1: Neighborhood motion maps of G U 2 , as label maps ,
as label maps, for ? ? (? 1 , ? 2 ) that differ from those for ? ? (0, ? 1 ) Each label (p, q) corresponds to the frame f ? p,q . Neighborhood motion maps which correspond to non-injective zones are marked by brown dashed frames. The edges of the neighborhood motion maps graph are marked by color line segments which connect different neighborhood motions maps (see Chapter 3 of Part I). The elements which have not changed with respect to the set ,
? 3 ) that differ from those for ? ? (? 1 , ? 2 ) Each label (p, q) corresponds to the frame f ? p,q . Neighborhood motion maps which correspond to non-injective zones are marked by brown dashed frames. The edges of the neighborhood motion maps graph are marked by color line segments which connect different neighborhood motions maps (see Chapter 3 of Part I). The elements which have not changed with respect to the set ,
Neighborhood motion maps for G U 2 (8-neighborhood case) (-4,4) (-3,4) (-2,4) (-1,4) (0,4), pp.4-6 ,
as label maps, for ? ? (? 3 , ? 4 ) that differ from those for ? ? (? 2 , ? 3 ) Each label (p, q) corresponds to the frame f ? p,q . Neighborhood motion maps which correspond to non-injective zones are marked by brown dashed frames. The edges of the neighborhood motion maps graph are marked by color line segments which connect different neighborhood motions maps (see Chapter 3 of Part I). The elements which have not changed with respect to the set ,
Neighborhood motion maps for G U 2 (8-neighborhood case) (-4,4) (-3,4) (-2,4) (-1,4) (0,4), pp.4-6 ,
1: The set of neighborhood motion maps M 1 , for rotation angles ? ? (? 0 , ? 1 ), visualized by the label map L U 1 (see Chapter ,
The set of neighborhood motion maps M 1 , for rotation angles ? ? (? 1 ,
The set of neighborhood motion maps M 1 , for rotation angles ? ? (? 2 ,
Object tracking, ACM Computing Surveys, vol.38, issue.4, 2006. ,
DOI : 10.1145/1177352.1177355
Halftoning by Rotating Non-Bayer Dispersed Dither Arrays. Milestone Series, pp.238-255, 1999. ,
DOI : 10.1117/12.207537
URL : http://diwww.epfl.ch/w3lsp/pub/papers/colour/SPIE95_RotatedNonBayer.pdf
Rotated dispersed dither, Proceedings of the 21st annual conference on Computer graphics and interactive techniques , SIGGRAPH '94, pp.123-130, 1994. ,
DOI : 10.1145/192161.192188
Digital topology: Introduction and survey, Computer Vision, Graphics, and Image Processing, vol.48, issue.3, pp.357-393, 1989. ,
DOI : 10.1016/0734-189X(89)90147-3
Digital Geometry: Geometric Methods for Digital Picture Analysis, 2004. ,
Hexagonal Image Processing: A Practical Approach Advances in Pattern Recognition, 2005. ,
Image Analysis and Mathematical Morphology, 1982. ,
Bijective Digitized Rigid Motions on Subsets of the Plane, Journal of Mathematical Imaging and Vision, vol.38, issue.4, pp.84-105, 2017. ,
DOI : 10.1145/1177352.1177355
URL : https://hal.archives-ouvertes.fr/hal-01497610
Honeycomb Geometry: Rigid Motions on the Hexagonal Grid, DGCI, pp.33-45, 2017. ,
DOI : 10.1016/j.tcs.2007.03.032
URL : https://hal.archives-ouvertes.fr/hal-01497608
Configurations induced by discrete rotations: periodicity and quasi-periodicity properties, Discrete Applied Mathematics, vol.147, issue.2-3, pp.325-343, 2005. ,
DOI : 10.1016/j.dam.2004.09.018
URL : https://doi.org/10.1016/j.dam.2004.09.018
On Colorations Induced by Discrete Rotations, DGCI, volume 2886 of Lecture Notes in Computer Science, pp.174-183, 2003. ,
DOI : 10.1007/978-3-540-39966-7_16
Graphics Gems. chapter A Fast Algorithm for General Raster Rotation, pp.179-195, 1990. ,
The Quasi-Shear rotation, DGCI, pp.307-314, 1996. ,
DOI : 10.1007/3-540-62005-2_26
URL : https://link.springer.com/content/pdf/10.1007%2F3-540-62005-2_26.pdf
Rotations in 2D and 3D Discrete Spaces, 2010. ,
URL : https://hal.archives-ouvertes.fr/tel-00596947
Using Pythagorean Triangles to Approximate Angles, The American Mathematical Monthly, vol.95, issue.6, pp.540-541, 1988. ,
DOI : 10.1080/00029890.1988.11972043
Reproduction Couleur par Trames Irrégulières et Semi-régulières, 1995. ,
Géométrie Discrète, Calcul en Nombres Entiers et Algorithmique ,
On Discrete Rotations, 5th International Workshop on Discrete Geometry for Computer Imagery, pp.161-174, 1995. ,
Cercles Discrets et Rotations Discrètes, 1992. ,
Modélisation Analytique Discrète d'Objets Géométriques. Habilitation à diriger des recherches, 2000. ,
Characterization of Bijective Discretized Rotations by Gaussian Integers ,
URL : https://hal.archives-ouvertes.fr/hal-01259826
Incremental and Transitive Discrete Rotations, IW- CIA, pp.199-213393, 2006. ,
DOI : 10.1007/11774938_16
URL : https://hal.archives-ouvertes.fr/hal-00016037
Topology-Preserving Conditions for 2D Digital Images Under Rigid Transformations, Journal of Mathematical Imaging and Vision, vol.8, issue.2, pp.418-433, 2014. ,
DOI : 10.1023/A:1008273227913
URL : https://hal.archives-ouvertes.fr/hal-00838183
Discrete rigid registration: A local graph-search approach, Discrete Applied Mathematics, vol.216, pp.461-481, 2017. ,
DOI : 10.1016/j.dam.2016.05.005
URL : https://hal.archives-ouvertes.fr/hal-01306035
On 2D constrained discrete rigid transformations, Annals of Mathematics and Artificial Intelligence, vol.21, issue.11, pp.163-193, 2014. ,
DOI : 10.1016/S0262-8856(03)00137-9
URL : https://hal.archives-ouvertes.fr/hal-00838184
Topology-Preserving Rigid Transformation of 2D Digital Images, IEEE Transactions on Image Processing, vol.23, issue.2, pp.885-897 ,
DOI : 10.1109/TIP.2013.2295751
URL : https://hal.archives-ouvertes.fr/hal-00795054
Geometric transformations on the hexagonal grid, IEEE Transactions on Image Processing, vol.4, issue.9, pp.1213-1222, 1995. ,
DOI : 10.1109/83.413166
Computing upper and lower bounds of rotation angles from digital images, Pattern Recognition, vol.42, issue.8, pp.1708-1717, 2009. ,
DOI : 10.1016/j.patcog.2008.12.027
URL : https://hal.archives-ouvertes.fr/hal-00622416
Characterization of Bijective Digitized Rotations on the Hexagonal Grid, Journal of Mathematical Imaging and Vision, vol.38, issue.4, 2018. ,
DOI : 10.1145/1177352.1177355
URL : https://hal.archives-ouvertes.fr/hal-01540772
Properties of Eisenstein Triples, Mathematics Magazine, vol.13, issue.1, pp.12-25, 2012. ,
DOI : 10.2307/3595782
Integer-sided Triangles with an Angle of 60 ? . The Mathematical Gazette, pp.261-266, 1982. ,
DOI : 10.2307/3615511
Introduction to Classical Geometries. Birkhäuser, 2007. ,
Discrete rotations and symbolic dynamics, Theoretical Computer Science, vol.380, issue.3, pp.276-285, 2007. ,
DOI : 10.1016/j.tcs.2007.03.032
Image registration methods: a survey, Image and Vision Computing, vol.21, issue.11, pp.977-1000, 2003. ,
DOI : 10.1016/S0262-8856(03)00137-9
Algorithms in Real Algebraic Geometry, 2005. ,
DOI : 10.1007/978-3-662-05355-3
URL : https://hal.archives-ouvertes.fr/hal-01083587
Quantifier elimination for real closed fields by cylindrical algebraic decompostion, ATFL, pp.134-183, 1975. ,
DOI : 10.1007/3-540-07407-4_17
On the computational complexity and geometry of the first-order theory of the reals. Part I: Introduction. Preliminaries. The geometry of semi-algebraic sets. The decision problem for the existential theory of the reals, Journal of Symbolic Computation, vol.13, issue.3, pp.255-299, 1992. ,
DOI : 10.1016/S0747-7171(10)80003-3
Semialgebraic Sard Theorem for Generalized Critical
Values, Journal of Differential Geometry, vol.56, issue.1, pp.67-92, 2000. ,
DOI : 10.4310/jdg/1090347525
URL : https://doi.org/10.4310/jdg/1090347525
Three-Dimensional Rotations by Three Shears, Graphical Models and Image Processing, vol.59, issue.2, pp.89-95, 1997. ,
DOI : 10.1006/gmip.1997.0420
URL : http://pm1.bu.edu/~tt/publ/rot-gmip.ps
3D Volume Rotation Using Shear Transformations, Graphical Models, vol.62, issue.4, pp.308-322, 2000. ,
DOI : 10.1006/gmod.2000.0525
URL : http://www-users.cs.umn.edu/~baoquan/papers/rot.pdf
3D discrete rotations using hinge angles, Theoretical Computer Science, vol.412, issue.15, pp.1378-1391, 2011. ,
DOI : 10.1016/j.tcs.2010.10.031
URL : https://hal.archives-ouvertes.fr/hal-00734881
Quadric Arrangement in Classifying Rigid Motions of a 3D Digital Image, CASC, pp.426-443, 2016. ,
DOI : 10.1016/S0262-8856(03)00137-9
URL : https://hal.archives-ouvertes.fr/hal-01334257
Bijectivity Certification of 3D Digitized Rotations, CTIC, pp.30-41, 2016. ,
DOI : 10.1007/978-0-85729-760-0
URL : https://hal.archives-ouvertes.fr/hal-01315226
A Mathematical Introduction to Robotic Manipulation, 52] K. Kanatani. Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics, 1994. ,
The Collected Mathematical Papers of Arthur Cayley, 1898. ,
Multi-resolution Methods for Modeling and Control of Dynamical Systems, 2008. ,
DOI : 10.1201/9781584887706
Largest Triangle with Vertices in the Unit Cube Mathematics Stack Exchange . URL https://math.stackexchange.com/q/44499, pp.2011-2017 ,
Unsolved Problems in Geometry, 1994. ,
DOI : 10.1007/978-1-4612-0963-8
Theory of Linear and Integer Programming, 1998. ,
Letter to the Editor, American Mathematical Monthly, vol.94, issue.8, pp.757-758, 1987. ,
A linear space algorithm for computing the hermite normal form, Proceedings of the 2001 international symposium on Symbolic and algebraic computation , ISSAC '01, pp.231-236, 2001. ,
DOI : 10.1145/384101.384133
URL : http://www-cse.ucsd.edu/~daniele/papers/HNFalg.ps
Fast computation of Hermite normal forms of random integer matrices, Journal of Number Theory, vol.130, issue.7, pp.1675-1683, 2010. ,
DOI : 10.1016/j.jnt.2010.01.017
URL : https://hal.archives-ouvertes.fr/hal-00798442
Topological Alterations of 3D Digital Images Under Rigid Transformations, 2014. ,
URL : https://hal.archives-ouvertes.fr/hal-01333586
Faster two-dimensional pattern matching with rotations, Theoretical Computer Science, vol.368, issue.3, pp.196-204, 2006. ,
DOI : 10.1016/j.tcs.2006.09.012
URL : https://doi.org/10.1016/j.tcs.2006.09.012
On the computation of an arrangement of quadrics in 3D, Computational Geometry, vol.30, issue.2, pp.145-164, 2005. ,
DOI : 10.1016/j.comgeo.2004.05.003
URL : https://hal.archives-ouvertes.fr/inria-00350858
Digital Homeomorphisms in Deformable Registration, IPMI, pp.211-222, 2007. ,
DOI : 10.1007/978-3-540-73273-0_18
Ehresmann Fibrations and Palais-Smale Conditions for Morphisms of Finsler Manifolds, The Annals of Mathematics, vol.146, issue.3, pp.647-691, 1997. ,
DOI : 10.2307/2952457
Quantitative Generalized Bertini-Sard Theorem for Smooth Affine Varieties, Discrete & Computational Geometry, vol.34, issue.4, pp.659-678, 2005. ,
DOI : 10.1007/s00454-005-1203-1
URL : https://hal.archives-ouvertes.fr/hal-00389073
On asymptotic critical values of a complex polynomial, Journal f??r die reine und angewandte Mathematik (Crelles Journal), vol.36, issue.565, pp.1-11, 2003. ,
DOI : 10.1515/crll.2003.101
Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 1996. ,
Topological Characterization of Finite Mappings, Bulletin of the Polish Academy of Sciences ? Mathematics, vol.49, issue.3, pp.279-283, 2001. ,
Properness Defects of Projections and Computation of at Least One Point in Each Connected Component of a Real Algebraic Set ,
URL : https://hal.archives-ouvertes.fr/inria-00099962
Properness defects of projection and minimal discriminant variety, Journal of Symbolic Computation, vol.46, issue.10, pp.1139-1157, 2011. ,
DOI : 10.1016/j.jsc.2011.05.013
URL : https://hal.archives-ouvertes.fr/hal-01148309
Efficient isolation of polynomial's real roots, Journal of Computational and Applied Mathematics, vol.162, issue.1, pp.33-50, 2004. ,
DOI : 10.1016/j.cam.2003.08.015
URL : https://doi.org/10.1016/j.cam.2003.08.015
Global optimization using interval analysis ? the multi-dimensional case, Numerische Mathematik, vol.14, issue.3, pp.247-270, 1980. ,
DOI : 10.1007/BF01396702
Interval Methods for Systems of Equations. Encyclopedia of Mathematics and its Applications, 1991. ,
Quadratic interval refinement for real roots, ACM Communications in Computer Algebra, vol.48, issue.1/2, pp.3-12, 2014. ,
DOI : 10.1145/2644288.2644291
URL : http://arxiv.org/pdf/1203.1227.pdf