An expression-invariant 3D face recognition approach is presented. Our basic assumption is that facial expressions can be modelled as isometries of the facial surface. This allows to construct expression-invariant representations of faces using the bending-invariant canonical forms approach. The result is an efficient and accurate face recognition algorithm, robust to facial expressions, that can distinguish between identical twins (the first two authors). We demonstrate a prototype system based on the proposed algorithm and compare its performance to classical face recognition methods. The numerical methods employed by our approach do not require the facial surface explicitly. The surface gradients field, or the surface metric, are sufficient for constructing the expression-invariant representation of any given face. It allows us to perform the 3D face recognition task while avoiding the surface reconstruction stage.

. This paper considers problems of distributed parameter estimation from data measurements on solutions of partial differential equations (PDEs). A nonlinear least squares functional is minimized to approximately recover the sought parameter function (i.e., the model). This functional consists of a data fitting term, involving the solution of a finite volume or finite element discretization of the forward differential equation, and a Tikhonov-type regularization term, involving the discretization of a mix of model derivatives. We develop a multigrid method for the resulting constrained optimization problem. The method directly addresses the discretized PDE system which defines a critical point of the Lagrangian. The discretization is cell-based. This system is strongly coupled when the regularization parameter is small. Moreover, the compactness of the discretization scheme does not necessarily follow from compact discretizations of the forward model and of the regularization term. We therefore employ a Marquardt-type modification on coarser grids. Alternatively, fewer grids are used and a preconditioned Krylov-space method is utilized on the coarsest grid. A collective point relaxation method (weighted Jacobi or a Gauss-Seidel variant) is used for smoothing. We demonstrate the efficiency of our method on a classical model problem from hydrology. 1.

A variety of new imaging modalities, such as optical diffusion tomography, require the inversion of a forward problem that is modeled by the solution to a 3-D partial differential equation. For these applications, image reconstruction is particularly difficult because the forward problem is both nonlinear and computationally expensive to evaluate. In this paper, we propose a general framework...

... In this paper we present a multigrid framework for MDS problems. We demonstrate the performance of our algorithm on dimensionality reduction and isometric embedding problems, two classical problems requiring efficient large-scale MDS. Simulation results show that the proposed approach significantly outperforms conventional MDS algorithms.

A variety of new imaging modalities, such as optical diffusion tomography, require the inversion of a forward problem that is modeled by the solution to a three-dimensional partial differential equation. For these applications, image reconstruction can be formulated as the solution to a non-quadratic optimization problem. In this paper, we discuss the use of nonlinear multigrid methods as both tools for optimization and algorithms for the solution of difficult inverse problems. In particular, we review some existing methods for directly formulating optimization algorithm in a multigrid framework, and we introduce a new method for the solution of general inverse problems which we call multigrid inversion. These methods work by dynamically adjusting the cost functionals at different scales so that they are consistent with, and ultimately reduce, the finest scale cost functional. In this way, the multigrid optimization methods can efficiently compute the solution to a desired fine scale optimization problem. Importantly, the multigrid inversion algorithm can greatly reduce computation because both the forward and inverse problems are more coarsely discretized at lower resolutions. An application of our method to optical diffusion tomography shows the potential for very large computational savings.

We discuss a multigrid approach to the optimization of systems governed by differential equations. Such optimization problems appear in many applications and are of a different nature than systems of equations. Our approach uses an optimization-based multigrid algorithm in which the multigrid algorithm relies explicitly on nonlinear optimization models as subproblems on coarser grids. Our goal is not to argue for a particular optimization-based multigrid algorithm, but instead to demonstrate how multigrid can be used to accelerate nonlinear programming algorithms. Furthermore, using several model problems we give evidence (both theoretical and numerical) that the optimization setting is well suited to multigrid algorithms. Some of the model problems show that the optimization problem may be more amenable to multigrid than the governing differential equation. In addition, we relate the multigrid approach to more traditional optimization methods as further justification for the use of an optimization-based multigrid algorithm.

contract CCR-0073357.

This paper considers problems of distributed parameter estimation from data measurements on solutions of dierential equations. A nonlinear least squares functional is minimized to approximately recover the sought parameter function (i.e., the model). This functional consists of a data tting term, involving the solution of a nite volume or nite element discretization of the forward dierential equation, and a Tikhonov-type regularization term, involving the discretization of a mix of model derivatives. The grid spacing of the model discretization, as well as the relative weight of the entire regularization term, aect the sort of regularization achieved. We investigate a number of questions arising regarding their relationship, including the degree of nonlinearity of the least squares functional. We also investigate the correct scaling of the regularization matrix, where we rigorously associate the practice of using unscaled regularization matrices with approximations of...

Abstract. Recent years have seen growth in the number of algorithms designed to solve challenging simulation-based nonlinear optimization problems. One such algorithm is the Trust-Region Parallel Direct Search (TRPDS) method developed by Hough and Meza. In this paper, we take advantage of the theoretical properties of TRPDS to make use of approximation models in order to reduce the computational cost of simulationbased optimization. We describe the extension, which we call mTRPDS, and present the results of a case study for two earth penetrator design problems. In the case study, we conduct computational experiments with an array of approximations within the mTRPDS algorithm and compare the numerical results to the original TRPDS algorithm and a trust-region method implemented using the speculative gradient approach described by Byrd, Schnabel, and Shultz. The results suggest new ways to improve the algorithm.

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