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Bertrand, Gilles

Cousty, Jean

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A uniqueness property of perfect fusion gridsJean Cousty, Gilles BertrandTo cite this version:Jean Cousty, Gilles Bertrand. A uniqueness property of perfect fusion grids. AlayranguesSylvie and Damiand Guillaume and Fuchs Laurent and Lienhardt Pascal. 1st Workshop onComputational Topology in Image Context (CTIC’08), Jun 2008, France. 6pp., 2008. <hal-00622348>HAL Id: hal-00622348https://hal-upec-upem.archives-ouvertes.fr/hal-00622348Submitted on 12 Sep 2011HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.A uniqueness property of perfect fusion grids on ZdJean Cousty1,2 and Gilles Bertrand21 Asclepios Research Project, INRIA Sophia Antipolis, France2 Universite´ Paris-Est, LABINFO-IGM, UMR CNRS 8049, A2SI-ESIEE, FranceE-mail addresses: Jean.Cousty@sophia.inria.fr, g.bertrand@esiee.frAbstract Region merging methods consist of improving an initial segmentation by mergingsome pairs of neighboring regions. In a graph, merging two regions is not straightforward.The perfect fusion graphs defined in a previous paper verify all the properties requested byregion merging algorithms. In this paper, we present a theorem which states that, in anydimension, the perfect fusion grids introduced previously are the only perfect fusion graphsbetween the adjacency relations which are the most frequently used in image analysis.Keywords region merging, perfect fusion graph, perfect fusion grid, adjacency relationIntroductionImage segmentation is the task of delineating objects of interest that appear in an image. In manycases, the result of such a process, also called a segmentation, is a set of connected regions lyingin a background which constitutes the separation between regions. To define regions, an image isoften considered as a graph whose vertex set is made of the pixels and whose edge set is given byan adjacency relation between them. Then, the regions are simply the connected components offoreground pixels (see for instance Fig. 1).A popular approach to image segmentation, called region merging [6], consists of progressivelymerging pairs of regions, starting from an initial segmentation that contains too many regions (see,for instance, Figs. 1a and b). Given a subset S of an image equipped with an adjacency relation,merging two neighboring regions (connected components) of S is not straightforward. A problemoccurs when we want to merge a pair of neighboring regions A and B of S and when each pointadjacent to these two regions is also adjacent to a third one, which is not wanted in the merging.Fig. 1c illustrates such a situation, where x is adjacent to regions A,B,C and y to A,B,D. Thisproblem has been identified in particular by T. Pavlidis (see [6], section 5.6: “When three regionsmeet”), and, as far as we know, has not been solved in general. A major contribution of [2] is thedefinition of a merging operation and the study of a class of graphs, called perfect fusion graphs,which verify all the properties required by region merging algorithms.Unfortunately, the graphs which are the most frequently used in image analysis (namely, thoseinduced by direct and indirect adjacency relations on Zd) are not perfect fusion graphs (see Section6 in [2]). In [2], we introduced a graph on Zd that we call the perfect fusion grid, which is indeeda perfect fusion graph, and which is “between” the graphs induced by the direct and indirectadjacencies. The main result (whose proof will be given in a forthcoming extended version [1]) inthis paper, is that the perfect fusion grid is the only such graph, in any dimension d ∈ N.1 Perfect fusion graphs1.1 Basic notions on graphsLet E be a set, we denote by 2E the set composed of all subsets of E. Let X ⊆ E, we write Xthe complementary set of X in E, i.e., , X = E \X . Let E′ be a set. The Cartesian product of Eby E′, denoted by E × E′, is the set made of all pairs (x, y) such that x ∈ E and y ∈ E′.1A BCDxyz(a) (b) (c)Figure 1: (a): Cross-section of a brain, after applying a gradient operator. (b): A segmentationof (a) (obtained by a watershed algorithm [5] using the adjacency Γ21). (c): A zoom on a partof (b); the graph induced by Γ21 is superimposed in gray.yzwxyzwxyzwxrpqs(a) (b) (c) (d) (e)Figure 2: (a) A graph (E,Γ) (which is not a PFG) and a set S ⊂ E (gray and white vertices).(b,c) Illustrations of region merging operations and the problem encountered by this operation[see text]. (d) Example of a PFG. (c) The graph GN used in Th. 2 to characterize the PFGs.A graph is a pair (E,Γ) where E is a set and Γ is a binary relation on E (i.e., Γ ⊆ E × E)which is anti-reflexive (for any x ∈ E, (x, x) /∈ Γ) and symmetric (for any x and y in E, (y, x) ∈ Γwhenever (x, y) ∈ Γ). Each element of E (resp. Γ) is called a vertex or a point (resp. an edge). Wewill also denote by Γ the map from E to 2E such that, for any x ∈ E,Γ(x) = {y ∈ E | (x, y) ∈ Γ}.Let x ∈ E, the set Γ(x) is called the neighborhood of x and if y ∈ Γ(x), we say that y is adjacentto x. If X ⊆ E, the neighborhood of X , denoted by Γ(X), is the set [∪x∈XΓ(x)] \X .Let (E,Γ) be a graph and X ⊆ E. A path in X is a sequence 〈x0, . . . , xℓ〉 such that xi ∈ X ,i ∈ [0, ℓ], and (xi−1, xi) ∈ Γ, i ∈ [1, ℓ]. The set X is connected if, for any x, y ∈ X , there existsa path in X from x to y. Let Y ⊆ X , we say that Y is a (connected) component of X if Y isconnected and maximal for this property, i.e., if Z = Y whenever Y ⊆ Z ⊆ X and Z connected.Important remark. From now on, when speaking about a graph G = (E,Γ), we assumethat E is connected and that G is locally finite, i.e., the set Γ(x) is finite for any x ∈ E.1.2 Region merging and perfect fusion graphsConsider the graph (E,Γ) depicted in Fig. 2a, where a subset S of E (white and gray vertices) iscomposed of four components. If we replace the set S by, for instance, the set S ∪ T where T ={x, y, z}, we obtain a set composed of three regions (see Fig. 2b). We can also say that we “mergedtwo components of S through T ”. This operation may be seen as an “elementary merging” in thesense that only two regions of S were merged while all other regions of S were preserved. On theopposite, replacing the set S by the set S ∪ T where T = {w} (see Fig. 2c), would merge threecomponents of S. This section recalls the definitions introduced in [2] related to such mergingoperations in graphs. Then, we remind the definition of the perfect fusion graphs, which are thegraphs in which any two neighboring regions can be merged through their common neighborhoodwhile preserving all other regions.2Let (E,Γ) be a graph and let S ⊆ E. Let A and B be two distinct components of S and T ⊆ S.We say that A and B can be merged (for S) through T if A and B are the only connectedcomponents of S adjacent to T and if A ∪B ∪ T is connected.In other words (see Property 21 in [2] for a formal proof), the two regions A and B can bemerged through T if and only if A ∪B ∪ T is a component of S ∪ T . More precisely, they can bemerged if and only if the components of S ∪ T are the same as the components of S exceptingthat A and B are replaced by A ∪B ∪ T . For instance, in Fig. 2a the two white components canbe merged through {x, y, z} but the two gray components cannot be merged through {z}.Let (E,Γ) be a graph, S ⊆ E, and let A and B be two distinct connected components of S.We set Γ(A,B) = Γ(A)∩ Γ(B) and we say that Γ(A,B) is the common neighborhood of A and B.If the common neighborhood of A and B is nonempty, A and B are said to be neighbors.Definition 1 (perfect fusion graph). Let (E,Γ) be a graph. We say that (E,Γ) is a perfectfusion graph (PFG) if, for any S ⊆ E, any two connected components A and B of S which areneighbors can be merged through Γ(A,B).The graph of Fig. 2a is not a PFG since the two gray components cannot be merged throughtheir common neighborhood {w}. On the other hand, the graph of Fig. 2d is a PFG.The definition of the PFGs is based on a condition which must be verified for all subsets ofthe vertex sets. This means, if we want to check whether a graph is a PFG, then, using thestraightforward method based on the definition, this will cost an exponential time with respectto the number of vertices. In fact, the PFGs can be recognized in a simpler way thanks to thefollowing conditions which can be checked independently in the neighborhood of each vertex.Let G = (E,Γ) be a graph and let X ⊆ E. The subgraph of G induced by X is the graph GX =(X,Γ ∩ [X ×X ]). In this case, we also say that GX is a subgraph of G.We denote by GN the graph of Fig. 2e.Theorem 2 (from Th. 41 in [2]). The three following statements are equivalent:i) (E,Γ) is a PFG;ii) the graph GN is not a subgraph of (E,Γ);iii) for any x ∈ E, any X ⊆ Γ(x) contains at most two connected components.Thanks to Th. 2, it can be verified that the graph (E,Γ) depicted in Fig. 2 is a PFG. Indeed, GNis not a subgraph of (E,Γ). Remark in particular that the subgraph induced by {p, q, r, s} is not GNsince it contains the edge (r, s).We finish this section by reminding that, through the notion of a line graph, the region mergingapproaches based on “separations” composed of edges (e.g., inter-pixel elements) also fall in thescope of perfect fusion graphs (see Prop. 29 in [2]).2 Cubical grids in arbitrary dimensionsDigital images are defined on (hyper-) rectangular subsets of Zd (with d ∈ N⋆). Therefore, forregion merging applications, Zd must be equipped with an adjacency relation reflecting the geomet-rical relationship between its elements. We provide, in this section, a set of definitions that allowsfor recovering the adjacency relations which are the most frequently used in 2- and 3-dimensionalimage analysis and permit to extend them to images of arbitrary dimension.Let Z be the set of integers. We define the families of setsH10 andH11 such thatH10 = {{a} | a ∈Z} and H11 = {{a, a+1} | a ∈ Z}. Let m ∈ [0, d]. A subset C of Zd which is the Cartesian productof exactly m elements of H11 and (d−m) elements of H10 is called a (m-)cube of Zd.Observe that an m-cube of Zd is a point if m = 0, a (unit) interval if m = 1, a (unit) squareif m = 2 and a (unit) cube if m = 3.Let C be a set of cubes of Zd. The binary relation induced by C is the set of all pairs (x, y)of Zd such that there exists a cube in C which contains both x and y.Definition 3 (m-adjacency). Let m ∈ [1, d]. The m-adjacency on Zd is the binary relation Γdminduced by the set of all m-cubes of Zd. If (x, y) ∈ Γdm, we say that x and y are m-adjacent.301 0100 0001 0100 00012340 1 2 3 410 10 1110 11 10 1111012340 1 2 3 4 012340 1 2 3 4(a) (b) (c)Figure 3: Chessboards and perfect fusion grids on Z2. (a): The map ψ; (b,c): two subgraphs of(Z2,Λ211) and (Z2,Λ210) induced by {0, . . . , 4}×{0, . . . , 4} with in gray the associated chessboards.Observe that two points of Zd are m-adjacent if and only if |xi − yi| ≤ 1 for any i ∈ [1, d]and (∑di=1 |xi − yi|) ≤ m. In the literature [4], Γ21, Γ22, Γ31, Γ32 and Γ33 are often referred to asthe 4-, 8-, 6- 18- and 26-adjacencies. On Zd, Γd1 and Γddare sometimes called direct and indirectadjacencies.The graphs induced by Γ21 are not in general PFGs. For instance, in Fig. 1c the two neighboringregions A and B cannot be merged through Γ21(A,B) = {x, y} and thus the graph is not a PFG.More generally, the graphs which are the most frequently used in image analysis (namely, thoseinduced by Γd1 and Γdd, with d = 2, 3) are not PFGs (see Sec. 6 in [2]).3 Perfect fusion gridsIn [2], we introduced a family of graphs on Zd, called the perfect fusion grids, which are indeedPFGs and which are “between” Γd1 and Γdd. In this section, we recall the definition of the perfectfusion grids, give a local characterization and then show that they are the only PFGs “between”the direct and indirect adjacency relations.The perfect fusion grids can be defined by the mean of chessboards. Intuitively, a chessboard Con Zd is a spanning set of d-cubes (i.e., ∪{C ∈ C} = Zd) such that the intersection of any twocubes in C is either empty or reduced to a point. For instance, the gray cubes shown in Figs. 3band c constitute subsets of the two chessboards on Z2. The perfect fusion grids are the graphsinduced by the chessboards on Zd (see, e.g., the graphs of Figs. 3b and c). Fig. 4a shows a set ofregions obtained in this grid by a watershed algorithm [3]. Note that the problems pointed out inthe introduction do not exist in this case: any pair of neighboring regions can be merged by simplyremoving from the black vertices the points which are adjacent to both regions (see Fig. 4b,c).Let Cd be the set of all d-cubes of Zd, we define the map ϕ from Cd to Zd, such that for anyC ∈ Cd, ϕ(C)i = min{xi | x ∈ C}, where ϕ(C)i is the i-th coordinate of ϕ(C). It may be seenthat C is equal to the Cartesian product: {ϕ(C)1, ϕ(C)1 + 1} × · · · × {ϕ(C)d, ϕ(C)d + 1}. Thus,clearly ϕ is a bijection and allows for indexing the cubes of Zd.Let B = {0, 1}. We set 0 = 1 and 1 = 0. A binary word of length d is an element in Bd. Ifb = (b1, . . . , bd) is in Bd, we define b as the binary word of Bd such that for any i ∈ [1, d], (b)i = (bi).We define the map ψ from Cd to Bd such that for any C ∈ Cd and any i ∈ [1, d], ψ(C)i is equalto [ϕ(C)i mod 2], that is the remainder in the integer division of ϕ(C)i by 2.Definition 4 (chessboard & perfect fusion grid). Let b ∈ Bd.We set Cdb= {C ∈ Cd | ψ(C) = b} and we say that the set Cdb∪ Cdbis a (global) chessboard on Zd.Let C be the chessboard on Zd defined by Cdb∪ Cdb. We denote by Λdbthe binary relation inducedby C and we say that the pair (Zd,Λdb) is a perfect fusion grid on Zd.Fig. 3 illustrates these definitions on Z2.Since the cardinality of Bd is 2d, from the previous definition, it can be deduced that thereare exactly 2d−1 distinct perfect fusion grids on Zd. However, any two perfect fusion grids are4BADCADE(a) (b) (c)(d) (e) (f)Figure 4: (a) A segmentation of Fig. 1a obtained on a perfect fusion grid. (b) A zoom on a part of(a); the regions A, B, C and D correspond to the ones of Fig. 1c; the corresponding perfect fusiongrid is shown in gray. (c) Same as (b) after having merged B and C to form a new region E.(b,c,d) Several steps, starting from (a), of a region merging process on a perfect fusion grid.equivalent up to a “binary translation”. More precisely, it is proved in [2] that for any b and b′in Bd, there exists t ∈ Bd such that, for any x and y in Zd, y ∈ Λdb(x) if and only if y+t ∈ Λdb′(x+t).3.1 Local characterizationCertain classes of graphs can be locally characterized, i.e., we can test if an arbitrary graph belongsto such a class by independently checking a condition in the neighborhood of each point. The nexttheorem states that the chessboards (hence the perfect fusion grids) can be locally characterized.We set U⋆ = {1,−1}. Let x ∈ Zd and u ∈ Ud, we denote by C(u, x) the d-cube of Zd definedby {x1, x1 + u1} × · · · × {xd, xd + ud} and we set Cˆ(u, x) = C(−u, x). In other words, C(u, x) isthe set of all points y such that, for any i ∈ [1, d], yi = xi or yi = xi + ui.Definition 5 (local chessboard). Let C be a set of d-cubes of Zd. We say that C is a localchessboard on Zd if, for any x ∈ Zd, there exists u ∈ Ud⋆such that C(u, x) and Cˆ(u, x) belong to Cand such that they are the only two elements in C which contain x.On Z2 (resp. Z3), a local chessboard C is a set of 2-cubes (resp. 3-cubes) such that, for anypoint x, the cubes of C which contain xmatch one of two (resp. four) configurations depicted in thefirst (resp. second) column of Fig. 5. Observe that this notion of a local chessboard correspondsexactly to the intuitive idea given in the introduction of the section. As assessed by the followingtheorem, we can indeed prove the equivalence between global and local chessboards.Theorem 6. Let C be a set of d-cubes of Zd. The set C is a chessboard on Zd if and only if C isa local chessboard on Zd.Furthermore, if C = Cdb∪Cdb(with b ∈ Bd), then for any x ∈ Zd, the only two d-cubes of C whichcontain x are defined by C(u, x) and Cˆ(u, x) with u ∈ Ud⋆, and ui = (−1)(xi−bi) for any i ∈ [1, d].From this local characterization of chessboards and by Th. 2.iii, it can be deduced that anyperfect fusion grid (Zd,Λdb) is indeed a PFG between Γd1 and Γdd(in the sense Γd1 ⊆ Λdb⊆ Γdd). Fur-521u = (1,1)u = (1,−1)xx123u = (1,1,1) u = (1,−1,1)u = (1,1,−1) u = (1,−1,−1)x xx xFigure 5: First (resp. second) column: local configurations in a local chessboard on Z2 (resp. Z3).In each column, we assume that Zd is oriented as shown by the arrows. Then, the element of u ∈ Ud⋆which “generates” each configuration as C(u, x ∪ Cˆ(u, x)) is written under the configuration.thermore, we point out that the second assertion of Th. 6 constitutes a practical way for computingthe neighborhood of a point in a perfect fusion grid, as it is often required in applications.3.2 Unicity theoremThe following theorem asserts that the only PFGs “between” Γd1 and Γddare the perfect fusiongrids. Since any two perfect fusion grids are equivalent up to a binary translation, this resultestablishes the uniqueness of the perfect fusion grid in any dimension d ∈ N.Theorem 7. The graph (Zd,Γd) is a PFG such that Γd1 ⊆ Γd ⊆ Γddif and only if it is a perfectfusion grid on Zd.In other words, the perfect fusion grid is, in any dimension, the only graph, “between” thedirect and indirect adjacencies, which guarantees that any two neighboring regions can be mergedthrough their common neighborhood while preserving all other regions. Fig. 4a,d,e,f show anexample in image segmentation of such a region merging procedure in a perfect fusion grid.References[1] J. Cousty and G. Bertrand. Uniqueness of the perfect fusion grid on Zn. 2008. In preparation.[2] J. Cousty, G. Bertrand, M. Couprie, and L. Najman. Fusion graphs: Merging properties andwatersheds. Journal of Mathematical Imaging and Vision, 30(1):87–104, 2008.[3] J. Cousty, M. Couprie, L. Najman, and G. Bertrand. Weighted fusion graphs: mergingproperties and watersheds. Discrete Applied Mathematics, 2008. In press, Preprint at:http://igm.univ-mlv.fr/LabInfo/rapportsInternes/2007/09.v2.pdf.[4] T.Y. Kong and A. Rosenfeld. Digital topology: Introduction and survey. Computer VisionGraphics and Image Processing, 48(3):357–393, 1989.[5] F. Meyer. Un algorithme optimal de ligne de partage des eaux. In Actes du 8e`me Congre`sAFCET, pages 847–859. Lyon-Villeurbanne, France, 1991.[6] T. Pavlidis. Structural Pattern Recognition. Springer Series in Electrophysics. Springer, 1977.Segmentation techniques, chapter 4–5.6

A uniqueness property of perfect fusion grids

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A uniqueness property of perfect fusion grids

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