Shape diagrams for 2D compact sets - Part III: convexity discrimination for analytic and discretized simply connected sets.

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convexity discrimination for analytic and discretized

simply connected sets.

Séverine Rivollier, Johan Debayle, Jean-Charles Pinoli

To cite this version:

Séverine Rivollier, Johan Debayle, Jean-Charles Pinoli. Shape diagrams for 2D compact sets - Part III: convexity discrimination for analytic and discretized simply connected sets.. Australian Journal of Mathematical Analysis and Applications, Austral Internet Publishing, 2010, 7 (2), Article 5, pp. 1-18. �hal-00550951�

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DISCRETIZED SIMPLY CONNECTED SETS

S. RIVOLLIER, J. DEBAYLE AND J.-C. PINOLI

Abstract. Shape diagrams are representations in the Euclidean plane introduced to study 3-dimensional and 2-dimensional compact convex sets. However, they can also been applied to more general compact sets than compact convex sets. A compact set is represented by a point within a shape diagram whose coordinates are morphometrical functionals defined as normalized ratios of geometrical functionals. Classically, the geometrical functionals are the area, the perimeter, the radii of the inscribed and circumscribed circles, and the minimum and maximum Feret diameters. They allow twenty-two shape diagrams to be built. Starting from these six classical geometrical functionals, a detailed comparative study has been performed in order to analyze the representation relevance and discrimination power of these twenty-two shape diagrams. The two first parts of this study are published in previous papers [8, 9]. They focus on analytic compact convex sets and analytic simply connected compact sets, respectively. The purpose of this paper is to present the third part, by focusing on the convexity discrimination for analytic and discretized simply connected compact sets.

1 Introduction

The Santalo’s shape diagrams [10] allow to represent a 2D compact convex set by a point in the Euclidean 2D plane from six geometrical functionals: the area, the perimeter, the radii of the inscribed and circumscribed circles, and the minimum and maximum Feret diameters [4]. The axes of each shape diagram are defined from a pair of geometric inequalities relating these functionals. Sometimes, the two geometric inequalities provide a complete system: for any range of numerical values satisfying them, there exists a compact convex set with these values for the geometrical functionals (in other words, a point within the 2D Santalo shape diagram describes a 2D compact convex set). This is not valid for all the Santalo shape diagrams.

This paper deals with the study of the convexity discrimination for shape di-agrams of 2D non-empty analytic and discretized simply connected compact sets. The two first parts [8, 9] of this study focus on the analytic compact convex sets and analytic simply connected compact sets, respectively. This third part presents an analysis of the convexity discrimination and extends the previous work to the discretized simply connected compact sets. The considered discretized simply con-nected sets are mapped onto points in these shape diagrams, and through disper-sion quantification and convexity discrimination, the shape diagrams are classified according to their ability to discriminate the simply connected compact sets.

Key words and phrases. Analytic and discretized simply connected compact sets, Convexity dis-crimination, Geometrical and morphometrical functionals, Shape diagrams, Shape discrimination.

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2 Shape convexity

The following study on the convexity discrimination first requires the definition of the shape convexity. A set is convex when the line segment which joins any two points in it lies totally within the set. In other terms, the shape convexity could be quantified with the probability that two points in the set lies totally within it.

Convexity parameters are commonly used in the analysis of shapes. The mea-surement value of the shape convexity of any set ranges between 0 and 1 (it is a probability). A convex set gives the value 1. Futhermore, the less the parameter value is high, the less the shape is convex. The convexity measurement can be com-puted, for instance, by the ratio A / AC where AC is the area of the convex hull of

the set, but this is not sensitive to boundary small variations (Figure 2.1). A second convexity parameter is the ratio PC/ P where PC is the Euclidean perimeter of the

convex hull of the set.

(a) A / AC= 1 and PC/ P = 1 (b) A / AC close to 1 (c) PC/ P far from 1

Figure 2.1: A / AC gives result values equal to 1 for the set (a) and close to 1 for the sets (b) and

(c). Nevertheless, the set (b) is far to check the convexity definition (the probability that two points

in the set lies totally within it is low). PC/ P gives result values equal to 1 for the set (a) and

far from 1 for the sets (b) and (c). Nevertheless, the set (c) is not very far to check the convexity definition.

Futhermore, a suitable convexity parameter is particularly required for discretized sets. For that purpose, the convexity parameter of Zunic and Rosin [12] will be used for the following study on the convexity discrimination. For all polygon S of the Euclidean 2D plane E2_{, it is defined by:}

c(S) = min

α∈[0,2π]

P2(R(S, α))

P1(S, α)

(2.1)

P1(S, α) denotes the perimeter of the set S, rotated by the angle α with the origin

as the center of rotation, in the sense of the l1 metric (l1(e) equals the sum of the

projections of the straight line segment e onto the coordinate axes). P2(R(S, α))

denotes the Euclidean perimeter of the minimal bounding rectangle R with edges parallel to the coordinate axes which includes the rotated set of S by the angle α.

This convexity parameter c has the following desirable properties: • its value is always a number within (0, 1]

• its value is 1 if and only if the measured set is convex • there are sets with its value arbitrarily close to 0

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• it is invariant under similitude transformations

• there is a simple and fast procedure for computing it.

3 Convexity discrimination for analytic simply connected sets

Observing the 2D compact set locations in the shape diagrams Dk, k ∈ J1, 30K \

(J7, 10K ∪ J17, 20K) for the families Fc

1 and F1sc [8, 9] (Figures 3.1 and 3.2), some

shape diagrams seem to stronger discriminate the convex shapes from the non-convex shapes than others.

Figure 3.1: Family Fc

1 of 2D analytic compact convex sets [8].

Figure 3.2: Family Fsc

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Equation 3.1 quantifies (by values ranging between 0 and 1) this convexity dis-crimination for each shape diagram Dk, k ∈ J1, 30K \ (J7, 10K ∪ J17, 20K). A high

(respectively low) resulting value means a weak (respectively strong) convexity dis-crimination. Convexity_discrimination(Dk) = 1 # {Fc 1 ∪ F1sc}   #Fc 1 X i=1 |c(i) (3.1) −c (argmin {dE(i, j)|j ∈ F1c})| + #Fsc 1 X i=1

|c(i) − c (argmin {dE(i, j)|j ∈ F1sc})|





where c(i) denotes the convexity value of the shape indexed by i in Figure 5.1, and dE is the Euclidean distance.

Figure 3.3 shows the results of this quantification for every shape diagrams Dk,

k ∈ J1, 30K \ (J7, 10K ∪ J17, 20K). The stronger convexity discrimination appears in the shape diagrams D24, D26, D27, and D29, that is in agreement with the visual

interpretation [9], and the weaker discrimination appears for the shape diagrams D4

and D14. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 D1 D2 D3 D4 D5 D6 _D11 D12 D13 D14 D15 D16 D21 D22 D23 D24 D25 D26 D27 D28 D29 D30

Figure 3.3: Convexity discrimination for the families Fc

1 and F

sc

1 .

These results have to be confirmed with more general sets. This is not easy to conclude with the restriction to analytic sets. For this reason, this study is extended to the discrete case.

4 Shape functionals for a discretized simply connected set

In this paper, the non-empty discretized simply connected compact sets in the discrete Euclidean 2-space E2

d are considered. They are represented by points. The

inter-point distance is the step discretization. The discretization is fine enough to preserve the simple connectivity [5]. For the characterization of these sets, six geometrical functionals are used allowing to define morphometrical functionals from geometric inequalities.

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4.1 Geometrical functionals For a discretized simply connected compact set in E2

d, let A, P, r, R, ω, d, denote the estimations of its area, its perimeter, the

radii of its inscribed and circumscribed circles, its minimum and maximum Feret diameters [4], respectively. Figure 4.1 illustrates these geometrical functionals for a discretized simply connected compact set represented in a point lattice [5]. For all non-empty discretized simply connected compact sets, these geometrical functionals are greater or equal than zero.

b b b b b b b b b b b b b b b b b b b b b b b b b r R ω d

Figure 4.1: Geometrical functionals of a discretized simply connected compact set: radii of inscribed (r) and circumscribed (R) circles, minimum (ω) and maximum (d) Feret diameters. The area A is given by the point number, and the perimeter is estimated thanks to the Cauchy-Crofton-Poincaré formula [2, 3, 6].

4.2 Geometric inequalities and morphometrical functionals Geometric inequalities and morphometrical functionals for analytic simply connected compact sets are referenced in [9]. There exists special cases where a discretized set has geometrical functional values that do not verify a geometric inequality, due to the estimation error (because of the discretization). This means that this set is a dis-cretization of an extremal set for the associated geometric inequality. Consequently, for practical reasons, the morphometrical functional value greater than one is re-duced to one. Thus, the twenty-two shape diagrams referenced in [9] can be used for discretized simply connected compact sets.

5 Shape diagrams for a basis

of various discretized simply connected compact sets

5.1 Shape diagrams Figure 5.1 illustrates seventy-eight discretized patterns constituting the family Fdsc

1 . It is assumed that the discretization process is fine

enough such that each discretized pattern is a discretized simply connected compact set on the points lattice [5]. Thus, the morphometrical functionals are computed and located in each shape diagram.

The discretized patterns are numerated from one to seventy-eight. The pattern number are mapped to its proper point in each shape diagram. Figure 5.2 illustrates several of these twenty-two shape diagrams, chosen according to the synthesized results of [9]. The color of the number is related to the convexity parameter value c of the associated set (dark red for a high c value, dark blue for a low c value). Moreover, the black curves indicate the convex domain boundary, if it is known [8].

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Sometimes, a dark red number appears slightly outside this domain. This is mainly due to the estimation error (discretization) of the geometrical functionals. This convexity range will enable to analyze the convexity discrimination within shape diagrams. 1 2 3 4 5 6 7 ₈ 9 10 11 12 13 14 ₁₅ 16 17 18 19 20 21 ₂₂ 23 24 ₂₅ 26 27 28 29 30 31 ₃₂ 33 34 35 36 37 38 39 40 41 ₄₂ 43 44 45 ₄₆ 47 48 49 50 51 ₅₂ 53 54 55 56 ₅₇ 58 59 ₆₀ 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 0 0.2 0.28 0.35 0.4 0.45 0.49 0.53 0.57 0.6 0.63 0.66 0.69 0.72 0.75 0.77 0.8 0.82 0.85 0.87 0.89 0.92 0.94 0.96 0.98 1 Figure 5.1: Family Fdsc

1 of seventy-eight 2D discretized simply connected compact sets (discretized

patterns). The color of each pattern number is related to the convexity parameter value using non-linear color-tones.

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0 0.2 0.28 0.35 0.4 0.45 0.49 0.53 0.57 0.6 0.63 0.66 0.69 0.72 0.75 0.77 0.8 0.82 0.85 0.87 0.89 0.92 0.94 0.96 0.98 1 Convexity 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 32 40 22 34 35 6 55 3 68 16 69 11 54 30 45 77 60 74 78 72 62 70 51 33 7 38 58 76 42 28 17 41 47 14 46 56 63 8 67 59 5 48 53 2921 25 52 37 64 31 49 27 61 50 26 43 19 44 15 73 1 36 71 23 65 2 4 18 75 24 39 13 9 66 20 57 10 12 D1: (ω, r, R) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 18 19 14 34 9 72 4 10 2062 30 68 28 60 55 61 63 58 15 7 73 76 26 56 40 65 1 49 16 31 3 46 6 8 74 67 64 3871 39 24 57 37 41 42 48 25 52 13 54 7577 53 32 45 23 21 29 78 22 27 36 33 5 44 47 43 17 59 51 2 69 66 35 50 11 12 70 D3: (r, A, R) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 60 10 47 27 69 67 70 7 72 31 37 2 35 76 62 57 46 1 20165 242134458335013154258 6632 6149 39 65 6 78 19 71 1856 1452 53 22 38 77 23417536 4512 684834 64 40115563 9₅₉29304217 74542851 ₄₃ 73 26 D4: (A, d, R) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 63 10 54 46 47 48 53 38 32 51 14 39 24 70 1 77 13 44 27 11 4 23 18 6 20 40 12 16 29 9 2 68 52 50 71 35 67 45 56 73 59 76 37 65 78 28 49 7 ₃ 58 8 5 75 41 22 31 19 43 25 69 21 60 15 42 30 74 55 33 64 61 36 17 72 57 26 62 34 66 D12: (ω, A, d) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 11 75 68 19 29 22 54 58 48 1 49 12 25 37 67 50 27 51 65 28 24 26 46 42 1641 31 ₇₁ 70 72 21 66 45 36 73 14 23 38 34 64 35 32 17 77 43 53 10 8 20 52 2 18 30 44 74 3 33 56 4 57 7 63 76 15 39 78 62 59 13 40 5 9 55 60 69 47 6 61 D13: (r, A, d) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 16 17 4 36 4418 58 607124 549 61 52 1022 2573 46 43 49 56 74 32 63 55 27 6 37 42 38 3 47 78 5969 12 51 ₆₆ 76 1140 50 7030 5772148282 12639 64 67 23 577534568 331529 8 31 6241 75 19 14 35 13 72 20 65 34 D15: (ω, R, d) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 35 73 24 63 47 12 8 27 72 20 39 15 14 33 75 11 68 46 3 34 44 55 29 66 77 71 53 31 32 52 74 43 62 2 10 59 18 28 22 30 ₄₂ 70 65 60 51 69 37 16 49 23 9 19 50 76 6 7 26 45 5 4 48 57 41 25 56 54 13 78 58 64 17 40 36 38 1 61 21 67 D21: (ω, r, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 11 15 36 48 35 31 46 29 63 6 55 25 40 30 33 42 5 19 24 20 7 39 70 74 41 61 1 76 16 56 47 67 4 78 50 3 22 53 32 49 43 73 71 37 60 75 23 45 65 2110 17 59 8 18 64 68 77 27 14 38 9 52 26 66 69 12 2 54 34 44 58 13 57 28 62 72 51 D23: (r, A, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 46 8 37 72 63 17 70 78 47 6058 52 25 27 56 39 73 12 50 20 54 53 11 23 71 4 55 64 57 18 34 9 43 29 62 51 35 19 32 1 31 67 77 41 28 15 44 14 49 16 10 40 42 68 2 13 21 74 75 24 65 6 66 30 45 26 5 22 33 76 59 69 61 7 36 3 38 48 D24: (A, R, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 23 37 52 22 16 71 74 50 65 11 78 3 28 25 15 18 41 45 63 53 59 8 38 7364 9 17 57 13 40 4 34 69 24 10 36 ₄₆ 31 70 39 14 51 54 49 21 6 35 12 58 76 20 66 19 2 29 56 43 7 44 62 47 77 67 32 30 75 72 68 1 48 55 42 26 33 60 27 5 61 D25: (ω, R, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5 40 66 11 62 37 26 3 51 50 1 77 25 70 75 1432 8 2127 61 52 19 68 57 56 47 76 38 72 35 2 71 9 48 12 1839 53 4342 10 6 33 67 64 24 4 23 60 13 78 74 30 55 31 41 20 49 46 7 34 73 69₄₄ 15 63 17 58 28 45 29 59 65 22 16 54 36 D30: (d, R, P) Figure 5.2: Family Fdsc

1 of discretized simply connected compact sets mapped into eleven shape

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0 0.2 0.28 0.35 0.4 0.45 0.49 0.53 0.57 0.6 0.63 0.66 0.69 0.72 0.75 0.77 0.8 0.82 0.85 0.87 0.89 0.92 0.94 0.96 0.98 1 Convexity 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D1: (ω, r, R) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D3: (r, A, R) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D4: (A, d, R) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D12: (ω, A, d) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D13: (r, A, d) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D15: (ω, R, d) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D21: (ω, r, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D23: (r, A, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D24: (A, R, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D25: (ω, R, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D30: (d, R, P) Figure 5.3: Family Fdsc

2 of discretized simply connected compact sets mapped into eleven shape

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5.2 Dispersion quantification and convexity discrimination A wider family Fdsc

2 of 1370 discretized patterns is considered (Kimia database [11], not

illustrated here). These are twenty deformations of each of the sixty-eight first sets of the family Fdsc

1 , plus the ten last sets of F1dsc; section 6 shows two examples of

these deformations). For each discretized pattern representing a discretized simply connected compact set, the morphometrical functionals are computed and located by a point in each shape diagram (Figure 5.3). The point color is associated to the convexity parameter value c, and the convex domain boundary, if it is known, is illustrated with black lines.

5.2.1 Dispersion quantification For each shape diagram, the dispersion of the locations of the 2D discretized simply connected compact sets of the family Fdsc

2 is

studied.

The spatial distribution of discretized simply connected compact sets locations in each shape diagram is characterized and quantified from algorithmic geometry using Delaunay’s graph (DG) and minimum spanning tree (MST) [1]. Some useful information about the disorder and the neighborhood relationships between sets can be deduced. From each geometrical model, it is possible to compute two values from the edge lengths, denoted µ (average) and σ (standard deviation) for DG or MST. The simple reading of the coordinates in the (µ, σ)-plane eanables to determine the type of spatial distribution of the discretized simply connected compact set (regular, random, cluster, . . . ) [7]. The decrease of µ and the increase of σ characterize the shift from a regular distribution toward random and cluster distributions.

Figure 5.4 represents both values of parameters of the twenty-two shape diagrams for each model, DG and MST.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 1 2 3 4 5 6 11 12 13 14 15 16 21 22 23 24 25 26 27 2829 30 average µ standard deviation σ DG 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 1 2 3 4 5 6 11 12 13 14 15 16 21 22 23 24 25 26 27 28 29 30 average µ standard deviation σ MST

Figure 5.4: Two dispersion quantifications for all shape diagrams applied on the family Fdsc

2 . For

each representation (according to the models DG and MST, respectively), indices k ∈ J1, 30K \

(J7, 10K ∪ J17, 20K) of the shape diagrams Dk is located according to its µ and σ values.

As in [9], the DG model allows to divide the shape diagrams into two groups, mainly according to the average µ value: the shape diagrams D4, D5, D6, D14, D15,

D16 and D30 have a weak average µ, thus the discretized simply connected compact

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this tendency for the average. Figure 5.4 shows also a distribution of these two groups according to the standard deviation σ. Note that the constitution of each of the two groups is the same as in [9].

5.2.2 Convexity discrimination Observing the colors dispersion in the shape diagrams of Figure 5.3, some shape diagrams seem to stronger discriminate the convex shapes from the non-convex shapes than others. This stronger discrimination is visually highlighted by a color gradient, from dark red to dark blue. Within a shape diagram, if the colors are irregularly distributed according to the color range, this means that convex sets and non-convex sets can not be discriminated.

Equation 5.1 quantifies (by values ranging between 0 and 1) this discrimination for each shape diagram Dk, k ∈ J1, 30K \ (J7, 10K ∪ J17, 20K). A high (resp. low)

resulting value means a weak (resp. strong) convexity discrimination. Convexity_discrimination(Dk) = (5.1) 1 #Fdsc 2 #Fdsc 2 X i=1

c(i) − c argmindE(i, j)|j ∈ F₂dsc

where c(i) denotes the convexity value of the shape indexed by i in Figure 5.1, and dE is the Euclidean distance.

Figure 5.5 shows the results of this quantification for every shape diagrams. The stronger convexity discrimination appears in the shape diagrams D24, D25, D26,

D27, D28 and D29, that is in agreement with the visual interpretation. The weaker

discrimination appears for the shape diagrams D4, D6, D14 and D16. Even if the

shape diagram D30 is not strong for the general shape discrimination [8, 9], it is

not weak for the convexity discrimination. In fact, these two discriminations are independant. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 D1 D2 D3 D4 D5 D6 _D11 D12 D13 D14 D15 D16 D21 D22 D23 D24 D25 D26 D27 D28 D29 D30

Figure 5.5: Convexity discrimination for the family Fdsc

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6 Shape diagrams for similar discretized simply connected compact sets

This section focuses on the discrimination of shapes that are visually similar enough, so that they can be considered as the same global shape. The shape di-agrams D4, D5, D6, D14, D15, D16 and D30 provide a weak shape discrimination,

whatever the visual similarity of the sets. Thus, they are not considered from this section and until the end of this paper. It remains the fifteen shape diagrams Dk,

k ∈ J1, 29K \ (J4, 10K ∪ J14, 20K). Let be three families Fdsc

3 , F4dsc and F5dsc of discretized patterns representing

triangles (Figure 6.1), disks (Figure 6.2) and crosses (Figure 6.3), respectively, that have undergone minor transformations, modifications, deformations. The morpho-metrical functionals are computed, and the patterns are located by a point in each shape diagram (Figures 6.1, 6.2 and 6.3). The color of the number is related to the convexity parameter value c, and the convex domain boundary, if it is known, is illustrated with black lines.

Figures 6.4, 6.5 and 6.6 give the dispersion quantification representation of the fifteen shape diagrams for each model, DG and MST (method described in subsub-section 5.2.1).

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0 0.2 0.28 0.35 0.4 0.45 0.49 0.53 0.57 0.6 0.63 0.66 0.69 0.72 0.75 0.77 0.8 0.82 0.85 0.87 0.89 0.92 0.94 0.96 0.98 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 14 3 13 2 19 8 5 4 9 12 20 11 1 15 18₁₀ 16 17 6 7 D1: (ω, r, R) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 11 16 56 3 8 1 10 17 20 15 18 14 4 13 12 7 2 9 19 D3: (r, A, R) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 16 20 17 6 10 2 89 15 19 1 18 14 7 4 5 11 3 13 12 D12: (ω, A, d) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 12 2 14 17 19 13 4 1 9 6 8 10 16 20 18 15 3 7 11 5 D13: (r, A, d) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 16 18 9 17 4 11 2 12 10 8 6 15 5 1 13 7 14 19 20 3 D21: (ω, r, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 11 6 2 5 20 9 4 19 13 14 8 18 1 3 16 71217 15 10 D23: (r, A, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 8 10 12 13 19 7 11 6 20 4 1 2 15 3 17 5 18 16 9 14 D24: (A, R, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 24 8 13 3 6 18 14 9 5 11 1216 20 17 7 15 10 1 19 D25: (ω, R, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 8 15 12 1116 4 5 2 10 1 18 14 7 9 6 19 3 13 17 20 D29: (r, d, P) Figure 6.1: Family Fdsc

3 of twenty 2D discretized simply connected compact sets with ’triangle’

shape, mapped into nine shape diagrams (chosen according to the results synthetized in [9]). The color of the number is related to the convexity parameter value.

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0 0.2 0.28 0.35 0.4 0.45 0.49 0.53 0.57 0.6 0.63 0.66 0.69 0.72 0.75 0.77 0.8 0.82 0.85 0.87 0.89 0.92 0.94 0.96 0.98 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 3 14 19 20 5 10 12 15 11 9 8 17 7 13 4 2 6 16 18 D1: (ω, r, R) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 7 2 6 1615 10 3 13 12 1 20 8 5 9 4 19 18 17 11 14 D3: (r, A, R) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 11 14 12 15 9 10 18 7 2 3 19 6 1 4 8 16 20 1713 5 D12: (ω, A, d) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 6 11 14 9 3 17 5 10 16 8 2 13 18 20 7 12 ₁₉ 4 1 15 D13: (r, A, d) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 18 9 17 3 16 20 19 15 4 8 1 6 5 11 13 12 10 7 14 D21: (ω, r, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5 9 11 19 16 10 17 6 12 3 20 15 14 2 7 4 13 1 8 18 D23: (r, A, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 20 1 8 15 2 17 7 19 4 3 109 16 13 6 12 14 11 5 18 D24: (A, R, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 7 1 16 5 9 15 8 12 10 17 13 11 4 2 6 18 20 3 14 19 D25: (ω, R, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 20 5 11 10 12 18 17 8 6 14 2 16 9 13 15 3 47 19 D29: (r, d, P) Figure 6.2: Family Fdsc

4 of twenty 2D discretized simply connected compact sets with ’disk’ shape,

mapped into nine shape diagrams (chosen according to the results synthetized in [9]). The color of the number is related to the convexity parameter value.

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0 0.2 0.28 0.35 0.4 0.45 0.49 0.53 0.57 0.6 0.63 0.66 0.69 0.72 0.75 0.77 0.8 0.82 0.85 0.87 0.89 0.92 0.94 0.96 0.98 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 17 11 18 8 13 19 920 14 16 4₁₀ 7 512 2 15 1 6 3 D1: (ω, r, R) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 16 9 7 19 14 18 118 5 13 6 4 15 12 17 220 10 13 D3: (r, A, R) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 12 14 1720 19 11 9₇ 10 6 1815 813 5 1 4 3 16 D12: (ω, A, d) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5 13 1216 17 3 6 182015 4 9 8 19 1 11 10 14 2 7 D13: (r, A, d) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 9 19 11 812 13 10 1 14 5 20 46 7 15 3 1618 17 2 D21: (ω, r, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 7 1 13 118 15 17 19 18 4 12 20 9 32 16 5 10 6 14 D23: (r, A, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 14 19 4 10 15 2 16 1 20 7 17 13918 8 6 5 3 12 11 D24: (A, R, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 11 17 14 3 20 2 4 1 9 19 131215 5 7 16 8 6 18 10 D25: (ω, R, P) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 3 20 6 1416 18 5 4 1291715 11 1013 2 7 19 1 8 D29: (r, d, P) Figure 6.3: Family Fdsc

5 of twenty 2D discretized simply connected compact sets with ’cross’ shape,

mapped into nine shape diagrams (chosen according to the results synthetized in [9]). The color of the number is related to the convexity parameter value.

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0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 1 2 3 11 12 13 21 22 23 24 25 2627 28 29 average µ standard deviation σ DG 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.02 0.04 0.06 0.08 0.1 0.12 1 2 3 11 12 13 21 22 23 24 25 26 27 28 29 average µ standard deviation σ MST

3 . For

0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 1 2 3 11 12 13 21 22 23 24 25 26 27 28 29 average µ standard deviation σ DG 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.02 0.04 0.06 0.08 0.1 0.12 1 2 3 11 12 13 21 22 23 24 25 26 27 28 29 average µ standard deviation σ MST

4 . For

0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 1 2 3 1112 13 21 22 23 24 25 26 27 28 29 average µ standard deviation σ DG 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 1 2 3 11 12 13 21 22 23 24 25 26 27 28 29 average µ standard deviation σ MST

5 . For

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The results obtained allows to reveal that the shape diagrams D2 and D12 do

not discriminate the sets in each family Fdsc

3 , F4dsc and F5dsc. This discrimination

appears little stronger for the shape diagrams D1 and D11. From a global vision, the

shape diagrams D21 and D22 strongest discriminate these sets.

7 Synthesis

To obtain a strong discrimination of 2D discretized simply connected compact sets, it is necessary to have both a strong dispersion and a strong convexity discrim-ination.

• The shape diagram D4, D5, D6, D14, D15 and D16 are excluded due to their

weak dispersion and convexity discrimination results.

• The shape diagram D30 presents a strong convexity discrimination although

its weak dispersion result.

• The dispersion quantification of the shape diagrams D1, D2, D3, D11, D12,

D13, D21, D22, D23, D24, D25, D26, D27, D28 and D29 gives strong values.

• The convexity discrimination is stronger for D24, D25, D26, D27, D28 and D29.

• The similar sets discrimination appears stronger for D21 and D22.

Futhermore, among the shape diagrams D24, D25, D26, D27, D28 and D29that

ob-tain the best results for dispersion quantification and convexity discrimination, only D24, D26 and D28 are based on known complete systems of inequalities. Observing

in details the representation of quantifications for these three shape diagrams, D24

retained for shape discrimination of analytic simply connected compact sets, is also retained for shape discrimination of discretized simply connected compact sets.

This analysis is summarized in Table 7.1.

Complete system Non-complete system of inequalities of inequalities Strong D24 , D26, D28 D25, D27, D29 discrimination Moderate _D₁ _{, D}₃ _{, D}₁₁_{, D}₁₂_{, D}₂₂_, D23 D2, D13 , D21 discrimination Weak _D₄ _{, D}₅_{, D}₆_{, D}₁₄_{, D}₁₅ _, D16, D30 discrimination

Table 7.1: Shape diagrams classification according to their quality of shape discrimination of dis-cretized simply connected compact sets and according to the completeness of associated systems of inequalities.

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8 Global synthesis for the three parts of this study

For each part of this study [8, 9], the synthesis gives the shape diagrams that provide the strongest shape discrimination. These are D12 and D24. Table 9.1

gathers the syntheses.

9 Conclusion

This paper has dealed with shape diagrams of 2D non-empty analytic and dis-cretized simply connected compact sets built from six geometrical functionals: the area, the perimeter, the radii of the inscribed and circumscribed circles, and the min-imum and maxmin-imum Feret diameters. Each set is represented by a point within a shape diagram whose coordinates are morphometrical functionals defined as normal-ized ratios of geometrical functionals. From existing morphometrical functionals for these sets, twenty-two shape diagrams can be built. A detailed comparative study has been performed in order to analyze the representation relevance and discrimi-nation power of these shape diagrams. It is based on the dispersion quantification and convexity discrimination from compact set locations in diagrams. Among all the shape diagrams, six present a strong convexity discrimination of sets, three are based on complete system of inequalities. Among these three diagrams, the shape diagram D24: (A, R, P) is retained for its representation relevance and discrimination power.

The purpose of this paper was to present the third part of a general comparative study of shape diagrams. The focus was placed on convexity discrimination of analytic and discretized simply connected compact sets. The two first parts [8, 9] was restricted to the analytic compact convex and simply connected compact sets, respectively. For an analytic set, the geometrical functionals were accurately calculated. For a discretized set, they are estimated. Thus, in the discrete case, the shape diagrams are based on estimated morphometrical functionals.

Actually, the authors work on the case of hollowed sets (analytic and discretized) which is not specifically considered in this paper.

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S . R IV O L L IE R , J. D E BA Y L E A N D J. -C . P IN O L I

Complete system of inequalities Non-complete system of inequalities

Analytic Analytic Analytic Analytic

compact simply connec- Convexity compact simply connec- Convexity

convex sets [8] ted compact sets [9]

discrimination convex sets [8] ted compact sets [9] discrimination Strong D3, D12, D22, D23 D3, D22, D23, D24, D26 D24, D26, D28 D2, D13 D13, D21, D27, D29 D25, D27, D29 discrimination D1, D7, D9, D11, D18, D24, D26, D28 D1, D11, D12, D28 D1, D3, D11, D12, D22, D23 D8, D17, D19, D21, D25, D27, D29 D2, D25 D2, D13,D21 Moderate discrimination Weak D4, D5, D6, D10, D14, D15, D16, D20, D30 D4, D5, D6, D14, D15, D16, D30 D4, D5, D6, D14, D15, D16, D30 D31 discrimination

Table 9.1: Shape diagrams classification according to their quality of shape discrimination and according to the completeness of associated systems of inequalities.

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