Registration, which aims to find an optimal 1-1 correspondence between shapes, is an important process in different research areas. Conformal mappings have been widely used to obtain a diffeomorphism between shapes that minimizes angular distortion. Conformal registrations are beneficial since it preserves the local geometry well. However, when landmark constraints are enforced, conformal mappings generally do not exist. This motivates us to look for a unique landmark matching quasi-conformal registration, which minimizes the conformality distortion. Under suitable condition on the landmark constraints, a unique diffeomporphism, called the Teichmller extremal mapping between two surfaces can be obtained, which minimizes the maximal conformality distortion. In this paper, we propose an efficient iterative algorithm, called the Quasi-conformal (QC) iterations, to compute the Teichmller mapping. The basic idea is to represent the set of diffeomorphisms using Beltrami coefficients (BCs), and look for an optimal BC associated to the desired Teichmller mapping. The associated diffeomorphism can be efficiently reconstructed from the optimal BC using the Linear Beltrami Solver (LBS). Using BCs to represent diffeomorphisms guarantees the diffeomorphic property of the registration. Using our proposed method, the Teichmller mapping can be accurately and efficiently computed within 10 seconds. The obtained registration is guaranteed to be bijective. The proposed algorithm can also be extended to compute Teichmller mapping with soft landmark constraints. We applied the proposed algorithm to real applications, such as brain

It is observed that three-dimensional Gödel black holes can be promoted to exact string theory backgrounds through an orbifold of an hyperbolic asymmetric marginal deformation of the SL(2,R) WZW model. Tachyons are found in the spectrum of long strings. Uplifting these solutions in type IIB supergravity, extremal black holes are shown to preserve one supersymmetry in accordance with the BTZ limit. We also make connections with some recently discussed warped black hole solutions of topologically massive gravity, showing that they actually correspond to quotients of spacelike squashed AdS_3.

Designing an unsupervised image denoising approach in practical applications is a challenging task due to the complicated data acquisition process. In the realworld case, the noise distribution is so complex that the simplified additive white Gaussian (AWGN) assumption rarely holds, which significantly deteriorates the Gaussian denoisers’ performance. To address this problem, we apply a deep neural network that maps the noisy image into a latent space in which the AWGN assumption holds, and thus any existing Gaussian denoiser is applicable. More specifically, the proposed neural network consists of the encoder-decoder structure and approximates the likelihood term in the Bayesian framework. Together with a Gaussian denoiser, the neural network can be trained with the input image itself and does not require any pre-training in other datasets. Extensive experiments on real-world noisy image datasets have shown that the combination of neural networks and Gaussian denoisers improves the performance of the original Gaussian denoisers by a large margin. In particular, the neural network+BM3D method significantly outperforms other unsupervised denoising approaches and is competitive with supervised networks such as DnCNN, FFDNet, and CBDNet.

this article we generalize the classical gradient estimate for the minimal surface equation to higher codimension. We consider a vector-valued function $u : \Omega \subset \math{R}^n \to \math{R}^m$ that satisfies the minimal surface system, see equation (1.1) in §1. The graph of u is then a minimal submanifold of $\math{R}^{n+m}$. We prove an a priori gradient bound under the assumption that the Jacobian of $ du: \math{R}^n \to \math{R}^m$ on any two dimensional subspace of $\mtah{R}^n$ is less than or equal to one. This assumption is automatically satisfied when $du$ is of rank one and thus the estimate covers the case when m=1, i.e., the original minimal surface equation. This is applied to Bernstein type theorems for minimal submanifolds of higher codimension.

The construction of local reduced-order models via multiscale basis functions has been an area of active research. In this paper, we propose online multiscale basis functions which are constructed using the offline space and the current residual. Online multiscale basis functions are constructed adaptively in some selected regions based on our error indicators. We derive an error estimator which shows that one needs to have an offline space with certain properties to guarantee that additional online multiscale basis function will decrease the error. This error decrease is independent of physical parameters, such as the contrast and multiple scales in the problem. The offline spaces are constructed using Generalized Multiscale Finite Element Methods (GMsFEM). We show that if one chooses a sufficient number of offline basis functions, one can guarantee that additional online multiscale basis functions will reduce