Generalized Gradients: Priors on Minimization Flows

Abstract

This paper tackles an important aspect of the variational problem underlying active contours: optimization by gradient flows. Classically, the definition of a gradient depends directly on the choice of an inner product structure. This consideration is largely absent from the active contours literature. Most authors, explicitely or implicitely, assume that the space of admissible deformations is ruled by the canonical L 2 inner product. The classical gradient flows reported in the literature are relative to this particular choice. Here, we investigate the relevance of using (i) other inner products, yielding other gradient descents, and (ii) other minimizing flows not deriving from any inner product. In particular, we show how to induce different degrees of spatial consistency into the minimizing flow, in order to decrease the probability of getting trapped into irrelevant local minima. We report numerical experiments indicating that the sensitivity of the active contours method to initial conditions, which seriously limits its applicability and efficiency, is alleviated by our application-specific spatially coherent minimizing flows. We show that the choice of the inner product can be seen as a prior on the deformation fields and we present an extension of the definition of the gradient toward more general priors.

Diffusion-based Regularisation Strategies for Variational Level Set Segmentation

“…Therefore it is mathematically correct to refer to ∇E as the L 2 -gradient and not simply the gradient. Unfortunately, the L 2 -gradient is, to put it simply, too local and therefore prone to lead into an undesired local minimum (see [6] or [20] for further details). Thus regularisation strategies are necessary to avoid these undesired local minima and one can classify them as either implicit or explicit.…”

Fast Musculoskeletal Registration Based on Shape Matching

“…The parameterization of elastic deformations from multiple global transforms has been successfully used in various domains such as shape editing [7], computer animation [8,9], computer vision [10] and image registration [11]. Spatially coherent (e.g.…”

“…Spatially coherent (e.g. quasi-rigid) flow can be designed through generalized gradients as shown by Charpiat et al [10] (Eulerian setting) and Eckstein et al [12] (Lagrangian setting). In [11], Arsigny et al fuse locally rigid or affine transforms into a global transform by infinitesimally averaging their logarithms.…”

“…We compare two techniques for estimating rigid transforms and simulate plasticity through reference positions update. Like generalized gradients [10,12], this enforces spatially coherent deformations. We use shape matching as a regularization step that filters out noise from external forces, and thus propose a modified force-based evolution scheme.…”

“…Using initial positions as a reference (x r = x(0)) [8,9], shape matching forces act as elastic forces, proportional to the strain. When updating reference positions at each time step (x r = x(t − dt)), the effect, similarly to a plastic deformation, is to enforce spatially coherent deformations like generalized gradients [10,12]. We will show how to combine the two approaches in Section 3.…”