We consider a general class of symmetric or Hermitian random band matrices $H=(h_{xy})_{x,y \in \llbracket 1,N\rrbracket^d}$ in any dimension $d\ge 1$, where the entries are independent, centered random variables with variances $s_{xy}=\mathbb E|h_{xy}|^2$. We assume that $s_{xy}$ vanishes if $|x-y|$ exceeds the band width $W$, and we are interested in the mesoscopic scale with $1\ll W\ll N$. Define the {\it{generalized resolvent}} of $H$ as $G(H,Z):=(H - Z)^{-1}$, where $Z$ is a deterministic diagonal matrix with entries $Z_{xx}\in \mathbb C_+$ for all $x$. Then we establish a precise high-probability bound on certain averages of polynomials of the resolvent entries. As an application of this fluctuation averaging result, we give a self-contained proof for the delocalization of random band matrices in dimensions $d\ge 2$. More precisely, for any fixed $d\ge 2$, we prove that the bulk eigenvectors of $H$ are delocalized in certain averaged sense if $N\le W^{1+\frac{d}{2}}$. This improves the corresponding results in \cite{HeMa2018} under the assumption $N\ll W^{1+\frac{d}{d+1}}$, and in \cite{ErdKno2013,ErdKno2011} under the assumption $N\ll W^{1+\frac{d}{6}}$. For 1D random band matrices, our fluctuation averaging result was used in \cite{PartII,PartI} to prove the delocalization conjecture and bulk universality for random band matrices with $N\ll W^{4/3}$.

Sharp edges are important shape features and their extraction has been extensively studied both on point clouds and surfaces. We consider the problem of extracting sharp edges from a sparse set of colour-and-depth (RGB-D) images. The noise-ridden depth measurements are challenging for existing feature extraction methods that work solely in the geometric domain (e.g. points or meshes). By utilizing both colour and depth information, we propose a novel feature extraction method that produces much cleaner and more coherent feature lines. We make two technical contributions. First, we show that intensity edges can augment the depth map to improve normal estimation and feature localization from a single RGB-D image. Second, we designed a novel algorithm for consolidating feature points obtained from multiple RGB-D images. By utilizing normals and ridge/valley types associated with the feature points, our algorithm is effective in suppressing noise without smearing nearby features.

We introduce a unified particle framework which integrates the phase-field method with multi-material simulation to allow modeling of both liquids and solids, as well as phase transitions between them. A simple elastoplastic model is used to capture the behavior of various kinds of solids, including deformable bodies, granular materials, and cohesive soils. States of matter or phases, particularly liquids and solids, are modeled using the non-conservative Allen-Cahn equation. In contrast, materials—made of different substances—are advected by the conservative Cahn-Hilliard equation. The distributions of phases and materials are represented by a phase variable and a concentration variable, respectively, allowing us to represent commonly observed fluid-solid interactions. Our multi-phase, multi-material system is governed by a unified Helmholtz free energy density. This framework provides the first method in computer graphics capable of modeling a continuous interface between phases. It is versatile and can be readily used in many scenarios that are challenging to simulate. Examples are provided to demonstrate the capabilities and effectiveness of this approach.

Personal videos often contain visual distractors, which are objects that are accidentally captured that can distract viewers from focusing on the main subjects. We propose a method to automatically detect and localize these distractors through learning from a manually labeled dataset. To achieve spatially and temporally coherent detection, we propose extracting features at the Temporal-Superpixel (TSP) level using a traditional SVM-based learning framework. We also experiment with end-to-end learning using Convolutional Neural Networks (CNNs), which achieves slightly higher performance than other methods. The classification result is further refined in a post-processing step based on graph-cut optimization. Experimental results show that our method achieves an accuracy of 81% and a recall of 86%. We demonstrate several ways of removing the detected distractors to improve the video quality, including video hole filling; video frame replacement; and camera path re-planning. The user study results show that our method can significantly improve the aesthetic quality of videos.

In this paper, based on an idea of Tian we establish a new sufficient condition for a positive integer n to be a congruent number in terms of the Legendre symbols for the prime factors of n. Our criterion generalizes previous results of Heegner, Birch–Stephens, Monsky, and Tian, and conjecturally provides a list of positive density of congruent numbers. Our method of proving the criterion is to give formulae for the analytic Tate–Shafarevich number $\mathcal{L}(n)$ in terms of the so-called genus periods and genus points. These formulae are derived from the Waldspurger formula and the generalized Gross–Zagier formula of Yuan–Zhang–Zhang.