Statistical analysis of genetic changes within cell nuclei that are far from the primary tumor would help determine whether such changes have occurred prior to tumor invasion. To determine whether the gene amplification in cells is morphologically and/or genetically related to the primary tumor requires quantitative evaluation of a large number of cell nuclei from continuous meaningful structures such as milk-ducts, tumors, etc., located relatively far from the primary tumor. To address this issue, we have designed an integrated image analysis software system for high-throughput segmentation of nuclei. Filters such as Beltrami flow-based reaction-diffusion, directional diffusion, etc., were used to pre-process the images resulting in a better segmentation. The accurate shape of the segmented nucleus was recovered using an iterative "shrink-wrap" operation. The study of two cases of ductal carcinoma in situ in breast tissue supports the biological observation regarding the existence of a preferential intraductal invasion, and therefore a common origin, between the primary tumor and the gene amplification in the cell-nuclei lining the ductal structures in the breast.

In many instances, numerical integration of space-scale PDEs is the most time consuming operation of image processing. This is because the scale step is limited by conditional stability of explicit schemes. In this work, we introduce the unconditionally stable semi-implicit linearized difference scheme that is fashioned after additive operator split (AOS) [1], [2] for Beltrami and the subjective surface computation. The Beltrami flow [3], [4], [5] is one of the most effective denoising algorithms in image processing. For gray-level images, we show that the flow equation can be arranged in an advection-diffusion form, revealing the edge-enhancing properties of this flow. This also suggests the application of AOS method for faster convergence. The subjective surface [6] deals with constructing a perceptually meaningful interpretation from partial image data by mimicking the human visual system. However, initialization of the surface is critical for the final result and its main drawbacks are very slow convergence and the huge number of iterations required. In this paper, we first show that the governing equation for the subjective surface flow can be rearranged in an AOS implementation, providing a near real-time solution to the shape completion problem in 2D and 3D. Then, we devise a new initialization paradigm where we first "condition" the viewpoint surface using the Fast-Marching algorithm. We compare the original method with our new algorithm on several examples of real 3D medical images, thus revealing the improvement achieved.

In this paper, we present a shape recovery technique in 2D and 3D with specific applications in visualizing and measuring anatomical shapes from medical images. This algorithm models extremely corrugated structures like the brain, is topologically adaptable, is robust, and runs in O(N log N) time where N is the total number of points in the domain. Our two-stage technique is based on the level set shape recovery scheme and the fast marching method for computing solutions to static Hamilton-Jacobi equations.

We introduce a new geometrical framework based on which natural flows for image scale space and enhancement are presented. We consider intensity images as surfaces in the (x, I) space. The image is, thereby, a two-dimensional (2-D) surface in three-dimensional (3-D) space for gray-level images, and 2-D surfaces in five dimensions for color images. The new formulation unifies many classical schemes and algorithms via a simple scaling of the intensity contrast, and results in new and efficient schemes. Extensions to multidimensional signals become natural and lead to powerful denoising and scale space algorithms.

A general geometrical framework for image processing is presented. We consider intensity images as surfaces in the (x, I) space. The image is thereby a two dimensional surface in three dimensional space for gray level images. The new formulation unifies many classical schemes, algorithms, and measures via choices of parameters in "master" geometrical measure. More important, it is a simple and efficient tool for the design of natural schemes for image enhancement, segmentation, and scale space. Here we give the basic motivation and apply the scheme to enhance images. We present the concept of an image as a surface in dimensions higher than the three dimensional intuitive space. This will help us handle movies, color, and volumetric medical images.

We introduce a new geometrical framework for image processing. This framework finds a seamless link between the TV-L/sub 1/ and the L/sub 2/ norms that are often used in image processing, based on the geometry of the image and its interpretation as a surface. It unifies most of the current "scale space" models for images by a simple selection of one parameter, yet more important, it enables one to introduce new methods to deal with images in a simple and natural way. A functional called "Polyakov (1981) action", borrowed from high energy physics, is shown to be useful for image enhancement in color, texture, volumetric medical data, movies, and more.

Level set methods have been used in a variety of settings for problems in computer vision and image processing. A related numerical methodology, known as "fast marching methods", has been developed to solve static Hamilton-Jacobi equations extremely quickly; the techniques rely on conversion to a static problem, and are based on a marriage between narrow band techniques for level set methods and fast sorting algorithms. We show the application of these techniques to a collection of problems, including image denoising and enhancement schemes based on curvature-controlled diffusion with automatic stopping and hierarchical scales, extremely fast shape-from-shading schemes, and shape recovery in medical imaging. Level set methods have also been applied to problems in image denoising and enhancement through curvature-controlled diffusion schemes. An extension of these techniques, known as fast marching methods, has been developed to solve static Hamilton-Jacobi equations which arise in aspects of computer vision. We discuss advances in both of these techniques for such problems.

We present a unified approach to noise removal, image enhancement, and shape recovery in images. The underlying approach relies on the level set formulation of the curve and surface motion, which leads to a class of PDE-based algorithms. Beginning with an image, the first stage of this approach removes noise and enhances the image by evolving the image under flow controlled by min/max curvature and by the mean curvature. This stage is applicable to both salt-and-pepper grey-scale noise and full-image continuous noise present in black and white images, grey-scale images, texture images, and color images. The noise removal/enhancement schemes applied in this stage contain only one enhancement parameter, which in most cases is automatically chosen. The other key advantage of our approach is that a stopping criteria is automatically picked from the image; continued application of the scheme produces no further change. The second stage of our approach is the shape recovery of a desired object; we again exploit the level set approach to evolve an initial curve/surface toward the desired boundary, driven by an image-dependent speed function that automatically stops at the desired boundary.

Shape modeling is an important constituent of computer vision as well as computer graphics research. Shape models aid the tasks of object representation and recognition. This paper presents a new approach to shape modeling which retains some of the attractive features of existing methods and overcomes some of their limitations. The authors' techniques can be applied to model arbitrarily complex shapes, which include shapes with significant protrusions, and to situations where no a priori assumption about the object's topology is made. A single instance of the authors' model, when presented with an image having more than one object of interest, has the ability to split freely to represent each object. This method is based on the ideas developed by Osher and Sethian (1988) to model propagating solid/liquid interfaces with curvature-dependent speeds. The interface (front) is a closed, nonintersecting, hypersurface flowing along its gradient field with constant speed or a speed that depends on the curvature. It is moved by solving a "Hamilton-Jacobi" type equation written for a function in which the interface is a particular level set. A speed term synthesized from the image is used to stop the interface in the vicinity of object boundaries. The resulting equation of motion is solved by employing entropy-satisfying upwind finite difference schemes. The authors present a variety of ways of computing the evolving front, including narrow bands, reinitializations, and different stopping criteria. The efficacy of the scheme is demonstrated with numerical experiments on some synthesized images and some low contrast medical images.>

A technique for constructing a canonical surface parameterization in terms of lines of curvature is presented. Two methods of computing the canonical invariant representation are also presented. In the first method, a static instance of the controlled continuity spline is used for the stabilizer. Ways to modify it to reflect a change of parameters to the lines of curvature are described. In the second method, the dynamic instance of the controlled continuity spline called the deformable model is used. A force field defined in terms of the principal vectors is synthesized and applied to the parameter curves of the deformable model to coerce them along the lines of curvature. In essence, any transformation of parameters requires a modification of the stabilizer in the first method, whereas in the second method, it is tantamount to synthesizing a new force field. Experimental results with real and synthetic range data are included.>

A canonical intrinsic parameterization that provides a consistent, invariant form for describing surfaces is defined and constructed using an elastically deformable model. The salient features of this method are that it provides a unified and general framework for reparameterization of a surface and easily allows for incorporation of multiview data sets. The canonical parameterization of the surface is defined in terms of the surface lines of curvature. Depth constraints are first imposed as an external force field on the deformable model that molds itself to be consistent with the data. Principal vectors computed from this conformed model surface are then imposed as a force field on the parameter curves of the model. The parameter curves deform to become tangential to the principal vectors thereby yielding an invariant surface parameterized by the lines of curvature. Extension of the canonical parametric grid to multiple views is demonstrated by incorporating depth and curvature constraints from multiple views.>