Results 1  10
of
31
Generating Differential Invariants
, 2007
"... The equivariant method of moving frames is used to specify systems of generating differential invariants for finitedimensional Lie group actions. ..."
Abstract

Cited by 36 (19 self)
 Add to MetaCart
(Show Context)
The equivariant method of moving frames is used to specify systems of generating differential invariants for finitedimensional Lie group actions.
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 ..."
Abstract

Cited by 24 (3 self)
 Add to MetaCart
(Show Context)
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
Differential invariants of a Lie group action: syzygies on a generating set
, 2009
"... ..."
(Show Context)
Rational invariants of a group action. Construction and rewriting
, 2007
"... Geometric constructions applied to a rational action of an algebraic group lead to a new algorithm for computing rational invariants. A finite generating set of invariants appears as the coefficients of a reduced Gröbner basis. The algorithm comes in two variants. In the first construction the ideal ..."
Abstract

Cited by 22 (8 self)
 Add to MetaCart
Geometric constructions applied to a rational action of an algebraic group lead to a new algorithm for computing rational invariants. A finite generating set of invariants appears as the coefficients of a reduced Gröbner basis. The algorithm comes in two variants. In the first construction the ideal of the graph of the action is considered. In the second one the ideal of a crosssection is added to the ideal of the graph. Zerodimensionality of the resulting ideal brings a computational advantage. In both cases, reduction with respect to the computed Gröbner basis allows us to express any rational invariant in terms of the generators.
Differential Invariants of Conformal and Projective Surfaces
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2007
"... We show that, for both the conformal and projective groups, all the differential invariants of a generic surface in threedimensional space can be written as combinations of the invariant derivatives of a single differential invariant. The proof is based on the equivariant method of moving frames. ..."
Abstract

Cited by 16 (13 self)
 Add to MetaCart
We show that, for both the conformal and projective groups, all the differential invariants of a generic surface in threedimensional space can be written as combinations of the invariant derivatives of a single differential invariant. The proof is based on the equivariant method of moving frames.
Classification of curves in 2D and 3D via affine integral signatures
 Acta. Appl. Math
"... We propose a robust classification algorithm for curves in 2D and 3D, under special and full groups of affine transformations. To each plane or spatial curve we assign a plane signature curve. Curves, equivalent under an affine transformation, have the same signature. The signatures introduced in th ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
(Show Context)
We propose a robust classification algorithm for curves in 2D and 3D, under special and full groups of affine transformations. To each plane or spatial curve we assign a plane signature curve. Curves, equivalent under an affine transformation, have the same signature. The signatures introduced in this paper are based on integral invariants, which behave much better on noisy images than classically known differential invariants. The comparison with other types of invariants is given in the introduction. Though the integral invariants for planar curves were known before, the affine integral invariants for spatial curves are proposed here for the first time. Using the inductive variation of the moving frame method we compute affine invariants in terms of Euclidean invariants. We present two types of signatures, the global signature and the local signature. Both signatures are independent of parameterization (curve sampling). The global signature depends on the choice of the initial point and does not allow us to compare fragments of curves, and is therefore sensitive to occlusions. The local signature, although is slightly more sensitive to noise, is independent of the choice of the initial point and is not sensitive to occlusions in an image. It helps establish local equivalence of curves. The robustness of these invariants and signatures in their application to the problem of classification of noisy spatial curves extracted from a 3D object is analyzed. 1
On the Algebra of Differential Invariants of a Lie Pseudo–Group
, 2005
"... In this paper we prove some basic theoretical results underlying the moving frame theory of pseudogroups developed in the first two papers in this series. The first result demonstrates that a pseudogroup that acts locally freely at some sufficiently high order acts locally freely at all subsequen ..."
Abstract

Cited by 9 (4 self)
 Add to MetaCart
In this paper we prove some basic theoretical results underlying the moving frame theory of pseudogroups developed in the first two papers in this series. The first result demonstrates that a pseudogroup that acts locally freely at some sufficiently high order acts locally freely at all subsequent orders, and thus can be completely analyzed with the moving frame method. The second result is an algorithmic version of the Tresse–Kumpera theorem that states that a locally freely acting pseudogroup admits a finite generating system of differential invariants. The results will be based on moving frame methods and Gröbner basis algorithms.
Rational invariants of scalings from Hermite normal forms.
, 2012
"... Scalings form a class of group actions on affine spaces that have both theoretical and practical importance. A scaling is accurately described by an integer matrix. Tools from linear algebra are exploited to compute a minimal generating set of rational invariants, trivial rewriting and rational sect ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
Scalings form a class of group actions on affine spaces that have both theoretical and practical importance. A scaling is accurately described by an integer matrix. Tools from linear algebra are exploited to compute a minimal generating set of rational invariants, trivial rewriting and rational sections for such a group action. The primary tools used are Hermite normal forms and their unimodular multipliers. With the same line of ideas, a complete solution to the scaling symmetry reduction of a polynomial system is also presented.