency (smooth) directions preferable to high frequencies, the resulting algorithm is significantly more stable than the original ones, even in the presence of high noise. In the same range of idea, gradient descent on shape spaces arising from the deformable template manifold structure provides "Diffeomorphic active contours" [106,107] that follow smooth gradient directions without topological changes. The same range of ideas led to "diffeomorphic mean curvature flow" for surfaces, as described in [108].…”