Level set methods have been widely used in image processing and computer vision. In conventional level set formulations, the level set function typically develops irregularities during its evolution, which may cause numerical errors and eventually destroy the stability of the evolution. Therefore, a numerical remedy, called reinitialization, is typically applied to periodically replace the degraded level set function with a signed distance function. However, the practice of reinitialization not only raises serious problems as when and how it should be performed, but also affects numerical accuracy in an undesirable way. This paper proposes a new variational level set formulation in which the regularity of the level set function is intrinsically maintained during the level set evolution. The level set evolution is derived as the gradient flow that minimizes an energy functional with a distance regularization term and an external energy that drives the motion of the zero level set toward desired locations. The distance regularization term is defined with a potential function such that the derived level set evolution has a unique forward-and-backward (FAB) diffusion effect, which is able to maintain a desired shape of the level set function, particularly a signed distance profile near the zero level set. This yields a new type of level set evolution called distance regularized level set evolution (DRLSE). The distance regularization effect eliminates the need for reinitialization and thereby avoids its induced numerical errors. In contrast to complicated implementations of conventional level set formulations, a simpler and more efficient finite difference scheme can be used to implement the DRLSE formulation. DRLSE also allows the use of more general and efficient initialization of the level set function. In its numerical implementation, relatively large time steps can be used in the finite difference scheme to reduce the number of iterations, while ensuring sufficient numerical accuracy. To demonstrate the effectiveness of the DRLSE formulation, we apply it to an edge-based active contour model for image segmentation, and provide a simple narrowband implementation to greatly reduce computational cost.

Accurate and fast image segmentation algorithms are of paramount importance for a wide range of medical imaging applications. Level set algorithms based on narrow band implementation have been among the most widely used segmentation algorithms. However, the accuracy of standard level set algorithms is compromised by the fact that their evolution schemes deteriorate the signed distance level set functions required for accurate computation of normals and curvatures. The most common remedy is to use an ad-hoc reinitialization step to rebuild the signed distance function frequently. Meanwhile, complex upwind finite difference schemes are required for stable evolution. They together make the overall computation expensive. In this paper, we propose a novel fast narrow band distance preserving level set evolution algorithm that eliminates the need for both reinitialization and complex upwind finite difference schemes. This is achieved by incorporating into a variational level set formulation with a signed distance preserving term that regularizes the evolution. As a result, stable, accurate, fast evolution could be obtained using a simple finite difference scheme within a very narrow band, defined as the union of all 3times3 pixel blocks around the zero crossing pixels. Also, our method allows the use of larger time step to speed up the convergence while ensuring accurate result, as well as the use of more general and computational efficient initial level set functions rather than the signed distance functions required by standard level set methods. The proposed algorithm has been applied on both synthetic and real images of different modalities with promising results

Active contours or snakes have been extensively utilized in handling image segmentation and classification problems. In traditional active contour models, snake initialization is performed manually by users, and topological changes, such as splitting of the snake, can not be automatically handled. In this paper, we introduce a new method to solve the snake initialization and splitting problem, based on an area segmentation approach: the external force field is segmented first, and then the snake initialization and splitting can be automatically performed by using the segmented external force field. Such initialization and splitting produces multiple snakes, each of which is within the capture range associated to an object and evolved to the object boundary. The external force used in this paper is a gradient vector flow with an edge-preserving property (EPGVF), which can prevent the snakes from passing over weak boundaries. To segment the external force field, we represent it with a graph, and a graph-theory approach can be taken to determine the membership of each pixel. Experimental results establish the effectiveness of the proposed approach.

In this paper, we present a new variational formulation for geometric active contours that forces the level set function to be close to a signed distance function, and therefore completely eliminates the need of the costly re-initialization procedure. Our variational formulation consists of an internal energy term that penalizes the deviation of the level set function from a signed distance function, and an external energy term that drives the motion of the zero level set toward the desired image features, such as object boundaries. The resulting evolution of the level set function is the gradient flow that minimizes the overall energy functional. The proposed variational level set formulation has three main advantages over the traditional level set formulations. First, a significantly larger time step can be used for numerically solving the evolution partial differential equation, and therefore speeds up the curve evolution. Second, the level set function can be initialized with general functions that are more efficient to construct and easier to use in practice than the widely used signed distance function. Third, the level set evolution in our formulation can be easily implemented by simple finite difference scheme and is computationally more efficient. The proposed algorithm has been applied to both simulated and real images with promising results.

The classification of 3 common breast lesions, fibroadenomas, cysts, and cancers, was achieved using computerized image analysis of tumor shape in conjunction with patient age. The process involved the digitization of 69 mammographic images using a video camera and a commercial frame grabber on a PC-based computer system. An interactive segmentation procedure identified the tumor boundary using a thresholding technique which successfully segmented 57% of the lesions. Several features were chosen based on the gross and fine shape describing properties of the tumor boundaries as seen on the radiographs. Patient age was included as a significant feature in determining whether the tumor was a cyst, fibroadenoma, or cancer and was the only patient history information available for this study. The concept of a radial length measure provided a basis from which 6 of the 7 shape describing features were chosen, the seventh being tumor circularity. The feature selection process was accomplished using linear discriminant analysis and a Euclidean distance metric determined group membership. The effectiveness of the classification scheme was tested using both the apparent and the leaving-one-out test methods. The best results using the apparent test method resulted in correctly classifying 82% of the tumors segmented using the entire feature space and the highest classification rate using the leaving-one-out test method was 69% using a subset of the feature space. The results using only the shape descriptors, and excluding patient age resulted in correctly classifying 72% using the entire feature space (except age), and 51% using a subset of the feature space.