A fundamental question for edge detection in noisy images is how faint can an edge be and still be detected. In this paper we offer a formalism to study this question and subsequently introduce computationally efficient multiscale edge detection algorithms designed to detect faint edges in noisy images. In our formalism we view edge detection as a search in a discrete, though potentially large, set of feasible curves. First, we derive approximate expressions for the detection threshold as a function of curve length and the complexity of the search space. We then present two edge detection algorithms, one for straight edges, and the second for curved ones. Both algorithms efficiently search for edges in a large set of candidates by hierarchically constructing difference filters that match the curves traced by the sought edges. We demonstrate the utility of our algorithms in both simulations and applications involving challenging real images. Finally, based on these principles, we develop an algorithm for fiber detection and enhancement. We exemplify its utility to reveal and enhance nerve axons in light microscopy images.

We present a bottom-up aggregation approach to image segmentation. Beginning with an image, we execute a sequence of steps in which pixels are gradually merged to produce larger and larger regions. In each step, we consider pairs of adjacent regions and provide a probability measure to assess whether or not they should be included in the same segment. Our probabilistic formulation takes into account intensity and texture distributions in a local area around each region. It further incorporates priors based on the geometry of the regions. Finally, posteriors based on intensity and texture cues are combined using “a mixture of experts” formulation. This probabilistic approach is integrated into a graph coarsening scheme, providing a complete hierarchical segmentation of the image. The algorithm complexity is linear in the number of the image pixels and it requires almost no user-tuned parameters. In addition, we provide a novel evaluation scheme for image segmentation algorithms, attempting to avoid human semantic considerations that are out of scope for segmentation algorithms. Using this novel evaluation scheme, we test our method and provide a comparison to several existing segmentation algorithms.

Multigrid solvers proved very efficient for solving massive systems of equations in various fields. These solvers are based on iterative relaxation schemes together with the approximation of the "smooth" error function on a coarser level (grid). We present two efficient multilevel eigensolvers for solving massive eigenvalue problems that emerge in data analysis tasks. The first solver, a version of classical algebraic multigrid (AMG), is applied to eigenproblems arising in clustering, image segmentation, and dimensionality reduction, demonstrating an order of magnitude speedup compared to the popular Lanczos algorithm. The second solver is based on a new, much more accurate interpolation scheme. It enables calculating a large number of eigenvectors very inexpensively.

We present a multigrid framework for the simulation of high-resolution elastic deformable models, designed to facilitate scalability on shared memory multiprocessors. We incorporate several state-of-the-art techniques from multigrid theory, while adapting them to the specific requirements of graphics and animation applications, such as the ability to handle elaborate geometry and complex boundary conditions. Our method supports simulation of linear elasticity and corotational linear elasticity. The efficiency of our solver is practically independent of material parameters, even for near-incompressible materials. We achieve simulation rates as high as 6 frames per second for test models with 256K vertices on an 8-core SMP, and 1.6 frames per second for a 2M vertex object on a 16-core SMP.

We present a parameter free approach that utilizes multiple cues for image segmentation. Beginning with an image, we execute a sequence of bottom-up aggregation steps in which pixels are gradually merged to produce larger and larger regions. In each step we consider pairs of adjacent regions and provide a probability measure to assess whether or not they should be included in the same segment. Our probabilistic formulation takes into account intensity and texture distributions in a local area around each region. It further incorporates priors based on the geometry of the regions. Finally, posteriors based on intensity and texture cues are combined using a mixture of experts formulation. This probabilistic approach is integrated into a graph coarsening scheme providing a complete hierarchical segmentation of the image. The algorithm complexity is linear in the number of the image pixels and it requires almost no user-tuned parameters. We test our method on a variety of gray scale images and compare our results to several existing segmentation algorithms.

We present an algorithm for edge detection suitable for both natural as well as noisy images. Our method is based on efficient multiscale utilization of elongated filters measuring the difference of oriented means of various lengths and orientations, along with a theoretical estimation of the effect of noise on the response of such filters. We use a scale adaptive threshold along with a recursive decision process to reveal the significant edges of all lengths and orientations and to localize them accurately even in low-contrast and very noisy images. We further use this algorithm for fiber detection and enhancement by utilizing stochastic completion-like process from both sides of a fiber. Our algorithm relies on an efficient multiscale algorithm for computing all "significantly different" oriented means in an image in O(N log rho), where N is the number of pixels, and p is the length of the longest structure of interest. Experimental results on both natural and noisy images are presented.

We present a novel multiscale approach that combines segmentation with classification to detect abnormal brain structures in medical imagery, and demonstrate its utility in detecting multiple sclerosis lesions in 3D MRI data. Our method uses segmentation to obtain a hierarchical decomposition of a multi-channel, anisotropic MRI scan. It then produces a rich set of features describing the segments in terms of intensity, shape, location, and neighborhood relations. These features are then fed into a decision tree-based classifier, trained with data labeled by experts, enabling the detection of lesions in all scales. Unlike common approaches that use voxel-by-voxel analysis, our system can utilize regional properties that are often important for characterizing abnormal brain structures. We provide experiments showing successful detections of lesions in both simulated and real MR images.

We present a novel approach that allows us to reliably compute many useful properties of a silhouette. Our approach assigns, for every internal point of the silhouette, a value reflecting the mean time required for a random walk beginning at the point to hit the boundaries. This function can be computed by solving Poisson's equation, with the silhouette contours providing boundary conditions. We show how this function can be used to reliably extract various shape properties including part structure and rough skeleton, local orientation and aspect ratio of different parts, and convex and concave sections of the boundaries. In addition to this, we discuss properties of the solution and show how to efficiently compute this solution using multigrid algorithms. We demonstrate the utility of the extracted properties by using them for shape classification and retrieval.

Texture segmentation is a difficult problem, as is apparent from camouflage pictures. A textured region can contain texture elements of various sizes, each of which can itself be textured. We approach this problem using a bottom-up aggregation framework that combines structural characteristics of texture elements with filter responses. Our process adaptively identifies the shape of texture elements and characterize them by their size, aspect ratio, orientation, brightness, etc., and then uses various statistics of these properties to distinguish between different textures. At the same time our process uses the statistics of filter responses to characterize textures. In our process the shape measures and the filter responses crosstalk extensively. In addition, a top-down cleaning process is applied to avoid mixing the statistics of neighboring segments. We tested our algorithm on real images and demonstrate that it can accurately segment regions that contain challenging textures.

Image segmentation is difficult because objects may differ from their background by any of a variety of properties that can be observed in some, but often not all scales. A further complication is that coarse measurements, applied to the image for detecting these properties, often average over properties of neighboring segments, making it difficult to separate the segments and to reliably detect their boundaries. Below we present a method for segmentation that generates and combines multiscale measurements of intensity contrast, texture differences, and boundary integrity. The method is based on our former algorithm SWA, which efficiently detects segments that optimize a normalized-cut like measure by recursively coarsening a graph reflecting similarities between intensities of neighboring pixels. In this process aggregates of pixels of increasing size are gradually collected to form segments. We intervene in this process by computing properties of the aggregates and modifying the graph to reflect these coarse scale measurements. This allows us to detect regions that differ by fine as well as coarse properties, and to accurately locate their boundaries. Furthermore, by combining intensity differences with measures of boundary integrity across neighboring aggregates we can detect regions separated by weak, yet consistent edges.

We introduce a fast, multiscale algorithm for image segmentation. Our algorithm uses modern numeric techniques to find an approximate solution to normalized cut measures in time that is linear in the size of the image with only a few dozen operations per pixel. In just one pass the algorithm provides a complete hierarchical decomposition of the image into segments. The algorithm detects the segments by applying a process of recursive coarsening in which the same minimization problem is represented with fewer and fewer variables producing an irregular pyramid. During this coarsening process we may compute additional internal statistics of the emerging segments and use these statistics to facilitate the segmentation process. Once the pyramid is completed it is scanned from the top down to associate pixels close to the boundaries of segments with the appropriate segment. The algorithm is inspired by algebraic multigrid (AMG) solvers of minimization problems of heat or electric networks. We demonstrate the algorithm by applying it to real images.

The detection of smooth curves in images and their completion over gaps are two important problems in perceptual grouping. We examine the notion of completion energy of curve elements, showing, and exploiting its intrinsic dependence on length and width scales. We introduce a fast method for computing the most likely completion between two elements, by developing novel analytic approximations and a fast numerical procedure for computing the curve of least energy. We then use our newly developed energies to find the most likely completions in images through a generalized summation of induction fields. This is done through multiscale procedures, i.e., separate processing at different scales with some interscale interactions. Such procedures allow the summation of all induction fields to be done in a total of only O(N log N) operations, where N is the number of pixels in the image. More important, such procedures yield a more realistic dependence of the induction field on the length and width scales: the field of a long element is very different from the sum of the fields of its composing short segments.

The detection of smooth curves in images and their completion over gaps are two important problems in perceptual grouping. In this paper we examine the notion of completion energy and introduce a fast method to compute the most likely completions in images. Specifically we develop two novel analytic approximations to the curve of least energy. In addition, we introduce a fast numerical method to compute the curve of least energy, and show that our approximations are obtained at early stages of this numerical computation. We then use our newly developed energies to find the most likely completions in images through a generalized summation of induction fields. Since in practice edge elements are obtained by applying filters of certain widths and lengths to the image, we adjust our computation to take these parameters into account. Finally, we show that, due to the smoothness of the kernel of summation, the process of summing induction fields can be run in time that is linear in the number of different edge elements in the image, or in O(N log N) where N is the number of pixels in the image, using multigrid methods.