Variational B-Spline Level-Set: A Linear Filtering Approach for Fast Deformable Model Evolution

In the field of image segmentation, most level-set-based active-contour approaches take advantage of a discrete representation of the associated implicit function. We present in this paper a different formulation where the implicit function is modeled as a continuous parametric function expressed on a B-spline basis. Starting from the active-contour energy functional, we show that this formulation allows us to compute the solution as a restriction of the variational problem on the space spanned by the B-splines. As a consequence, the minimization of the functional is directly obtained in terms of the B-spline coefficients. We also show that each step of this minimization may be expressed through a convolution operation. Because the B-spline functions are separable, this convolution may in turn be performed as a sequence of simple 1-D convolutions, which yields an efficient algorithm. As a further consequence, each step of the level-set evolution may be interpreted as a filtering operation with a B-spline kernel. Such filtering induces an intrinsic smoothing in the algorithm, which can be controlled explicitly via the degree and the scale of the chosen B-spline kernel. We illustrate the behavior of this approach on simulated as well as experimental images from various fields.

I. Introduction

Level-set-based formulations have become a well-established tool in the field of image processing [1]–[3]. In image segmentation, level-set-based methods correspond to a class of deformable models where the shape to be recovered is captured by propagating an interface represented by the zero level-set of a smooth function which is usually called the level-set function. The evolution of the interface is generally derived through a variational formulation: the segmentation problem is expressed as the minimization of an energy functional that reflects the properties of the objects to be recovered. Formally, the minimization of this functional provides the evolution of the level-set function as a time-dependent partial differential equation (PDE) that is usually solved using finite-difference methods. These numerical schemes have been developed to obtain an accurate and unique solution; they involve upwind differencing, essentially nonoscillatory schemes borrowed from the numerical solutions of conservation laws and Hamilton–Jacobi equations [2].