3D discrete rotations using hinge angles

Thibault, Yohan; Sugimoto, Akihiro; Kenmochi, Yukiko

doi:10.1016/j.tcs.2010.10.031

Cited by 5 publications

(4 citation statements)

References 17 publications

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“…These are non-trivial questions, and their difficulty increases with the dimension of the Cartesian grid [46]. Indeed, most of these works deal with Z 2 [4, 5, 7, 8, 25, 33, 38-40, 42-45, 50, 54]; fewer with Z 3 [41,49,56] or Z n [18].…”

Section: Introductionmentioning

confidence: 99%

Homotopic Affine Transformations in the 2D Cartesian Grid

et al. 2022

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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Section: Introductionmentioning

confidence: 99%

Homotopic Affine Transformations in the 2D Cartesian Grid

et al. 2022

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“…Translations [5,19], rotations [1,2,6,13,27,28,31,37] and more generally rigid motions [22-26, 29, 32] in the Cartesian grids have been studied with various purposes: describing the combinatorial structure of these transformations with respect to R n vs. Z n [5,6,19,22,30,38], guaranteeing their bijectivity [1,2,13,27,31,32,37] or transitivity [28] in Z n , preserving geometrical properties [24,25] and, less frequently, ensuring their topological invariance [23,26] in Z n . These are non-trivial questions, and their difficulty increases with the dimension of the Cartesian grid [29].…”

Section: Introductionmentioning

confidence: 99%

“…These are non-trivial questions, and their difficulty increases with the dimension of the Cartesian grid [29]. Indeed, most of these works deal with Z 2 [1, 2, 5, 6, 13, 19, 22, 23, 25-28, 32, 37]; fewer with Z 3 [24,31,38].…”

Section: Introductionmentioning

confidence: 99%

Homotopic Digital Rigid Motion: An Optimization Approach on Cellular Complexes

Passat

Ngo

Kenmochi

2021

Lecture Notes in Computer Science

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Topology preservation is a property of rigid motions in R 2 , but not in Z 2 . In this article, given a binary object X ⊂ Z 2 and a rational rigid motion R, we propose a method for building a binary object X R ⊂ Z 2 resulting from the application of R on a binary object X. Our purpose is to preserve the homotopy-type between X and X R . To this end, we formulate the construction of X R from X as an optimization problem in the space of cellular complexes with the notion of collapse on complexes. More precisely, we define a cellular space H by superimposition of two cubical spaces F and G corresponding to the canonical Cartesian grid of Z 2 where X is defined, and the Cartesian grid induced by the rigid motion R, respectively. The object X R is then computed by building a homotopic transformation within the space H, starting from the cubical complex in G resulting from the rigid motion of X with respect to R and ending at a complex fitting X R in F that can be embedded back into Z 2 .

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“…However, there are few algorithms available for generating all the transformed images from a given image patch. Algorithms known to us are: 2D rotations [21]; 3D rotations around a given rational axis [30,31]; 2D rigid motions [17,23] and 2D affine transformations [10]. However, none of them can be applied to 3D rigid motions.…”

Section: Introductionmentioning

confidence: 99%

Quadric Arrangement in Classifying Rigid Motions of a 3D Digital Image

Pluta

Moroz

Kenmochi

et al. 2016

Computer Algebra in Scientific Computing

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International audienceRigid motions are fundamental operations in image processing. While bijective and isometric in $\mathbb{R}^3$, they lose these properties when digitized in $\mathbb{Z}^3$. To understand how the digitization of 3D rigid motions affects the topology and geometry of a chosen image patch, we classify the rigid motions according to their effect on the image patch. This classification can be described by an arrangement of hypersurfaces in the parameter space of 3D rigid motions of dimension six. However, its high dimensionality and the existence of degenerate cases make a direct application of classical techniques, such as cylindrical algebraic decomposition or critical point method, difficult. We show that this problem can be first reduced to computing sample points in an arrangement of quadrics in the 3D parameter space of rotations. Then we recover information about remaining three parameters of translation. We implemented an ad-hoc variant of state-of-the-art algorithms and applied it to an image patch of cardinality $7$. This leads to an arrangement of 81 quadrics and we recovered the classification in less than one hour on a machine equipped with 40 cores

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3D discrete rotations using hinge angles

Cited by 5 publications

References 17 publications

Homotopic Affine Transformations in the 2D Cartesian Grid

Homotopic Affine Transformations in the 2D Cartesian Grid

Homotopic Digital Rigid Motion: An Optimization Approach on Cellular Complexes

Quadric Arrangement in Classifying Rigid Motions of a 3D Digital Image

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