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Characterization of Bijective Digitized Rotations on the Hexagonal Grid

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Characterization of Bijective Digitized Rotations on the Hexagonal Grid Characterization of bijective digitized rotations on the hexagonal grid Kacper Pluta, Tristan Roussillon, David Cœurjolly, Pascal Romon, Yukiko Kenmochi, Victor Ostromoukhov To cite this version: Kacper Pluta, Tristan Roussillon, David Cœurjolly, Pascal Romon, Yukiko Kenmochi, et al.. Char- acterization of bijective digitized rotations on the hexagonal grid. 2017. hal-01540772v1 HAL Id: hal-01540772 https://hal.archives-ouvertes.fr/hal-01540772v1 Preprint submitted on 16 Jun 2017 (v1), last revised 30 Nov 2017 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Noname manuscript No. (will be inserted by the editor) Characterization of bijective digitized rotations on the hexagonal grid Kacper Pluta · Tristan Roussillon · David Cœurjolly · Pascal Romon · Yukiko Kenmochi · Victor Ostromoukhov the date of receipt and acceptance should be inserted later Abstract Digitized rotations on discrete spaces are tention than ones defined on the hexagonal grid. usually defined as the composition of a Euclidean Indeed, the square grid is predominant over other rotation and a rounding operator; they are in gen- grids in fields such as image processing, mostly eral not bijective. Nevertheless, it is well known because of its common use by image acquisition that digitized rotations defined on the square grid devices, even though it is burdened with funda- are bijective for some specific angles. This infinite mental topological problems – one has to chose family of angles has been characterized by Nou- between different connectivity relations for objects vel and Rémila and more recently by Roussillon of interest and their complements [8, 9]. and Cœurjolly. In this article, we characterize bijec- tive digitized rotations on the hexagonal grid using On the contrary, the hexagonal grid is free from arithmetical properties of the Eisenstein integers. these problems since it possesses the following properties: equidistant neighbors—each hexagon has six equidistant neighbors; uniform connectiv- ity—there is only one type of connectivity [10,17]. 1 Introduction Still, memory addressing for representing objects Rotations defined on discrete spaces are simple yet defined on hexagonal grids and sampling of contin- crucial operations in many image processing appli- uous signals with the hexagonal grids, are relatively cations involving 2D data. While rotations defined less efficient. However, several approaches to tackle these issues have been provided [10]. on R2 are isometric and bijective, their digitized cousins, in general, do not preserve distances and Concerning the characterization of the bijec- are not bijective. Over last 20 years, digitized ro- tive digitized rotations on the square grid, the most tations on the square grid have attracted more at- important studies are the following: Andres and Kacper Pluta and Yukiko Kenmochi Jacob were the first to give a sufficient condition Université Paris-Est, LIGM, CNRS-ESIEE Paris, France for bijectivity [7]; Nouvel and Rémila prove its Kacper Pluta and Pascal Romon necessity [12] using their framework introduced Université Paris-Est, LAMA, France in [11]; more recently Roussillon and Cœurjolly Tristan Roussillon, David Cœurjolly and Victor Ostro- provided another proof which is based on the arith- moukhov metical properties of Gaussian integers [16]. It is Université de Lyon, CNRS, LIRIS, UMR5205, F-69622, also worth to mention that Pluta et al. used an ex- France tension of the framework proposed originally by 2 Kacper Pluta et al. Nouvel and Rémila [11], to characterize bijective digitized rigid motions [14]. Studies related to digitized rotations on the p (− 1 ; 3 ) hexagonal grid are less numerous: Her—while work- 2 2 (1; 0) ing with the hexagonal grid represented by cube coordinate system in 3D—showed how to derive a rotation matrix such that it is simpler than a 3D rotation matrix obtained in a direct way [6]; Pluta et al. provided a framework to study local alter- (a) (b) ations of discrete points on the hexagonal grid un- der digitized rigid motions. They also character- Fig. 1 Two variations of hexagonal grids, called pointy topped (a) and flat topped (b) hexagonal grids which are ized Eisenstein rational rotations on the hexagonal π equivalent up to rotation by (2k + 1) 6 ; k 2 Z. Arrows repre- grid which, together with their framework, allowed sent the bases of the underlying lattices them to compare the loss of information induced by digitized rigid motions with the both grids [15]. In this article, we consider an approach similar !2 + ! + 1 = 0. Thus H is the Voronoi tessellation to the one proposed by Roussillon and Cœurjolly associated with the lattice Λ in C. The elements of [16], based on complex numbers. Indeed points in Λ are called Eisenstein integers [2,5]; they are the Z2 can be identified with Gaussian integers, and complex numbers of the form rational rotations correspond to multiplications by complex numbers of unit modulus with rational α := a + b!; real and imaginary parts. For the hexagonal lattice, we use Eisenstein integers instead of Gaussian inte- where a; b 2 Z. gers [2, 5]. First, we discuss the density of rational rotations on the hexagonal grid, which were char- acterized in [15]. That is similar to the study of 2.1 Properties of Eisenstein integers rational rotations on the square grid – angles given by right triangles of integer sides [1]. Then, we The Eisenstein integers form a Euclidean ring Z[!] prove that there exist two subsets of Eisenstein in- in C; in particular they possess similar properties to tegers yielding bijective digitized rotations. This the ordinary integers: division, prime factorization differs from the square case, where only one subset and greatest common divisor are well defined. Let is involved. We also show that bijective digitized us consider α := a + b! 2 Λ; rotations on the hexagonal grid are more numerous – The conjugate of α is given as α¯ = (a − b) − b!. than their counterparts on the square grid for angles – The squared modulus of α is given as jαj2 := relatively close to multiple of π k, k 2 angle. 3 Z α · α¯ = a2 − ab + b2. – The units of Λ are Υ := {±1; ±!; ±!¯ g, where !¯ = !2. 2 Eisenstein integers – α is divisible by β if there exists φ 2 Λ such that α = β · φ. A hexagonal grid H is a grid formed by a tessella- – Given α, β 2 Λ, a greatest common divisor tion of the complex plane C by hexagons of side gcd(α, β) = φ 2 Λ is defined as a largest Eisen- length p1 . Figure 1 shows two examples of such 3 stein integer (up to multiplications by units) that hexagonal grids. Hereafter, without lack of general- divides both α and β; every common divisor of ity, we consider pointy topped hexagonal grid (see α and β also divides φ. By a largest Eisenstein Figure 1). In such a setting we consider centroids of integer, we mean one of a largest modulus. hexagons, i.e., the points of thep underlying lattice – α is said to be an Eisenstein prime if its divisors 1 3 Λ := Z⊕Z!, where ! := − 2 + 2 i, is a solution of are only of the form υ · α such that υ 2 Υ. Characterization of bijective digitized rotations on the hexagonal grid 3 Moreover, as for the ordinary complex numbers, the product of two Eisenstein integers has a geometric interpretation. Let α, β 2 Λ, then α · β is equal to a p −1 3b combination of a rotation by angle θ := tan 2a−b p 3b and the scaling by jαj. Note that sin θ = 2jαj and 2a−b cos θ = 2jαj . 2.2 Eisenstein triples Eisenstein triples (a; b; c) 2 Z3 are triples of inte- gers such that jαj2 = a2 − ab + b2 = c2: + ≤ Thanks to symmetry, hereafter, we consider only Fig. 2 Visualisation of Λρ for 0 < a; b 2000. Note that the axes were scaled for a better visualisation effect and the positive primitive Eisenstein triples, i.e., 0 < a < distance between two consecutive axes’ ticks is 30 c < b, gcd(a; b; c) = 1 and either a + b + c . 0 (mod 3) or 2b − a + c . 0 (mod 3) [4]. The later condition, is that we excludes triples which lead π to rotations by k 3 ; k 2 Z, e.g., (1; 1; 1). Note that, if gcd(a; b; c) = d 2 Z+ the rotation by a + b! re- θ0 duces to that given by the primitive Eisenstein triple θ 1 d (a; b; c), with the scale factor. Let us then denote the set of positive primitive Eisenstein triples by + Eρ and consider the subset n p o + 2 2 + b c b Λρ := α := a + b! 2 Λ (a; b; a − ab + b ) 2 Eρ : + A part of Λρ is illustrated by Figure 2. Moreover, we denote by Eρ and Λρ, respectively, the set of primitive Eisenstein triples and a subset of Λ which + is defined in a similar way to Λρ . Lemma 1 ([4]) Positive integers a; b and c form b − a a a pair of primitive Eisenstein triples a; b; c and ( ) Fig. 3 Geometric interpretation of a pair of Eisenstein p (b − a; b; c), if and only if there exist s; t 2 Z, 0 < −1 3a triples, (a; b; c) and (b − a; b; c), where θ = tan 2b−a s < t; gcd(s; t) = 1 and t − s .
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