Characterization of Bijective Discretized Rotations

Nouvel, Bertrand; Rémila, Éric

doi:10.1007/978-3-540-30503-3_19

Cited by 25 publications

(39 citation statements)

References 2 publications

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“…When G is dense (see Figure 2(a)), the reasoning of Nouvel and Rémila, originally used to discard 2D digitized irrational rotations as being bijective [13], shows that a corresponding 3D digitized rotation cannot be bijective as well. What differs from the 2D case is the possible existence of non-dense G with a dense factor (see Figure 2(b)).…”

Section: Propositionmentioning

confidence: 99%

“…The contributions known to us were geared toward understanding these alterations in Z 2 : Andres and Jacob provided some necessary conditions under which 2D digitized rotations are bijective [5]; Andres proposed quasi-shear rotations which are bijective but possibly generate errors, particularly for angles around π/2 [1]; Nouvel and Rémila studied the discrete structure induced by digitized rotations that are not bijective but generate no error [12,14]; moreover, they characterized the set of 2D bijective digitized rotations [13]. More recently, Roussillon and Coeurjolly used arithmetic properties of the Gaussian integers to give a different proof of the conditions for bijectivity of 2D digitized rotations [17].…”

Section: Introductionmentioning

confidence: 99%

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Bijectivity Certification of 3D Digitized Rotations

Pluta

Romon²,

Kenmochi

et al. 2016

Computational Topology in Image Context

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Section: Propositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bijectivity Certification of 3D Digitized Rotations

Pluta

Romon²,

Kenmochi

et al. 2016

Computational Topology in Image Context

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“…However, in the discrete spaces (i.e., Z n ), their analogues, namely discrete rotations, are more complex. The induced challenges are not simply due to highdimensionality: indeed, even in Z 2 , discrete rotations raise many difficulties, deriving mainly from their non-necessary bijectivity [1]. In this context, discrete rotations -and the closely related discrete rigid tansformations -have been widely investigated.…”

Section: Discrete Rotations and Discrete Rigid Transformationsmentioning

confidence: 99%

Efficient Neighbourhood Computing for Discrete Rigid Transformation Graph Search

Kenmochi¹,

Ngo

Talbot³

et al. 2014

Advanced Information Systems Engineering

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International audienceRigid transformations are involved in a wide variety of image processing applications, including image registration. In this context, we recently proposed to deal with the associated optimization problem from a purely discrete point of view, using the notion of discrete rigid transformation (DRT) graph. In particular, a local search scheme within the DRT graph to compute a locally optimal solution without any numerical approximation was formerly proposed. In this article, we extend this study, with the purpose to reduce the algorithmic complexity of the proposed optimization scheme. To this end, we propose a novel algorithmic framework for just-in-time computation of sub-graphs of interest within the DRT graph. Experimental results illustrate the potential usefulness of our approach for image registration

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“…The Pythagorean angles, as seen in [NR04] or [Vos93], are such that α = arctan(a/b) where a and b are issued from a Pythagorean triple (a, b, c) ∈ N 3 (such that a 2 + b 2 = c 2 ). c will be called the radius of the Pythagorean angle.…”

Section: Fundamental Lemmasmentioning

confidence: 99%

Incremental and Transitive Discrete Rotations

Nouvel

Rémila

2006

Lecture Notes in Computer Science

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A discrete rotation algorithm can be apprehended as a parametric map f α from Z[i] to Z[i], whose resulting permutation "looks like" the map induced by an Euclidean rotation. For this kind of algorithm, to be incremental means to compute successively all the intermediate rotated copies of an image for angles in-between 0 and a destination angle. The discretized rotation consists in the composition of an Euclidean rotation with a discretization; the aim of this article is to describe an algorithm which computes incrementally a discretized rotation. The suggested method uses only integer arithmetic and does not compute any sine nor any cosine. More precisely, its design relies on the analysis of the discretized rotation as a step function: the precise description of the discontinuities turns to be the key ingredient that will make the resulting procedure optimally fast and exact. A complete description of the incremental rotation process is provided, also this result may be useful in the specification of a consistent set of definitions for discrete geometry.

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Characterization of Bijective Discretized Rotations

Cited by 25 publications

References 2 publications

Bijectivity Certification of 3D Digitized Rotations

Bijectivity Certification of 3D Digitized Rotations

Efficient Neighbourhood Computing for Discrete Rigid Transformation Graph Search

Incremental and Transitive Discrete Rotations

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