2,032 research outputs found

    Joint Image Reconstruction and Segmentation Using the Potts Model

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    We propose a new algorithmic approach to the non-smooth and non-convex Potts problem (also called piecewise-constant Mumford-Shah problem) for inverse imaging problems. We derive a suitable splitting into specific subproblems that can all be solved efficiently. Our method does not require a priori knowledge on the gray levels nor on the number of segments of the reconstruction. Further, it avoids anisotropic artifacts such as geometric staircasing. We demonstrate the suitability of our method for joint image reconstruction and segmentation. We focus on Radon data, where we in particular consider limited data situations. For instance, our method is able to recover all segments of the Shepp-Logan phantom from 77 angular views only. We illustrate the practical applicability on a real PET dataset. As further applications, we consider spherical Radon data as well as blurred data

    Entropy evaluation sheds light on ecosystem complexity

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    Preserving biodiversity and ecosystem stability is a challenge that can be pursued through modern statistical mechanics modeling. Here we introduce a variational maximum entropy-based algorithm to evaluate the entropy in a minimal ecosystem on a lattice in which two species struggle for survival. The method quantitatively reproduces the scale-free law of the prey shoals size, where the simpler mean-field approach fails: the direct near neighbor correlations are found to be the fundamental ingredient describing the system self-organized behavior. Furthermore, entropy allows the measurement of structural ordering, that is found to be a key ingredient in characterizing two different coexistence behaviors, one where predators form localized patches in a sea of preys and another where species display more complex patterns. The general nature of the introduced method paves the way for its application in many other systems of interest.Comment: 13 pages, 5 figure

    Fast multi-dimensional scattered data approximation with Neumann boundary conditions

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    An important problem in applications is the approximation of a function ff from a finite set of randomly scattered data f(xj)f(x_j). A common and powerful approach is to construct a trigonometric least squares approximation based on the set of exponentials {e2πikx}\{e^{2\pi i kx}\}. This leads to fast numerical algorithms, but suffers from disturbing boundary effects due to the underlying periodicity assumption on the data, an assumption that is rarely satisfied in practice. To overcome this drawback we impose Neumann boundary conditions on the data. This implies the use of cosine polynomials cos(πkx)\cos (\pi kx) as basis functions. We show that scattered data approximation using cosine polynomials leads to a least squares problem involving certain Toeplitz+Hankel matrices. We derive estimates on the condition number of these matrices. Unlike other Toeplitz+Hankel matrices, the Toeplitz+Hankel matrices arising in our context cannot be diagonalized by the discrete cosine transform, but they still allow a fast matrix-vector multiplication via DCT which gives rise to fast conjugate gradient type algorithms. We show how the results can be generalized to higher dimensions. Finally we demonstrate the performance of the proposed method by applying it to a two-dimensional geophysical scattered data problem

    Semantic 3D Reconstruction with Finite Element Bases

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    We propose a novel framework for the discretisation of multi-label problems on arbitrary, continuous domains. Our work bridges the gap between general FEM discretisations, and labeling problems that arise in a variety of computer vision tasks, including for instance those derived from the generalised Potts model. Starting from the popular formulation of labeling as a convex relaxation by functional lifting, we show that FEM discretisation is valid for the most general case, where the regulariser is anisotropic and non-metric. While our findings are generic and applicable to different vision problems, we demonstrate their practical implementation in the context of semantic 3D reconstruction, where such regularisers have proved particularly beneficial. The proposed FEM approach leads to a smaller memory footprint as well as faster computation, and it constitutes a very simple way to enable variable, adaptive resolution within the same model

    A Dynamically Diluted Alignment Model Reveals the Impact of Cell Turnover on the Plasticity of Tissue Polarity Patterns

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    The polarisation of cells and tissues is fundamental for tissue morphogenesis during biological development and regeneration. A deeper understanding of biological polarity pattern formation can be gained from the consideration of pattern reorganisation in response to an opposing instructive cue, which we here consider by example of experimentally inducible body axis inversions in planarian flatworms. Our dynamically diluted alignment model represents three processes: entrainment of cell polarity by a global signal, local cell-cell coupling aligning polarity among neighbours and cell turnover inserting initially unpolarised cells. We show that a persistent global orienting signal determines the final mean polarity orientation in this stochastic model. Combining numerical and analytical approaches, we find that neighbour coupling retards polarity pattern reorganisation, whereas cell turnover accelerates it. We derive a formula for an effective neighbour coupling strength integrating both effects and find that the time of polarity reorganisation depends linearly on this effective parameter and no abrupt transitions are observed. This allows to determine neighbour coupling strengths from experimental observations. Our model is related to a dynamic 88-Potts model with annealed site-dilution and makes testable predictions regarding the polarisation of dynamic systems, such as the planarian epithelium.Comment: Preprint as prior to first submission to Journal of the Royal Society Interface. 25 pages, 6 figures, plus supplement (18 pages, contains 1 table and 7 figures). A supplementary movie is available from https://dx.doi.org/10.6084/m9.figshare.c388781

    Development of Hybrid Deterministic-Statistical Models for Irradiation Influenced Microstructural Evolution.

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    Ion irradiation holds promise as a cost-effective approach to developing structured nano--porous and nano--fiberous semiconductors. Irradiation of certain semiconductors leads to the development of these structures, with exception of the much desired silicon. Hybrid deterministic-statistical models were developed to better understand the dominating mechanisms during structuring. This dissertation focuses on the application of hybrid models to two different radiation damage behavior: (1) precipitate evolution in a binary two-phase system and (2) void nucleation induced nano--porous structuring. Phenomenological equations defining the deterministic behavior were formulated by considering the expected kinetic and phenomenological behavior. The statistical component of the models is based on the Potts Monte Carlo (PMC) method. It has been demonstrated that hybrid models efficiently simulate microstructural evolution, while retaining the correct kinetics and physics. The main achievement was the development of computational methods to simulate radiation induced microstructural evolution and highlight which processes and materials properties could be essential for nano--structuring. Radiation influenced precipitate evolution was modeled by coupling a set of non-linear partial differential equations to the PMC model. The simulations considered the effects of dose rate and interfacial energy. Precipitate growth becomes retarded with increased damage due to diffusion of the radiation defects countering capillarity driven precipitate growth. The effects of grain boundaries (GB) as sinks was studied by simulating precipitate growth in an irradiated bi-crystalline matrix. Qualitative comparison to experimental results suggest that precipitate coverage of the GB is due to kinetic considerations and increased interfacial energy effects. Void nucleation induced nano--porous/fiberous structuring was modeled by coupling rate theory equations, kinetic Monte Carlo swelling algorithm and the PMC model. Point defect (PD) diffusivities were parameterized to study their influence on nano--structuring. The model showed that PD kinetic considerations are able to describe the formation of nano--porous structures. As defects diffuse faster, void nucleation becomes limited due to the fast removal of the defects. It was shown that as the diffusivities' ratio diverges from unity, the microstructures become statistically similar and uniform. Consequently, the computational results suggest that nano--pore structuring require interstitials that are much faster than the slow diffusing vacancies, which accumulate and cluster into voids.PhDNuclear Engineering and Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111424/1/efrainhr_1.pd

    A Replica Inference Approach to Unsupervised Multi-Scale Image Segmentation

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    We apply a replica inference based Potts model method to unsupervised image segmentation on multiple scales. This approach was inspired by the statistical mechanics problem of "community detection" and its phase diagram. Specifically, the problem is cast as identifying tightly bound clusters ("communities" or "solutes") against a background or "solvent". Within our multiresolution approach, we compute information theory based correlations among multiple solutions ("replicas") of the same graph over a range of resolutions. Significant multiresolution structures are identified by replica correlations as manifest in information theory overlaps. With the aid of these correlations as well as thermodynamic measures, the phase diagram of the corresponding Potts model is analyzed both at zero and finite temperatures. Optimal parameters corresponding to a sensible unsupervised segmentation correspond to the "easy phase" of the Potts model. Our algorithm is fast and shown to be at least as accurate as the best algorithms to date and to be especially suited to the detection of camouflaged images.Comment: 26 pages, 22 figure

    Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

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    This article is a mini-review about electrical current flows in networks from the perspective of statistical physics. We briefly discuss analytical methods to solve the conductance of an arbitrary resistor network. We then turn to basic results related to percolation: namely, the conduction properties of a large random resistor network as the fraction of resistors is varied. We focus on how the conductance of such a network vanishes as the percolation threshold is approached from above. We also discuss the more microscopic current distribution within each resistor of a large network. At the percolation threshold, this distribution is multifractal in that all moments of this distribution have independent scaling properties. We will discuss the meaning of multifractal scaling and its implications for current flows in networks, especially the largest current in the network. Finally, we discuss the relation between resistor networks and random walks and show how the classic phenomena of recurrence and transience of random walks are simply related to the conductance of a corresponding electrical network.Comment: 27 pages & 10 figures; review article for the Encyclopedia of Complexity and System Science (Springer Science

    Graph Spectral Image Processing

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    Recent advent of graph signal processing (GSP) has spurred intensive studies of signals that live naturally on irregular data kernels described by graphs (e.g., social networks, wireless sensor networks). Though a digital image contains pixels that reside on a regularly sampled 2D grid, if one can design an appropriate underlying graph connecting pixels with weights that reflect the image structure, then one can interpret the image (or image patch) as a signal on a graph, and apply GSP tools for processing and analysis of the signal in graph spectral domain. In this article, we overview recent graph spectral techniques in GSP specifically for image / video processing. The topics covered include image compression, image restoration, image filtering and image segmentation
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