One-Shot Integral Invariant Shape Priors for Variational Segmentation

Abstract

Abstract. We match shapes, even under severe deformations, via a smooth reparametrization of their integral invariant signatures. These robust signatures and correspondences are the foundation of a shape energy functional for variational image segmentation. Integral invariant shape templates do not require registration and allow for significant deformations of the contour, such as the articulation of the object's parts. This enables generalization to multiple instances of a shape from a single template, instead of requiring several templates for searching or training. This paper motivates and presents the energy functional, derives the gradient descent direction to optimize the functional, and demonstrates the method, coupled with a data term, on real image data where the object's parts are articulated.

Image segmentation with one shape prior — A template-based formulation

“…The most common shape representations C used in the literature include parametric shape representations: [0, l(C)] → R [16,26], signed distance functions (SDFs) φ : Ω → R [6,10,4], binary characteristic functions u : Ω → {0, 1} [11,13], or very recent work on relaxed characteristic functions u : Ω → [0, 1] [19,29]. Typical transformations T discussed in the shape-prior segmentation is limited to parametric global transformation, include rigid, similarity or more general projective transformations [21].…”

“…When the shape is represented parametrically, as in [16,26], there are many shape descriptors that are invariant to certain geometric transformations T . For example, curvature (either integral or differential) is an invariant shape measure with respect to rigid transformations [16], and tangent angle is an invariant shape measure with respect to translations [26]. However, the shape distances D based on these shape descriptors are not usually invariant to these geometric transformations, since correspondence between the shapes affects these distances.…”

“…Fix T and update C. This becomes a standard image segmentation problem and common optimization methods include level-set-based gradient descent [6,10,4,16,5], discrete graph cuts [11,13] or the recent convex continuous cut method [19,29]. The drawback of level-set gradient-based methods include numerical issues, slow convergence and local optimality at convergence.…”

Shape Reconstruction with A Priori Knowledge Based on Integral Invariants

“…In contrast, we propose using integral invariants to explain and handle the variability of a shape. This concept has been applied successfully to segmentation with priors [20] but not yet for more general inverse and ill-posed problems.…”

“…All invariants, however, that are based on differentiation-thus, in particular, curvature-suffer from an inherent sensitivity regarding noise. As a remedy, it has been suggested to replace differentiation by integration in such a way that the ensuing integral invariants still carry geometrical information about the object [5,19,20,22]. These invariants have proven to be successful for object classification [19] and geometry processing [14].…”