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.