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101
Moving coframes. I. A practical algorithm
 Acta Appl. Math
, 1998
"... Abstract. This is the first in a series of papers devoted to the development and applications of a new general theory of moving frames. In this paper, we formulate a practical and easy to implement explicit method to compute moving frames, invariant differential forms, differential invariants and in ..."
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Cited by 116 (30 self)
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Abstract. This is the first in a series of papers devoted to the development and applications of a new general theory of moving frames. In this paper, we formulate a practical and easy to implement explicit method to compute moving frames, invariant differential forms, differential invariants and invariant differential operators, and solve general equivalence problems for both finitedimensional Lie group actions and infinite Lie pseudogroups. A wide variety of applications, ranging from differential equations to differential geometry to computer vision are presented. The theoretical justifications for the moving coframe algorithm will appear in the next paper in this series.
Joint invariant signatures
 Found. Comput. Math
, 1999
"... Dedicated to the memory of Gian–Carlo Rota Abstract. A new, algorithmic theory of moving frames is applied to classify joint invariants and joint differential invariants of transformation groups. Equivalence and symmetry properties of submanifolds are completely determined by their joint signatures, ..."
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Cited by 47 (25 self)
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Dedicated to the memory of Gian–Carlo Rota Abstract. A new, algorithmic theory of moving frames is applied to classify joint invariants and joint differential invariants of transformation groups. Equivalence and symmetry properties of submanifolds are completely determined by their joint signatures, which are parametrized by a suitable collection of joint invariants and/or joint differential invariants. A variety of fundamental geometric examples are developed in detail. Applications to object recognition problems in computer vision and the design of invariant numerical approximations are indicated.
Numerically invariant signature curves
 Int. J. Computer Vision
"... Abstract. Corrected versions of the numerically invariant expressions for the affine and Euclidean signature of a planar curve given in [1] are presented. The new formulas are valid for fine but otherwise arbitrary partitions of the curve. We also give numerically invariant expressions for the four ..."
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Cited by 47 (2 self)
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Abstract. Corrected versions of the numerically invariant expressions for the affine and Euclidean signature of a planar curve given in [1] are presented. The new formulas are valid for fine but otherwise arbitrary partitions of the curve. We also give numerically invariant expressions for the four differential invariants parametrizing the three dimensional version of the Euclidean signature curve, namely the curvature, the torsion and their derivatives with respect to arc length. 1.
Integral invariants for shape matching
 PAMI
, 2006
"... Abstract—For shapes represented as closed planar contours, we introduce a class of functionals which are invariant with respect to the Euclidean group and which are obtained by performing integral operations. While such integral invariants enjoy some of the desirable properties of their differential ..."
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Cited by 42 (2 self)
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Abstract—For shapes represented as closed planar contours, we introduce a class of functionals which are invariant with respect to the Euclidean group and which are obtained by performing integral operations. While such integral invariants enjoy some of the desirable properties of their differential counterparts, such as locality of computation (which allows matching under occlusions) and uniqueness of representation (asymptotically), they do not exhibit the noise sensitivity associated with differential quantities and, therefore, do not require presmoothing of the input shape. Our formulation allows the analysis of shapes at multiple scales. Based on integral invariants, we define a notion of distance between shapes. The proposed distance measure can be computed efficiently and allows warping the shape boundaries onto each other; its computation results in optimal point correspondence as an intermediate step. Numerical results on shape matching demonstrate that this framework can match shapes despite the deformation of subparts, missing parts and noise. As a quantitative analysis, we report matching scores for shape retrieval from a database. Index Terms—Integral invariants, shape, shape matching, shape distance, shape retrieval. Ç 1
Geometric foundations of numerical algorithms and symmetry
 Appl. Alg. Engin. Commun. Comput
"... Abstract. This paper outlines a new general construction, named “multispace”, that forms the proper geometrical foundation for the numerical analysis of differential equations — in direct analogy with the role played by jet space as the basic object underlying the geometry of differential equations ..."
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Cited by 36 (15 self)
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Abstract. This paper outlines a new general construction, named “multispace”, that forms the proper geometrical foundation for the numerical analysis of differential equations — in direct analogy with the role played by jet space as the basic object underlying the geometry of differential equations. Application of the theory of moving frames leads to a general framework for constructing symmetrypreserving numerical approximations to differential invariants and invariant differential equations.
MaurerCartan Forms and the Structure of Lie Pseudo–Groups
, 2005
"... This paper begins a series devoted to developing a general and practical theory of moving frames for infinitedimensional Lie pseudogroups. In this first, preparatory part, we present a new, direct approach to the construction of invariant Maurer–Cartan forms and the Cartan structure equations fo ..."
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Cited by 25 (10 self)
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This paper begins a series devoted to developing a general and practical theory of moving frames for infinitedimensional Lie pseudogroups. In this first, preparatory part, we present a new, direct approach to the construction of invariant Maurer–Cartan forms and the Cartan structure equations for a pseudogroup. Our approach is completely explicit and avoids reliance on the theory of exterior differential systems and prolongation. The second paper [60] will apply these constructions in order to develop the moving frame algorithm for the action of the pseudogroup on submanifolds. The third paper [61] will apply Gröbner basis methods to prove a fundamental theorem on the freeness of pseudogroup actions on jet bundles, and a constructive version of the finiteness theorem of Tresse and Kumpera for generating systems of differential invariants and also their syzygies. Applications of the moving frame method include practical algorithms for constructing complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, analyzing invariant variational principles, and solving equivalence and symmetry problems arising in geometry and physics.
A survey of moving frames
 Computer Algebra and Geometric Algebra with Applications. Volume 3519 of Lecture Notes in Computer Science, 105–138
, 2005
"... Abstract. This article presents the equivariant method of moving frames for finitedimensional Lie group actions, surveying a variety of applications, including geometry, differential equations, computer vision, numerical analysis, the calculus of variations, and invariant flows. 1. Introduction. Acc ..."
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Cited by 24 (3 self)
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Abstract. This article presents the equivariant method of moving frames for finitedimensional Lie group actions, surveying a variety of applications, including geometry, differential equations, computer vision, numerical analysis, the calculus of variations, and invariant flows. 1. Introduction. According to Akivis, [1], the method of moving frames originates in work of the Estonian mathematician Martin Bartels (1769–1836), a teacher of both Gauss and Lobachevsky. The field is most closely associated with Élie Cartan, [21], who forged earlier contributions by Darboux, Frenet, Serret, and Cotton into a powerful tool for analyzing the geometric
Symmetries of Polynomials
 J. Symb. Comp
"... New algorithms for determining discrete and continuous symmetries of polynomials  also known as binary forms in classical invariant theory  are presented. Implementations in Mathematica and Maple are discussed and compared. The results are based on a new, comprehensive theory of moving frames ..."
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Cited by 24 (17 self)
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New algorithms for determining discrete and continuous symmetries of polynomials  also known as binary forms in classical invariant theory  are presented. Implementations in Mathematica and Maple are discussed and compared. The results are based on a new, comprehensive theory of moving frames that completely characterizes the equivalence and symmetry properties of submanifolds under general Lie group actions. This work was partially supported by NSF Grant DMS 9803154. 1 Introduction. The purpose of this paper is to explain the detailed implementation of a new algorithm for determining the symmetries of polynomials (binary forms). The method was first described in the second author's new book [24], and the present paper adds details and refinements. We shall demonstrate that the symmetry group of both real and complex binary forms can be completely determined by solving two simultaneous bivariate polynomial equations, which are based on two fundamental covariants of the for...
Moving frames and singularities of prolonged group actions
 Selecta Math. (N.S
"... Abstract. The prolongation of a transformation group to jet bundles forms the geometric foundation underlying Lie’s theory of symmetry groups of differential equations, the theory of differential invariants, and the Cartan theory of moving frames. Recent developments in the moving frame theory have ..."
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Cited by 23 (13 self)
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Abstract. The prolongation of a transformation group to jet bundles forms the geometric foundation underlying Lie’s theory of symmetry groups of differential equations, the theory of differential invariants, and the Cartan theory of moving frames. Recent developments in the moving frame theory have necessitated a detailed understanding of the geometry of prolonged transformation groups. This paper begins with a basic review of moving frames, and then focuses on the study of both regular and singular prolonged group orbits. Highlights include a corrected version of the basic stabilization theorem, a discussion of “totally singular points, ” and geometric and algebraic characterizations of totally singular submanifolds, which are those that admit no moving frame. In addition to applications to the method of moving frames, the paper includes a generalized Wronskian lemma for vectorvalued functions, and methods for the solution to Lie determinant equations.
The Complex Representation of Algebraic Curves and its Simple Exploitation for Pose Estimation and Invariant Recognition
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2000
"... New representations are introduced for handling 2D algebraic curves (implicit polynomial curves) of arbitrary degree in the scope of computer vision applications. These representations permit fast accurate poseindependent shape recognition under Euclidean transformations with a complete set of inva ..."
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Cited by 19 (0 self)
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New representations are introduced for handling 2D algebraic curves (implicit polynomial curves) of arbitrary degree in the scope of computer vision applications. These representations permit fast accurate poseindependent shape recognition under Euclidean transformations with a complete set of invariants, and fast accurate poseestimation based on all the polynomial coefficients. The latter is accomplished by a new centering of a polynomial based on its coefficients, followed by rotation estimation by decomposing polynomial coefficient space into a union of orthogonal subspaces for which rotations within two dimensional subspaces or identity transformations within one dimensional subspaces result from rotations in x,y measureddata space. Angles of these rotations in the two dimensional coefficient subspaces are proportional to each other and are integer multiples of the rotation angle in the x,y data space. By recasting this approach in terms of a complex variable, i.e, x+iy=z and complex polynomialcoefficients, further conceptual and computational simplification results. Application to shapebased indexing into databases is presented to illustrate the usefulness and the robustness of the complex representation of algebraic curves.