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20
ThreeDimensional Face Recognition
, 2005
"... An expressioninvariant 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 expressioninvariant representations of faces using the bendinginvariant canonical forms approach. The re ..."
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Cited by 103 (22 self)
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An expressioninvariant 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 expressioninvariant representations of faces using the bendinginvariant 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 expressioninvariant representation of any given face. It allows us to perform the 3D face recognition task while avoiding the surface reconstruction stage.
A Multigrid Method For Distributed Parameter Estimation Problems
 Trans. Numer. Anal
, 2001
"... . 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 ..."
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Cited by 39 (13 self)
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. 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 Tikhonovtype 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 cellbased. 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 Marquardttype modification on coarser grids. Alternatively, fewer grids are used and a preconditioned Krylovspace method is utilized on the coarsest grid. A collective point relaxation method (weighted Jacobi or a GaussSeidel variant) is used for smoothing. We demonstrate the efficiency of our method on a classical model problem from hydrology. 1.
A General Framework for Nonlinear Multigrid Inversion
 IEEE Trans. on Image Processing
, 2005
"... 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 3D partial differential equation. For these applications, image reconstruction is particularly difficult because the forward problem is both non ..."
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Cited by 13 (5 self)
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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 3D 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...
MODEL PROBLEMS FOR THE MULTIGRID OPTIMIZATION OF SYSTEMS GOVERNED BY DIFFERENTIAL EQUATIONS
, 2005
"... 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 optimizationbased multigrid algorithm in which the multigrid algori ..."
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Cited by 12 (1 self)
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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 optimizationbased 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 optimizationbased 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 optimizationbased multigrid algorithm.
Multigrid Multidimensional Scaling
 NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
, 2000
"... ... 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 largescale MDS. Simulation results show that the proposed approach significan ..."
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Cited by 10 (5 self)
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... 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 largescale MDS. Simulation results show that the proposed approach significantly outperforms conventional MDS algorithms.
Multigrid algorithms for optimizations and inverse problems
 Proc. the SPIE/IS&T Conference on Computational Imaging 2003
, 2003
"... 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 threedimensional partial differential equation. For these applications, image reconstruction can be formulated as the solution to a nonquadrati ..."
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Cited by 6 (4 self)
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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 threedimensional partial differential equation. For these applications, image reconstruction can be formulated as the solution to a nonquadratic 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.
An OcTree Method for Parametric Image Registration
, 2006
"... Already for reasonable sized 3D images, image registration becomes a computationally intensive task. Here, we introduce and explore the concept of OcTree’s for registration which drastically reduces the number of processed data and thus the computational costs. We show how to map the registration pr ..."
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Cited by 4 (1 self)
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Already for reasonable sized 3D images, image registration becomes a computationally intensive task. Here, we introduce and explore the concept of OcTree’s for registration which drastically reduces the number of processed data and thus the computational costs. We show how to map the registration problem onto an OcTree and present a suitable optimization technique. Furthermore, we demonstrate the performance of the new approach by academic as well as real life examples. These examples indicate that the computational time can be reduced by a factor of 10 compared with standard approaches. 1
Recursive trustregion methods for multiscale nonlinear optimization
 SIAM J. Optim
"... Abstract. A class of trustregion methods is presented for solving unconstrained nonlinear and possibly nonconvex discretized optimization problems, like those arising in systems governed by partial differential equations. The algorithms in this class make use of the discretization level as a mean o ..."
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Cited by 4 (1 self)
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Abstract. A class of trustregion methods is presented for solving unconstrained nonlinear and possibly nonconvex discretized optimization problems, like those arising in systems governed by partial differential equations. The algorithms in this class make use of the discretization level as a mean of speeding up the computation of the step. This use is recursive, leading to true multilevel/multiscale optimization methods reminiscent of multigrid methods in linear algebra and the solution of partialdifferential equations. A simple algorithm of the class is then described and its numerical performance is shown to be numerically promising. This observation then motivates a proof of global convergence to firstorder stationary points on the fine grid that is valid for all algorithms in the class.
A LINE SEARCH MULTIGRID METHOD FOR LARGESCALE NONLINEAR OPTIMIZATION ∗
, 2008
"... Abstract. We present a line search multigrid method for solving discretized versions of general unconstrained infinite dimensional optimization problems. At each iteration on each level, the algorithm computes either a “direct search ” direction on the current level or a “recursive search ” directio ..."
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Cited by 2 (0 self)
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Abstract. We present a line search multigrid method for solving discretized versions of general unconstrained infinite dimensional optimization problems. At each iteration on each level, the algorithm computes either a “direct search ” direction on the current level or a “recursive search ” direction from coarser level models. Introducing a new condition that must be satisfied by a backtracking line search procedure, the “recursive search ” direction is guaranteed to be a descent direction. Global convergence is proved under fairly minimal requirements on the minimization method used at all grid levels. Using a limited memory BFGS quasiNewton method to produce the “direct search ” direction, preliminary numerical experiments show that our line search multigrid approach is promising.