We propose a deep learning method for single image super-resolution (SR). Our method directly learns an end-to-end mapping between the low/high-resolution images. The mapping is represented as a deep convolutional neural network (CNN) [15] that takes the low-resolution image as the input and outputs the high-resolution one. We further show that traditional sparse-coding-based SR methods can also be viewed as a deep convolutional network. But unlike traditional methods that handle each component separately, our method jointly optimizes all layers. Our deep CNN has a lightweight structure, yet demonstrates state-of-the-art restoration quality, and achieves fast speed for practical on-line usage.

/pdf/learning-a-deep-convolutional-network-for-image-super-46oz0wm3qh.pdf

Learning a Deep Convolutional Network for Image Super-Resolution

We address the problem of image upscaling in the form of single image super-resolution based on a dictionary of low- and high-resolution exemplars. Two recently proposed methods, Anchored Neighborhood Regression (ANR) and Simple Functions (SF), provide state-of-the-art quality performance. Moreover, ANR is among the fastest known super-resolution methods. ANR learns sparse dictionaries and regressors anchored to the dictionary atoms. SF relies on clusters and corresponding learned functions. We propose A+, an improved variant of ANR, which combines the best qualities of ANR and SF. A+ builds on the features and anchored regressors from ANR but instead of learning the regressors on the dictionary it uses the full training material, similar to SF. We validate our method on standard images and compare with state-of-the-art methods. We obtain improved quality (i.e. 0.2–0.7 dB PSNR better than ANR) and excellent time complexity, rendering A+ the most efficient dictionary-based super-resolution method to date.

https://lirias.kuleuven.be/retrieve/333249

A+: Adjusted Anchored Neighborhood Regression for Fast Super-Resolution

Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

Progress in Mathematics

The concept of prior probability for signals plays a key role in the successful solution of many inverse problems. Much of the literature on this topic can be divided between analysis-based and synthesis-based priors. Analysis-based priors assign probability to a signal through various forward measurements of it, while synthesis-based priors seek a reconstruction of the signal as a combination of atom signals. The algebraic similarity between the two suggests that they could be strongly related; however, in the absence of a detailed study, contradicting approaches have emerged. While the computationally intensive synthesis approach is receiving ever-increasing attention and is notably preferred, other works hypothesize that the two might actually be much closer, going as far as to suggest that one can approximate the other. In this paper we describe the two prior classes in detail, focusing on the distinction between them. We show that although in the simpler complete and undercomplete formulations the two approaches are equivalent, in their overcomplete formulation they depart. Focusing on the l1 case, we present a novel approach for comparing the two types of priors based on high-dimensional polytopal geometry. We arrive at a series of theoretical and numerical results establishing the existence of an unbridgeable gap between the two.

/pdf/analysis-versus-synthesis-in-signal-priors-3y43v8aa6n.pdf

Analysis versus synthesis in signal priors

Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

This introductory textbook puts forth a clear and focused point of view on the differential geometry of curves and surfaces. Following the modern point of view on differential geometry, the book emphasizes the global aspects of the subject. The excellent collection of examples and exercises (with hints) will help students in learning the material. Advanced undergraduates and graduate students will find this a nice entry point to differential geometry. In order to study the global properties of curves and surfaces, it is necessary to have more sophisticated tools than are usually found in textbooks on the topic. In particular, students must have a firm grasp on certain topological theories. Indeed, this monograph treats the Gauss - Bonnet theorem and discusses the Euler characteristic. The authors also cover Alexandrov's theorem on embedded compact surfaces in R3 with constant mean curvature. The last chapter addresses the global geometry of curves, including periodic space curves and the four-vertices theorem for plane curves that are not necessarily convex. Besides being an introduction to the lively subject of curves and surfaces, this book can also be used as an entry to a wider study of differential geometry. It is suitable as the text for a first-year graduate course or an advanced undergraduate course.

Curves and surfaces

/pdf/real-hypersurfaces-of-a-complex-hyperbolic-space-3m47bvnw02.pdf

Real hypersurfaces of a complex hyperbolic space

Given a real hypersurface of a complex hyperbolic space #x2102;?Hn,we construct a principal circle bundle over it which is a Lorentzian hypersurface of the anti-De Sitter space H12n+1.Relations between the respective second fundamental forms are obtained permitting us to classify a remarkable family of real hypersurfaces of ℂHn.

On some real hypersurfaces of a complex hyperbolic space

Pour l'espace de De Sitter S 1 n+1 , dans le cas compact, on demontre que des hypersurfaces completes du genre espace de courbure moyenne constante H sont ombilicales

Sebastián Montiel

Papers

Curves and surfaces

Real hypersurfaces of a complex hyperbolic space

On some real hypersurfaces of a complex hyperbolic space

An integral inequality for compact spacelike hypersurfaces in De Sitter space and applications to the case of constant mean curvature

Unicity of constant mean curvature hypersurfaces in some riemannian manifolds