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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
Automatic Solution of Jigsaw Puzzles
"... We present a method for automatically solving apictorial jigsaw puzzles that is based on an extension of the method of differential invariant signatures. Our algorithms are designed to solve challenging puzzles, without having to impose any restrictive assumptions on the shape of the puzzle, the sha ..."
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Cited by 7 (6 self)
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We present a method for automatically solving apictorial jigsaw puzzles that is based on an extension of the method of differential invariant signatures. Our algorithms are designed to solve challenging puzzles, without having to impose any restrictive assumptions on the shape of the puzzle, the shapes of the individual pieces, or their intrinsic arrangement. As a demonstration, the method was successfully used to solve two commercially available puzzles. Keywords: jigsaw puzzle, curvature, Euclidean signature, bivertex arc, piece fitting, piece locking
Rotations + reflections:
, 2014
"... Next to the concept of a function, which is the most important concept pervading the whole of mathematics, the concept of a group is of the greatest significance in the various branches of mathematics and its applications. — P.S. Alexandroff Groups Definition. A group G is a set with a binary operat ..."
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Next to the concept of a function, which is the most important concept pervading the whole of mathematics, the concept of a group is of the greatest significance in the various branches of mathematics and its applications. — P.S. Alexandroff Groups Definition. A group G is a set with a binary operation (g, h) #− → g · h ∈ G satisfying • Associativity: g · (h · k) = (g · h) · k • Identity: g · e = g = e · g • Inverse: g · g−1 = e = g−1 · g Examples: • G = R — addition • G = R+ — multiplication • G = GL(n) = {detA ' = 0} — matrix multiplication • G = SO(n) = {AT = A−1, detA = +1} — rotation group Groups Definition. A group G is a set with a binary operation (g, h) #− → g · h ∈ G satisfying • Associativity: g · (h · k) = (g · h) · k • Identity: g · e = g = e · g • Inverse: g · g−1 = e = g−1 · g Examples: • G = R — addition • G = R+ — multiplication • G = GL(n) = {detA ' = 0} — matrix multiplication • G = SO(n) = {AT = A−1, detA = +1} — rotation group Symmetry Definition. A symmetry of a set S is a transformation that preserves it: g · S = S! ! The set of symmetries forms a group, called the symmetry group of the set S. Symmetry Definition. A symmetry of a set S is a transformation that preserves it: g · S = S! ! The set of symmetries forms a group GS, called the symmetry group of the set S.
Modern Developments in the Theory and Applications of Moving Frames
, 2014
"... Abstract. This article discusses recent advances in the general equivariant approach to the method of moving frames, concentrating on finitedimensional Lie group actions. A few of the many applications — to geometry, invariant theory, differential equations, and image processing — are presented. ..."
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Abstract. This article discusses recent advances in the general equivariant approach to the method of moving frames, concentrating on finitedimensional Lie group actions. A few of the many applications — to geometry, invariant theory, differential equations, and image processing — are presented.
Surface Reconstruction Using Differential Invariant Signatures
, 2014
"... This thesis addresses the problem of reassembling a broken surface. Three dimensional curve matching is used to determine shared edges of broken pieces. In practice, these pieces may have different orientation and position in space, so edges cannot be directly compared. Instead, a differential inva ..."
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This thesis addresses the problem of reassembling a broken surface. Three dimensional curve matching is used to determine shared edges of broken pieces. In practice, these pieces may have different orientation and position in space, so edges cannot be directly compared. Instead, a differential invariant signature is used to make the comparison. A similarity score between edge signatures determines if two pieces share an edge. The Procrustes algorithm is applied to find the translations and rotations that best fit shared edges. The method is implemented in Matlab, and tested on a broken spherical surface. i ii Acknowledgments I would like to thank Professor Robert Thompson and Yiwen Hu for their vision and contributions to this project. iii