A collaborative convex framework for factoring a data matrix <i>X</i> into a nonnegative product <i>AS</i> , with a sparse coefficient matrix <i>S</i> , is proposed. We restrict the columns of the dictionary matrix <i>A</i> to coincide with certain columns of the data matrix <i>X</i> , thereby guaranteeing a physically meaningful dictionary and dimensionality reduction. We use <i>l</i> _1, regularization to select the dictionary from the data and show that this leads to an exact convex relaxation of <i>l</i> ₀ in the case of distinct noise-free data. We also show how to relax the restriction-to- <i>X</i> constraint by initializing an alternating minimization approach with the solution of the convex model, obtaining a dictionary close to but not necessarily in <i>X</i> . We focus on applications of the proposed framework to hyperspectral endmember and abundance identification and also show an application to blind source separation of nuclear magnetic resonance data.

Watermarking algorithms provide a way of hiding or embedding some bits of information in a watermark. In the case of watermarking a 3D model, many algorithms employ a so-called indexed localization scheme. In this paper, we propose an optimization framework with two new steps for such watermarking algorithms to improve their capacity and invisibility. The first step is to find an optimal layout of invariant units to improve capacity. The second step is to rearrange the correspondence between the watermark units and the invariant units to improve invisibility. Experimental tests show that by using this framework, the capacity and invisibility of watermarking algorithms can be greatly improved.

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The alternating direction multiplier method (ADMM) is widely used in computer graphics for solving optimization problems that can be nonsmooth and nonconvex. It converges quickly to an approximate solution, but can take a long time to converge to a solution of high‐accuracy. Previously, Anderson acceleration has been applied to ADMM, by treating it as a fixed‐point iteration for the concatenation of the dual variables and a subset of the primal variables. In this paper, we note that the equivalence between ADMM and Douglas‐Rachford splitting reveals that ADMM is in fact a fixed‐point iteration in a lower‐dimensional space. By applying Anderson acceleration to such lower‐dimensional fixed‐point iteration, we obtain a more effective approach for accelerating ADMM. We analyze the convergence of the proposed acceleration method on nonconvex problems, and verify its effectiveness on a variety of computer graphics including geometry processing and physical simulation.

This paper presents a new approach for reconstructing solids with planar, quadric and toroidal surfaces from three-view engineering drawings. By applying geometric theory to 3-D reconstruction, our method is able to remove restrictions placed on the axes of curved surfaces by existing methods. The main feature of our algorithm is that it combines the geometric properties of conics with affine properties to recover a wider range of 3-D edges. First, the algorithm determines the type of each 3-D candidate conic edge based on its projections in three orthographic views, and then generates that candidate edge using the conjugate diameter method. This step produces a wire-frame model that contains all candidate vertices and candidate edges. Next, a maximum turning angle method is developed to find all the candidate faces in the wire-frame model. Finally, a general and efficient searching technique is proposed for finding valid solids from the candidate faces; the technique greatly reduces the searching space and the backtracking incidents. Several examples are given to demonstrate the efficiency and capability of the proposed algorithm.