A novel scheme for the detection of object boundaries is presented. The technique is based on active contours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric is defined by the image content. This geodesic approach for object segmentation allows to connect classical "snakes" based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved, allowing stable boundary detection when their gradients suffer from large variations, including gaps. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. The scheme was implemented using an efficient algorithm for curve evolution. Experimental results of applying the scheme to real images including objects with holes and medical data imagery demonstrate its power. The results may be extended to 3D object segmentation as well.

The eikonal equation and variants of it are of significant interest for problems in computer vision and image processing. It is the basis for continuous versions of mathematical morphology, stereo, shape-from-shading and for recent dynamic theories of shape. Its numerical simulation can be delicate, owing to the formation of singularities in the evolving front and is typically based on level set methods. However, there are more classical approaches rooted in Hamiltonian physics which have yet to be widely used by the computer vision community. In this paper we review the Hamiltonian formulation, which offers specific advantages when it comes to the detection of singularities or shocks. We specialize to the case of Blum's grass fire flow and measure the average outward ux of the vector field that underlies the Hamiltonian system. This measure has very different limiting behaviors depending upon whether the region over which it is computed shrinks to a singular point or a non-singular one. Hence, it is an effective way to distinguish between these two cases. We combine the ux measurement with a homotopy preserving thinning process applied in a discrete lattice. This leads to a robust and accurate algorithm for computing skeletons in 2D as well as 3D, which has low computational complexity. We illustrate the approach with several computational examples.

We undertake to develop a general theory of two-dimensional shape by elucidating several principles which any such theory should meet. The principles are organized around two basic intuitions: first, if a boundary were changed only slightly, then, in general, its shape would change only slightly. This leads us to propose an operational theory of shape based on incremental contour deformations. The second intuition is that not all contours are shapes, but rather only those that can enclose "physical" material. A theory of contour deformation is derived from these principles, based on abstract conservation principles and Hamilton-Jacobi theory. These principles are based on the work of Sethian [82, 86], the Osher-Sethian level set formulation [65], the classical shock theory of Lax [53, 54], as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of Gage-Hamilton-Grayson [32, 37]. The result is a characterization of th...

The class of geometric deformable models, so-called level sets, has brought tremendous impact to medical imagery due to its capability to preserve topology and fast shape recovery. In an effort to facilitate a clear and full understanding of these powerful state-of-the-art applied mathematical tools, this paper is an attempt to explore these geometric methods, their implementations and integration of regularization terms to improve the robustness of these topologically independent propagating curves/surfaces. This paper first presents the origination of the level sets, followed by the taxonomy tree of level sets. We then derive the fundamental equation of curve/surface evolution and zero-level curves/surfaces. The paper then focuses on the first core class of level sets, the so-called level sets "without regularizers". The next section is devoted on a second kind, so-called level sets "with regularizers". In this class, we present four kinds of systems on the design of the regularizers. Next, the paper presents a third kind of level sets, so-called the "bubble-based" techniques. An entire section is dedicated to optimization and quantification techniques for shape recovery when used with the level sets. Finally, the paper concludes with 22 general merits and four demerits on level sets and the future of level sets in medical image segmentation. We present the applications of level sets to complex shapes likethehuman cortex acquired via MRI for neurological image analysis.

A new framework for computing the Euclidean distance and weighted distance from the boundary of a given digitized shape is presented. The distance is calculated with sub--pixel accuracy. The algorithm is based on an equal distance contour evolution process. The moving contour is embedded as a level set in a time varying function of higher dimension. This representation of the evolving contour makes possible the use of an accurate and stable numerical scheme, due to Osher and Sethian [22]. The relation between the classical shape from shading problem and the weighted distance transform is presented, as well as an algorithm that calculates the geodesic distance transform on surfaces.

Abstract—In this paper, we use partial-differential-equationbased filtering as a preprocessing and post processing strategy for computer-aided cytology. We wish to accurately extract and classify the shapes of nuclei from confocal microscopy images, which is a prerequisite to an accurate quantitative intranuclear (genotypic and phenotypic) and internuclear (tissue structure) analysis of tissue and cultured specimens. First, we study the use of a geometry-driven edge-preserving image smoothing mechanism before nuclear segmentation. We show how this filter outperforms other widely-used filters in that it provides higher edge fidelity. Then we apply the same filter, with a different initial condition, to smooth nuclear surfaces and obtain sub-pixel accuracy. Finally we use another instance of the geometrical filter to correct for misinterpretations of the nuclear surface by the segmentation algorithm. Our prefiltering and post filtering nicely complements our initial segmentation strategy, in that it provides substantial and measurable improvement in the definition of the nuclear surfaces. Index Terms—Cytology, differential geometry, dynamic surfaces, image processing, level sets, Riemannian geometry, segmentation, surface evolution. I.

Abstract—We propose an efficient and deterministic algorithm for computing the one-dimensional dilation and erosion (max and min) sliding window filters. For a p-element sliding window, our algorithm computes the 1D filter using 1:5 þ oð1Þ comparisons per sample point. Our algorithm constitutes a deterministic improvement over the best previously known such algorithm, independently developed by van Herk [25] and by Gil and Werman [12] (the HGW algorithm). Also, the results presented in this paper constitute an improvement over the Gevorkian et al. [9] (GAA) variant of the HGW algorithm. The improvement over the GAA variant is also in the computation model. The GAA algorithm makes the assumption that the input is independently and identically distributed (the i.i.d. assumption), whereas our main result is deterministic. We also deal with the problem of computing the dilation and erosion filters simultaneously, as required, e.g., for computing the unbiased morphological edge. In the case of i.i.d. inputs, we show that this simultaneous computation can be done more efficiently then separately computing each. We then turn to the opening filter, defined as the application of the min filter to the max filter and give an efficient algorithm for its computation. Specifically, this algorithm is only slightly slower than the computation of just the max filter. The improved algorithms are readily generalized to two dimensions (for a rectangular window), as well as to any higher finite dimension (for a hyperbox window), with the number of comparisons per window remaining constant. For the sake of concreteness, we also make a few comments on implementation considerations in a contemporary programming language. Index Terms—Mathematical morphology, running maximum filter, min-max filter, computational efficiency. æ 1

Poor convergence to concave shapes is a main limitation of snakes as a standard segmentation and shape modelling technique. The gradient of the external energy of the snake represents a force that pushes the snake into concave regions, as its internal energy increases when new inexion points are created. In spite of the improvement of the external energy by the gradient vector ow technique, highly non convex shapes can not be obtained, yet. In the present paper, we develop a new external energy based on the geometry of the curve to be modelled. By tracking back the deformation of a curve that evolves by minimum curvature ow, we construct a distance map that encapsulates the natural way of adapting to non convex shapes. The gradient of this map, which we call curvature vector ow (CVF), is capable of attracting a snake towards any contour, whatever its geometry. Our experiments show that, any initial snake condition converges to the curve to be modelled in optimal time.

The eikonal equation and variants of it are of significant interest for problems in computer vision and image processing. It is the basis for continuous versions of mathematical morphology, stereo, shape-from-shading and for recent dynamic theories of shape. Its numerical simulation can be delicate, owing to the formation of singularities in the evolving front, and is typically based on level set methods introduced by Osher and Sethian. However, there are more classical approaches rooted in Hamiltonian physics, which have received little consideration in the computer vision literature. Here the front is interpreted as minimizing a particular action functional. In this context, we introduce a new algorithm for simulating the eikonal equation, which offers a number of computational advantages over the earlier methods. In particular, the locus of shocks is computed in a robust and efficient manner. We illustrate the approach with several numerical examples.

Numerical analysis of conservation laws plays an important role in the implementation of curve evolution equations. This paper reviews the relevant concepts in numerical analysis and the relation between curve evolution, Hamilton-Jacobi partial differential equations, and differential conservation laws. This close relation enables us to introduce finite difference approximations, based on the theory of conservation laws, into curve evolution. It is shown how curve evolution serves as a powerful tool for image analysis, and how these mathematical relations enable us to construct efficient and accurate numerical schemes. Some examples demonstrate the importance of the CFL condition as a necessary condition for the stability of the numerical schemes. 1 Introduction Recently, researchers in the field of image processing and computer vision started to pay attention to new ways of analyzing and representing two-dimensional, stationary or moving images, via planar curve evolution. In fact, a...