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199
Topologically Adaptable Snakes
 Medical Image Analysis
, 1995
"... This paper presents a topologically adaptable snakes model for image segmentation and object representation. The model is embedded in the framework of domain subdivision using simplicial decomposition. This framework extends the geometric and topological adaptability of snakes while retaining all of ..."
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Cited by 202 (5 self)
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This paper presents a topologically adaptable snakes model for image segmentation and object representation. The model is embedded in the framework of domain subdivision using simplicial decomposition. This framework extends the geometric and topological adaptability of snakes while retaining all of the features of traditionalsnakes, such as user interaction, and overcoming many of the limitations of traditionalsnakes. By superposing a simplicial grid over the image domain and using this grid to iteratively reparameterize the deforming snakes model, the model is able to flow into complex shapes, even shapes with significant protrusions or branches, and to dynamically change topology as necessitated by the data. Snakes can be created and can split into multiple parts or seamlessly merge into other snakes. The model can also be easily converted to and from the traditional parametric snakes model representation. We apply a 2D model to various synthetic and real images in order to segment ...
Numerical methods for image registration
, 2004
"... In this paper we introduce a new framework for image registration. Our formulation is based on consistent discretization of the optimization problem coupled with a multigrid solution of the linear system which evolve in a GaussNewton iteration. We show that our discretization is helliptic independ ..."
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Cited by 170 (26 self)
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In this paper we introduce a new framework for image registration. Our formulation is based on consistent discretization of the optimization problem coupled with a multigrid solution of the linear system which evolve in a GaussNewton iteration. We show that our discretization is helliptic independent of parameter choice and therefore a simple multigrid implementation can be used. To overcome potential large nonlinearities and to further speed up computation, we use a multilevel continuation technique. We demonstrate the efficiency of our method on a realistic highly nonlinear registration problem. 1 Introduction and problem setup Image registration is one of today’s challenging image processing problems. Given a socalled reference R and a socalled template image T, the basic idea is to find a “reasonable ” transformation such that a transformed version of the template image becomes “similar ” to the reference image. Image registration
Solving Systems of Polynomial Equations
 American Mathematical Society, CBMS Regional Conferences Series, No 97
, 2002
"... Abstract. One of the most classical problems of mathematics is to solve systems of polynomial equations in several unknowns. Today, polynomial models are ubiquitous and widely applied across the sciences. They arise in robotics, coding theory, optimization, mathematical biology, computer vision, gam ..."
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Cited by 162 (10 self)
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Abstract. One of the most classical problems of mathematics is to solve systems of polynomial equations in several unknowns. Today, polynomial models are ubiquitous and widely applied across the sciences. They arise in robotics, coding theory, optimization, mathematical biology, computer vision, game theory, statistics, machine learning, control theory, and numerous other areas. The set of solutions to a system of polynomial equations is an algebraic variety, the basic object of algebraic geometry. The algorithmic study of algebraic varieties is the central theme of computational algebraic geometry. Exciting recent developments in symbolic algebra and numerical software for geometric calculations have revolutionized the field, making formerly inaccessible problems tractable, and providing fertile ground for experimentation and conjecture. The first half of this book furnishes an introduction and represents a snapshot of the state of the art regarding systems of polynomial equations. Afficionados of the wellknown text books by Cox, Little, and O’Shea will find familiar themes in the first five chapters: polynomials in one variable, Gröbner
Parallel LagrangeNewtonKrylovSchur methods for PDEconstrained optimization. Part I: The KrylovSchur solver
 SIAM J. Sci. Comput
, 2000
"... Abstract. Large scale optimization of systems governed by partial differential equations (PDEs) is a frontier problem in scientific computation. The stateoftheart for such problems is reduced quasiNewton sequential quadratic programming (SQP) methods. These methods take full advantage of existin ..."
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Cited by 78 (11 self)
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Abstract. Large scale optimization of systems governed by partial differential equations (PDEs) is a frontier problem in scientific computation. The stateoftheart for such problems is reduced quasiNewton sequential quadratic programming (SQP) methods. These methods take full advantage of existing PDE solver technology and parallelize well. However, their algorithmic scalability is questionable; for certain problem classes they can be very slow to converge. In this twopart article we propose a new method for steadystate PDEconstrained optimization, based on the idea of full space SQP with reduced space quasiNewton SQP preconditioning. The basic components of the method are: Newton solution of the firstorder optimality conditions that characterize stationarity of the Lagrangian function; Krylov solution of the KarushKuhnTucker (KKT) linear systems arising at each Newton iteration using a symmetric quasiminimum residual method; preconditioning of the KKT system using an approximate state/decision variable decomposition that replaces the forward PDE Jacobians by their own preconditioners, and the decision space Schur complement (the reduced Hessian) by a BFGS approximation or by a twostep stationary method. Accordingly, we term the new method LagrangeNewtonKrylov Schur (LNKS). It is fully parallelizable, exploits the structure of available parallel algorithms for the PDE forward problem, and is locally quadratically convergent. In the first part of the paper we investigate the effectiveness of the KKT linear system solver. We test the method on two optimal control problems in which the flow is described by the steadystate Stokes equations. The
Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components
, 2001
"... In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposi ..."
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Cited by 65 (31 self)
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In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component. As byproducts, the computation also determines the degree of each component and an upper bound on itsmultiplicity. The bound issharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroesof a finite number of polynomials.
Sparse Elimination and Applications in Kinematics
, 1994
"... This thesis proposes efficient algorithmic solutions to problems in computational algebra and computational algebraic geometry. Moreover, it considers their application to different areas where algebraic systems describe kinematic and geometric constraints. Given an arbitrary system of nonlinear mul ..."
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Cited by 47 (10 self)
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This thesis proposes efficient algorithmic solutions to problems in computational algebra and computational algebraic geometry. Moreover, it considers their application to different areas where algebraic systems describe kinematic and geometric constraints. Given an arbitrary system of nonlinear multivariate polynomial equations, its resultant serves in eliminating variables and reduces root finding to a linear eigenproblem. Our contribution is to describe the first efficient and general algorithms for computing the sparse resultant. The sparse resultant generalizes the classical homogeneous resultant and exploits the structure of the given polynomials. Its size depends only on the geometry of the input Newton polytopes. The first algorithm uses a subdivision of the Minkowski sum and produces matrix...
Complexity Theory and Numerical Analysis
 Acta Numerica
, 1996
"... this paper asserts: ..."
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Newton’s method with deflation for isolated singularities of polynomial systems
 Theor. Comp. Sci. 359
"... We present a modification of Newton’s method to restore quadratic convergence for isolated singular solutions of polynomial systems. Our method is symbolicnumeric: we produce a new polynomial system which has the original multiple solution as a regular root. We show that the number of deflation sta ..."
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Cited by 33 (9 self)
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We present a modification of Newton’s method to restore quadratic convergence for isolated singular solutions of polynomial systems. Our method is symbolicnumeric: we produce a new polynomial system which has the original multiple solution as a regular root. We show that the number of deflation stages is bounded by the multiplicity of the isolated root. Our implementation performs well on a large class of applications. 2000 Mathematics Subject Classification. Primary 65H10. Secondary 14Q99, 68W30. Key words and phrases. Newton’s method, deflation, numerical homotopy algorithms, symbolicnumeric computations. 1