We view the fundamental edge integration problem for object segmentation in a geometric variational framework. First we show that the classical zero-crossings of the image Laplacian edge detector as suggested by Marr and Hildreth, inherently provides optimal edge-integration with regard to a very natural geometric functional. This functional accumulates the inner product between the normal to the edge and the gray level image-gradient along the edge. We use this observation to derive new and highly accurate active contours based on this functional and regularized by previously proposed geodesic active contour geometric variational models. 1.

Inflammatory disease is initiated by leukocytes (white blood cells) rolling along the inner surface Hning of small blood vessels called postcapillary venules. Studying the number and velocity of rolling leukocytes is essential to understanding and successfully treating inflammatory diseases. Potential inhibitors of leukocyte recruitment can be screened by leukocyte rolling assays and successful inhibitors validated by intravital microscopy. In this paper we present an active contour or snake-based technique to automatically track the movement of the leukocytes. The novelty of the proposed method Hes in the energy functional that constrains the shape and size of the active contour. This paper introduces a significant enhancement over existing gradientbased snakes in the form of a modified gradient vector flow. Using the gradient vector flow, we can track leukocytes rolling at high speeds that are not amenable to tracking with the existing edge-based techniques. We also propose a new energy based implicit sampling method of the points on the active contour that replaces the computationally expensive explicit method. To enhance the performance of this shape and size constrained snake model we have coupled it with Kalman f'fiter, so that during coasting (when the leukocytes are completely occluded or obscured), the tracker may infer the location of the center of the leukocyte. Finally we have compared the performance of the proposed snake tracker with that of the correlation and centroid-based trackers. The proposed snake tracker results in superior performance measures such as reduced error in locating the leukocyte under tracking and improvements in the percentage of frames successfully tracked. For screening and drug validation, the tracker shows promise as an automat...

Abstract. All previous geometric active contour models that have been formulated as gradient flows of various energies use the same L 2-type inner product to define the notion of gradient. Recent work has shown that this inner product induces a pathological Riemannian metric on the space of smooth curves. However, there are also undesirable features associated with the gradient flows that this inner product induces. In this paper, we reformulate the generic geometric active contour model by redefining the notion of gradient in accordance with Sobolev-type inner products. We call the resulting flows Sobolev active contours. Sobolev metrics induce favorable regularity properties in their gradient flows. In addition, Sobolev active contours favor global translations, but are not restricted to such motions; they are also less susceptible to certain types of local minima in contrast to traditional active contours. These properties are particularly useful in tracking applications. We demonstrate the general methodology by reformulating some standard edge-based and regionbased active contour models as Sobolev active contours and show the substantial improvements gained in segmentation.

There are currently two main types of active contours: 1) parametric active contours, which represent contours explicitly as parameterized curves; and 2) geometric active contours, which represent contours implicitly as level sets of two-dimensional scalar functions. In this paper, we derive an explicit mathematical relationship between the general formulations of parametric and geometric active contours. Based on this relationship and the results of two recent parametric active contours, we propose two new geometric active contours. Using both simulated and real images, we show that the proposed algorithms have an improved performance over both existing parametric and geometric active contours. 1 Introduction Active contours [9], a physically-motivated model that can deform itself to recover object shape from digital images, have been extensively researched in the past decade (see [14] for a recent survey on this topic). Current active contours can be classified as either parametric...

In this paper we present a graph cuts based active contours (GCBAC) approach to object segmentation. GCBAC approach is a combination of the iterative deformation idea of active contours and the optimization tool of graph cuts. It differs from traditional active contours in that it uses graph cuts to iteratively deform the contour and its cost function is defined as the summation of edge weights on the cut. The resulting contour at each iteration is the global optimum within a contour neighborhood (CN) of the previous result. Since this iterative algorithm is shown to converge, the final contour is the global optimum within its own CN. The use of contour neighborhood alleviates the well-known bias of the minimum cut in favor of a shorter boundary. GCBAC approach easily extends to the segmentation of three and higher dimensional objects, and is suitable for interactive correction. Experimental results on selected data sets and performance analysis are provided.

Globally minimal surfaces by continuous maximal flows In this paper we address the computation of globally minimal curves and surfaces for image segmentation and stereo reconstruction. We present a solution, simulating a continuous maximal flow by a novel system of partial differential equations. Existing methods are either grid-biased (graph-based methods) or sub-optimal (active contours and surfaces). The solution simulates the flow of an ideal fluid with isotropic velocity constraints. Velocity constraints are defined by a metric derived from image data. An auxiliary potential function is introduced to create a system of partial differential equations. It is proven that the algorithm produces a globally maximal continuous flow at convergence, and that the globally minimal surface may be obtained trivially from the auxiliary potential. The bias of minimal surface methods toward small objects is also addressed. An efficient implementation is given for the flow simulation. The globally minimal surface algorithm is applied to segmentation in 2D and 3D as well as to stereo matching. Results in 2D agree with an existing minimal contour algorithm for planar images. Results in 3D segmentation and stereo matching demonstrate that the new algorithm is robust and free from grid bias. I.

Abstract—We outline an Eulerian framework for computing the thickness of tissues between two simply connected boundaries that does not require landmark points or parameterizations of either boundary. Thickness is defined as the length of correspondence trajectories, which run from one tissue boundary to the other, and which follow a smooth vector field constructed in the region between the boundaries. A pair of partial differential equations (PDEs) that are guided by this vector field are then solved over this region, and the sum of their solutions yields the thickness of the tissue region. Unlike other approaches, this approach does not require explicit construction of any correspondence trajectories. An efficient, stable, and computationally fast solution to these PDEs is found by careful selection of finite differences according to an upwinding condition. The behavior and performance of our method is demonstrated on two simulations and two magnetic resonance imaging data sets in two and three dimensions. These experiments reveal very good performance and show strong potential for application in tissue thickness visualization and quantification. Index Terms—Correspondence trajectory, numerical methods, partial differential equations (PDEs), thickness. I.

This paper describes a new approach to adaptive estimation of parametric deformable contours based on B-spline representations. The problem is formulated in a statistical framework with the likelihood function being derived from a region -based image model. The parameters of the image model, the contour parameters, and the B-spline parameterization order (i.e., the number of control points) are all considered unknown. The parameterization order is estimated via a minimum description length (MDL) type criterion. A deterministic iterative algorithm is developed to implement the derived contour estimation criterion. The result is an unsupervised parametric deformable contour: it adapts its degree of smoothness/complexity (number of control points) and it also estimates the observation (image) model parameters. The experiments reported in the paper, performed on synthetic and real (medical) images, confirm the adequacy and good performance of the approach.

Abstract—The paper presents a novel stochastic active contour scheme (STACS) for automatic image segmentation designed to overcome some of the unique challenges in cardiac MR images such as problems with low contrast, papillary muscles, and turbulent blood flow. STACS minimizes an energy functional that combines stochastic region-based and edge-based information with shape priors of the heart and local properties of the contour. The minimization algorithm solves, by the level set method, the Euler-Lagrange equation that describes the contour evolution. STACS includes an annealing schedule that balances dynamically the weight of the different terms in the energy functional. Three particularly attractive features of STACS are: 1) ability to segment images with low texture contrast by modeling stochastically the image textures; 2) robustness to initial contour and noise because of the utilization of both edge and region-based information; 3) ability to segment the heart from the chest wall and the undesired papillary muscles due to inclusion of heart shape priors. Application of STACS to a set of 48 real cardiac MR images shows that it can successfully segment the heart from its surroundings such as the chest wall and the heart structures (the left and right ventricles and the epicardium.) We compare STACS ’ automatically generated contours with manually-traced contours, or the “gold standard, ” using both area and edge similarity measures. This assessment demonstrates very good and consistent segmentation performance of STACS. Index Terms—Active contour, cardiac magnetic resonance imaging (cardiac MRI), chamfer method, energy minimization, image segmentation, level set, shape and area similarities, stochastic model, stochastic relaxation. I.

Active contour and surface models, also known as deformable models, constitute a class of powerful segmentation techniques. Geometric deformable models implemented via level-set methods have advantages over parametric ones due to their intrinsic behavior, parameterization independence, and ease of implementation. However, a long claimed advantage of geometric deformable models --- the ability to automatically handle topology changes --- turns out to be a liability in applications where the objects to be segmented have a known topology that must be preserved. In this paper, we present a geometric deformable model that preserves topology using the simple point concept from digital topology. This algorithm maintains the other advantages of standard geometric deformable models including sub-pixel accuracy and production of nonintersecting curves (or surfaces). Several experiments on simulated and real data are provided to demonstrate the performance of the proposed algorithm.