We develop a new method for deriving local laws for a large class of random matrices. It is applicable to many matrix models built from sums and products of deterministic or independent random matrices. In particular, it may be used to obtain local laws for matrix ensembles that are \emph{anisotropic} in the sense that their resolvents are well approximated by deterministic matrices that are not multiples of the identity. For definiteness, we present the method for sample covariance matrices of the form Q:=TXX∗T∗, where T is deterministic and X is random with independent entries. We prove that with high probability the resolvent of Q is close to a deterministic matrix, with an optimal error bound and down to optimal spectral scales. As an application, we prove the edge universality of Q by establishing the Tracy-Widom-Airy statistics of the eigenvalues of Q near the soft edges. This result applies in the single-cut and multi-cut cases. Further applications include the distribution of the eigenvectors and an analysis of the outliers and BBP-type phase transitions in finite-rank deformations; they will appear elsewhere. We also apply our method to Wigner matrices whose entries have arbitrary expectation, i.e. we consider W+A where W is a Wigner matrix and a Hermitian deterministic matrix. We prove the anisotropic local law for W+A and use it to establish edge universality.

We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion and show that microscopic universality follows from mesoscopic statistics

3D models are now widely available for use in various applications. The demand for automatic model analysis and understanding is ever increasing. Model segmentation is an important step towards model understanding, and acts as a useful tool for different model processing applications, e.g. reverse engineering and modeling by example. We extend a random walk method used previously for image segmentation to give algorithms for both interactive and automatic model segmentation. This method is extremely efficient, and scales almost linearly with the number of faces, and the number of regions. For models of moderate size, interactive performance is achieved with commodity PCs. We demonstrate that this method can be applied to both triangle meshes and point cloud data. It is easy to implement, robust to noise in the model, and yields results suitable for downstream applications for both graphical and engineering models.

In this report, we propose a new semi-fragile watermarking algorithm for the authentication of 3D models based on integral invariants. To do so, we embed a watermark image by modifying the integral invariants of some of the vertices. Basically, we shift a vertex and its neighbors in order to change the integral invariants. To extract the watermark, test all the vertices for the embedded information, and combine them to recover the watermark image. How many parts can the watermark image be recovered would help us to make the authentication decision. Experimental test shows that this method is robust against rigid transform and noise attack, and useful to test purposely attack besides transferring noise and geometrical transforming noise. An additional contribution of this paper is a new algorithm for computing two kinds of integral invariants.

Gaseous fluids may move slowly, as smoke does, or at high speed, such as occurs with explosions. High-speed gas flow is always accompanied by low-speed gas flow, which produces rich visual details in the fluid motion. Realistic visualization involves a complex dynamic flow field with both low and high speed fluid behavior. In computer graphics, algorithms to simulate gaseous fluids address either the low speed case or the high speed case, but no algorithm handles both efficiently. With the aim of providing visually pleasing results, we present a hybrid algorithm that efficiently captures the essential physics of both low- and high-speed gaseous fluids. We model the low speed gaseous fluids by a grid approach and use a particle approach for the high speed gaseous fluids. In addition, we propose a physically sound method to connect the particle model to the grid model. By exploiting complementary strengths and avoiding weaknesses of the grid and particle approaches, we produce some animation examples and analyze their computational performance to demonstrate the effectiveness of the new hybrid method.