In the context of variational image segmentation, we propose a new finite-dimensional implicit surface representation. The key idea is to span a subset of implicit functions with linear combinations of spatially-localized kernels that follow image features. This is achieved by replacing the Euclidean distance in conventional Radial Basis Functions with non-Euclidean, image-dependent distances. For the minimization of an objective region-based criterion, this representation yields more accurate results with fewer control points than its Euclidean counterpart. If the user positions these control points, the non-Euclidean distance enables to further specify our localized kernels for a target object in the image. Moreover, an intuitive control of the result of the segmentation is obtained by casting inside/outside labels as linear inequality constraints. Finally, we discuss several algorithmic aspects needed for a responsive interactive workflow. We have applied this framework to 3D medical imaging and built a real-time prototype with which the segmentation of whole organs is only a few clicks away.

We describe a novel variational segmentation algorithm designed to split an image in two regions based on their intensity distributions. A functional is proposed to integrate the unknown probability density functions of both regions within the optimization process. The method simultaneously performs segmentation and non-parametric density estimation. It does not make any assumption on the underlying distributions, hence it is flexible and can be applied to a wide range of applications. Although a boundary evolution scheme may be used to minimize the functional, we choose to consider an alternative formulation with a membership function. The latter has the advantage of being convex in each variable, so that the minimization is faster and less sensitive to initial conditions. Finally, to improve the accuracy and the robustness to low-frequency artifacts, we present an extension for the more general case of local space-varying probability densities. The approach readily extends to vectorial images and 3D volumes, and we show several results on synthetic and photographic images, as well as on 3D medical data.