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## Invariants of objects and their images under surjective maps (2015)

### Citations

4702 |
Multiple view geometry in computer vision.
- Hartley, Zisserman
- 2000
(Show Context)
Citation Context ...ons model pinhole cameras, while parallel projections provide a good approximation for a pinhole camera when the distance between a camera and an object is significantly greater than the object depth =-=[21]-=-. The formulation of Section 2 19 does not entirely cover these examples, since the associated group action of the affine group on R3 is not projectable. To handle such cases, in general, we identify ... |

1575 |
Applications of Lie Groups to Differential Equations,
- Olver
- 1993
(Show Context)
Citation Context ...variants Given 0 ≤ k ≤ ∞, let Jk(M, p) be the k-th order extended jet bundle consisting of equivalence classes of p-dimensional submanifolds of M under the equivalence relation of k-th order contact, =-=[38]-=-. In particular J0(M, p) = M . When l ≥ k ≥ 0, we use pilk : Jl(M, p)→ Jk(M, p) to denote the standard projection. Given a surjective map Π: M → N , let JkΠ(M, p) ⊂ Jk(M, p) be the open dense subset c... |

1496 |
3-D Computer Vision, A Geometric Viewpoint.
- Faugeras
- 1993
(Show Context)
Citation Context ...ge processing, and covers a broad spectrum of fundamental issues in computer vision, including stereo vision, structure from motion, shape from shading, projective invariants, etc.; see, for example, =-=[2, 4, 14, 16, 20, 32, 48]-=-. Our focus on differential invariants is motivated by the method of differential invariant signatures, [11], used to classify objects up to group transformations, including rigid motions, and equi-af... |

322 |
Geometric Partial Differential Equations and Image Processing.
- Sapiro
- 2001
(Show Context)
Citation Context ...ge processing, and covers a broad spectrum of fundamental issues in computer vision, including stereo vision, structure from motion, shape from shading, projective invariants, etc.; see, for example, =-=[2, 4, 14, 16, 20, 32, 48]-=-. Our focus on differential invariants is motivated by the method of differential invariant signatures, [11], used to classify objects up to group transformations, including rigid motions, and equi-af... |

282 |
An Invitation to 3-D Vision. From Images to Geometric Models
- Ma, Soatto, et al.
- 2004
(Show Context)
Citation Context ...ge processing, and covers a broad spectrum of fundamental issues in computer vision, including stereo vision, structure from motion, shape from shading, projective invariants, etc.; see, for example, =-=[2, 4, 14, 16, 20, 32, 48]-=-. Our focus on differential invariants is motivated by the method of differential invariant signatures, [11], used to classify objects up to group transformations, including rigid motions, and equi-af... |

249 |
Vorlesungen über Differentialgeometrie
- Blaschke
- 1923
(Show Context)
Citation Context ...pressions as their planar counterparts η and Dξ. Nonetheless, we will be using hats to emphasize that the former are defined on M , and to be consistent with the notation of Section 2.5. 3In Blaschke =-=[6]-=-, as well as in some other sources, the equi-affine curvature is defined to be 1/3 of the expression µ in (2.28). Our choice, however, leads to simpler numerical factors in the subsequent expressions.... |

161 | Classical Invariant Theory - Olver - 1999 |

119 | The Variational Bicomplex
- Anderson
- 1989
(Show Context)
Citation Context ...n Section 2.6. 2.5 Invariant differential forms and differential operators Turning to differential forms, we assume the reader is familiar with the basic variational bicomplex structure on jet space, =-=[1, 17, 29]-=-. As usual, for certain technical reasons, it is preferable to work on the infinite jet bundle even though all calculations are performed on jet bundles of finite order. As above, we introduce local c... |

114 | Visual Motion of Curves and Surfaces
- Cipolla, Giblin
- 2000
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98 | Differential and numerically invariant signature curves applied to object recognition
- Calabi, Olver, et al.
- 1998
(Show Context)
Citation Context ...n, shape from shading, projective invariants, etc.; see, for example, [2, 4, 14, 16, 20, 32, 48]. Our focus on differential invariants is motivated by the method of differential invariant signatures, =-=[11]-=-, used to classify objects up to group transformations, including rigid motions, and equi-affine, affine, centro-affine, and projective maps. Our analysis is founded on the method of equivariant movin... |

95 |
Moving coframes
- Fels, Olver
- 1999
(Show Context)
Citation Context ...o group transformations, including rigid motions, and equi-affine, affine, centro-affine, and projective maps. Our analysis is founded on the method of equivariant moving frames, as first proposed in =-=[17]-=-, and we will assume that the reader is familiar with the basic techniques. See [33, 43] for recent surveys of the method and many of its applications. In [23, 24], an algebraic interpretation of the ... |

85 | Méthode du Repère Mobile, la Théorie des Groupes Continus, et les Espaces Généralisés, Exposés de Géométrie - Cartan, La - 1935 |

79 |
The C–spectral sequence, Lagrangian formalism and conservation laws
- Vinogradov
- 1984
(Show Context)
Citation Context ...e horizontal forms, and hence the induced splitting, depend upon the choice of independent variable local coordinates. A more 8 intrinsic approach is based on filtrations and the C spectral sequence, =-=[50, 51]-=-; however, this extra level of abstraction is unnecessary in what follows. We use piH to denote the projection of a one-form onto its horizontal component, so dHF̂ = piH(dF̂ ) for any differential fun... |

46 | Joint invariant signatures
- Olver
(Show Context)
Citation Context ... curves under projections. Extensions of the method to signatures parametrized by joint invariants and joint differential invariants, also known as semi-differential invariants, [36], can be found in =-=[41]-=-. A wide range of image processing applications includes jigsaw puzzle assembly, [22], recognition of DNA supercoils, [49], distinguishing malignant from benign breast cancer tumors, [19], recovering ... |

41 | Invariant Euler-Lagrange equations and the invariant variational bicomplex
- Kogan, Olver
(Show Context)
Citation Context ...n Section 2.6. 2.5 Invariant differential forms and differential operators Turning to differential forms, we assume the reader is familiar with the basic variational bicomplex structure on jet space, =-=[1, 17, 29]-=-. As usual, for certain technical reasons, it is preferable to work on the infinite jet bundle even though all calculations are performed on jet bundles of finite order. As above, we introduce local c... |

32 |
Foundations of semi-differential invariants
- Moons, Pauwels, et al.
- 1995
(Show Context)
Citation Context ...espondence problem for curves under projections. Extensions of the method to signatures parametrized by joint invariants and joint differential invariants, also known as semi-differential invariants, =-=[36]-=-, can be found in [41]. A wide range of image processing applications includes jigsaw puzzle assembly, [22], recognition of DNA supercoils, [49], distinguishing malignant from benign breast cancer tum... |

29 | Smooth and algebraic invariants of a group action: local and global constructions
- Hubert, Kogan
(Show Context)
Citation Context ...iant moving frames, as first proposed in [17], and we will assume that the reader is familiar with the basic techniques. See [33, 43] for recent surveys of the method and many of its applications. In =-=[23, 24]-=-, an algebraic interpretation of the equivariant moving frame was developed, leading to an algorithm for constructing a generating set of rational invariants along with a set of algebraic invariants, ... |

28 | Moving frames for Lie pseudo–groups
- Olver, Pohjanpelto
(Show Context)
Citation Context ...ve maps, we will concentrate on finite-dimensional Lie group actions, although our analysis can, in principle, be extended to infinite-dimensional Lie pseudo-groups, using the techniques developed in =-=[45, 46]-=-. We will distinguish between projectable group actions, in which the group transformations respect the surjective map’s fibers, and the more general non-projectable actions. In the former case, there... |

24 | Differential invariant algebras of Lie pseudo-groups
- Olver, Pohjanpelto
(Show Context)
Citation Context ...ve maps, we will concentrate on finite-dimensional Lie group actions, although our analysis can, in principle, be extended to infinite-dimensional Lie pseudo-groups, using the techniques developed in =-=[45, 46]-=-. We will distinguish between projectable group actions, in which the group transformations respect the surjective map’s fibers, and the more general non-projectable actions. In the former case, there... |

23 | On reconstructing n-point configurations from the distribution of distances or
- Kemper, Boutin
(Show Context)
Citation Context ...in 2D and 3D images, [18]. Further applications of the moving framebased signatures include classical invariant theory, [5, 27, 28, 40], symmetry and equivalence of polygons and point configurations, =-=[8, 25]-=-, geometry of curves and surfaces in homogeneous spaces, with applications to Poisson structures and integrable systems, [34, 35], the design and analysis of geometric integrators and symmetry-preserv... |

21 | Symmetries of polynomials
- Berchenko, Olver
(Show Context)
Citation Context ... [20], and construction of integral invariant signatures for object recognition in 2D and 3D images, [18]. Further applications of the moving framebased signatures include classical invariant theory, =-=[5, 27, 28, 40]-=-, symmetry and equivalence of polygons and point configurations, [8, 25], geometry of curves and surfaces in homogeneous spaces, with applications to Poisson structures and integrable systems, [34, 35... |

19 | Computation of canonical forms for ternary cubics
- Kogan, Maza, et al.
- 2002
(Show Context)
Citation Context ... [20], and construction of integral invariant signatures for object recognition in 2D and 3D images, [18]. Further applications of the moving framebased signatures include classical invariant theory, =-=[5, 27, 28, 40]-=-, symmetry and equivalence of polygons and point configurations, [8, 25], geometry of curves and surfaces in homogeneous spaces, with applications to Poisson structures and integrable systems, [34, 35... |

19 |
A Practical Guide to the Invariant Calculus
- Mansfield
- 2010
(Show Context)
Citation Context ...fine, and projective maps. Our analysis is founded on the method of equivariant moving frames, as first proposed in [17], and we will assume that the reader is familiar with the basic techniques. See =-=[33, 43]-=- for recent surveys of the method and many of its applications. In [23, 24], an algebraic interpretation of the equivariant moving frame was developed, leading to an algorithm for constructing a gener... |

16 | Projective curvature and integral invariants
- Hann, Hickman
(Show Context)
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15 | Computation of invariants of Lie algebras by means of moving frames
- Boyko, Patera, et al.
(Show Context)
Citation Context ...grators and symmetry-preserving numerical schemes, [26, 37, 47], the determination of Casimir invariants of Lie algebras and the classification of subalgebras, with applications in quantum mechanics, =-=[7]-=-, and many more. In our analysis of the behavior of invariants under surjective maps, we will concentrate on finite-dimensional Lie group actions, although our analysis can, in principle, be extended ... |

13 | Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds, Annales de l’Institut Fourier,
- Beffa
- 2008
(Show Context)
Citation Context ...28, 40], symmetry and equivalence of polygons and point configurations, [8, 25], geometry of curves and surfaces in homogeneous spaces, with applications to Poisson structures and integrable systems, =-=[34, 35]-=-, the design and analysis of geometric integrators and symmetry-preserving numerical schemes, [26, 37, 47], the determination of Casimir invariants of Lie algebras and the classification of subalgebra... |

12 |
problèmes d’équivalence, in: Oeuvres Complètes
- Cartan, Les
- 1953
(Show Context)
Citation Context ...an be mapped to each other by a transformation belonging to a prescribed group or pseudo-group action. Élie Cartan’s solution to the equivalence problem for submanifolds under transformation groups, =-=[13]-=-, is based on the functional interrelationships among the associated differential invariants. Cartan’s result was reformulated through the introduction of the classifying submanifold, [39], subsequent... |

12 | Two algorithms for a moving frame construction
- Kogan
(Show Context)
Citation Context ...ection. Assume now that there is a subgroup G̃ ⊂ G that is isomorphic with the quotient group [G]. In this case, G factors as a product G = GN · G̃, and we can use inductive construction developed in =-=[30]-=- to determine the moving frame and the invariants. (More generally, one can apply the general recursive algorithm in [44] directly to the subgroup GN without requiring the existence of a suitable subg... |

9 | Structure from motion: theoretical foundations of a novel approach using custom built invariants
- Bazin, Boutin
(Show Context)
Citation Context ...udes jigsaw puzzle assembly, [22], recognition of DNA supercoils, [49], distinguishing malignant from benign breast cancer tumors, [19], recovering structure of three-dimensional objects from motion, =-=[3]-=-, classification of pro1Identification of the required differential invariants can be facilitated and systematized through the equivariant moving frame calculus and, specifically, the recurrence formu... |

9 | Classification of curves in 2D and 3D via affine integral signatures - Feng, Kogan, et al. |

9 | Symmetry preserving numerical schemes for partial differential equations and their numerical tests
- Rebelo, Valiquette
(Show Context)
Citation Context ...d surfaces in homogeneous spaces, with applications to Poisson structures and integrable systems, [34, 35], the design and analysis of geometric integrators and symmetry-preserving numerical schemes, =-=[26, 37, 47]-=-, the determination of Casimir invariants of Lie algebras and the classification of subalgebras, with applications in quantum mechanics, [7], and many more. In our analysis of the behavior of invarian... |

8 | Rational invariants of a group action
- Hubert, Kogan
(Show Context)
Citation Context ...iant moving frames, as first proposed in [17], and we will assume that the reader is familiar with the basic techniques. See [33, 43] for recent surveys of the method and many of its applications. In =-=[23, 24]-=-, an algebraic interpretation of the equivariant moving frame was developed, leading to an algorithm for constructing a generating set of rational invariants along with a set of algebraic invariants, ... |

7 |
Geometric integration via multi-space, Regular and Chaotic Dynamics 9
- Kim, Olver
- 2004
(Show Context)
Citation Context ...d surfaces in homogeneous spaces, with applications to Poisson structures and integrable systems, [34, 35], the design and analysis of geometric integrators and symmetry-preserving numerical schemes, =-=[26, 37, 47]-=-, the determination of Casimir invariants of Lie algebras and the classification of subalgebras, with applications in quantum mechanics, [7], and many more. In our analysis of the behavior of invarian... |

7 | Lectures on moving frames, in: Symmetries and Integrability of Difference Equations - Olver - 2011 |

6 | Polygon recognition and symmetry detection
- Boutin
(Show Context)
Citation Context ...in 2D and 3D images, [18]. Further applications of the moving framebased signatures include classical invariant theory, [5, 27, 28, 40], symmetry and equivalence of polygons and point configurations, =-=[8, 25]-=-, geometry of curves and surfaces in homogeneous spaces, with applications to Poisson structures and integrable systems, [34, 35], the design and analysis of geometric integrators and symmetry-preserv... |

6 | Automatic solution of jigsaw puzzles
- Hoff, Olver
(Show Context)
Citation Context ...nt invariants and joint differential invariants, also known as semi-differential invariants, [36], can be found in [41]. A wide range of image processing applications includes jigsaw puzzle assembly, =-=[22]-=-, recognition of DNA supercoils, [49], distinguishing malignant from benign breast cancer tumors, [19], recovering structure of three-dimensional objects from motion, [3], classification of pro1Identi... |

6 | Moving frames and differential invariants in centro–affine geometry
- Olver
(Show Context)
Citation Context ...1743. 2Supported in part by NSF Grant DMS 11–08894. 1 investigations. We will, in particular, derive relatively simple formulas relating the centroaffine invariants of a space curve, as classified in =-=[42]-=-, to the projective curvature invariant of its projections. The relationship between three-dimensional objects and their two-dimensional images under projection is a problem of major importance in ima... |

5 |
Image Processing for Cinema
- Bertalmı́o
- 2014
(Show Context)
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5 |
Inductive approach to Cartan’s moving frames method with applications to classical invariant theory
- Kogan
- 2000
(Show Context)
Citation Context ... [20], and construction of integral invariant signatures for object recognition in 2D and 3D images, [18]. Further applications of the moving framebased signatures include classical invariant theory, =-=[5, 27, 28, 40]-=-, symmetry and equivalence of polygons and point configurations, [8, 25], geometry of curves and surfaces in homogeneous spaces, with applications to Poisson structures and integrable systems, [34, 35... |

5 |
Hamiltonian Structures on the space of differential invariants of curves in flat semisimple homogenous manifolds
- Beffa, G
(Show Context)
Citation Context ...28, 40], symmetry and equivalence of polygons and point configurations, [8, 25], geometry of curves and surfaces in homogeneous spaces, with applications to Poisson structures and integrable systems, =-=[34, 35]-=-, the design and analysis of geometric integrators and symmetry-preserving numerical schemes, [26, 37, 47], the determination of Casimir invariants of Lie algebras and the classification of subalgebra... |

4 |
Object-image metrics for generalized weak perspective projection, in: Statistics and Analysis of Shapes
- Arnold, Stiller, et al.
- 2006
(Show Context)
Citation Context |

4 |
Research works of Romanian mathematicians on centro-affine geometry
- Cruceanu
- 2005
(Show Context)
Citation Context .... This action forms the foundation of affine geometry, and, for this reason, the previous linear action of GL(n) is sometimes referred to as the centroaffine group, underlying centro-affine geometry, =-=[15, 42]-=-. We also consider the action of the projective group PGL(n) = GL(n)/ {λ I | 0 6= λ ∈ R } on the projective space RPn−1 along with its local, linear fractional action on the dense open subset Rn−1 ⊂ R... |

4 |
Signature curves statistics of DNA supercoils
- Shakiban, Lloyd
- 2004
(Show Context)
Citation Context ...invariants, also known as semi-differential invariants, [36], can be found in [41]. A wide range of image processing applications includes jigsaw puzzle assembly, [22], recognition of DNA supercoils, =-=[49]-=-, distinguishing malignant from benign breast cancer tumors, [19], recovering structure of three-dimensional objects from motion, [3], classification of pro1Identification of the required differential... |

3 |
Object-image correspondence for curves under central and parallel projections, in
- Burdis, Kogan
- 2012
(Show Context)
Citation Context ...ubmanifold can also be classified by the dimension and, in the case of discrete symmetries, the index of its associated signature. Differential invariant signatures of families of curves were used in =-=[9, 10]-=- to establish a novel algorithm for solving the object-image correspondence problem for curves under projections. Extensions of the method to signatures parametrized by joint invariants and joint diff... |

3 | Recursive moving frames
- Olver
(Show Context)
Citation Context ...s as a product G = GN · G̃, and we can use inductive construction developed in [30] to determine the moving frame and the invariants. (More generally, one can apply the general recursive algorithm in =-=[44]-=- directly to the subgroup GN without requiring the existence of a suitable subgroup G̃.) These constructions allow one to determine the formulae relating the invariants and invariant differential form... |

2 | Global Lie–Tresse theorem - Kruglikov, Lychagin - 2011 |

1 | Object-image correspondence for algebraic curves under projections
- Burdis, Kogan, et al.
(Show Context)
Citation Context ...ubmanifold can also be classified by the dimension and, in the case of discrete symmetries, the index of its associated signature. Differential invariant signatures of families of curves were used in =-=[9, 10]-=- to establish a novel algorithm for solving the object-image correspondence problem for curves under projections. Extensions of the method to signatures parametrized by joint invariants and joint diff... |

1 |
Diagnosing breast cancer with symmetry of signature curves
- Grim, Shakiban
- 2014
(Show Context)
Citation Context ... be found in [41]. A wide range of image processing applications includes jigsaw puzzle assembly, [22], recognition of DNA supercoils, [49], distinguishing malignant from benign breast cancer tumors, =-=[19]-=-, recovering structure of three-dimensional objects from motion, [3], classification of pro1Identification of the required differential invariants can be facilitated and systematized through the equiv... |

1 |
On post–Lie algebras, Lie–Butcher series and moving
- Munthe–Kaas, Lundervold
(Show Context)
Citation Context ...d surfaces in homogeneous spaces, with applications to Poisson structures and integrable systems, [34, 35], the design and analysis of geometric integrators and symmetry-preserving numerical schemes, =-=[26, 37, 47]-=-, the determination of Casimir invariants of Lie algebras and the classification of subalgebras, with applications in quantum mechanics, [7], and many more. In our analysis of the behavior of invarian... |