We propose a variational approach to obtain superresolution images from multiple low-resolution frames extracted from video clips. First the displacement between the lowresolution frames and the reference frame are computed by an optical flow algorithm. Then a low-rank model is used to construct the reference frame in high-resolution by incorporating the information of the low-resolution frames. The model has two terms: a 2-norm data fidelity term and a nuclear-norm regularization term. Alternating direction method of multipliers is used to solve the model. Comparison of our methods with other models on synthetic and real video clips show that our resulting images are more accurate with less artifacts. It also provides much finer and discernable details.
We propose to combine cepstrum and nonlinear time–frequency (TF) analysis
to study multiple component oscillatory signals with time-varying frequency and
amplitude and with time-varying non-sinusoidal oscillatory pattern. The concept of
cepstrum is applied to eliminate the wave-shape function influence on the TF analysis,
and we propose a new algorithm, named de-shape synchrosqueezing transform (deshape
SST). The mathematical model, adaptive non-harmonic model, is introduced
and the de-shape SST algorithm is theoretically analyzed. In addition to simulated
signals, several different physiological, musical and biological signals are analyzed to
illustrate the proposed algorithm.
We investigate the energy landscape of the mixed even p-spin model with Ising spin configurations.
We show that for any given energy level between zero and the maximal energy, with
overwhelming probability there exist exponentially many distinct spin configurations such that
their energies stay near this energy level. Furthermore, their magnetizations and overlaps are
concentrated around some fixed constants. In particular, at the level of maximal energy, we
prove that the Hamiltonian exhibits exponentially many orthogonal peaks. This improves the
results of Chatterjee  and Ding-Eldan-Zhai , where the former established a logarithmic
size of the number of the orthogonal peaks, while the latter proved a polynomial size. Our second
main result obtains disorder chaos at zero temperature and at any external field. As a byproduct,
this implies that the fluctuation of the maximal energy is superconcentrated when the external
field vanishes and obeys a Gaussian limit law when the external field is present.
We present a framework for generating street networks and parcel layouts. Our goal is the generation of high-quality layouts that can be used for urban planning and virtual environments. We propose a solution based on hierarchical domain splitting using two splitting types: streamline-based splitting, which splits a region along one or multiple streamlines of a cross field, and template-based splitting, which warps pre-designed templates to a region and uses the interior geometry of the template as the splitting lines. We combine these two splitting approaches into a hierarchical framework, providing automatic and interactive tools to explore the design space.
For the purpose of isogeometric analysis, one of the most common ways is to construct structured hexahedral meshes, which have regular tensor product structure, and fit them by volumetric T-Splines. This theoretic work proposes a novel surface quadrilateral meshing method, colorable quad-mesh, which leads to the structured hexahedral mesh of the enclosed volume for high genus surfaces.
The work proves the equivalence relations among colorable quad-meshes, finite measured foliations and Strebel differentials on surfaces. This trinity theorem lays down the theoretic foundation for quadrilateral/hexahedral mesh generation, and leads to practical, automatic algorithms.
The work proposes the following algorithm: the user inputs a set of disjoint, simple loops on a high genus surface, and specifies a height parameter for each loop; a unique Strebel differential is computed with the combinatorial type and the heights prescribed by the user’s input; the Strebel differential assigns a flat metric on the surface and decomposes the surface into cylinders; a colorable quad-mesh is generated by splitting each cylinder into two quadrilaterals, followed by subdivision; the surface cylindrical decomposition is extended inward to produce a solid cylindrical decomposition of the volume; the hexadhedral meshing is generated for each volumetric cylinder and then glued together to form a globally consistent hex-mesh.
The method is rigorous, geometric, automatic and conformal to the geometry. This work focuses on the theoretic aspects of the framework, the algorithmic details and practical evaluations will be given in the future expositions.
Min ZhangStony Brook UniversityRen GuoOregon State UniveristyWei ZengSchool of Computing and Information Sciences, Florida International UniversityFeng LuoRutgers UniversityShing Tung YauHarvard UniversityXianfeng GuStony Brook Univerisity
Computational GeometryDifferential GeometryGeometric Modeling and ProcessingConvex and Discrete Geometry mathscidoc:1612.01001
Ricci ﬂow deformsthe Riemannian metric proportionallyto the curvature, such that the curvatureevolves accordingto a heat diffusion process and eventually becomes constant everywhere. Ricci ﬂow has demonstrated its great potential by solving various problems in many ﬁelds, which can be hardly handled by alternative methods so far. This work introduces the uniﬁed theoretic framework for discrete Surface Ricci Flow, including all the common schemes: Tangential Circle Packing, Thurston’s Circle Packing, Inversive Distance Circle Packing and Discrete Yamabe Flow. Furthermore, this work also introduces a novel schemes, Virtual Radius Circle Packing and the Mixed Type schemes, under the uniﬁed framework. This work gives explicit geometric interpretation to the discrete Ricci energies for all the schemes with all back ground geometries, and the corresponding Hessian matrices. The uniﬁed frame work deepens our understanding to the the discrete surface Ricci ﬂow theory, and has inspired us to discover the new schemes, improved the ﬂexibility and robustness of the algorithms, greatly simpliﬁed the implementation and improved the efﬁciency. Experimental results show the uniﬁed surface Ricci ﬂow algorithms can handle general surfaces with different topologies, and is robust to meshes with different qualities, and is effective for solving real problems.
3D scene modeling has long been a fundamental problem in computer graphics and computer vision. With the popularity of consumer-level RGB-D cameras, there is a growing interest in digitizing real-world indoor 3D scenes. However, modeling indoor
3D scenes remains a challenging problem because of the complex structure of interior objects and poor quality of RGB-D data acquired by consumer-level sensors. Various methods have been proposed to tackle these challenges. In this survey, we provide an overview of recent advances in indoor scene modeling techniques, as well as public datasets and code libraries which can facilitate experiments and evaluation.
In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (A special case are ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.)
Sir Isaac Newton PRS was an English physicist and mathematician (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time and as a key figure in the scientific revolution.