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Equivalence Between Digital Well-Composedness and Well-Composedness in the Sense of Alexandrov on N-D Cubical Grids Nicolas Boutry, Laurent Najman, Thierry Géraud

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Equivalence Between Digital Well-Composedness and Well-Composedness in the Sense of Alexandrov on N-D Cubical Grids Nicolas Boutry, Laurent Najman, Thierry Géraud Equivalence between Digital Well-Composedness and Well-Composedness in the Sense of Alexandrov on n-D Cubical Grids Nicolas Boutry, Laurent Najman, Thierry Géraud To cite this version: Nicolas Boutry, Laurent Najman, Thierry Géraud. Equivalence between Digital Well-Composedness and Well-Composedness in the Sense of Alexandrov on n-D Cubical Grids. Journal of Mathematical Imaging and Vision, Springer Verlag, 2020, 62 (9), pp.1285-1333. 10.1007/s10851-020-00988-z. hal- 02990817 HAL Id: hal-02990817 https://hal.archives-ouvertes.fr/hal-02990817 Submitted on 5 Nov 2020 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. JMIV manuscript No. (will be inserted by the editor) Equivalence between Digital Well-Composedness and Well-Composedness in the sense of Alexandrov on n-D Cubical Grids Nicolas Boutry · Laurent Najman · Thierry G´eraud Received: date / Accepted: date Abstract Among the different flavors of well-compo- 1 Introduction sednesses on cubical grids, two of them, called respec- tively digital well-composedness (DWCness) and well- composedness in the sense of Alexandrov (AWCness), are known to be equivalent in 2D and in 3D. The for- mer means that a cubical set does not contain critical configurations when the latter means that the boundary of a cubical set is made of a disjoint union of discrete surfaces. In this paper, we prove that this equivalence holds in n-D, which is of interest because today im- ages are not only 2D or 3D but also 4D and beyond. The main benefit of this proof is that the topological properties available for AWC sets, mainly their sepa- Fig. 1: Equivalence between DWCness in Z2 and AWC- ration properties, are also true for DWC sets, and the ness in the 2D Khalimsky grid H2: The 3 first sets properties of DWC sets are also true for AWC sets: in black in the raster scan order are DWC (they do an Euler number locally computable, equivalent con- not contain any critical configuration); their analog in nectivities from a local or global point of view... This H2 are AWC (their dark grey boundaries are simple result is also true for gray-level images thanks to cross- closed curves). At the opposite, the last set (circled in section topology, which means that the sets of shapes red) which is not DWC in Z2 leads to an analog in H2 of DWC gray-level images make a tree like the ones of whose boundary is not a simple closed curve (see the AWC gray-level images. self-crossing of the boundary of this set in H2 in red) and then is not AWC in H2. Keywords digital topology, discrete surfaces, well- composed images In 1995, Latecki introduced in [23] the notion of Nicolas Boutry well-composedness as an elegant manner to get rid of EPITA Research and Development Laboratory (LRDE) the connectivity paradoxes well-known in digital topol- 14-16 rue Voltaire, FR-94276 Le Kremlin-Bic^etre,France 2 E-mail: [email protected] ogy. Roughly speaking, a set in Z is said to be well- composed if its connectivities are equivalent, that is, Laurent Najman Universit´eParis-Est, LIGM, Equipe´ A3SI, ESIEE its set of connected components is the same whatever E-mail: [email protected] the chosen connectivity. This definition has then been Thierry G´eraud extended to dimension 3 in [24] and in n-D in 2015 EPITA Research and Development Laboratory (LRDE) in [10]; at the same moment, this definition of well-com- 14-16 rue Voltaire, FR-94276 Le Kremlin-Bic^etre,France posedness has been renamed digital well-composedness E-mail: [email protected] (DWCness). 2 N. Boutry et al. Later, in 2013, Najman and G´eraud[28] introduced 2.1 Digital topology and DWCness a new notion of well-composedness: a subset X in a Khalimsky grid, a topological analog of the cubical grid n but with combinatorial properties, is said to be n Z B Orthonormal canonical basis of Z well-composed in the sense of Alexandrov (AWC) if its th n xi i coordinate of x 2 Z boundary is made of a disjoint union of discrete (n−1)- ∗ n N2n(x) 2n-neighborhood of x 2 Z surfaces (the definition of a discrete surface is formally ∗ n n N n (x) (3 − 1)-neighborhood of x 2 recalled later). This definition is used to be able to char- 3 −1 Z F (f 1; : : : ; f k) ⊆ acterize gray-level images defined on a Khalimsky grid B × The Cartesian product and whose set of shapes makes a tree; then, we can call fkg when k 2 ; \tree of shapes" this last set. This hierarchical repre- L(k) Z fk − 1=2; k + 1=2g when k 62 ; sentation is known to be powerful to make image seg- Z S(z) S(z) = ×i2 1;n L(zi) mentation or image filtering thanks to shapings (a fil- 8 J K 9 tering of the shapes of a given image based on its tree < X = S(z; F) z + λ f i λ 2 f0; 1g; 8i 2 1; k of shapes). i i : i2 1;k J K; J K n These two definitions, even if they seem very differ- antagS(q) The antagonist of q 2 Z in the block S ent, are related: it has been observed that in 2D and in 3D, a digital set is DWC if and only if it is AWC Table 1: Notations relative to digital topology. (see Figure 1). In other words, AWCness and DWCness are equivalent in 2D/3D on cubical grids. However, the question about their equivalence in 4D and beyond still remains an open question. For sake of completion, we propose here to prove that they are indeed equivalent in n-D, n ≥ 2 for sets and that it holds for gray-level images. A study of the relation between the different flavors can be found in [9] and a list of some of the im- portant properties of AWCness and DWCness can be found in [11]. The plan is the following: Section 2 recalls the ba- Fig. 2: The two classical neighborhoods used in digital sics in matter of digital topology; Section 3 recalls some n topology when the dimension n of space is 2. The point Z n mathematical background relative to 2 and H ; Sec- p is depicted in black, and its neighborhoods are de- tion 4 recalls how AWCness implies DWCness in n-D; picted in blue: the 2n-neighborhood is on the left side Section 5 presents some lemmas, propositions and nota- when the (3n − 1)-neighborhood is on the right side. tions necessary in the next section; Section 6 proves that We can observe that the 2n-neighborhood is made of 4 DWCness implies AWCness in n-D; Section 7 extends points and that the (3n − 1)-neighborhood is made of 8 the proofs seen before from sets to gray-level images; points (because we are in 2D). Section 8 shows some possible applications of well-com- posedness for gray-level images; Section 9 concludes the paper; Sections A, B, C and D contains the proofs of The following notations are detailed in Table 1. Let the preceding assertions. B = fe1; : : : ; eng be the canonical basis of Zn. We use 1 the notation xi, where i belongs to 1; n , to deter- Note that a star has been added in the title of each th n J K mine the i coordinate of x 2 Z . We recall that the assertion which seems to us crucial to understand the 1 n L -norm of a point x 2 Z is denoted by k:k1 and proof of the equivalence between AWCness and DWC- P is equal to i2 1;n jxij where j:j is the absolute value. ness. 1 J K Also, the L -norm is denoted by k:k1 and is equal to n maxi2 1;n jxij. For a given point x 2 Z , an element J K ∗ n of the set N2n(x) = fy 2 Z ; kx − yk1 = 1g (resp. of ∗ n 2 Digital topology the set N3n−1(x) = fy 2 Z ; kx − yk1 = 1g) is a 2n- neighbor (resp. a (3n − 1)-neighbor) of x (see Figure 2). Let us recall the mathematical background necessary to 1 As usual, for any a; b 2 Z, with a ≤ b, the notation with define digital well-composedness and well-composedness double brackets a; b means the set of all the integers in the in the sense of Alexandrov. interval [a; b] J K Equivalence between DWCness and AWCness on n-D Cubical Grids 3 For any z 2 Zn and any F = (f 1; : : : ; f k) ⊆ B, we denote by S(z; F) the set: 8 9 < X i = z + λif λi 2 f0; 1g; 8i 2 1; k : : i2 1;k J K; J K We call this set the block associated with the pair (z; F); P f its center is z + f2F 2 , and its dimension, denoted by dim(S), is equal to k. We denote by S(c) the block n Z centered at c 2 2 .
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