A novel method is proposed for performing multilabel, interactive image segmentation. Given a small number of pixels with user-defined (or predefined) labels, one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach one of the prelabeled pixels. By assigning each pixel to the label for which the greatest probability is calculated, a high-quality image segmentation may be obtained. Theoretical properties of this algorithm are developed along with the corresponding connections to discrete potential theory and electrical circuits. This algorithm is formulated in discrete space (i.e., on a graph) using combinatorial analogues of standard operators and principles from continuous potential theory, allowing it to be applied in arbitrary dimension on arbitrary graphs

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Random Walks for Image Segmentation

This paper describes two digital implementations of a new mathematical transform, namely, the second generation curvelet transform in two and three dimensions. The first digital transformation is based on unequally spaced fast Fourier transforms, while the second is based on the wrapping of specially selected Fourier samples. The two implementations essentially differ by the choice of spatial grid used to translate curvelets at each scale and angle. Both digital transformations return a table of digital curvelet coefficients indexed by a scale parameter, an orientation parameter, and a spatial location parameter. And both implementations are fast in the sense that they run in O(n^2 log n) flops for n by n Cartesian arrays; in addition, they are also invertible, with rapid inversion algorithms of about the same complexity. Our digital transformations improve upon earlier implementations—based upon the first generation of curvelets—in the sense that they are conceptually simpler, faster, and far less redundant. The software CurveLab, which implements both transforms presented in this paper, is available at http://www.curvelet.org.

/pdf/fast-discrete-curvelet-transforms-1z0gmpoaao.pdf

Fast Discrete Curvelet Transforms

Distance or similarity measures are essential to solve many pattern recognition problems such as classification, clustering, and retrieval problems. Various distance/similarity measures that are applicable to compare two probability density functions, pdf in short, are reviewed and categorized in both syntactic and semantic relationships. A correlation coefficient and a hierarchical clustering technique are adopted to reveal similarities among numerous distance/similarity measures.

Comprehensive Survey on Distance/Similarity Measures between Probability Density Functions

An imaging technique able to resolve and identify all individual atoms in non-periodic solids would be a very useful tool for materials analysis. Annular dark-field (ADF) imaging in an aberration-corrected scanning transmission electron microscope optimized for low voltage operation allows such an analysis, as shown by Ondrej Krivanek and co-workers. The technique was used to examine a monolayer of boron nitride, in which it revealed individual atomic substitutions involving carbon and oxygen impurity atoms. Careful analysis of the data enables the construction of a detailed map of the atomic structure, with all the atoms of the four species resolved and identified. An imaging technique that could identify all the individual atoms, including defects, in a material would be a useful tool. Here an electron-microscopy approach to the problem, based on annular dark-field imaging, is described. A monolayer of boron nitride was studied, and three types of atomic substitution were identified. Careful analysis of the data enabled the construction of a detailed map of the atomic structure. Direct imaging and chemical identification of all the atoms in a material with unknown three-dimensional structure would constitute a very powerful general analysis tool. Transmission electron microscopy should in principle be able to fulfil this role, as many scientists including Feynman realized early on1. It images matter with electrons that scatter strongly from individual atoms and whose wavelengths are about 50 times smaller than an atom. Recently the technique has advanced greatly owing to the introduction of aberration-corrected optics2,3,4,5,6,7,8. However, neither electron microscopy nor any other experimental technique has yet been able to resolve and identify all the atoms in a non-periodic material consisting of several atomic species. Here we show that annular dark-field imaging in an aberration-corrected scanning transmission electron microscope optimized for low voltage operation can resolve and identify the chemical type of every atom in monolayer hexagonal boron nitride that contains substitutional defects. Three types of atomic substitutions were found and identified: carbon substituting for boron, carbon substituting for nitrogen, and oxygen substituting for nitrogen. The substitutions caused in-plane distortions in the boron nitride monolayer of about 0.1 A magnitude, which were directly resolved, and verified by density functional theory calculations. The results demonstrate that atom-by-atom structural and chemical analysis of all radiation-damage-resistant atoms present in, and on top of, ultra-thin sheets has now become possible.

Atom-by-atom structural and chemical analysis by annular dark-field electron microscopy

The minimum lscr1-norm solution to an underdetermined system of linear equations y=Ax is often, remarkably, also the sparsest solution to that system. This sparsity-seeking property is of interest in signal processing and information transmission. However, general-purpose optimizers are much too slow for lscr1 minimization in many large-scale applications.In this paper, the Homotopy method, originally proposed by Osborne et al. and Efron et al., is applied to the underdetermined lscr1-minimization problem min parxpar1 subject to y=Ax. Homotopy is shown to run much more rapidly than general-purpose LP solvers when sufficient sparsity is present. Indeed, the method often has the following k-step solution property: if the underlying solution has only k nonzeros, the Homotopy method reaches that solution in only k iterative steps. This k-step solution property is demonstrated for several ensembles of matrices, including incoherent matrices, uniform spherical matrices, and partial orthogonal matrices. These results imply that Homotopy may be used to rapidly decode error-correcting codes in a stylized communication system with a computational budget constraint. The approach also sheds light on the evident parallelism in results on lscr1 minimization and orthogonal matching pursuit (OMP), and aids in explaining the inherent relations between Homotopy, least angle regression (LARS), OMP, and polytope faces pursuit.

/pdf/fast-solution-of-ell-1-norm-minimization-problems-when-the-36p60aek8y.pdf

Fast Solution of $\ell _{1}$ -Norm Minimization Problems When the Solution May Be Sparse

Advances in Imaging and Electron Physics merges two long-running serials, Advances in Electronics and Electron Physics and Advances in Optical and Electron Microscopy. The series features extended articles on the physics of electron devices (especially semiconductor devices), particle optics at high and low energies, microlithography, image science, and digital image processing, electromagnetic wave propagation, electron microscopy, and the computing methods used in all these domains. * Contains contributions from leading authorities on the subject matter* Informs and updates on all the latest developments in the field of imaging and electron physics* Provides practitioners interested in microscopy, optics, image processing, mathematical morphology, electromagnetic fields, electron, and ion emission with a valuable resource* Features extended articles on the physics of electron devices (especially semiconductor devices), particle optics at high and low energies, microlithography, image science, and digital image processing

Peter W. Hawkes

Papers

Advances in Imaging and Electron Physics