Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity

A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAP-EM algorithm. A dual mathematical interpretation of the proposed framework with structured sparse estimation is described, which shows that the resulting piecewise linear estimate stabilizes the estimation when compared to traditional sparse inverse problem techniques. This interpretation also suggests an effective dictionary motivated initialization for the MAP-EM algorithm. We demonstrate that in a number of image inverse problems, including inpainting, zooming, and deblurring, the same algorithm produces either equal, often significantly better, or very small margin worse results than the best published ones, at a lower computational cost.Comment: 30 page

Quantum Local Quench, AdS/BCFT and Yo-Yo String

We propose a holographic model for local quench in 1+1 dimensional Conformal Field Theory (CFT). The local quench is produced by joining two identical CFT's on semi-infinite lines. When these theories have a zero boundary entropy, we use the AdS/Boundary CFT proposal to describe this process in terms of bulk physics. Boundaries of the original CFT's are extended in AdS as dynamical surfaces. In our holographic picture these surfaces detach from the boundary and form a closed folded string which can propagate in the bulk. The dynamics of this string is governed by the tensionless Yo-Yo string solution and its subsequent evolution determines the time dependence after quench. We use this model to calculate holographic Entanglement Entropy (EE) of an interval as a function of time. We propose how the falling string deforms Ryu-Takayanagi's curves. Using the deformed curves we calculate EE and find complete agreement with field theory results.Comment: 20 pages, 13 figures, discussion improved, Version to appear in JHE

On the remote sensing of oceanic and atmospheric convection in the Greenland Sea by synthetic aperture radar

In this paper we discuss characteristic properties of radar signatures of oceanic and atmospheric convection features in the Greenland Sea. If the water surface is clean (no surface films or ice coverage), oceanic and atmospheric features can become visible in radar images via a modulation of the surface roughness, and their radar signatures can be very similar. For an unambiguous interpretation and for the retrieval of quantitative information on current and wind variations from radar imagery with such signatures, theoretical models of current and wind phenomena and their radar imaging mechanisms must be utilized. We demonstrate this approach with the analysis of some synthetic aperture radar (SAR) images acquired by the satellites ERS-2 and RADARSAT-1. In once case, an ERS-2 SAR image an a RADARSAT-1 ScanSAR image exhibit pronounced cell-like signatures with length scales on the order of 10-20 km and modulation depths of about 5-6 dB and 9-10 dB, respectively. Simulations with a numerical SAR imagaing model and various input current and wind fields reveal that the signatures in both images can be expained consistently by wind variations on the order of±2.5 ms, but not by surface current variations on realistic orders of magnitude. Accordingly, the observed features must be atmospheric convection cells. This is confirmed by visible typical cloud patterns in a NOAA AVHRR image of the test scenario. In another case, the presence of an oceanic convective chimney is obvious from in situ data, but no signatures of it are visible in an ERS-2 SAR image. We show by numerical simulations with an oceanic convection model and our SAR imaging model that this is consistent with theoretical predictions, since the current gradients associated with the observed chimney are not sufficiently strong to give rise to significant signatures in an ERS-2 SAR image under the given conditions. Further model results indicate that it should be generally difficult to observe oceanic convection features in the Greenland Sea with ERS-2 or RADARSAT-1 SAR, since their signatures resulting from pure wave-current interaction will be too weak to become visible in the noisy SAR images in most cases. This situation will improve with the availability of future high-resolution SARs such as RADARSAT-2 SAR in fine resolution mode (2004) and TerraSAR-X (2005) which will offer significantly reduced speckle noise fluctuations at comparable spatial resolutions and thus a much better visibility of small image variations on spatial scales on the order of a few hundred meters

Steps and terraces at quasicrystal surfaces. Application of the 6d-polyhedral model to the analysis of STM images of i-AlPdMn

6-d polyhedral models give a periodic description of aperiodic quasicrystals. There are powerful tools to describe their structural surface properties. Basis of the model for icosahedral quasicrystals are given. This description is further used to interpret high resolution STM images of the surface of i-AlPdMn which surface preparation was followed by He diffraction. It is found that both terrace structure and step-terrace height profiles in STM images can be consistently interpreted by the described model

Networks for Nonlinear Diffusion Problems in Imaging

A multitude of imaging and vision tasks have seen recently a major transformation by deep learning methods and in particular by the application of convolutional neural networks. These methods achieve impressive results, even for applications where it is not apparent that convolutions are suited to capture the underlying physics. In this work we develop a network architecture based on nonlinear diffusion processes, named DiffNet. By design, we obtain a nonlinear network architecture that is well suited for diffusion related problems in imaging. Furthermore, the performed updates are explicit, by which we obtain better interpretability and generalisability compared to classical convolutional neural network architectures. The performance of DiffNet tested on the inverse problem of nonlinear diffusion with the Perona-Malik filter on the STL-10 image dataset. We obtain competitive results to the established U-Net architecture, with a fraction of parameters and necessary training data