Here we study the extreme visual recovery problem, in which over 90 of pixel values in a given image are missing. Existing low rank-based algorithms are only effective for recovering data with at most 90 missing values. Thus, we exploit visual data's smoothness property to help solve this challenging extreme visual recovery problem. Based on the Discrete Cosine Transformation (DCT), we propose a novel DCT norm that involves all pixels and produces smooth estimations in any view. Our theoretical analysis shows that the total variation (TV) norm, which only achieves local smoothness, is a special case of the proposed DCT norm. We also develop a new visual recovery algorithm by minimizing the DCT and nuclear norms to achieve a more visually pleasing estimation. Experimental results on a benchmark image dataset demonstrate that the proposed approach is superior to state-of-the-art methods in terms of peak signal-to-noise ratio and structural similarity.
Since the @math is isotropic and not differentiable, an anisotropic version is proposed in @cite_9 that is easier to minimize:
@cite_9
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In this paper, we consider the issue of localization in anisotropic sensor networks. Anisotropic networks are differentiated from isotropic networks in that they possess properties that vary according to the direction of measurement. Anisotropic characteristics result from various factors such as the geographic shape of the region (non-convex region), the different node densities, the irregular radio patterns, and the anisotropic terrain conditions. In order to characterize anisotropic features, we devise a linear mapping method that transforms proximity measurements between sensor nodes into a geographic distance embedding space by using the truncated singular value decomposition-based (TSVD-based) pseudo-inverse technique. This transformation retains as much topological information as possible and reduces the effect of measurement noises on the estimates of geographic distances. We show via simulation that the proposed localization method outperforms DV-hop, DV-distance (D. Niculescu, 2001), and MDS-map (Y. , 2003), and makes robust and accurate estimates of sensor locations in both isotropic and anisotropic sensor networks. The Ginzburg-Landau functional is a phase transition model which is suitable for clustering or classification type problems. We study the asymptotics of a sequence of Ginzburg-Landau functionals with anisotropic interaction potentials on point clouds @math where @math denotes the number data points. In particular we show the limiting problem, in the sense of @math -convergence, is related to the total variation norm restricted to functions taking binary values; which can be understood as a surface energy. We generalize the result known for isotropic interaction potentials to the anisotropic case and add a result concerning the rate of convergence. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . We propose, analyze, and test an alternating minimization algorithm for recovering images from blurry and noisy observations with total variation (TV) regularization. This algorithm arises from a new half-quadratic model applicable to not only the anisotropic but also the isotropic forms of TV discretizations. The per-iteration computational complexity of the algorithm is three fast Fourier transforms. We establish strong convergence properties for the algorithm including finite convergence for some variables and relatively fast exponential (or @math -linear in optimization terminology) convergence for the others. Furthermore, we propose a continuation scheme to accelerate the practical convergence of the algorithm. Extensive numerical results show that our algorithm performs favorably in comparison to several state-of-the-art algorithms. In particular, it runs orders of magnitude faster than the lagged diffusivity algorithm for TV-based deblurring. Some extensions of our algorithm are also discussed. Convex and continuous energy formulations for low level vision problems enable efficient search procedures for the corresponding globally optimal solutions. In this work we extend the well-established continuous, isotropic capacity-based maximal flow framework to the anisotropic setting. By using powerful results from convex analysis, a very simple and efficient minimization procedure is derived. Further, we show that many important properties carry over to the new anisotropic framework, e.g. globally optimal binary results can be achieved simply by thresholding the continuous solution. In addition, we unify the anisotropic continuous maximal flow approach with a recently proposed convex and continuous formulation for Markov random fields, thereby allowing more general smoothness priors to be incorporated. Dense stereo results are included to illustrate the capabilities of the proposed approach. In this paper, we propose a generic framework for 3D surface remeshing. Based on a metric-driven Discrete Voronoi Diagram construction, our output is an optimized 3D triangular mesh with a user-defined vertex budget. Our approach can deal with a wide range of applications, from high-quality mesh generation to shape approximation. By using appropriate metric constraints, the method generates isotropic or anisotropic elements. Based on point sampling, our algorithm combines the robustness and theoretical strength of Delaunay criteria with the efficiency of an entirely discrete geometry processing. Besides the general described framework, we show the experimental results using isotropic, quadric-enhanced isotropic, and anisotropic metrics, which prove the efficiency of our method on large meshes at a low computational cost. This paper discusses a generalization of the function transformation scheme used in Coleman, Shalloway, and Wu [Comput. Optim. Appl., 2 (1993), pp. 145--170; J. Global Optim., 4 (1994), pp. 171--185] and Shalloway [Global Optimization, C. Floudas and P. Pardalos, eds., Princeton University Press, 1992, pp. 433--477; Global Optim., 2 (1992), pp. 281--311] for global energy minimization applied to the molecular conformation problem. A mathematical theory for the method as a special continuation approach to global optimization is established. We show that the method can transform a nonlinear objective function into a class of gradually deformed, but smoother'' or easier'' functions. An optimization procedure can then be successively applied to the new functions to trace their solutions back to the original function. Two types of transformation are defined: isotropic and anisotropic. We show that both transformations can be applied to a large class of nonlinear partially separable functions, including energy functions for molecular conformation. Methods to compute the transformation for these functions are given. The Green’s function of the time dependent radiative transfer equation for the semi-infinite medium is derived for the first time by a heuristic approach based on the extrapolated boundary condition and on an almost exact solution for the infinite medium. Monte Carlo simulations performed both in the simple case of isotropic scattering and of an isotropic point-like source, and in the more realistic case of anisotropic scattering and pencil beam source, are used to validate the heuristic Green’s function. Except for the very early times, the proposed solution has an excellent accuracy (>98 for the isotropic case, and >97 for the anisotropic case) significantly better than the diffusion equation. The use of this solution could be extremely useful in the biomedical optics field where it can be directly employed in conditions where the use of the diffusion equation is limited, e.g. small volume samples, high absorption and or low scattering media, short source-receiver distances and early times. Also it represents a first step to derive tools for other geometries (e.g. slab and slab with inhomogeneities inside) of practical interest for noninvasive spectroscopy and diffuse optical imaging. Moreover the proposed solution can be useful to several research fields where the study of a transport process is fundamental. In this study, Green’s function of the two-dimensional radiative transfer equation is derived for an infinitely extended anisotropically scattering medium, which is illuminated by a unidirectional source distribution. In the steady-state domain, the final results, which are based on eigenvalues and eigenvectors, are given analytically apart from the eigenvalues. For the time-dependent case an additional numerical inverse Fourier transform is required. The obtained solutions were successfully validated with another exact analytical solution in the time domain for isotropically scattering and with the Monte Carlo method for anisotropically scattering media. An inverse finite element method was developed to determine the anisotropic properties of bovine cornea from an in vitro inflation experiment. The experiment used digital image correlation (DIC) to measure the three-dimensional surface geometry and displacement field of the cornea at multiple pressures. A finite element model of a bovine cornea was developed using the DIC measured surface geometry of the undeformed specimen. The model was applied to determine five parameters of an anisotropic hyperelastic model that minimized the error between the measured and computed surface displacement field and to investigate the sensitivity of the measured bovine inflation response to variations in the anisotropic properties of the cornea. The results of the parameter optimization revealed that the collagen structure of bovine cornea exhibited a high degree of anisotropy in the limbus region, which agreed with recent histological findings, and a transversely isotropic central region. The parameter study showed that the bovine corneal response to the inflation experiment was sensitive to the shear modulus of the matrix at pressures below the intraocular pressure, the properties of the collagen lamella at higher pressures, and the degree of anisotropy in the limbus region. It was not sensitive to a weak collagen anisotropy in the central region. Purpose – The purpose of this paper is to suggest an alternative to the widely used Times Cited criterion for analysing citation networks. The approach involves taking account of the natures of the papers that cite a given paper, so as to differentiate between papers that attract the same number of citations.Design methodology approach – ArticleRank is an algorithm that has been derived from Google's PageRank algorithm to measure the influence of journal articles. ArticleRank is applied to two datasets – a citation network based on an early paper on webometrics, and a self‐citation network based on the 19 most cited papers in the Journal of Documentation – using citation data taken from the Web of Knowledge database.Findings – ArticleRank values provide a different ranking of a set of papers from that provided by the corresponding Times Cited values, and overcomes the inability of the latter to differentiate between papers with the same numbers of citations. The difference in rankings between Times Cited ... Theoretical Foundations of Anisotropic Diffusion in Image Processing. A frequent problem in low-level vision consists of eliminating noise and small-scale details from an image while still preserving or even enhancing the edge structure. Nonlinear anisotropic diffusion filtering may be one possibility to achieve these goals. The objective of the present paper is to review the author’s results on a scale-space interpretation of a class of diffusion filters which comprises also several nonlinear anisotropic models. It is demonstrated that these models—which use an adapted diffusion tensor instead of a scalar diffusivity—offer advantages over isotropic filters. Most of the restoration and scale-space properties carry over from the continuous to the discrete case. Applications are presented ranging from preprocessing of medical images and postprocessing of fluctuating numerical data to visualizing quality relevant features for the grading of wood surfaces and fabrics. Sweat pores on fingerprints have proven to be discriminative features and have recently been successfully employed in automatic fingerprint recognition systems (AFRS), where the extraction of fingerprint pores is a critical step. Most of the existing pore extraction methods detect pores by using a static isotropic pore model; however, their detection accuracy is not satisfactory due to the limited approximation capability of static isotropic models to various types of pores. This paper presents a dynamic anisotropic pore model to describe pores more accurately by using orientation and scale parameters. An adaptive pore extraction method is then developed based on the proposed dynamic anisotropic pore model. The fingerprint image is first partitioned into well-defined, ill-posed, and background blocks. According to the dominant ridge orientation and frequency on each foreground block, a local instantiation of appropriate pore model is obtained. Finally, the pores are extracted by filtering the block with the adaptively generated pore model. Extensive experiments are performed on the high resolution fingerprint databases we established. The results demonstrate that the proposed method can detect pores more accurately and robustly, and consequently improve the fingerprint recognition accuracy of pore-based AFRS. A class of fourth-order partial differential equations (PDEs) are proposed to optimize the trade-off between noise removal and edge preservation. The time evolution of these PDEs seeks to minimize a cost functional which is an increasing function of the absolute value of the Laplacian of the image intensity function. Since the Laplacian of an image at a pixel is zero if the image is planar in its neighborhood, these PDEs attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image. Piecewise planar images look more natural than step images which anisotropic diffusion (second order PDEs) uses to approximate an observed image. So the proposed PDEs are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion, while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion. Although both approaches seem to be comparable in removing speckles in the observed images, speckles are more visible in images processed by the proposed PDEs, because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges. Speckles can be easily removed by simple algorithms such as the one presented in this paper. This paper describes a partial differential equation for denoising images. The proposed method is demonstrably superior to anisotropic diffusion (and its many variations) for denoising images that are approximately piecewise constant. The method relies on an equation that is the level-set equivalent of the anisotropic diffusion equation proposed by Perona and Malik (1990). This proposed equation has come up in the literature, but has failed to be fully utilized due to a lack of analysis and the need for a stable, accurate numerical implementation. Our analysis shows that the proposed method is more aggressive than anisotropic diffusion at enhancing and preserving edges, and is less sensitive to the edge contrast parameter. Empirical results confirm these advantages, and show that for certain classes of images, one should always prefer the proposed method over anisotropic diffusion. We present an algorithm for denoising triangulated models based on L0 minimization. Our method maximizes the flat regions of the model and gradually removes noise while preserving sharp features. As part of this process, we build a discrete differential operator for arbitrary triangle meshes that is robust with respect to degenerate triangulations. We compare our method versus other anisotropic denoising algorithms and demonstrate that our method is more robust and produces good results even in the presence of high noise. Efficient path-planning algorithms are a crucial issue for modern autonomous underwater vehicles. Classical path-planning algorithms in artificial intelligence are not designed to deal with wide continuous environments prone to currents. We present a novel Fast Marching (FM)-based approach to address the following issues. First, we develop an algorithm we call FM* to efficiently extract a 2-D continuous path from a discrete representation of the environment. Second, we take underwater currents into account thanks to an anisotropic extension of the original FM algorithm. Third, the vehicle turning radius is introduced as a constraint on the optimal path curvature for both isotropic and anisotropic media. Finally, a multiresolution method is introduced to speed up the overall path-planning process It is not uncommon in the data anonymization literature to oppose the "old" @math k -anonymity model to the "new" differential privacy model, which offers more robust privacy guarantees. Yet, it is often disregarded that the utility of the anonymized results provided by differential privacy is quite limited, due to the amount of noise that needs to be added to the output, or because utility can only be guaranteed for a restricted type of queries. This is in contrast with @math k -anonymity mechanisms, which make no assumptions on the uses of anonymized data while focusing on preserving data utility from a general perspective. In this paper, we show that a synergy between differential privacy and @math k -anonymity can be found: @math k -anonymity can help improving the utility of differentially private responses to arbitrary queries. We devote special attention to the utility improvement of differentially private published data sets. Specifically, we show that the amount of noise required to fulfill @math ? -differential privacy can be reduced if noise is added to a @math k -anonymous version of the data set, where @math k -anonymity is reached through a specially designed microaggregation of all attributes. As a result of noise reduction, the general analytical utility of the anonymized output is increased. The theoretical benefits of our proposal are illustrated in a practical setting with an empirical evaluation on three data sets. In this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a Poisson process. We show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion. In general the resulting diffusion tensor is anisotropic, and we give necessary and sufficient conditions under which it is isotropic. We also give a method to construct approximations of arbitrary high order for large times. In a second paper (Part II) we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion. Depending on the strength of the bias, it may lead to an anisotropic diffusion equation, to a drift term in the flux, or to both. Our analysis generalizes and simplifies previous derivations that lead to the classical Patlak--Keller--Segel--Alt... Spectral methods have recently gained popularity in many domains of computer graphics and geometry processing, especially shape processing, computation of shape descriptors, distances, and correspondence. Spectral geometric structures are intrinsic and thus invariant to isometric deformations, are efficiently computed, and can be constructed on shapes in different representations. A notable drawback of these constructions, however, is that they are isotropic, i.e., insensitive to direction. In this paper, we show how to construct direction-sensitive spectral feature descriptors using anisotropic diffusion on meshes and point clouds. The core of our construction are directed local kernels acting similarly to steerable filters, which are learned in a task-specific manner. Remarkably, while being intrinsic, our descriptors allow to disambiguate reflection symmetries. We show the application of anisotropic descriptors for problems of shape correspondence on meshes and point clouds, achieving results significantly better than state-of-the-art methods.