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On the Continuity of Rotation Representations in Neural Networks [article]

Yi Zhou, Connelly Barnes, Jingwan Lu, Jimei Yang, Hao Li
2020 arXiv   pre-print
We also present continuous representations for the general case of the n-dimensional rotation group SO(n).  ...  While our main focus is on rotations, we also show that our constructions apply to other groups such as the orthogonal group and similarity transforms.  ...  We then investigate in Section 4.2 some continuous rotation representations. For the n dimensional rotation group SO(n), we present a continuous n 2 − n dimensional representation.  ... 
arXiv:1812.07035v4 fatcat:ym4w36c6vnfxrektpggsjjubja

Interpolating Rotations with Non-abelian Kuramoto Model on the 3-Sphere [chapter]

Zinaid Kapić, Aladin Crnkić
2021 Lecture Notes in Networks and Systems  
In addition, in order to gain a better insight, we have provided one short video illustrating the rotation of an object between two positions.  ...  The paper presents a novel method for interpolating rotations based on the non-Abelian Kuramoto model on sphere S3.  ...  Furthermore, to gain better insight, we have provided one short video demonstrating the continuous motion of that object.  ... 
doi:10.1007/978-3-030-90055-7_48 fatcat:6z5htpkd55cl3mupwfrx6god2e

Lifting smooth curves over invariants for representations of compact Lie groups, III [article]

Andreas Kriegl, Mark Losik, Peter W. Michor, Armin Rainer
2005 arXiv   pre-print
Any sufficiently often differentiable curve in the orbit space V/G of a real finite-dimensional orthogonal representation G → O(V) of a finite group G admits a differentiable lift into the representation  ...  Finite reflection groups and finite rotation groups in dimensions two and three are discussed in detail.  ...  characterization of finite rotation groups in R 3 (e.g  ... 
arXiv:math/0504101v1 fatcat:ve5gbjor6nbh5p6krfi5vmjyka

Two Dimensional Yau-Hausdorff Distance with Applications on Comparison of DNA and Protein Sequences

Kun Tian, Xiaoqian Yang, Qin Kong, Changchuan Yin, Rong L. He, Stephen S.-T. Yau, Yang Zhang
2015 PLoS ONE  
We propose a new approach, the Yau-Hausdorff method, which considers all translations and rotations when seeking the best match of graphical curves of DNA or protein sequences.  ...  The Yau-Hausdorff method can be used for measuring the similarity of DNA sequences based on two important tools: the Yau-Hausdorff distance and graphical representation of DNA sequences.  ...  It has potential broad application prospects in measuring the similarity of 2 or 3-dimensional curves, such as shape matching, image retrieval, and comparison of 3-dimensional protein structures.  ... 
doi:10.1371/journal.pone.0136577 pmid:26384293 pmcid:PMC4575136 fatcat:nkd3ta5orzfnzoxiun4qnddvca

Classification of three-dimensional thoracic deformities in adolescent idiopathic scoliosis from a multivariate analysis

Samuel Kadoury, Hubert Labelle
2011 European spine journal  
(T) and normal lordosis (L), with high rotation of plane of maximum curvature (55 cases), (3) hypo-kyphotic (T) and hyperlordosis (L) (21 cases) and (4) hyper-kyphotic (T) with strong vertebral rotation  ...  from the cohort of 3D spine reconstructions of Lenke Type-1 patients when compared with previous findings on the 3D classification of spinal deformities.  ...  At the end of the process, for each high-dimensional spine model, we obtained a simplified d-dimensional representation of that model denoted as x i which we can plot in {1, 2, 3,…, d} dimensions, based  ... 
doi:10.1007/s00586-011-2004-2 pmid:21879413 pmcid:PMC3252447 fatcat:qo5ygddtcngcdbys72oam247ma

Toponomic Quantum Computation [article]

C. Chryssomalakos, L. Hanotel, E. Guzmán-González, E. Serrano-Ensástiga
2022 arXiv   pre-print
We identify a special class of subspaces, for which a sequence of rotations results in a non-abelian holonomy of a topological nature, so that it is invariant under any SO(3)-perturbation.  ...  Holonomic quantum computation makes use of non-abelian geometric phases, associated to the evolution of a subspace of quantum states, to encode logical gates.  ...  ACKNOWLEDGEMENTS CC, LH, and ESE would like to acknowledge partial financial support from the DGAPA PAPIIT project IN111920 of UNAM.  ... 
arXiv:2202.01973v1 fatcat:diwexk3qinfufoa27i7qpbxjde

Supergroup approach to the Hubbard model

V. M. Zharkov, V. S. Kirchanov
2011 Theoretical and mathematical physics  
We reveal a relation of even Hubbard operators to the spinor representation of the generators of the rotation group of four-dimensional spaces.  ...  We propose a procedure for constructing a matrix representation of translation generators, yielding a curved space on which dynamic superfields are defined.  ...  Acknowledgments One of the authors (V. M. Zh.) is grateful to V. V. Kiselev and A. B. Borisov for the discussion on the problems touched upon here.  ... 
doi:10.1007/s11232-011-0015-0 fatcat:puo5mmvnofgafmzwz7cbtpkihe

Perceptual-cognitive universals as reflections of the world

Roger N. Shepard
2001 Behavioral and Brain Sciences  
inverse from the three-dimensional linear group of terrestrial transformations of the invariant solar source. (3) Kinds of objects support optimal generalization and categorization when represented, in  ...  fullest and simplest way when represented as points and connecting geodesic paths in the six-dimensional manifold jointly determined by the Euclidean group of threedimensional space and the symmetry group  ...  subgroup SO(3), and, more generally, helical curves in the full six-dimensional Euclidean group E+.  ... 
doi:10.1017/s0140525x01000012 fatcat:5lomrhbzyraopg3p7bkd4bw3ti

Perceptual-cognitive universals as reflections of the world

Roger N. Shepard
1994 Psychonomic Bulletin & Review  
inverse from the three-dimensional linear group of terrestrial transformations of the invariant solar source. (3) Kinds of objects support optimal generalization and categorization when represented, in  ...  fullest and simplest way when represented as points and connecting geodesic paths in the six-dimensional manifold jointly determined by the Euclidean group of threedimensional space and the symmetry group  ...  subgroup SO(3), and, more generally, helical curves in the full six-dimensional Euclidean group E+.  ... 
doi:10.3758/bf03200759 pmid:24203412 fatcat:ooqrxkrkoja6nlm2wsyg7jzshq

Rolling Rotations for Recognizing Human Actions from 3D Skeletal Data

Raviteja Vemulapalli, Rama Chellappa
2016 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)  
Since 3D rotations are members of the special orthogonal group SO 3 , our skeletal representation becomes a point in the Lie group SO 3 × . . . × SO 3 , which is also a Riemannian manifold.  ...  Then, using this representation, we model human actions as curves in this Lie group.  ...  Acknowledgements: This research was supported by a MURI from the US Office of Naval Research under the grant 1141221258513.  ... 
doi:10.1109/cvpr.2016.484 dblp:conf/cvpr/VemulapalliC16 fatcat:aee7dkysqvfafniflslk5yco2m

New Topological Configurations in the Continuous Heisenberg Spin Chain: Lower Bound for the Energy

Rossen Dandoloff
2015 Advances in Condensed Matter Physics  
If, for example, one end of the space curve is rotated by an angle of 2πrelative to the other, the Frenet frame traces out a noncontractible loop inSO(3)and this defines a new class of topological spin  ...  In order to study the spin configurations of the classical one-dimensional Heisenberg model, we map the normalized unit vector, representing the spin, on a space curve.  ...  group (3) .  ... 
doi:10.1155/2015/954524 fatcat:t7x3nppvk5dg7nqefqh757fblu

Diffeomorphic sulcal shape analysis for cortical surface registration

Shantanu H. Joshi, Ryan P. Cabeen, Anand A. Joshi, Roger P. Woods, Katherine L. Narr, Arthur W. Toga
2010 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition  
We propose the analysis of sulcal and gyral patterns by representing them by continuous open curves in ℝ 3 .  ...  The space of such curves, also termed as the shape manifold is equipped with a Riemannian L 2 metric on the tangent space, and shows desirable properties while matching shapes of sulci.  ...  Acknowledgments This research was partially supported by the National Institute of Health through the National Center for Research Resources (NCRR) for Center for Computational Biology (CCB), Grant U54  ... 
doi:10.1109/cvpr.2010.5540177 pmid:21076690 pmcid:PMC2979342 dblp:conf/cvpr/JoshiCJWNT10 fatcat:v4pjmxttejfwpagl3ogwfbfxia

An Efficient and Robust Technique for Region Based Shape Representation and Retrieval

Dengsheng Zhang, Melissa Chen Yi Lim
2007 6th IEEE/ACIS International Conference on Computer and Information Science (ICIS 2007)  
Specifically, we compute a one dimensional signature function from a two dimensional region shape and use it for region shape representation and retrieval.  ...  As the result of their complex computations, the retrieval performance of these techniques is not robust.  ...  In the test of scaling and rotation invariance, the PR curve of PRS is always above the 3 other curves and is significantly higher than the others.  ... 
doi:10.1109/icis.2007.55 dblp:conf/ACISicis/ZhangL07 fatcat:wjwd4lcmrjhmhhl2zdpgcg3o2u

Human Action Recognition by Representing 3D Skeletons as Points in a Lie Group

Raviteja Vemulapalli, Felipe Arrate, Rama Chellappa
2014 2014 IEEE Conference on Computer Vision and Pattern Recognition  
Since 3D rigid body motions are members of the special Euclidean group SE(3), the proposed skeletal representation lies in the Lie group SE(3)×. . .×SE(3), which is a curved manifold.  ...  Since classification of curves in this Lie group is not an easy task, we map the action curves from the Lie group to its Lie algebra, which is a vector space.  ...  Acknowledgements: This research was supported by a MURI from the US Office of Naval Research under the grant 1141221258513.  ... 
doi:10.1109/cvpr.2014.82 dblp:conf/cvpr/VemulapalliAC14 fatcat:gwn7cqiasfaw5fgarahn7husx4

Frequency Distribution and Dispersion Curves of Crystal Vibrations of Perdeuterated Polyethylene

Teizo Kitagawa, Tatsuo Miyazawa
1970 Polymer journal  
Frequency dispersion curves for boundaries of the first Brillouin zone were calculated and were assigned to symmetry classes of k groups on the basis of eigenvectors in symmetry co-ordinate space.  ...  Degeneracies and compatibility relations of frequency dispersion curves were derived from irreducible representations of k groups.  ...  For the F(n, n, o,) point, the k group is again C2 u and irreducible representations are one-dimensional.  ... 
doi:10.1295/polymj.1.471 fatcat:fuobq2oa4bc5noboa6ype3opti
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