Keywords: particle size distribution, (PSD), image analysis, 2-D images of particles, batch crystallization process, EZProbe sensor, segmentation algorithms, restoration algorithms

This article presents a new in situ method to monitor the particle size distribution (PSD) during batch solution crystallization processes. Using a new in situ imaging probe, the "EZProbe sensor," real time acquisition of 2-D images of particles during the batch process is now possible. To analyze these images, a novel image analysis method is carried out. First, segmentation and restoration algorithms are performed to identify the particles and thereafter geometrical particle measurements are achieved to obtained the PSD of the batch crystallization process over time. Satisfactory measurements are obtained provided that the overall solid concentration does not exceed a threshold above which too many overlapping crystals make discrimination between particles impossible. K Ke ey yw wo or rd

I. Introduction

Solution crystallization processes are widely used in the process industry and notably in the pharmaceutical industry as separation and purification operations and are expected to produce solids with desirable properties. In particular, in terms of pharmaceutical solid ingredients, the size and the shape of crystals are known to have a considerable impact on the final quality of drugs (i.e., bioavailability, stability during storage, ease of processing, etc.). To measure the particle size distribution (PSD); it is well established that conventional monitoring techniques, such as laser diffraction (LD), ultrasonic attenuation spectroscopy (UAS), or focused-beam reflectance measurement (FBRM) do not provide reliable in-line estimates. Major difficulties arise from the use of LD techniques since it requires highly diluted samples of rather ideally shaped particles. Indeed, "ideally shaped" means here that the particles, to fit the theoretical models used to process LD measurements, should be as close as possible to spheres and exhibit rather simple distributions [START_REF] Keck | Size analysis of submicron particles by laser diffractomery-90% of the published measurements are false[END_REF] (i.e., multimodal distributions should be very cautiously analyzed). The main disadvantage of UAS is that it requires a large set of accurate physical data related to the liquid and particle phases, [START_REF] Mougin | Sensitivity of particle sizing by ultrasonic attenuation spectroscopy to material properties[END_REF] which is almost prohibitive for routine industrial applications. Finally, the main disadvantage of FBRM is that it does not actually measures the PSD but the chord length distribution (CLD). One must therefore convert the measured CLD into its corresponding PSD, which is an illposed problem, even though successful experimental applications have been reported for spheres [START_REF] Hukkanen | Measurement of particle size distribution in suspension polymerization using is situ laser backscattering[END_REF] and octahedrons [START_REF] Worlitschek | Restoration of PSD from chord length distribution data using the method of projections onto convex sets[END_REF]. Various 2-D image based methods also exist, [START_REF] Calderon De Anda | Multi-scale segmentation image analysis for the in-process monitoring of particle shape with batch crystallisers[END_REF][START_REF] Larsen | An alforithm for analyzing noisy, in situ images of high-aspect-ratio crystals to monitor particle size distribution[END_REF][START_REF] Zhou | Critical evaluation of image processing approaches for real-time crystal size measurements[END_REF] but most of them require a priori knowledge and/or assumptions about the shape of the particles. Therefore, the latter methods cannot easily handle crystallization phenomena such as aggregation or breakage of particles. Moreover, those methods do not take into account particles outside the focal plane. Contrary to those methods, this study does not make any assumption about the shape of the particles. Therefore, it can handle particles of any shape, and moreover, it restores out-of-focus particles to maximize the data to be analyzed and to acquire an accurate particle characterization. In the first section, the experimental system used to monitor the crystallization process is explained. In the second section, the image analysis algorithm is developed (image segmentation, restoration and measurement). Finally, typical characterization results are presented before concluding. ] and filmed a restricted volume within the suspension. The system acquires video sequences of 2-D projections of 3-D particles with the following specifications: 25 frames/ s, each frame is 640 x 480 pixels 8-bit grayscale bit depth representing 1.3 x 1 mm. In addition to the in situ camera, the process was monitored using in situ temperature and supersaturation measurements; the latter were performed using attenuated total reflection (ATR) Fourier transform infrared (FTIR) spectroscopy [START_REF] Dunuwila | An investigation of the applicability of attenuated total reflection infrared spectroscopy for measurement of solubility and supersaturation of aqueous citric acid solutions[END_REF][START_REF] Lewiner | On-line ATR FTIR measurement of supersaturation during solution crystallization processes. Calibration and applications on three solute/solvent systems[END_REF]. I II I. .2 2. . I Im ma ag ge e A Ac cq qu ui is si it ti io on n The new imaging probe enables real-time acquisition of 2-D images of the particles generated during crystallization processes. Figure 2 shows some typical images of citric acid particles during crystallization at different times. To quantitatively characterize the particles, some difficulties must be considered in the different image processing steps: particle clustering, shape heterogeneity, anisotropy, particles outside the focal plane, etc. However, because the image analysis method was designed without any assumption about the shape of the particles and it restores blurred particles, it can handle these phenomena. The next section explains the image analysis method, which was designed so as to deal with these difficulties. Im ma ag ge e S Se eg gm me en nt ta at ti io on n First, for each image of the video sequence [Figure 3(a)], all particle regions were isolated, as displayed in [Figure 3(b)]. The method here consists of applying the watershed algorithm [START_REF] Beucher | The watershed transformation applied to image segmentation[END_REF] constrained by the h-minima [START_REF] Beucher | The watershed transformation applied to image segmentation[END_REF] of the image. Then, the spatial support of each particle was detected [Figure 3(c)] by applying on each delimited area the automatic thresholding Journal of Electronic Imaging, 2010, 19(3), 0312071-7, Note: Special Section on Quality Control by Artificial Vision, doi: 10.1117/1.3462800 proposed by Gonzalez and Woods [START_REF] Gonzalez | Digital Image Processing[END_REF]. Second, the particles in the focal plane (in-focus particles) were discriminated from those outside the focal plane (blurred-particles) using a focus measurement [START_REF] Yap | Image focus measurement based on Chebyshev moments[END_REF][START_REF] Kautsky | A new wavelet based measure of image focus[END_REF][START_REF] Wee | Measure of image sharpness using eigenvalues[END_REF] calculated locally: the variance focus measurement [START_REF] Subbarao | Focusing techniques[END_REF] defined as

II. Materials, Experimental Setup, and Image Acquisition

III. Image Analysis Method: PSD Imaging

[ 2 1 1 1 ( , ) N M x y ] I x y µ MN = = - ∑∑ (1)
where is the gray-level image

2 : I Ω ⊂ → Z Z ( , ) x y ∈ Ω , M x N
is the size of the neighborhood, and

1 1 1 ( ) ( , N M x y ) µ I x y MN = = = ∑ ∑
is the mean gray level in the neighborhood.

Then, performing a reconstruction process between the "watershed image" and the "variance image", out-of-focus regions and in-focus regions were determined, respectively [Figure 3(d)]. Therefore, at the end of the segmentation two elements of information were obtained: the spatial support and the degree of focus of each particle/region [START_REF] Presles | Suivi in situ de la distribution en taille et en forme de particules lors de leurs cristallisations en solution[END_REF].

3 I II II I. .2 2. . I Im ma ag ge e R Re es st to or ra at ti io on n To maximize the data to be analyzed and to achieve an accurate particle characterization, a restoration step was performed: particles outside the focal plane were restored. To do so, a blind deconvolution technique was applied to the detected blurred particles/clustered particles. Due to the presence of "ringing effects" in classical blind deconvolution algorithms, [START_REF] Kundur | Blind image deconvolution[END_REF][19][START_REF] Fish | Blind deconvolution by means of the Richardson-Lucy algorithm[END_REF] a new blind deconvolution algorithm based on the image characteristics was developed [START_REF] Presles | Monitoring the particle size distribution using image analysis during batch crystallization processes[END_REF]. About the latter, it is assumed that a particle 1-D profile can be modeled by a perfect square function [Figure 4]. However in practice, it is assumed that only the edge of an in-focus particle is perfect. If holes occur inside the particle image, they are filled. Moreover, it is hypothesized that on each region:

The particle image is noise free and the blur is spatially invariant; that is, g(x)= f(x)h(x), where g is the gray-level image of the blurred particle resulting from a convolution of f, the gray level image of the original particle, and h, the point spread function (PSF).

The blur is weak in the sense that max(g)=max(f).

The PSF h is isotropic and normalized so that

( ) 1 h x dx ∞ -∞ = ∫ .
Under these hypotheses and using Fubini's theorem, it can be demonstrated that

( ) ( ) g x dx f x dx ∞ ∞ -∞ -∞ = ∫ ∫
. Consequently, the gray level image f of the original particle can be approximated by optimization:

argmin ( ) ( ) p f p x dx g x dx ∞ ∞ -∞ -∞ = - ∫ ∫ , ( 2
)
where p belongs to the set of the square functions of "height" max(g). To compute the solution of Equation ( 2), f was calculated by iteratively eroding in the sense of mathematical morphology [START_REF] Soille | Morphological Image Analysis: Principles and Applications[END_REF] the spatial support of g (p 0 , p 1 , etc.) [Figure 5]. The blind deconvolution algorithm already described was applied to the already detected blurred particles. The focused particles were not processed. Figure 6 shows the results for a few selected particles from the original image [Figure 3 . Then, the proposed segmentation step was applied on these images with or without the deconvolution step, and in each case, the normalized root mean square error (NRMSE) was calculated according to σ [Figure 7(c)]. As it can be noticed in [Figure 7(c)], whatever the object is and whatever σ is, the value of the NRMSE is less important when the algorithm is applied with the restoration step than without it. I II II I. .4 4. . I Im ma ag ge e M Me ea as su ur re em me en nt t As the spatial support of each particle is now well defined (blurred particles in particular), an accurate PSD analysis is possible. To do so, each particle projected area was measured and for each projected area, the diameter of the sphere with the same projected area was calculated. Then, the PSD of the particles was constructed by calculating the normalized histogram of the diameter values. Hence, the particle size is equal to the diameter of the sphere with an equivalent projected area.

IV. Validation and Results

I

IV V. .1 1. . V
Va al li id da at ti io on n t th hr ro ou ug gh h C Co om mp pu ut te er r S Si im mu ul la at ti io on n To validate the PSD imaging method explained in the previous section, an experiment was performed on simulated images. A thousand isotropic particles with random size diameters D (2 pixels≤D≤60 pixels) were computationally generated [hence, their PSD was known (PSD 1) ]. Then, each particle was blurred artificially by a normalized gaussian PSF with a random standard deviation 1,5 σ ∈ . Applying the PSD imaging method already explained, the PSD of the particles was recovered (PSD 2 ). To achieve reliable statistical data, this process was repeated 100 times and each time, to quantify the error between the two PSDs, the normalized L 1 e 1 Err) between the two cumulat ve size distributions (CSD 1 and CSD 2 ) rror (NL i was calculated { }

1 1 2 100 / NL Err CSD CSD CSD ⎡ ⎤ = - ⎣ ⎦ ∫ 1 ∫
. Figure 8 shows an example of CSDs obtained with this simulation. As one can notice, CSD 2 fits better CSD 1 for particles with diameters larger than 12 pixels. Indeed, for small particles, the hypothesis max(g)=max(f) has a greater probability to be unverified. However, on average, the normalized L 1 error was only about 2.49% with a standard deviation of 0.18%. I IV V. .2 2. . E Ex xp pe er ri im me en nt ta al l V Va al li id da at ti io on n To validate the PSD imaging method explained in the previous section, an experiment was performed using polydisperse silica beads. The PSD of the silica beads was obtained using two different methods: the PSD imaging method presented here and Coulter counter measurements. The Coulter counter is a rather reliable offline device that is currently used for particle sizing and counting, e.g., in the field of biology [23] (blood cells, microscopic algae, etc.). It detects the change in electrical conductance of a small aperture as a fluid containing particles/cells is drawn through. The particle size measured with the Coulter counter is the diameter of the sphere whose volume is equal to that of the particle. The measured particles in this experiment are spherical beads, therefore the diameters measured with the PSD imaging method are comparable to the diameters measured with the Coulter counter. The PSD obtained with the Coulter counter is expressed in terms of volume percentage and for convenience, the bins size was chosen as 4 µm. By calculating for each diameter obtained with the PSD imaging method the volume of the corresponding bead, it is also possible to express the PSD of the silica beads in terms of volume percentage. Therefore, the PSDs obtained with these two techniques are comparable. The image processing PSD was calculated on 1000 particles. First, one can notice in Figure 9 that the two PSDs look very similar and more particularly that the main modes obtained with the two different methods are both located around 90 µm. Moreover, the value of the normalized L 1 error between the two cumulative size distributions is only about 3%. I IV V. .3 3. . R Re es su ul lt ts s Figure 10(b) shows a typical result of PSD (PSD is expressed in terms of percentage in number of particles) obtained after processing the raw image displayed in Figure 10(a) using the PSD imaging method explained in the previous section. Both the PSD in the reactor and in the measurement cell are assumed to be homogeneously distributed due to stirring, which maintains the particles in suspension. Five minutes after seeding (i.e., after crystallization the begun), it can be observed that the majority of the particles still exhibit a diameter (the measured diameter corresponds to the diameter of the sphere with an equivalent projected area) smaller than 100 µm [Figure 10(b)]. However, note that a peak is present around 167 µm, which corresponds to the aggregated particle present in the bottom center of the image [Figure 10(a)]. Since only few particles can be observed in each image, an improvement is obtained after averaging the PSD. It is usually considered that correct particle sampling requires at least 1000 particles to be measured. Therefore, to link the averaging of the PSD to the crystallization process, a particle counter was implemented in the PSD imaging method. Once the particle counter is above the threshold of 1000 particles, an average PSD is computed. Figure 11(a) . This is clearly because the set of particles processed to yield the "instantaneous PSD" at t=5 min 23 s is too small to achieve an accurate measurement. In addition to the size distribution, note that shape parameters (elongation, convexity, etc.) could be calculated for each particle, providing information about possible anisotropy of the particles and aggregation processes.

V. Conclusions

During this work, a novel image processing method, called PSD imaging, enabling us to estimate the time-varying PSD of particles in suspension was developed and evaluated through crystallization experiments performed with citric acid in water. We acquired 2-D images using a new in situ imaging probe and treated them using an image analysis method composed of three steps. A segmentation step enabled us to identify each particle in an image (spatial support and focused/nonfocused particles). Thereafter, a restoration step was designed that enabled us to maximize the number of particles that could be processed in a reliable way. Such a goal was reached by means of the deconvolution of blurred particles. Finally, the last step (particle measurement) enabled us to obtain the PSD over time (up to moderate solid concentration). Experimental validation was performed through computer simulation and silica beads experiments. The proposed PSD imaging method was implemented in MATLAB and all the computations were performed on a personal computer with a 2-GHz CPU and 512 Mb of RAM running windows XP. With such computer setup and after a few optimizations, the PSD imaging method required about 1s to complete an image. It is clear that such a processing time is too long to enable real-time PSD calculation. However, a conversion to the C language and a multi-CPU optimization could drastically improve it.

I

  II I. .1 1. . M Ma at te er ri ia al ls s a an nd d E Ex xp pe er ri im me en nt ta al l S Se et tu up p Crystallization experiments were performed with citric acid particles crystallizing in water in a 3-L lab-scale batchjacketed crystallizer, equipped with a profiled pale propeller (Mixel TT) and four baffles. The stirring rate was set to 250 rpm. The temperature of the crystallizing suspension was controlled by means of hydroalcoholic fluid circulating in the jacket. Isothermal desupersaturation crystallization experiments were started though seeding: sieved citric acid particles were introduced in the crystallizer maintained under supersaturated conditions. To monitor the experiments a new in situ imaging probe, the "EZProbe sensor" [Figure 1(a)], developed at the University of Lyon (Lyon 1) was used. A CCD camera connected to an acquisition interface records transmitted light images. The interface retrieves the video data, compresses it if necessary, and sends it to a computer [Figure 1(b)]. The probe was immersed into the reactor, as shown in [Figure 1(b)

  (a)]. I II II I. .3 3. . R Re es st to or ra at ti io on n V Va al li id da at ti io on n To validate the proposed deconvolution algorithm, an image of an isotropic object [Figure 7(a)] and an image of an anisotropic one [Figure 7(b)] were blurred artificially by a normalized Gaussian PSF with a standard deviation 1,5 σ ∈

  (PSD 1 ) shows the average PSD of citric acid particles computed after about 2 min of crystallization and Figure 11(b) (PSD 2 ) shows the average PSD of citric acid particles computed after about 5 min of crystallization. As one can notice, due to the development of both crystal nucleation and growth, the PSD spreads and the growth of the initial seed particles leads to the emergence of new PSD classes: [140µm, 200µm] [Figure 11(b)]. Note that the average PSD computed after about 5 min of crystallization is quite different from the PSD calculated at t=5 min 23 s [Figure 10(b)]

Figures 8 F 9 F

 89 Figures

F

  Fi ig gu ur re e 7 7: : N NR RM MS SE E a as s a a f fu un nc ct ti io on n o of f s si ig gm ma a f fo or r ( (a a) ) a an n i is so ot tr ro op pi ic c a an nd d ( (b b) ) a an n a an ni is so ot tr ro op pi ic c o ob bj je ec ct ts s a an nd d ( (c c) ) c ca al lc cu ul la at te ed d w wi it th h a an nd d w wi it th ho ou ut t t th he e d de ec co on nv vo ol lu ut ti io on n s st te ep p f fo or r t th he e t tw wo o o ob bj je ec ct ts s. . F Fi ig gu ur re e 8 8: : I In n r re ed d ( (s so ol li id d l li in ne e) ) t th he e r re ea al l C CS SD D o of f t th he e p pa ar rt ti ic cl le es s ( (C CS SD D1 1) ); ; i in n b bl lu ue e ( (d da as sh he ed d l li in ne e) ) t th he e r re ec co ov ve er re ed d C CS SD D ( (C CS SD D2 2) ). . ( (C Co ol lo or r o on nl li in ne e o on nl ly y) ). .F Fi ig gu ur re e 9 9: : P PS SD Ds s o of f t th he e p po ol ly yd di is sp pe er rs se e s si il li ic ca a b be ea ad ds s o ob bt ta ai in ne ed d w wi it th h ( (a a) ) t th he e P PS SD D i im ma ag gi in ng g m me et th ho od d a an nd d ( (b b) ) t th he e C Co ou ul lt te er r c co ou un nt te er r m me et th ho od d. . P PS SD Ds s a ar re e e ex xp pr re es ss se ed d i in n t te er rm ms s o of f v vo ol lu um me e p pe er rc ce en ntta ag ge e. . F Fi ig gu ur re e 1 10 0: : P PS SD D o of f a ac ci id d c ci it tr ri ic c p pa ar rt ti ic cl le es s pr ro oc ce es ss se ed d 2 2--D D i im ma ag ge e a an nd d ( (b b) ) P PS SD D i in n n nu um mb be er r. . T Th he e P PS SD D i is s e ex xp pr re es ss se ed d i in n t te er rm ms s o of f p pe er rc ce en nt ta ag ge e o of f n nu um mb be er r o of f p pa ar rt ti ic cl le es s. . F Fi ig gu ur re e 1 11 1: : A Av ve er ra ag ge e P PS SD D a af ft te er r a ab bo ou ut t ( cr ry ys st ta al ll li iz za at ti io on n. . T Th he e P PS SD Ds s a ar re e e ex xp pr re es ss se ed d i in n t te er rm ms s o of f p pe er rc ce en nt ta ag ge e i in n n nu um mb be er r o of f p pa ar rt ti ic cl le es s. .

Acknowledgements

We greatly acknowledge the French research agency ANR for the support granted to the project "IPAPI" (Improving the Properties of Active Pharmaceutical Ingredients), reference 07-BLAN-0183.


S Rivollier
email: rivollier@emse.fr

J Debayle
email: debayle@emse.fr

J.-C Pinoli
email: pinoli@emse.fr

SHAPE DIAGRAMS FOR 2D COMPACT SETS -PART I: ANALYTIC CONVEX SETS

Keywords: Analytic compact convex sets, Geometrical and morphometrical functionals, Shape diagrams, Shape discrimination

. They are focused on analytic simply connected sets and convexity discrimination for analytic and discretized simply connected sets, respectively.

Introduction

The Blaschke's shape diagram [START_REF] Blaschke | Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts[END_REF][START_REF] Blaschke | Eine frage über konvexe körper[END_REF] allows to represent a 3D compact convex set by a point in the Euclidean 2D plane from three geometrical functionals: the volume, the surface area and the integral of mean curvature. The axes of this shape diagram are defined from geometric inequalities relating these functionals. These geometric inequalities do not provide a complete system: for any range of numerical values satisfying them, a compact convex set with these values for the geometrical functionals does not necessarily exist (in other words, a point within the 2D Blaschke shape diagram does not necessarily describe a 3D compact convex set).

Following the approach of Blaschke, Santalo [START_REF] Santaló | Sobre los sistemas completos de desigualdades entre tres elementos de una figura convexa plana[END_REF] considered the shape diagrams of 2D compact convex sets from six geometrical functionals: the area, the perimeter, the radii of the inscribed and circumscribed circles, and the minimum and maximum Feret diameters [START_REF] Feret | La grosseur des grains des matières pulvérulentes[END_REF]. Several studies on these shape diagrams have been performed [START_REF] Böröczky | Optimizing area and perimeter of convex sets for fixed circumradius and inradius[END_REF][START_REF] Hernández Cifre | Is there a planar convex set with given width, diameter, and inradius ?[END_REF][START_REF] Hernández Cifre | The missing boundaries of the Santaló diagrams for the cases (d,ω,R) and (ω,R,r)[END_REF][START_REF] Hernández Cifre | Complete Systems of Inequalities[END_REF][START_REF] Hernández Cifre | Optimizing the perimeter and the area of convex sets with fixed diameter and circumradius[END_REF][START_REF] Santaló | Sobre los sistemas completos de desigualdades entre tres elementos de una figura convexa plana[END_REF], but they were mainly restricted to the complete systems of inequalities for some triplets of these geometrical functionals.

This paper focuses on the study of shape diagrams for a wide range of 2D nonempty analytic compact convex sets. This study is not limited to complete systems of inequalities. The considered compact convex sets are mapped onto points in these shape diagrams, and through dispersion and overlapping quantifications, the shape diagrams are classified according to their ability to discriminate the compact convex sets.

Shape functionals

In this paper, the non-empty analytic compact convex sets in the Euclidean 2space E 2 are considered. A set will be called analytic if its boundary is piecewise defined by explicit functions in such a way that the geometrical functionals enumerated below can be calculated. These geometrical functionals are determinated in order to characterize the sets. They are related by the so-called geometric inequalities, which allow to define morphometrical functionals.

Geometrical functionals

For a compact convex set in E 2 , let A, P, r, R, ω, d, denote its area, its perimeter, the radii of its inscribed and circumscribed circles, its minimum and maximum Feret diameters [START_REF] Feret | La grosseur des grains des matières pulvérulentes[END_REF], respectively. Figure 2.1 illustrates some of these geometrical functionals. For a compact convex set, these six geometrical functionals are greater than zero. The line segments provide null values for A, r and ω, and the points for P, R and d.

Geometric inequalities

For a compact convex set in E 2 , the relationships between these geometrical functionals are constrained by the geometric inequalities [START_REF] Bonnesen | Theorie der konvexen körper[END_REF][START_REF] Hillock | Inequalities for lattice constrained planar convex sets[END_REF][START_REF] Osserman | The isoperimetric inequality[END_REF][START_REF] Osserman | Bonnesen style isoperimetric inequalities[END_REF][START_REF] Scott | A family of inequalities for convex sets[END_REF][START_REF] Scott | Inequalities for convex sets[END_REF][START_REF] Siegel | An isoperimetric theorem in plane geometry[END_REF][START_REF] Yaglom | Convex figures[END_REF] referenced in the second column of Table 2.1. These inequalities link geometrical functionals by pairs and determine the so-called extremal compact convex sets that satisfy the corresponding equalities (Table 2.1, fourth column). Futhermore, they allow to determinate morphometrical functionals.

Morphometrical functionals

The morphometrical functionals are invariant under similitude transformations (consequently, they do not depend on the global size of the compact convex set) and are defined as ratios between geometrical functionals. In these ratios, the units of the numerator and the denominator are dimensionally homogeneous and the result has therefore no unit. Moreover, a normalization by a constant value (scalar multiplication) allows to have a ratio that ranges in [0, 1]. For each morphometrical functional, the scalar value depends directly on the associated geometric inequality. These morphometrical functionals are referenced in the third column of Table 2.1.

These morphometrical functionals are classified according to their concrete meanings namely:

• roundness: 4π A / P 2 , 4 A /π d 2 , π r 2 / A and A /π R 2 ;

• circularity: 2π r / P, P /2π R, r / R, 2 r / d and ω /2 R;

• diameter constancy: π ω / P, P /π d and ω / d;

• thinness: 2 d / P and 4 R / P;

• equilateral triangularity: ω /3 r and ω 2 / √ 3 A.

The morphometrical functional √ 3 R / d expresses both the equilateral triangularity and the diameter constancy. The ratios 2 r / ω and d /2 R do not have concrete meaning, they are equal to one for some different compact convex sets.

For example, the morphometrical functionals values are represented in Figure 2.2 for some 2D elementary analytic compact convex sets represented in Figure 2.3. These elementary compact convex sets constitute the family F c 1 : • line segments; • equilateral triangles; • Reuleaux triangles [START_REF] Eggleston | A proof of Blaschke's theorem on the Reuleaux triangle[END_REF][START_REF] Reuleaux | The Kinematics of Machinery: Outlines of a Theory of Machines[END_REF]: curves of constant diameter constructed by taking the three points at the corners of an equilateral triangle and connecting each pair of points by a circular arc centered at the remaining point; • squares; • "Reuleaux" squares: curves by taking the four edge middles of a square and connecting each pair of opposite points by a circular arc centered at the remaining edge middle; • disks; • semi-disks; • isosceles rectangle triangles;

• "1/2" diamonds: diamonds with π/3 and π/6 angles; • "1/2" rectangles: rectangles with length/width=1/2; • "1/2" ellipses: ellipses with major axis length/minor axis length=1/2; • regular pentagons; • regular hexagons. For instance, let be a Reuleaux triangle constructed from an equilateral triangle of edge length denoted l. The Reuleaux triangle has a constant Feret diameter: d = ω. Futhermore,

ω = l, d = l, r = l 1 - 1 √ 3 , R = l √ 3 , P = πl, A = l 2 π - √ 3 2
.

All these geometrical functionals are expressed in function of l, term which disappears in the computation of the morphometrical functionals. For example,

A π R 2 = 3 π - √ 3 2π , P π d = 1, 2 d P = 2/π, ω 2 √ 3 A = 2 √ 3(π - √ 3
) .

The line segments are particular compact convex sets in the sense that some morphometrical functionals are zero-valued and some other ones are undefined because A, r, ω are equal to zero. Figure 2.2 does not differenciate the roundness of the line segments which is null, and their equilateral triangularity which is undefined.

Table 2.1 synthesizes the geometrical and morphometrical functionals, the geometric inequalities and the extremal 2D analytic compact convex sets.

Geometrical

Geometric Morphometrical Extremal functionals inequalities functionals sets r, R r ≤ R r / R C ω, R ω ≤ 2 R ω /2 R C A, R A ≤ π R 2 A /π R 2 C P, R P ≤ 2π R P /2π R C d, R d ≤ 2 R d /2 R X r, d 2 r ≤ d 2 r / d C ω, d ω ≤ d ω / d W A, d 4 A ≤ π d 2 4 A /π d 2 C P, d P ≤ π d P /π d W R, d √ 3 R ≤ d √ 3 R / d Z r, P 2π r ≤ P 2π r / P C ω, P π ω ≤ P π ω / P W A, P 4π A ≤ P 2 4π A / P 2 C d, P 2 d ≤ P 2 d / P L R, P 4 R ≤ P 4 R / P L r, A π r 2 ≤ A π r 2 / A C ω, A ω 2 ≤ √ 3 A ω 2 / √ 3 A T r, ω 2 r ≤ ω 2 r / ω X ω, r ω ≤ 3 r ω /3 r T
Extremal sets are the sets for which an inequality becomes an equality.

C the disks T the equilateral triangles W the constant width compact convex sets L the line segments X some compact convex sets Z every compact convex set of diameter d containing an equilateral triangle of side-length d Table 2.1: Shape functionals for compact convex sets. A, P, r, R, ω, d, denote the area, perimeter, radii of the inscribed and circumscribed circles, minimum and maximum Feret diameters [START_REF] Feret | La grosseur des grains des matières pulvérulentes[END_REF], respectively.

Shape diagrams

From these morphometrical functionals, 2D shape diagrams can be defined. They enable to represent the morphology of any analytic compact convex sets in the Euclidean 2D plane from two morphometrical functionals (that is to say from three geometrical functionals because the two denominators use the same geometrical functionals).

Definition

Let be any triplet of the considered six geometrical functionals (A, P, r, R, ω, d) and (M 1 , M 2 ) be some particular morphometrical functionals valued in [0, 1] 2 (Table 3.1). A shape diagram D is represented in the plane domain [0, 1] 2 (whose axis coordinates are the morphometrical functionals M 1 and M 2 ) where any 2D compact set S is mapped onto a point (x, y). Note that if M 1 or M 2 is in {ω / √ 3 A, π r 2 / A, 2 r / ω, ω /3 r}, the line segments can not be mapped onto a point because they provide null values for A, r and ω. In other terms, a shape diagram D is obtained from the following mapping:

D : K(E 2 ) → [0, 1] 2 S → (x, y)
where K(E 2 ) denotes the compact sets of the Euclidean 2D plane. Using the morphometrical functionals listed in Table 2.1, thirty-one shape diagrams are defined, denoted (D k ) k∈ 1,31 , respectively.

Shape diagrams

Axes coordinates

D 1 : (ω, r, R) x = ω /2 R y = r / R D 2 : (ω, A, R) x = ω /2 R y = A /π R 2 D 3 : (r, A, R) x = r / R y = A /π R 2 D 4 : (A, d, R) x = A /π R 2 y = d /2 R D 5 : (ω, d, R) x = ω /2 R y = d /2 R D 6 : (r, d, R) x = r / R y = d /2 R D 7 : (A, P, R) x = A /π R 2 y = P /2π R D 8 : (ω, P, R) x = ω /2 R y = P /2π R D 9 : (r, P, R) x = r / R y = P /2π R D 10 : (P, d, R) x = P /2π R y = d /2 R D 11 : (ω, r, d) x = ω / d y = 2 r / d D 12 : (ω, A, d) x = ω / d y = 4 A /π d 2 D 13 : (r, A, d) x = 2 r / d y = 4 A /π d 2 D 14 : (A, R, d) x = 4 A /π d 2 y = √ 3 R / d D 15 : (ω, R, d) x = ω / d y = √ 3 R / d D 16 : (r, R, d) x = 2 r / d y = √ 3 R

Property:

The mapping which associates a 2D analytic compact convex set in E 2 to a point in a shape diagram (D k ) k∈ 1,31 is neither injective neither surjective.

Proof: Non-injectivity: in each shape diagram, there exists points on which several 2D compact convex sets are mapped.

• Let be an ellipse with major and minor axes of length equal to 2 and 1 respectively, and a semi-disk of radius value equal to 1. These two sets have five similar geometrical functionals values:

A = π/2, ω = 1, d = 2,
4 R 2 + r 2 R 2 R 2 -r 2 and π 3(R + r) -(R +3 r)(3 R + r) (Ramanujan)
, respectively. Notice that other approximations of the ellipse perimeter exist [START_REF] Ramanujan | Ramanujan's Collected Works[END_REF][START_REF] Villarino | A note on the accuracy of Ramanujan's approximative formula for the perimeter of an ellipse[END_REF]. When R = 1, the equation

4 R 2 + r 2 R 2 R 2 -r 2 = π 3(R + r) -(R +3 r)(3 R + r)
has a unique solution r in R + . Thus, an ellipse and a diamond with the same perimeter values, and the same inscribed and circumscribed radii values are found. Futhermore, for these two sets, ω = 2 r and d = 2 R. Therefore, these two sets have five similar geometrical functionals values: R, d, r, ω, P.

Consequently, this ellipse and this diamond are mapped onto the same point in the shape diagrams D

Complete systems of inequalities

A system of (two) geometric inequalities associated to a shape diagram is complete if and only if for any range of geometrical functionals values satisfying those conditions, a 2D compact convex set with these geometrical functionals values exists [START_REF] Hernández Cifre | The missing boundaries of the Santaló diagrams for the cases (d,ω,R) and (ω,R,r)[END_REF][START_REF] Santaló | Sobre los sistemas completos de desigualdades entre tres elementos de una figura convexa plana[END_REF]. In other words, such a system is complete if and only if the mapping which associates a 2D analytic compact convex set in E 2 to a point in a shape diagram (D k ) k∈ 1,31 can be surjective by restricting the arrival set. For a shape diagram, each of the two associated inequalities determines a part of the convex domain boundary (the domain in which all compact convex sets are mapped). These two inequalities determine the whole boundary of the convex domain if and only if they form a complete system. The compact convex sets mapped onto the boundary points are the extremal compact convex sets of each considered inequality. ), the completeness of systems of inequalities has been proved [START_REF] Böröczky | Optimizing area and perimeter of convex sets for fixed circumradius and inradius[END_REF][START_REF] Hernández Cifre | Is there a planar convex set with given width, diameter, and inradius ?[END_REF][START_REF] Hernández Cifre | The missing boundaries of the Santaló diagrams for the cases (d,ω,R) and (ω,R,r)[END_REF][START_REF] Hernández Cifre | Complete Systems of Inequalities[END_REF][START_REF] Hernández Cifre | Optimizing the perimeter and the area of convex sets with fixed diameter and circumradius[END_REF][START_REF] Santaló | Sobre los sistemas completos de desigualdades entre tres elementos de una figura convexa plana[END_REF]. Figure 4.1 illustrates the convex domain boundary for ten of them.

Shape diagrams dispersion quantification

Shape diagrams for compact convex sets

For the family F c 1 of thirteen compact convex sets (Figure 2.3), the morphometrical functionals are straightforwardly computed. Each compact convex set i ∈ 1, 13 is represented by one point denoted P i,k , in each shape diagram D k , for k ∈ 1, 31 .

Figure 4.2 illustrates several of these thirty-one shape diagrams, chosen according to the results synthetized in section 6. Remember that the shape diagrams are included in [0, 1] 2 . For a better visualization of the shapes drawn on a point of abscissa or ordinate equal to 0 or 1, the shape diagrams are illustrated in [-0.06, 1.04] 2 . Whatever the morphometrical functional, the extremal value 1 is reached for extremal compact convex sets. Thus, in each shape diagram, there is at least one compact convex set mapped to a point of abscissa or ordinate equal to 1. Moreover, the extremal value 0 is not always reached.

The dispersion of compact convex set locations within each shape diagram will be studied after the analysis of similarities between shape diagrams.

Similarity

The fact that ω = 2 r for some compact convex sets implies that shape diagrams (ω, x 1 , x 2 ) are similar to shape diagrams (r, x 1 , x 2 ) where x 1 ∈ {d, R, P, A} and x 2 ∈ {d, R, P}, that is to say D In the same way, the fact that d = 2 R for some compact convex sets implies that:

• shape diagrams (x 1 , x 2 , R) are similar to shape diagrams (x 1 , x 2 , d) where

x 1 ∈ {ω, r, A} and

x 2 ∈ {r, A, P} (D 1 ∼ D 11 , D 2 ∼ D 12 , D 3 ∼ D 13 , D 7 ∼ D 17 , D 8 ∼ D 18 , D 9 ∼ D 19 ); • shape diagrams (x 1 , R, P) are similar to shape diagrams (x 1 , d, P) where x 1 ∈ {ω, r, A} (D 24 ∼ D 27 , D 25 ∼ D 28 , D 26 ∼ D 29 ).
An algorithm of hierarchical classification [START_REF] Diday | Croisements, ordres et ultramétriques[END_REF] based on distances between shape diagrams allows to justify many of these similarities and to find other ones. Let k 1 , k 2 ∈ 1, 31 , the distance between shape diagrams D k 1 and D k 2 , based on the Euclidean distance d E , is defined by Equation 4.1.

d E (D k 1 , D k 2 ) = 1 13 i∈ 1,13 d E (P i,k 1 , P i,k 2 ) (4.1) For all k 1 ∈ 1, 30 and k 2 ∈ k 1 + 1, 31 , the distances d E (D k 1 , D k 2 ) are computed.
Among them, the minimum distance value gives the best similarity between two shape diagrams. From these two shape diagrams, a mean shape diagram is built. To each step of the algorithm, two shape diagrams are similar up to the distance computed and they are gathered to build a mean shape diagram. The algorithm can be run until all the shape diagrams are gathered. For instance, if the algorithm is stopped before the distance value reaches 0.2, the following classification of shape diagrams is obtained:

• D 1 , D 11 , D 2 , D 12 • D 21 , D 22 , D 23 • D 3 , D 13 • D 7 , D 17 , D 9 , D 19 • D 4 , D 5 , D 6 • D 8 , D 18 • D 24 , D 27 , D 26 D 29 • D 14 , D 15 , D 16 • D 25 , D 28 • D 10 , D 20 • D 30 • D 31

Dispersion quantification

For each shape diagram, the dispersion of the locations of 2D analytic compact convex sets of the family F c 1 is studied. The spatial distribution of compact convex sets locations in each shape diagram is characterized and quantified from algorithmic geometry using Delaunay's graph (DG) and minimum spanning tree (MST) [START_REF] Boissonnat | Algorithmic geometry[END_REF]. 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 compact convex set range (regular, random, cluster, . . . ) [START_REF] Raby | Quantitative cell dispersion analysis: new test to measure tumor cell aggressiveness[END_REF]. The decrease of µ and the increase of σ characterize the shift from a regular distribution toward random and cluster distributions, respectively. MST enables to visually distinguish five groups of shape diagrams, according to their dispersion measurements (µ,σ):

• D 22 • D 1 , D 11 , D 2 , D 12 , D 21 , D 23 • D 10 , D 20 , D 30 • D 4 , D 14 , D 7 , D 17 , D 31
• the sixteen others. DG also extracts the shape diagrams D 10 , D 20 , D 30 , that have both low µ and σ values. These three shape diagrams have a weak dispersion. Visually, the compact convex sets locations form a cluster (see Figure 4.2 for D 20 and D 30 ). The shape diagrams D 21 and D 22 have both high µ and σ values, that is to say a strong dispersion. Visually, the compact convex sets are located within a large domain within [0, 1] 2 and are well spaced from each other.

Finally, these statements are in agreement with those obtained for the similarities between shape diagrams.

A shape diagram with a strong dispersion (both high µ and σ values), for both DG and MST, guarantees a strong discrimination of shapes: the compact convex sets are located within a large domain in [0, 1] 2 and are well spaced from each other. However, remember that d = 2 R and ω = 2 r for some compact convex sets. This explains that the compact convex sets of the family F c 1 -except for equilateral triangles, Reuleaux triangles, pentagons where d = 2 R and ω = 2 r and isosceles rectangle triangles where ω = 2 r -are mapped onto a point on a line for some shape diagrams, namely:

• diagonal line from (0, 0) to (1, 1) on shape diagrams 1 , and the compact convex set class allowing to switch from one to the other using one degree of freedom. For example, the line segment goes to the square through rectangles whose elongation decreases (or through diamonds, . . . ). Therefore, several classes of "compact convex sets with one degree of freedom" could be defined. Thus, a curve denoted C i,k from each compact convex set class C i is created in each shape diagram D k , k ∈ 1, 31 . This process is used for some pairs of compact convex sets of the family F c 1 . The tendency of various curves are observed.

(x 1 , x 2 , x 3 ) where (x 1 , x 2 ) ∈ {(ω,
These analytic compact convex sets verify shape properties which are preserved only under similitude transformations and under the variation of one parameter t ∈ R. These are, for example, isosceles triangles (the "isosceles" property is preserved when the top angle varies between 0 and π), rectangles (the ratio width/length varies between 0 and 1), . . . In other terms, the degree of freedom is the parameter t ∈ R.

Twenty-three analytic compact convex set classes (chosen according to the faculty to compute analytically their geometrical (and morphometrical) functional values), gathered in four families, are considered: When it reaches the lower and upper bounds, the rectangle becomes a line segment and a square, respectively. C 6 -Symmetrical disk segments: They are the intersection of a disk and a symmetrically placed strip. The extremal compact convex sets are the line segment and the disk. C 7 -Rectangles terminated by semi-disks: The ratio between the width and the length of the rectangle varies in [0, 1]. Semi-disks are placed on two opposite edges. When the ratio reaches the lower and upper bounds, the resulting compact convex set becomes a line segment and a disk, respectively. C 8 -Ellipses: The ratio between the minor and major axes lengths varies in [0, 1]. When it reaches the lower and upper bounds, the ellipse becomes a line segment and a disk, respectively. C 9 -Symmetrical lens: They are the intersection of two congruent circular disks. The extremal compact convex sets are the point and the disk. C 10 -Symmetrical disks with two peaks: They are the convex hulls of a disk and two symmetrically placed points. The disk radius value varies between 0 and the half of the line segment length. The extremal compact convex sets are the line segment and the disk, respectively. C 11 -Diamonds: The ratio between the top angles varies in [0, 1]. When it reaches the lower and upper bounds, the diamond becomes a line segment and a square, respectively. C 12 -Rectangles terminated by equilateral triangles: The ratio between the width and the length of the rectangle varies in [0, 1]. Equilateral triangles are placed on two opposite edges. When the ratio reaches the lower and upper bounds, the resulting compact convex set becomes a line segment and a "1/2" diamond, respectively. C 13 -Rectangles terminated by isosceles rectangle triangles: The ratio between the width and the length of the rectangle varies in [0, 1]. Isosceles rectangle triangles are placed on two opposite edges. When the ratio reaches the lower and upper bounds, the resulting compact convex set becomes a line segment and a square, respectively.

• family F c 2.1 ⊇ {C i } i∈
• Family F c 2.3 : C 14 -Yamanouti triangles [START_REF] Yamanouti | Notes on closed convex sets[END_REF]: The radius value varies between the height and the edge length of an equilateral triangle. The extremal compact convex sets are the equilateral triangle and the Reuleaux triangle, respectively. C 15 -Disk-equilateral triangle intersections: They are the intersections of an equilateral triangle and a disk centered on the center of mass with a radius varying between the radii of inscribed and circumscribed circles. The extremal compact convex sets are the disk and the equilateral triangle, respectively. C 16 -Dilated (disk) equilateral triangles: They are obtained by the dilation of an equilateral triangle with a disk of radius included in [0, +∞[. When the radius value reaches the lower bound, the resulting compact convex set becomes an equilateral triangle. When the radius tends to infinity, the resulting compact convex set tends to the disk. C 17 -Regular odd polygons: They are regular polygons with a positive odd edge number. In the shape diagrams, the real positions of these compact convex sets do not describe a curve because it is not continuous (the edge number is necessarily an integer) but they are along a curve. Theorically, the extremal compact convex sets are the point and the disk. In the following shape diagrams, only the curve from the equilateral triangle to the disk is drawn. C 18 -Reuleaux odd polygons: They are Reuleaux polygons with a positive odd edge number. In the shape diagrams, the real positions of these compact convex sets do not describe a curve because it is not continuous (the edge number is necessarily an integer) but they are along a curve. Theorically, the extremal compact convex sets are the point and the disk. In the following shape diagrams, only the curve from the Reuleaux triangle to the disk is drawn.

• Family F c 2.4 : C 19 -"Yamanouti" squares: They are defined in a similar way as the Yamanouti triangles, but from the middle of the opposite edges. The extremal compact convex sets are the square and the "Reuleaux" square. C 20 -Disk-square intersections: They are the intersections of a square and a disk centered on the center of mass with a radius varying between the radii of inscribed and circumscribed circles. The extremal compact convex sets are the disk and the square, respectively. C 21 -Dilated (disk) squares: They are obtained by the dilation of a square with a disk of radius included in [0, +∞[. When the radius value reaches the lower bound, the resulting compact convex set becomes a square. When the radius tends to infinity, the resulting compact convex set tends to the disk. C 22 -Regular even polygons: They are regular polygons with a positive even edge number. In the shape diagrams, the real positions of these compact convex sets do not describe a curve because it is not continuous (the edge number is necessarily an integer) but they are along a curve. Theorically, the extremal compact convex sets are the line segment and the disk. In the following shape diagrams, only the curve from the square to the disk is drawn. C 23 -"Reuleaux" even polygons: They are defined in a similar way as the Reuleaux odd polygons, but from the middle of the opposite edges. In the shape diagrams, the real positions of these compact convex sets do not describe a curve because it is not continuous (the edge number is necessarily an integer) but they are along a curve. Theorically, the extremal compact convex sets are the line segment and the disk. In the following shape diagrams, only the curve from the "Reuleaux" square to the disk is drawn. For the compact convex sets of each class, the morphometrical functionals are computed. In each shape diagram D k , k ∈ 1, 31 , a compact convex set class i ∈ 1, 23 is represented by a parametric curve C i,k (t) since the class is infinite and bounded by the two extremal sets. For the considered compact convex set classes, the extremal compact convex set are in the family F c 1 (Figure 2.3). For example, up to a similitude transformation, an infinite class of rectangles exists, from the line segment to the square.

Figures 5.5, 5.6, 5.7 and 5.8 illustrate some of these shape diagrams, chosen according to the results synthetized in section 6. 2 of analytic compact convex sets with one degree of freedom and two symmetrical axes mapped into eleven shape diagrams (chosen according to the results synthetized in section 6). :"Reuleaux" even polygons C 23,k :"Reuleaux" even polygons C 23,k :"Reuleaux" even polygons C 23,k :"Reuleaux" even polygons C 23,k :"Reuleaux" even polygons C 23,k :"Reuleaux" even polygons C 23,k :"Reuleaux" even polygons C 23,k :"Reuleaux" even polygons C 23,k :"Reuleaux" even polygons C 23,k :"Reuleaux" even polygons C [START_REF] Scott | Inequalities for convex sets[END_REF] of analytic compact convex sets with one degree of freedom and an even number strictly greater to 2 of symmetrical axes mapped into eleven shape diagrams (chosen according to the results synthetized in section 6).

Overlapping quantification

An overlapping of curves is visible in some shape diagrams. Its quantification is based on a discretization of the spatial domain [0, 1] 2 of the shape diagrams. Let n ∈ N * , the discretization of a curve C i,k , i ∈ 1, 23 , k ∈ 1, 31 , is denoted C i,k : 0, n 2 → {0, 1} and defined as following, ∀(x, y) ∈ 0, n 2 : Finally, Equation 5.1 quantifies (by a measurement ranging between 0 and 1) the overlapping of all curves for each shape diagram D k , k ∈ 1, 31 . A high (resp. low) value for this ratio means a strong (resp. weak) overlapping. This quantification depends on the discretization level n. Thus, when the computation is done for various n values, not only the curves overlapping is considered but also the curves proximity (small n value).

C i,k (x, y) =    1 if ∃t 0 ∈ R | x -1 2 1 n ≤ C i,k (t 0 ) → u≤ x + 1 2 1 n y -1 2 1 n ≤ C i,k (t 0 ) → v ≤ y + 1
Overlapping

n (D k ) = 1 - (x,y)∈ 0,n 2 D max k (x, y) (x,y)∈ 0,n 2 D sum k (x, y) (5.1)

Synthesis

To obtain a strong discrimination of 2D analytic compact convex sets, it is necessary to have both a strong dispersion and a weak overlapping.

• This analysis is summarized in Table 6.1. of compact convex sets with one degree of freedom and an even number strictly greater to 2 of symmetrical axes. 6.1: Shape diagrams classification according to their quality of shape discrimination of analytic compact convex sets and according to the completeness of associated systems of inequalities.

In this paper, only some shape diagrams have been illustrated. The choice was based on the results of shape discrimination (dispersion and overlapping studies) and on the results of similarities between shape diagrams (subsection 4.2). The aim was to illustrate dissimilar shape diagrams with different qualities of shape discrimination, and shape diagrams with different completeness of associated systems of inequalities. The framed shape diagrams of Table 6.1 are those illustrated throughout this paper.

Conclusion

This paper has dealed with shape diagrams of 2D non-empty analytic compact convex sets built from six geometrical functionals: the area, the perimeter, the radii of the inscribed and circumscribed circles, and the minimum and maximum Feret diameters. Each such a set is represented by a point within a shape diagram whose coordinates are morphometrical functionals defined as normalized ratios of geometrical functionals. From existing morphometrical functionals for these sets, thirty-one shape diagrams can be built. A detailed comparative study has been performed in order to analyze the representation relevance and discrimination power of these shape diagrams. It is based on the dispersion and overlapping quantifications from compact convex set locations in diagrams. Among all the shape diagrams, six present a strong shape discrimination of sets, four are based on complete system of inequalities. Among these four diagrams, the shape diagram D 12 : (ω, A, d) is retained for its representation relevance and discrimination power.

This paper reports the first part of a general comparative study of shape diagrams. The focus was placed on analytic compact convex sets. However, most of these shape diagrams can also been applied to more general compact sets than compact convex sets. The second and third parts of the comparative study are published

Figure 2 . 1 :

 21 Figure 2.1: Geometrical functionals of a compact convex set: radii of inscribed (r) and circumscribed (R) circles, minimum (ω) and maximum (d) Feret diameters.

Figure 2 . 2 :

 22 Figure 2.2: Values of several morphometrical functionals for elementary 2D compact convex sets.

Figure 2 . 3 :

 23 Figure 2.3: Family F c 1 of 2D analytic compact convex sets.

  / d D 17 : (A, P, d) x = 4 A /π d 2 y = P /π d D 18 : (ω, P, d) x = ω / d y = P /π d D 19 : (r, P, d) x = 2 r / d y = P /π d D 20 : (P, R, d) x = P /π d y = √ 3 R / d D 21 : (ω, r, P) x = π ω / P y = 2π r / P D 22 : (ω, A, P) x = π ω / P y = 4π A / P 2 D 23 : (r, A, P) x = 2π r / P y = 4π A / P 2 D 24 : (A, R, P) x = 4π A / P 2 y = 4 R / P D 25 : (ω, R, P) x = π ω / P y = 4 R / P D 26 : (r, R, P) x = 2π r / P y = 4 R / P D 27 : (A, d, P) x = 4π A / P 2 y = 2 d / P D 28 : (ω, d, P) x = π ω / P y = 2 d / P D 29 : (r, d, P) x = 2π r / P y = 2 d / P D 30 : (d, R, P) x = 2 d / P y = 4 R / P D 31 : (ω, r, A) x = ω 2 / √ 3 A y = π r 2 / A

  r = 0.5, R = 1. Thus, this ellipse and the semi-disk are mapped onto the same point in the shape diagrams D 1 , D 2 , D 3 , D 4 , D 5 , D 6 , D 11 , D 12 , D 13 , D 14 , D 15 , D 16 , D 31 . • The diamond and ellipse perimeters are expressed in function of r and R by

√ 3 A ≤ 9 r 2 √ 3 A 3 4 2 . 2 .

 323322 . In the shape diagrams D 1 , D 11 , D 21 , on all points above the line of equation y = x or below the line of equation y = 2 3 x, none compact convex set are mapped. In the shape diagram D 31 , on all points above the line of equation y = π √ x or below the line of equation y = π In the shape diagrams D 2 , D 12 , D 22 , on all points below the line curve of equation y = 4 π √ 3 x 2 , none compact convex set are mapped. • π r 2 ≤ A implies r In the shape diagrams D 3 , D 13 , D 23 , on all points below the line curve of equation y = x 2 , none compact convex set are mapped.

  For twenty-one among the thirty-one shape diagrams (D 1 , D 3 , D 4 , D 5 , D 6 , D 7 , D 9 , D 10 , D 11 , D 12 , D 14 , D 15 , D 16 , D 18 , D 20 , D 22 , D 23 , D 24 , D 26 , D 28 , D 30

1 D 1 DFigure 4 . 1 :

 1141 Figure 4.1: Convex domains of ten shape diagrams for which complete systems of inequalities have been established. For a given shape diagram, the bordered region represents the convex domain in which all compact convex sets lay.

1 D 1 D 1 DFigure 4 . 2 :

 11142 Figure 4.2: Family F c1 of analytic compact convex sets mapped into eleven shape diagrams (chosen according to the results synthetized in section 6).

  2 ∼ D 3 , D 5 ∼ D 6 , D 8 ∼ D 9 , D 12 ∼ D 13 , D 15 ∼ D 16 , D 18 ∼ D 19 , D 22 ∼ D 23 , D 25 ∼ D 26 , D 28 ∼ D 29 where ∼ denotes a strong similarity between shape diagrams.

Figure 4 . 3 :

 43 Figure 4.3: The first twenty-three steps of an algorithm of hierarchical classification based on distances between the shape diagrams. To each step, two shape diagrams are similar up to the distance value, whose the scale is indicated on the right.

Figure 4 .Figure 4 . 4 :

 444 Figure 4.4: Two dispersion quantifications for all shape diagrams applied on the family F c 1 . For each representation (according to the models DG and MST, respectively), indices k ∈ 1, 31 of the shape diagrams D k are located according to their µ and σ values.

  r), (d, R)} and x 3 ∈ {A, R, d, P} (D 1 , D 11 , D 21 , D 30 , D 31 ) • horizontal line from (0, 1) to (1, 1) on shape diagrams (x 1 , d, R) where x 1 ∈ {ω, r, P, A} (D 4 , D 5 , D 6 , D 10 ) • horizontal line from (0, (3)/2) to (1, (3)/2) on shape diagrams (x 1 , R, d) where x 1 ∈ {ω, r, P, A} (D 14 , D 15 , D 16 , D 20 ) For a shape diagram, the fact that some compact convex sets are mapped onto a point on a line does not yield to allow a strong discrimination of the compact convex sets as quantified and illustrated in Figure 4.4. 5 Shape diagrams overlapping quantification 5.1 Shape diagrams for compact convex sets with one degree of freedom Let two 2D analytic compact convex sets of the family F c

1 , 4 :

 14 four classes of compact convex sets with one symmetrical axis (Figure 5.1), • family F c 2.2 ⊇ {C i } i∈ 5,13 : nine classes of compact convex sets with two symmetrical axes (Figure 5.2), • family F c 2.3 ⊇ {C i } i∈ 14,18 : five classes of compact convex sets with an odd number strictly greater to 1 of symmetrical axes (Figure 5.3), • family F c 2.4 ⊇ {C i } i∈ 19,23 : five classes of compact convex sets with an even number strictly greater to 2 of symmetrical axes (Figure 5.4). There are: • Family F c 2.1 : C 1 -Isosceles triangles: The top angle varies in [0, π]. When it reaches the bounds, the isosceles triangle becomes a line segment. C 2 -Angular sectors: The angle varies in [0, π]. When it reaches the lower and upper bounds, the angular sector becomes a line segment and a semidisk, respectively. C 3 -Ungula: They are based on angular sectors whose angle varies in [0, π]. The extremal compact convex sets are the point (null angle) and the semidisk (angle equals to π). C 4 -Semi-symmetrical disks with two peaks: They are the convex hulls of a semi-disk and two symmetrically placed points. The disk radius value varies between 0 and the half of the line segment length. The extremal compact convex sets are the line segment and the semi-disk, respectively. • Family F c 2.2 : C 5 -Rectangles: The ratio between the width and the length varies in [0, 1].

Figure 5 . 1 :

 51 Figure 5.1: Family F c 2.1 of 2D analytic compact convex sets with one degree of freedom and one symmetrical axis.

Figure 5 . 2 :

 52 Figure 5.2: Family F c 2.2 of 2D analytic compact convex sets with one degree of freedom and two symmetrical axes.

Figure 5 . 3 :

 53 Figure 5.3: Family F c 2.3 of 2D analytic compact convex sets with one degree of freedom and an odd number (strictly greater to 1) of symmetrical axes.

Figure 5 . 4 :

 54 Figure 5.4: Family F c 2.4 of 2D analytic compact convex sets with one degree of freedom and an even number (strictly greater to 2) of symmetrical axes.

  k :Semi-symmetrical disks with two peaks C 4,k :Semi-symmetrical disks with two peaks C 4,k :Semi-symmetrical disks with two peaks C 4,k :Semi-symmetrical disks with two peaks C 4,k :Semi-symmetrical disks with two peaks C 4,k :Semi-symmetrical disks with two peaks C 4,k :Semi-symmetrical disks with two peaks C 4,k :Semi-symmetrical disks with two peaks C 4,k :Semi-symmetrical disks with two peaks C 4,k :Semi-symmetrical disks with two peaks C

1 D 1 D 1 DFigure 5 . 6 :

 11156 Figure 5.6: Family F c2.2 of analytic compact convex sets with one degree of freedom and two symmetrical axes mapped into eleven shape diagrams (chosen according to the results synthetized in section 6).

C 20 ,

 20 k :Disk-square intersections C 20,k :Disk-square intersections C 20,k :Disk-square intersections C 20,k :Disk-square intersections C 20,k :Disk-square intersections C 20,k :Disk-square intersections C 20,k :Disk-square intersections C 20,k :Disk-square intersections C 20,k :Disk-square intersections C 20,k :Disk-square intersections C

vFigure 5 . 9 :

 59 Figure 5.9: "Intensity" discretized shape diagram D 1 : (ω, r, R) (n = 100) with the twenty-three curves representing the compact convex set classes with one degree of freedom.

Figure 6 .

 6 Figure 6.1 illustrates the discretized shape diagrams (n = 100) with overlapping intensities, and Figure 6.2 shows the quantification of curves overlapping (representing the compact convex set classes) for some shape diagrams, according to the n values 100 and 1000. Several informations can be extracted from these graphs: • The graphs representing the 2D analytic compact convex set classes of the families F c 2.2 and F c 2.4 show a strong overlap (from n = 1000) for the shape diagrams D 1 , D 4 , D 5 , D 6 , D 10 , D 11 , D 14 , D 15 , D 16 , D 20 , D 21 , D 30 and D 31 . • The graphs representing the 2D analytic compact convex set classes of the family F c 2.3 show a strong overlap (from n = 1000) for the shape diagram D 1 . They are close to each other (strong overlap from n = 100) in the shape diagrams D 17 , D 18 , D 19 , D 27 , D 28 and D 29 .

  The shape diagrams D 10 , D 20 and D 30 are excluded due to their weak dispersion and overlapping results, whatever the considered compact convex sets. • In the shape diagrams D 4 , D 5 , D 6 , D 14 , D 15 and D 16 , only the family F c 2.3 shows a weak overlapping. Futhermore, their dispersion quantification is moderate and even somewhat below. • In the shape diagram D 31 , the families of F c 2.1 and F c 2.3 show a weak overlapping. Futhermore, its dispersion quantification is moderate. • In the shape diagrams D 7 , D 8 , D 9 , D 17 , D 18 , D 19 , D 24 , D 25 , D 26 , D 27 , D 28 and D 29 , all the families considered in this paper show a weak overlapping. But in D 17 , D 18 , D 19 , D 27 , D 28 and D 29 , the compact convex set classes of F c 2.3 are located close together. However, the dispersion quantification of all these shape diagrams are moderate. • The dispersion quantification of the shape diagrams D 1 , D 2 , D 3 , D 11 , D 12 , D 13 , D 21 , D 22 and D 23 gives strong values, particularly for D 22 . In the shape diagram D 1 , only the family F c 2.1 shows a weak overlapping, and in the shape diagrams D 11 and D 21 , only the families of F c 2.1 and F c 2.3 show a weak overlapping. It remains the shape diagrams D 2 , D 3 , D 12 , D 13 , D 22 , D 23 that, in addition to their strong dispersion, provide a weak overlapping of the compact convex set classes considered in this paper. Futhermore, among the shape diagrams D 2 , D 3 , D 12 , D 13 , D 22 and D 23 that obtain the best results for dispersion and overlapping quantifications, only D 3 , D 12 , D 22 and D 23 are based on known complete systems of inequalities. Observing in details the representation of quantifications for these four shape diagrams, D 12 is retained for shape discrimination of analytic compact convex sets.

Figure 6 . 1 :

 61 Figure 6.1: "Intensity" discretized shape diagram D sum k (n = 100) with the twenty-three curves representing the compact convex set classes with one degree of freedom.

  the family F c 2.1 of compact convex sets with one degree of freedom and one symmetrical axis. the family F c 2.2 of compact convex sets with one degree of freedom and two symmetrical axes. the family F c 2.3of compact convex sets with one degree of freedom and an odd number strictly greater to 1 of symmetrical axes.

  Overlapping quantification for the twentythree classes of compact convex sets with one degree of freedom.

Figure 6 . 2 :

 62 Figure 6.2: Overlapping quantification for the thirty-one shape diagrams D k , k ∈ 1, 31 , with n = 100 in red, and n = 1000 in blue.

Table 3 .

 3

1: Axes coordinates of the thirty-one shape diagrams for 2D compact convex sets.

  1 , D 5 , D 6 , D 8 , D 9 , D 10 , D 11 , D 15 , D 16 , D 18 , D 19 , D 20 , D 21 , D 25 , D 26 , D 28 , D 29 , D 30 . • The diamond and ellipse areas are expressed in function of P and d by d 2 √ , D 7 , D 10 , D 14 , D 17 , D 20 , D 24 , D 27 , D 30 . has a unique solution P in R + . Thus, an ellipse and a rectangle with the same area and perimeter values, and the same minimum Feret diameter values are found. Futhermore, for these two sets, ω = 2 r. Therefore, these two sets have four similar geometrical functionals values: r, ω, P, A. Consequently, this ellipse and this rectangle are mapped onto the same point in the shape diagrams D 21 , D 22 , D 23 , D 31 . In the shape diagrams D 7 , D 8 , D 9 , D 17 , D 18 , D 19 , D 24 , D 25 , D 26 , D 27 , D 28 , D 29 , D 30 , on all points below the line of equation y = 2/π, none compact convex set are mapped. In the shape diagrams D 10 , D 20 , D 30 , on all points at the left of the line of equation x = 2/π, none compact convex set are mapped. In the shape diagrams D 4 , D 5 , D 6 , D 14 , D 15 , D 16 , D 10 , D 20 , on all points below the line of equation y =

	Non-surjectivity: in each shape diagram, there exists points on which none 2D
	compact convex set are mapped.
	• 4 R ≤ P and P ≤ 2π R imply 2 π ≤ 4 R P and 2 π ≤ P 2π R . 2 d ≤ P and P ≤ π d imply 2 π ≤ 2 d P and 2 π ≤ P π d .
	•	√	3 R ≤ d and d ≤ 2 R imply	√ 2 ≤ 3 √ 3/2, none compact convex set are mapped. √ 3 R d and √ 3 2 R . 2 ≤ d
	• 2 r ≤ ω implies r R ≤ ω 2 R and 2 r d ≤ ω d and 2π r P ≤ π ω P and π r 2 A ≤ π ω 2 4 A . ω ≤ 3 r implies ω 2 R ≤ 3 r 2 R and ω d ≤ 3 r d and π ω P ≤ 3π r P and ω 2
			P 2 -4 d 2 2 P	and
			d 12 3 P -2π d -3 P 2 +6π P d -5π 2 d 2 (Ramanujan), respectively. When d = 2, the equation
			d 2 P 2 -4 d 2 2 P	=	d 12	3 P -2π d -3 P 2 +6π P d -5π 2 d 2
			has a unique solution P in R + . Thus, an ellipse and a diamond with the same
			area and perimeter values, and the same maximum Feret diameter values are
			found. Futhermore, for these two sets, d = 2 R. Therefore, these two sets
			have four similar geometrical functionals values: R, d, P, A. Consequently,
			this ellipse and this diamond are mapped onto the same point in the shape
			diagrams D 4 • The rectangle and ellipse areas are expressed in function of P and ω by ω (P /2 -ω) and ω 12 3 P -2π ω + 3 P 2 +6π P ω -5π 2 ω 2 (Ramanujan), re-spectively. When ω = 2, the equation
			ω (P /2 -ω) =	ω 12	3 P -2π ω + 3 P 2 +6π P ω -5π 2 ω 2

  Figure 5.7: Family F c2.3 of analytic compact convex sets with one degree of freedom and an odd number strictly greater to 1 of symmetrical axes mapped into eleven shape diagrams (chosen according to the results synthetized in section 6).

			C 14,k :Yamanouti triangles C 14,k :Yamanouti triangles C 14,k :Yamanouti triangles C 14,k :Yamanouti triangles C 14,k :Yamanouti triangles C 14,k :Yamanouti triangles C 14,k :Yamanouti triangles C 14,k :Yamanouti triangles C 14,k :Yamanouti triangles C 14,k :Yamanouti triangles C 14,k :Yamanouti triangles C 19,k :"Yamanouti" squares C 19,k :"Yamanouti" squares C 19,k :"Yamanouti" squares C 19,k :"Yamanouti" squares C 19,k :"Yamanouti" squares C 19,k :"Yamanouti" squares C 19,k :"Yamanouti" squares C 19,k :"Yamanouti" squares C 19,k :"Yamanouti" squares C 19,k :"Yamanouti" squares C 19,k :"Yamanouti" squares			1											1
				C 15,k :Disk-equilateral triangle intersections C 15,k :Disk-equilateral triangle intersections C 15,k :Disk-equilateral triangle intersections C 15,k :Disk-equilateral triangle intersections C 15,k :Disk-equilateral triangle intersections C 15,k :Disk-equilateral triangle intersections C 15,k :Disk-equilateral triangle intersections C 15,k :Disk-equilateral triangle intersections C 15,k :Disk-equilateral triangle intersections C 15,k :Disk-equilateral triangle intersections C 15,k :Disk-equilateral triangle intersections	0.9											0.9
				C 16,k :Dilated (disk) equilateral triangles C 16,k :Dilated (disk) equilateral triangles C 16,k :Dilated (disk) equilateral triangles C 16,k :Dilated (disk) equilateral triangles C 16,k :Dilated (disk) equilateral triangles C 16,k :Dilated (disk) equilateral triangles C 16,k :Dilated (disk) equilateral triangles C 16,k :Dilated (disk) equilateral triangles C 16,k :Dilated (disk) equilateral triangles C 16,k :Dilated (disk) equilateral triangles C 16,k :Dilated (disk) equilateral triangles	0.8											0.8
				C 17,k :Regular odd polygons C 17,k :Regular odd polygons C 17,k :Regular odd polygons C 17,k :Regular odd polygons C 17,k :Regular odd polygons C 17,k :Regular odd polygons C 17,k :Regular odd polygons C 17,k :Regular odd polygons C 17,k :Regular odd polygons C 17,k :Regular odd polygons C 17,k :Regular odd polygons			0.7											0.7
				C 18,k :Reuleaux odd polygons C 18,k :Reuleaux odd polygons C 18,k :Reuleaux odd polygons C 18,k :Reuleaux odd polygons C 18,k :Reuleaux odd polygons C 18,k :Reuleaux odd polygons C 18,k :Reuleaux odd polygons C 18,k :Reuleaux odd polygons C 18,k :Reuleaux odd polygons C 18,k :Reuleaux odd polygons C 18,k :Reuleaux odd polygons			0.6											0.6
											0.5											0.5
											0.4											0.4
				Line segments Line segments Line segments Line segments Line segments Line segments Line segments Line segments Line segments Line segments Line segments
											0.3											0.3
				Equilateral triangles Equilateral triangles Equilateral triangles Equilateral triangles Equilateral triangles Equilateral triangles Equilateral triangles Equilateral triangles Equilateral triangles Equilateral triangles Equilateral triangles
											0.2											0.2
				Reuleaux triangles Reuleaux triangles Reuleaux triangles Reuleaux triangles Reuleaux triangles Reuleaux triangles Reuleaux triangles Reuleaux triangles Reuleaux triangles Reuleaux triangles Reuleaux triangles
											0.1											0.1
				Disks Disks Disks Disks Disks Disks Disks Disks Disks Disks Disks						0											0
				Regular pentagons Regular pentagons Regular pentagons Regular pentagons Regular pentagons Regular pentagons Regular pentagons Regular pentagons Regular pentagons Regular pentagons Regular pentagons					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
															D 1 : (ω, r, R)							D 4 : (A, d, R)
	1										1											1
	0.9										0.9											0.9
	0.8										0.8											0.8
	0.7										0.7											0.7
	0.6										0.6											0.6
	0.5										0.5											0.5
	0.4										0.4											0.4
	0.3										0.3											0.3
	0.2										0.2											0.2
	0.1										0.1											0.1
	0										0											0
	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	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
				D 9 : (r, P, R)						D 12 : (ω, A, d)							D 13 : (r, A, d)
	1										1											1
	0.9										0.9											0.9
	0.8										0.8											0.8
	0.7										0.7											0.7
	0.6										0.6											0.6
	0.5										0.5											0.5
	0.4										0.4											0.4
	0.3										0.3											0.3
	0.2										0.2											0.2
	0.1										0.1											0.1
	0										0											0
	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	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
				D 16 : (r, R, d)						D 20 : (P, R, d)							D 21 : (ω, r, P)
	1										1											1
	0.9										0.9											0.9
	0.8										0.8											0.8
	0.7										0.7											0.7
	0.6										0.6											0.6
	0.5										0.5											0.5
	0.4										0.4											0.4
	0.3										0.3											0.3
	0.2										0.2											0.2
	0.1										0.1											0.1
	0										0											0
	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	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
			D 25 : (ω, R, P)						D 30 : (d, R, P)							D 31 : (ω, r, A)

  , D 12 , D 22 , D 23 D 2 , D 13 discrimination Moderate D 1 , D 7 , D 9 , D 11 , D 18 , D 24 , D 26 , D 28 D 8 , D 17 , D 19 , D 21 , D 25 , D 27 , D 29 discrimination Weak D 4 , D 5 , D 6 , D 10 , D 14 , D 15 , D 16 , D 20 , D 30 D 31 discrimination

	Complete system	Non-complete system
	of inequalities	of inequalities
	Strong D 3 Table

in two following papers [START_REF] Rivollier | Shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF][START_REF] Rivollier | Shape diagrams for 2D compact sets -Part III: convexity discrimination for analytic and discretized simply connected sets[END_REF]. They are focused on analytic simply connected compact sets and convexity discrimination for analytic and discretized simply connected compact sets, respectively.

Acknowledgments

The authors would like to thank M.A. Hernández Cifre for her constructive comments.