Monte Carlo renormalization-group calculation for the d=3 Ising model using a modified transformation.

We present a simple approach to high-accuracy calculations of critical properties for the three-dimensional Ising model, without prior knowledge of the critical temperature. The iterative method uses a modified block-spin transformation with a tunable parameter to improve convergence in the Monte Carlo renormalization group trajectory. We found experimentally that the iterative method enables the calculation of the critical temperature simultaneously with a critical exponent.

On Detection of Faint Edges in Noisy Images.

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.

Surprising convergence of the Monte Carlo renormalization group for the three-dimensional Ising model.

We present a surprisingly simple approach to high-accuracy calculations of the critical properties of the three-dimensional Ising model. The method uses a modified block-spin transformation with a tunable parameter to improve convergence in the Monte Carlo renormalization group. The block-spin parameter must be tuned differently for different exponents to produce optimal convergence.

Locally Refined Multigrid Solution of the All-Electron Kohn-Sham Equation.

We present a fully numerical multigrid approach for solving the all-electron Kohn-Sham equation in molecules. The equation is represented on a hierarchy of Cartesian grids, from coarse ones that span the entire molecule to very fine ones that describe only a small volume around each atom. This approach is adaptable to any type of geometry. We demonstrate it for a variety of small molecules and obtain high accuracy agreement with results obtained previously for diatomic molecules using a prolate-spheroidal grid. We provide a detailed presentation of the numerical methodology and discuss possible extensions of this approach.

Image segmentation by probabilistic bottom-up aggregation and cue integration.

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.

Efficient multilevel eigensolvers with applications to data analysis tasks.

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.

Automatic segmentation and classification of multiple sclerosis in multichannel MRI.

We introduce a multiscale approach that combines segmentation with classification to detect abnormal brain structures in medical imagery, and demonstrate its utility in automatically detecting multiple sclerosis (MS) lesions in 3-D multichannel magnetic resonance (MR) images. Our method uses segmentation to obtain a hierarchical decomposition of a multichannel, anisotropic MR scans. It then produces a rich set of features describing the segments in terms of intensity, shape, location, neighborhood relations, and anatomical context. These features are then fed into a decision forest classifier, trained with data labeled by experts, enabling the detection of lesions at 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 on two types of real MR images: a multichannel proton-density-, T2-, and T1-weighted dataset of 25 MS patients and a single-channel fluid attenuated inversion recovery (FLAIR) dataset of 16 MS patients. Comparing our results with lesion delineation by a human expert and with previously extensively validated results shows the promise of the approach.

Prior knowledge driven multiscale segmentation of brain MRI.

We present a novel automatic multiscale algorithm applied to segmentation of anatomical structures in brain MRI. The algorithm which is derived from algebraic multigrid, uses a graph representation of the image and performs a coarsening process that produces a full hierarchy of segments. Our main contribution is the incorporation of prior knowledge information into the multiscale framework through a Bayesian formulation. The probabilistic information is based on an atlas prior and on a likelihood function estimated from a manually labeled training set. The significance of our new approach is that the constructed pyramid, reflects the prior knowledge formulated. This leads to an accurate and efficient methodology for detection of various anatomical structures simultaneously. Quantitative validation results on gold standard MRI show the benefit of our approach.

Shape representation and classification using the poisson equation.

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.

Geometry of intensive scalar dissipation events in turbulence.

The maxima of the scalar dissipation rate in turbulence appear in the form of sheets and correspond to the potentially most intensive scalar mixing events. Their cross section extension determines a locally varying diffusion scale of the mixing process and extends the classical Batchelor picture of one mean diffusion scale. The distribution of the local diffusion scales is analyzed for different Reynolds and Schmidt numbers with a fast multiscale technique applied to very high-resolution simulation data. The scales always take values across the whole Batchelor range and beyond. Furthermore, their distribution is traced back to the distribution of the contractive short-time Lyapunov exponent of the flow.

Hierarchy and adaptivity in segmenting visual scenes.

Finding salient, coherent regions in images is the basis for many visual tasks, and is especially important for object recognition. Human observers perform this task with ease, relying on a system in which hierarchical processing seems to have a critical role. Despite many attempts, computerized algorithms have so far not demonstrated robust segmentation capabilities under general viewing conditions. Here we describe a new, highly efficient approach that determines all salient regions of an image and builds them into a hierarchical structure. Our algorithm, segmentation by weighted aggregation, is derived from algebraic multigrid solvers for physical systems, and consists of fine-to-coarse pixel aggregation. Aggregates of various sizes, which may or may not overlap, are revealed as salient, without predetermining their number or scale. Results using this algorithm are markedly more accurate and significantly faster (linear in data size) than previous approaches.

Atlas guided identification of brain structures by combining 3D segmentation and SVM classification.

This study presents a novel automatic approach for the identification of anatomical brain structures in magnetic resonance images (MRI). The method combines a fast multiscale multi-channel three dimensional (3D) segmentation algorithm providing a rich feature vocabulary together with a support vector machine (SVM) based classifier. The segmentation produces a full hierarchy of segments, expressed by an irregular pyramid with only linear time complexity. The pyramid provides a rich, adaptive representation of the image, enabling detection of various anatomical structures at different scales. A key aspect of the approach is the thorough set of multiscale measures employed throughout the segmentation process which are also provided at its end for clinical analysis. These features include in particular the prior probability knowledge of anatomic structures due to the use of an MRI probabilistic atlas. An SVM classifier is trained based on this set of features to identify the brain structures. We validated the approach using a gold standard real brain MRI data set. Comparison of the results with existing algorithms displays the promise of our approach.

Inverse Monte Carlo renormalization group transformations for critical phenomena.

We introduce a computationally stable inverse Monte Carlo renormalization group transformation method that provides a number of advantages for the calculation of critical properties. We are able to simulate the fixed point of a renormalization group for arbitrarily large lattices without critical slowing down. The log-log scaling plots obtained with this method show remarkable linearity, leading to accurate estimates for critical exponents. We illustrate this method with calculations in two- and three-dimensional Ising models for a variety of renormalization group transformations.