" In the context of polymer crystallization under high and constant cooling rates, a new survey is presented. The growth kinetics of spherulites from the molten state and the corresponding temperature of crystallization under similar cooling rates are considered. An industrial grade of isotactic polypropylene (iPP) is investigated in the range of slow (from 1 to 10 ° C/min), of relatively moderate (from 30 to 500 ° C/min) and of high (from 500 to 1600 ° C/min) constant cooling rates. The growth kinetics for the α,modification of iPP the temperature of crystallization is in agreement with the data published by Janeschitz,Kriegl [Macromolecules, 2006]. To understand the growth mechanism, the Hoffman equation is used and then extrapolated to lower temperatures of crystallization. Finally, to describe the non,isothermal crystallization, the equation of Ozawa is used. The benefits to use the spherulitic growth kinetics to extrapolate data from alternative methods, i.e., differential scanning calorimetry (DSC), are briefly discussed.

# $ % " Industrial Processes, Liquid to Solid Transition, Isotactic Polypropylene, Kinetics, Polarized Optical Microscopy, Differential Scanning Calorimetry

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Liquid to solid transition of polymers at high cooling rates has to be well documented in order to be modeled and implemented in simulation codes describing the relationships between polymers and thermo,mechanical conditions encountered in processing conditions, like in injection,molding [START_REF] Haudin | Numerical and physical modelling of polymer crystallization[END_REF]. If we focus the work on thermal effects, the attractive point supposes to adopt cooling procedures comparable with those undergone by the polymer during processing and to reproduce them. An approach is to consider that crystallization mechanism understanding requires a minimum of information on growth kinetics at the spherulitic scale. The present study is concerned with the crystallization behavior under high and constant cooling rates from 50 to 1600 ° C/min. For that purpose, a novel approach involving a new home,made set,up named « Polymer High Cooling -Optics » is introduced. The optical set, up recently developed in our laboratory [START_REF] Da Passano | [END_REF] has been revisited [START_REF] Brisebourg | # $[END_REF]. It allows us to determine the temperature of crystallization together with the spherulitic growth kinetics constant cooling rates. The extraction of the parameters is done by classical methods. A comparison with literature data is established. In order to model the non,isothermal crystallization from Ozawa equation, the benefits to use the spherulitic growth kinetics to extrapolate data from an additional method, i.e., DSC, are briefly discussed.

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At the spherulitic scale and in quiescent conditions, polymer crystallization is described by means of nucleation and growth, or in a more global manner by the overall kinetics. The nucleation and growth steps are used for the overall description, e.g., in the approaches developed by Kolmogoroff, Avrami and Evans [START_REF] Haudin | Numerical and physical modelling of polymer crystallization[END_REF]. These approaches are generally used in isothermal conditions. Description of non,isothermal crystallization uses Ozawa equation [START_REF] Ozawa | Kinetics of non,isothermal crystalliza[END_REF], which is derived from Evans theory [START_REF] Evans | The laws of expanding circles and spheres in relation to the lateral growth of surface films and the grain,size of metals[END_REF]. Attempts to describe growth mechanism were several times proposed by Hoffman and Lauritzen [START_REF] Lauritzen | Theory of formation of polymer crystals with folded chains in dilute solution[END_REF].

Their model still remains the only one satisfactory.

' ' ) *&

Techniques for measuring rates of crystallization have been the subject of reviews. The phase change has to be monitored from a direct observation of the structure development or from an effect resulting from the structure development, e.g., a thermal effect. To observe the structure development under high cooling rates, special hot stages must be constructed and appropriate methods proposed. For instance, Ding & Spruiell [START_REF] Ding | An experimental method for studying nonisothermal crystallization of polymers at very high cooling rates[END_REF] proposed a light,depolarizing microscopy (LDM) technique for studying crystallization of a polymer film, 150 Lm in thickness, at average cooling rates up to 5000 ° C/min. Brucato, Piccarolo and Titomanlio proposed a general experimental route involving the recording of the thermal history of a thin polymer film, 50,100 Lm in thickness, and the detection of depolarized light intensities in the course of polymer crystallization [START_REF] Lamberti | Modeling the interactions between light and crystallization polymer during fast cooling[END_REF]. Very high cooling rates up to 2000 ° C/s were attained. The structure development can be characterized from thermal analysis by means of Differential Scanning Calorimetry (DSC). For instance, Pijpers have shown that DSC can be used at rates up to 500 K/min [START_REF] Pijpers | High,speed calorimetry for the study of the kinetics of (de)vitrification, crystallization, and melting of macromolecules[END_REF] to study the rate,dependent phenomena ; but no more investigation were done. To attain higher cooling rates, an ultra,fast thin film calorimeter, or nano, calorimeter, was proposed by Adamovsky & Schick [START_REF] Adamovsky | Ultra,fast isothermal calorimetry using thin film sensors[END_REF][START_REF] Gradys | Crystallization of polypropylene at various cooling rates[END_REF]. Rates of more than 1000 K/s can be attained, both in cooling and in heating modes. observation of the crystallization is achieved. The polymer is placed between a pair of thin microscope glasses. It is heated and cooled down by a gaseous flowing supplied at a constant temperature. The thermo,sensor is inserted into the polymer film to eliminate lag time between the polymer and the heat transfer medium. Careful attention was paid to locate the thermo,sensor very near to the optical observation area. These positions ensure a realistic polymer temperature. Relatively stable temperature conditions for the sample were provided as illustrated in Figure 1. The thermal protocol was: rapidly heating the sample to about 215 ° C, about 90 s holding at 215 ° C to erase the thermal history and to avoid its degradation, enabling thus the of the sample for many cycles, cooling at various cooling rates . + ' + 8 3 38 4 2 A Labview program was especially written in order to program the cooling rates, the temperature profile of the sample and the monitoring of optical observation. Two cooling modes are possible: a proportional one using a rotating throttle to control constant cooling rates at relatively moderate values (from 30 to 500 ° C/min) and an 'all,or,none' one using an 'open,close' valve for cold flux to attain high cooling rates (from 500 to 1600 ° C/min).

+ )

& : % &

Different films of iPP were investigated in a broad range of slow (from 3 to 10 ° C/min) with a conventional Mettler hot,stage, of relatively moderate (from 30 to 500 ° C/min) and of high (from 500 to 1600 ° C/min) cooling rates. Figure 2 shows the structure development investigated with the newly developed « Polymer High Cooling -Optics » set,up. The two cooling processes are illustrated, i.e., relatively moderate rates as shown at 144 ° C/min (Figure 2a) and high cooling rates as shown at 1462 ° C/min (Figure 2b). As expected at first glance, when the cooling rate increases, the crystallization occurs at lower temperature, about 30 ° C lower at 1462 ° C/min compared to 144 ° C/min. The nuclei are randomly distributed over the area of observation; their density rapidly increases with time and with cooling rate. The size distribution of crystallized entities decreases with increasing cooling rates. The growing entities have a radial growth direction. During crystallization, no drastic birefringent contrast was observed, α,modification is predominant.

With the present « Polymer High Cooling -Optics » set, up, it is possible to discern and to follow the spherulitic crystallization at high cooling rate up to 1600 ° C/min. Figure 3 plots the growth kinetics against the temperature of crystallization . The newly obtained spherulite growth rates are plotted with the ones collected by Janeschitz,Kriegl [START_REF] Janeschitz | Phases of flow,induced crystallization of i,PP: how remote pieces of the puzzle appear to fit[END_REF]. The growth kinetics follows the bell,shaped dependence with the temperature of crystallization. It is remarkable to observe that with only one method, the growth kinetics in the range 65,125 ° C can be well investigated. The information collected from the literature data do not show a so broad range of investigated resulting from one study. At relatively moderate cooling rates, the growth kinetics increases with undercooling up to around 90 ° C. For high cooling rates, attains 65 ° C, the growth kinetics seems to gradually decrease. This tendency is not very accentuated probably due to the limitation of use of the set,up.

According to the Hoffman and Lauritzen theory [START_REF] Lauritzen | Theory of formation of polymer crystals with folded chains in dilute solution[END_REF], as written in equation ( 1), Figure 4 is the activation energy for the transportation of segments of molecules across the melt/solid surface boundary and is usually given by a universal value of 1500 cal/mol. ∞ is the temperature where the molecular motion ceases with ∞ = -30. is the universal gas constant.

is a pre,exponential term.

is the nucleation parameter. It was estimated to 8.01 10 5 K 2 from relatively moderate constant cooling rate investigation and was in agreement with literature data [START_REF] Xu | Equilibrium melting and undercooling dependence of the spherulitic growth rate of isotactic polypropylene[END_REF]. From now, the benefits are to use the growth kinetics the temperature of crystallization curve to extrapolate data from alternative differential scanning calorimetry methods. Cooling rates extended from 1 to 500 ° C/min were examined. The calculated overall kinetics allows us to experimentally extend the Ozawa model to 55 ° C, a temperature which approaches the process temperatures. The Ozawa model can then be extrapolated theoretically at low temperature thanks to the estimate of the initial number of nuclei per unit volume from the DSC experiments and thanks to Hoffman & Lauritzen theory. At present, the control of the kinetics of crystallization from the melt at higher cooling rates, higher than 1600 ° C/min, would be of interest especially to conclude or not on the bell,shape of growth kinetics. It would be needed to think about a second generation of « Polymer High Cooling -Optics » set,up.

Nevertheless, additional analysis of the acquired data, i.e., optimization of the growth kinetics, estimation of the density of nuclei when it is possible, would be used for implementation into new crystallization models and then into numerical simulation.

# $ %

Figure 1 :

 1 Figure 1: Temperature-time evolution of iPP sample at a constant programmed cooling rate of 144 ° C/min and at a high cooling rate of 1462 ° C/min using the mode 'all-ornone'.

Figure 2 :

 2 Figure 2: Development of iPP spherulites at a constant cooling rate a) 144 ° C/min and b) 1462 ° C/min. Two different iPP films were used.

Figure 3 :

 3 Figure 3: Compilation of iPP α-spherulites growth rates versus the crystallization temperatures. Red open circles are from the present investigation at high and relatively moderate cooling rates (from 50 to 1600 °C/min); da ta at slow cooling rates are from an investigation using a conventional Mettler hot-stage (at 3 and 10 ° C/min). Open symbols are from Lovinger et al., 1977 (rhombs), Padden & Keith, 1959 (circles), Von Falkai, 1960 (triangles), Olley & Bassett, 1989 (stars). Closed symbols are from: β-transcrystallization, Ratajski, 1993 (closed squares), microtome slices, Ratajski & Janeschitz-Kriegl, 1996 (closed circles), DSC, Paulik, 1993 (closed triangles pointed upward) and Ratajski & Janeschitz-Kriegl, 1996 (closed diamonds), incubation times, Ratajski & Janeschitz-Kriegl, 1996 (closed triangles pointed downward). Literature data are from reference[START_REF] Janeschitz | Phases of flow,induced crystallization of i,PP: how remote pieces of the puzzle appear to fit[END_REF].

  plots the growth kinetics as a function of the temperature of crystallization and the degree of undercooling ∆ with equilibrium melting temperature.

  and ∞ are the Vogel,Fulcher,Tamman,Hesse (VFTH) parameters.

Figure 4 :

 4 Figure 4: Spherulite growth rates of iPP versus the crystallization temperatures T c based on the Lauritzen & Hoffman secondary nucleation theory with 0 = 208 ° C and = -21 ° C.

&

  The major task of the present paper is to tackle the polymer growth kinetics under high cooling rates. A newly developed and revisited set,up, named « Polymer High Cooling -Optics », first generation in date, was employed. It allows us to simultaneously follow the polymer temperature and the radial growth of spherulites during cooling rates up to 1600 ° C/min. The spherulitic growth rate follows the typical bell,shaped dependence with the temperature of crystallization in a broad range. The benefits are to use the newly plotted growth kinetics the temperature of crystallization to extrapolate the Ozawa model constructed for the overall kinetics of crystallization.

Acknowledgements are presented to the CEMEF technical team for their support during the revisit of the set,up. Perkin,Elmer company is also acknowledged for specific DSC experiments


S Milgram

T Livache

Y Roupioz
email: yoann.roupioz@cea.fr

S Cortes

M-B Villiers

P N Marche

Real time and label-free analysis of cellular activity on chip

Biochips for cellular applications are of considerable interest to both fundamental research and diagnostic fields. In this study, we demonstrate the use of Surface Plasmon Resonance imaging (SPRi) for the analysis of blood cell activity on biochip. This method previously used to detect antigen-antibody interactions was adapted for the detection of antibodies secreted from B-cell hybridoma using specific antigens grafted on the chip by electropolymerization. For the first time, living cells were maintained in culture on the biochip and the kinetic of their secretion was monitored by SPRi. Unlike traditional methods such as ELISA or ELISPOT, this novel technique permits a realtime and label-free analysis. As a result, antibodies secreted were detected only few minutes after the loading of cells on the chip. Thus, this sensor provides a promising tool for cells phenotype analysis and could be useful in analyzing other secreting cells, like T-cells, widely involved in the inflammatory response and several pathologies.

INTRODUCTION

The miniaturization of biosensors, allowing a rapid analysis of small sample volumes, has generated a considerable interest in the diagnostic and clinical research which involves biological samples both rare and delicate. Several studies address this topic, developing microarrays for the direct analysis of live cells [START_REF] Chen | Molecular and functional analysis using live cell microarrays[END_REF]. Blood analysis is crucial in these fields. Phenotype and secretory activities of blood cells are specially studied to characterize immune responses. In these aim, capture and characterization of living lymphocytes have been developed on a miniature cytometry plateform [START_REF] Zhu | A miniature cytometry platform for capture and characterization of T-lymphocytes from human blood[END_REF]. Other studies deal with microarrays specially dedicated to the detection of cellular metabolites [START_REF] Jones | Multifunctional protein microarrays for cultivation of cells and immunodetection of secreted cellular products[END_REF][START_REF] Zhu | A microdevice for multiplexed detection of T-cell-secreted cytokines[END_REF]. However, these methods, as other traditional techniques for detecting secreted metabolites, involve fluorescent staining and end-point analysis.

Label-free analysis is possible using other detection system like Surface Plasmon Resonance imaging (SPRi). SPR-based sensors have previously been used to detect interactions between many different types of molecules like DNA, peptide or protein [START_REF] Guedon | Characterization and optimization of a real-time, parallel, label-free, polypyrrole-based DNA sensor by surface plasmon resonnance imaging[END_REF][START_REF] Grosjean | A polypyrrole protein microarray for antibody-antigen interaction studies using a labelfree detection process[END_REF]. More recently, this method was used to detect live cells [START_REF] Hide | Real-Time Analysis of Ligand-Induced Cell Surface and Intracellular Reactions of Living Mast Cells Using a Surface Plasmon Resonance-Based Biosensor[END_REF][START_REF] Suraniti | Real-time detection of lymphocytes binding on an antibody chip using SPR imaging[END_REF]. In this way, Localized Surface Plasmon Resonance (LSPR) and Grating-Coupled Surface Plasmon Resonance imaging (GCSPRi) allow the label-free detection of specific cells or cellular metabolites [START_REF] Endo | Label-free cell-based assay using localized surface plasmon resonance biosensor[END_REF][START_REF] G.-B. Jin | Cytometry on a chip: Cellular phenotypic and functional analysis using grating-coupled surface plasmon resonance[END_REF].

Although label-free analysis was demonstrated in these studies, there is no method described in literature that allows a real-time analysis of cell secreted products. In this study, we developed a new miniature sensor for cell analysis in a both label free and real-time manner. The experimental setup using SPRi as detection method was adapted to maintain B-cell hybridomas in culture on chip and detect antibodies secreted from these cells (Fig. 1).

II. MATERIALS AND METHODS

A. Biochip preparation

Prisms bearing a gold surface (50nm thick) (Fig. 1b) were provided by GenOptics (Orsay, France) and prepared as described previously [START_REF] Guedon | Characterization and optimization of a real-time, parallel, label-free, polypyrrole-based DNA sensor by surface plasmon resonnance imaging[END_REF].

N-hydroxysuccinimidyl 6-(pyrrol-yl)-caproate (NHSpyrrole) were prepared as previously described [START_REF] Grosjean | A polypyrrole protein microarray for antibody-antigen interaction studies using a labelfree detection process[END_REF] and conjugated to proteins in phosphate-buffered saline (PBS) pH 7,4 overnight at 4°C using NHS-pyrrole/protein molar ratio of 10. Conjugated proteins were then purified by ultracentrigugation on 3 kDa MWCO membranes and diluted at different concentration (5, 10 or 20µM) in spotting buffer (50mM phosphate buffer, 50mM NaCl and 10% glycerol, pH 6.8) containing 20mM of free pyrrole. Proteins were then grafted on chip by electrocopolymerization of free pyrrole and pyrrole-modified protein (100ms, 2V ddp).

B. Cells culture

The mouse hybridoma cell line F10 producing monoclonal anti-lysozyme (anti-HEL) [START_REF] Tello | Three-dimensional structure and thermodynamics of antigen binding by anti-lysozyme antibodies[END_REF] was grown in DMEM supplemented with 10% fetal bovine serum, non-essential amino acids (0.1mM.mL -1 ), sodium pyruvate (1mM.mL -1 ), 50µM 2-mercaptoethanol, 50 U/mL penicillin and 50mg.mL -1 streptomycin. The cultures were incubated at 37°C in a humidified 5% CO 2 incubator. Cells were centrifuged 5 minutes at 200g prior to the experiments and the cell pellets were resuspended at 5.10 5 cells.mL -1 in serum-free RPMI medium buffered with Hepes at 28mM.

C. ELISA titration of anti-HEL IgG secreted by hybridoma cells

Secretory activities of cells used during SPRi experiment were characterized by ELISA assay. Samples of cellular suspension were incubated at 37°C in 24-well plate. At different times, the supernatant was collected and anti-HEL immunoglobulin G (IgG) secreted from cells were quantified by ELISA.

Plates were coated with HEL 10µg.mL -1 in PBS for 2h at 37°C. The ELISA was performed as described by Villiers et al. [START_REF] Villiers | Involvement of the Zn-Binding region of tetanus toxin in B and T recognition. Influence of Zn fixation[END_REF] and bound antibodies were revealed with peroxidaseconjugated rabbit anti-mouse IgG (Sigma) using TMB (3,3',5,5'-tetramethylbenzidine) as substrate. Standard curves were established using purified monoclonal anti-HEL antibody F10.

D. SPRi detection of antibody secretion

Cells incubations were carried out at 37°C in a peek chamber held in place on the chip (Fig. 1). After saturation of the chip surface with RPMI supplemented with 10% foetal bovine serum, B-cells hybridomas suspended in serum-free medium were loaded onto the chamber and their secretory activity was monitored by SPRi. Protein-protein interactions of the secreted IgG and the sensor grafted antigen on the chip surface induce a variation of the refractive index near the gold surface, triggering a variation of reflectivity monitored using a 12-bit CCD [START_REF] Guedon | Characterization and optimization of a real-time, parallel, label-free, polypyrrole-based DNA sensor by surface plasmon resonnance imaging[END_REF].

E. Fluorescent detection of antigen-antibody interaction

At the end of the SPRi experiment, the biochip was washed in PBS. IgG bound to the chip were revealed with a biotinylated anti-mouse antibody, followed by a Streptavidin-Phycoerythrin conjugation step. Fluorescence was observed using an epi-fuorescence microscope (BX, Olympus) equipped with Peltier cooled CCD camera.

III. RESULTS AND DISCUSSION

B-cell hybridoma F10 was chosen for its ability to secret monoclonal anti-HEL IgG. Biochips were functionalized by electropolymerization using HEL-pyrrole conjugate solution at 5, 10 or 20µM. Cytochrome C was chosen as a negative controle because of its molecular weigh close to that of HEL. The control spots corresponding without HEL were made from a solution of electropolymerisation containing 20µM of cytochrome C-pyrrole conjugate, in order to have a steric hindrance similar to that of HEL spots. Cells were loaded in the sample chamber as described in Fig. 1. Binding between HEL antigen grafted on the chip and anti-HEL IgG secreted from cells was detected by SPRi and the kinetic of cellular secretion was recorded for eight hours by both SPRi and ELISA titration of cellular supernatants.

A reflectivity variation can be observed in real time on the SPRi image only few minutes after cells injection (Fig. 2A). Thanks to the imaging setup, reflectivity variations of the surface were observed and the averaged grey level of pixels contained in the image of HEL spots were shown to increase overtime. These direct observations were plotted to allow comparison within all the samples spotted on the biochip (Fig. 3A).

The specificity of the reaction was verified thanks to the cytochrome C control spots. No reflectivity variation was observed on these spots. This specificity was also confirmed at the end of the experiment by counter staining of the chip surface with anti-mouse IgG biotin conjugate followed by a Streptavidin-Phycoerythrin conjugation step. It revealed a very specific labeling of the HEL spots (Fig. 2B). On the other hand, anti-HEL IgG released from B-cells were titrated by ELISA (Fig. 3B). After 30 minutes of incubation, the concentration of IgG in cellular supernatant was 51.3 ± 5.6 ng.mL -1 and increased linearly with the time reaching a concentration of 579.1 ± 47.7 ng.ml -1 after eight hours of incubation.

ELISA was used as a reference technique for titration of cellular metabolites. It allowed the characterisation of secretory activity of F10 hybridoma cells in our experimental conditions. Moreover, ELISA results can be compared to SPRi results in order to evaluate the efficiency of SPRi-based sensor detection method.

Firstly, these results allowed to verify that the secretory activity of B-cells is correctly maintained in our experimental conditions. Besides, when we compare both types of results, a difference of shape was observed between both curves. In the case of ELISA, the time-course curve obtained was linear. The SPRi curves, unlike the ELISA curve, showed a signal increase much higher for short incubation times, with a slope much greater than for ELISA curves. In a second time, the SPRi slopes decreased and reached a plateau. This phenomenon can be explained by the intrinsic nature of the detection system: the surface is functionalized with a finite number of antigen molecules which can be seen as vacant sites. Interactions taking place on the surface are directly linked to the vacant site densities and thus explain the differences of plateau levels when almost all vacant sites are occupied. The saturation of spots in our experiments was verified by injection of purified IgG anti-HEL at the end of the experiment. No increase of reflectivity was observed, signifying that spots are well saturated (data not shown).

Another interesting feature is the low concentration of metabolites titrated by ELISA in supernatant after thirty minutes incubation. At this time, a great specific SPR signal was observed and ELISA results indicated a concentration of about 50 ng.mL -1 of secreted anti-HEL IgG, corresponding to 0.35 nM in bulk medium. In classical SPRi experiments studying molecular interaction, the typical detection limit is about 10 nM. Thus, IgG in solution at 0.3 nM should not be detected with a classic SPRi detection system. In our case, live secreting cells are directly loaded onto the sensor and sediment near the gold surface. The local exploration of secreted proteins in the intimate vicinity of the cells gives access to locally high concentrations of secreted products. The localization of cells close to the chip surface allowed us to access to protein concentrations above the SPR current LOD and thus showed the performance of SPR-based sensors for cellular secretion detection. Thus, in our conditions, the secretory activity of B-cells can be detected by SPRi with a good specificity and with sensitivity better than ELISA for early cellular activities.

IV. CONCLUSION

To conclude, we shown that our method allows detect very quickly metabolites secreted from cells, maintaining cells in culture on chip without loss of their activity.

The techniques routinely used involve most often several steps of analysis: collection of cells, lysis or fixing of them, immunostaining with different steps of wash and staining, and finally, observation. For the first time, we propose a one-step imaging method able to detect in real time and without label secretory activity of live cells. Moreover, the system shows a great flexibility and could be adapt to multiparametric detection, allowing simultaneous analysis of different metabolites. This quick and easy method could be a tool very useful for cell phenotype analysis. Currently, the SPR-based sensor is being adapted for the study of T-cells behaviour.

Figure 1 .

 1 Figure 1. (A) Scheme of the SPR imaging setup.(B) Photograph of a goldcoated chip.

Figure 2 .

 2 Figure 2. (A) SPR image of a biochip containing HEL spots (5, 10 or 20µM) or cytochromeC control spot (HEL 0µM) after different times of exposure to secreting cells. (B) Image of the same spots observed at the end point of the experiment by fluorescence microscopy after staining with antimouse IgG (Fc specific).

Figure 3 .

 3 Figure 3. (A) SPR monitoring of the reflectivity variations of different spots functionalized with Cytochrome C 20µM (gray dotted line), HEL 5µM (light gray line), 10µM (dark gray line) or 20µM (black line). Cells were loaded on chip at time 0. (B) ELISA titration of anti-HEL IgG secreted by cells in culture medium at different times.


Séverine Rivollier
email: rivollier@emse.fr

Johan Debayle
email: debayle@emse.fr

Jean-Charles Pinoli
email: pinoli@emse.fr

Shape representation and analysis of 2D compact sets by shape diagrams

Keywords: Shape diagrams, Pattern analysis, Convex and noconvex sets

Shape diagrams are shape representations in the Euclidean plane introduced for studying 3D and 2D compact sets. A compact set is represented by a point within a shape diagram whose coordinates are morphological functionals defined from geometrical functionals and inequalities. Classically, the geometrical functionals for 2D sets are the area, the perimeter, the radii of the inscribed and circumscribed circles, and the minimum and maximum Feret diameters. The purpose of this paper is to present a particular shape diagram for which mathematical properties have been well-defined and to analyse the shape of several families of 2D sets for the discrimination of convex and non convex sets as well as the discrimination of similar sets.

I. 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 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 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] Santaló | Sobre los sistemas completos de desigualdades entre tres elementos de una figura convexa plana[END_REF], [START_REF] Hernandez Cifre | The missing boundaries of the santaló diagrams for the cases (d, ω, r) and (ω, R, r)[END_REF], [START_REF] Hernandez Cifre | Complete systems of inequalities[END_REF].

The purpose of this paper is to present a particular shape diagram for which properties have been well-defined [START_REF] Hernandez Cifre | Complete systems of inequalities[END_REF] and to analyse several families of 2D sets for shape discrimination. The geometrical and morphological functionals (morphological functionals are invariant under similitude transformations and are defined as ratios between geometrical functionals; for instance, their concrete meanings are circularity, thinness, . . . ) of 2D compact sets are exposed in the next section. From these functionals, a specific shape diagram is studied for the discrimination of convex and non convex sets as well as the discrimination of similar sets. The final section offers an application related to this study.

II. SHAPE REPRESENTATION

In this paper, the non-empty compact convex sets in the Euclidean 2D space E 2 are considered. Several geometrical functionals are determinated in order to characterize the sets. They are related by the so-called geometric inequalities, which allow to define morphological functionals and thereafter shape diagrams.

A. Geometrical functionals and inequalities

For a compact 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 1 illustrates some of these geometrical functionals. For a compact set, these six geometrical functionals are greater than zero. The line segments and curves provide null values for A, r and ω, and the points for P, R and d.

For a compact set in E 2 , the relationships between these geometrical functionals are constrained by the geometric inequalities [START_REF] Bonnesen | Theorie des konvexen körper[END_REF], [START_REF] Yaglom | Convex figures[END_REF], [START_REF] Osserman | The isoperimetric inequality[END_REF] referenced in the second column of Table 1. Some geometric inequalities (that are not considered in this paper) are restricted to convex sets. These inequalities link geometrical functionals by pairs and determine the socalled extremal compact sets that satisfy the corresponding equalities (Table 1, fourth column). Futhermore, they allow to determinate morphological functionals.

B. Morphological functionals

The morphological functionals are invariant under similitude transformations (consequently, they do not depend on the global size of the compact set) and are defined as ratios between geometrical functionals. In these ratios, the units of the numerator and the denominator are dimensionaly 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 morphological functional, the scalar value depends directly on the associated inequality. These morphological functionals are referenced in the third column of Table 1. They 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, r / R, 2
≤ R r / R C ω, R ω ≤ 2 R ω /2 R C A, R A ≤ π R 2 A /π R 2 C d, R d ≤ 2 R d /2 R Y 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 R, d √ 3 R ≤ d √ 3 R / d Z

C. Shape diagrams

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

Let be any 3-uplet of the considered six geometrical functionals (A, P, r, R, ω, d) and (M 1 , M 2 ) be some particular morphological functionals valued in [0, 1] 2 . A shape diagram D is represented in this plane domain [0, 1] 2 (whose axis coordinates are the morphological 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 {π r 2 / A, 2 r / ω}, the line segments and curves can not be mapped onto a point because they provide null values for A and ω. Mathematically, 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 morphological functionals listed in Table 1, twenty-two shape diagrams are defined, denoted (D k ) k∈ 1,22 , respectively (Table 2).

Table 2. The twenty two shape diagrams axes coordinates for 2D compact sets.

Shape diagrams Axes coordinates

D1 : (ω, r, R) x = ω /2 R y = r / R D2 : (ω, A, R) x = ω /2 R y = A /π R 2 D3 : (r, A, R) x = r / R y = A /π R 2 D4 : (A, d, R) x = A /π R 2 y = d /2 R D5 : (ω, d, R) x = ω /2 R y = d /2 R D6 : (r, d, R) x = r / R y = d /2 R D7 : (ω, r, d) x = ω / d y = 2 r / d D8 : (ω, A, d) x = ω / d y = 4 A /π d 2 D9 : (r, A, d) x = 2 r / d y = 4 A /π d 2 D10 : (A, R, d) x = 4 A /π d 2 y = √ 3 R / d D11 : (ω, R, d) x = ω / d y = √ 3 R / d D12 : (r, R, d) x = 2 r / d y = √ 3 R / d D13 : (ω, r, P)
x = π ω / P y = 2π r / P D14 : (ω, A, P)

x = π ω / P y = 4π A / P 2 D15 : (r, A, P)

x = 2π r / P y = 4π A / P 2 D16 : (A, R, P)

x = 4π A / P 2 y = 4 R / P D17 : (ω, R, P) x = π ω / P y = 4 R / P D18 : (r, R, P) x = 2π r / P y = 4 R / P D19 : (A, d, P) x = 4π A / P 2 y = 2 d / P D20 : (ω, d, P) x = π ω / P y = 2 d / P D21 : (r, d, P) x = 2π r / P y = 2 d / P D22 : (d, R, P) x = 2 d / P y = 4 R / P
More specifically, the mathematical properties of the shape diagram D 8 have been well-studied [START_REF] Hernandez Cifre | Complete systems of inequalities[END_REF] (the choice of this shape diagram is based on the results synthetized in [START_REF] Rivollier | A comparative study of shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF], [START_REF] Rivollier | A comparative study of shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF], [START_REF] Rivollier | A comparative study of shape diagrams for 2D compact sets -Part III: convexity discrimination for analytic and discretized simply connected sets[END_REF]). For example, this shape diagram is based on a complete system of geometrical inequalities: for any 3-uplet (r, A, d) satisfying those conditions, a 2D compact convex set with these geometrical functionals values exists [START_REF] Santaló | Sobre los sistemas completos de desigualdades entre tres elementos de una figura convexa plana[END_REF], [START_REF] Hernandez Cifre | The missing boundaries of the santaló diagrams for the cases (d, ω, r) and (ω, R, r)[END_REF]. In other words, the mapping that associates a 2D compact convex set in E 2 to a point in this shape diagram can be surjective by restricting the arrival set. 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. The compact convex sets mapped onto the boundary points are the extremal compact convex sets of each considered inequality. Figure 3c shows the convex domain of the shape diagram (complete system) D 8 .

D. Illustration

Figure 3 illustrates the shape diagram D 8 for the family F 1 of compact sets (Figure 2). Note that the two axes represent the diameter constance (x-axis) and the roundness (y-axis) of a shape. The compact sets of the family F 1 are chosen according to the faculty to compute analytically their geometrical (and morphological) functional values. Semi-disks Semi-disks Semi-disks Semi-disks Semi-disks Semi-disks Semi-disks Semi-disks Semi-disks Semi-disks Semi-disks "1/2" semi-rings "1/2" semi-rings "1/2" semi-rings "1/2" semi-rings "1/2" semi-rings "1/2" semi-rings "1/2" semi-rings "1/2" semi-rings "1/2" semi-rings "1/2" semi-rings "1/ In the next section, different families of compact sets will be studied through this shape diagram D .

III. SHAPE ANALYSIS

In this section, shape discrimination is studied for convex and non convex sets and also for sets with similar (visually) shapes.

A. Discrimination of convex and non convex sets

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. The parameter of Zunic and Rosin [START_REF] Zunic | A new convexity measure for polygons[END_REF] is used for computing the shape convexity. It ranges between 0 and 1.

Figure 4 illustrates seventy-eight binary images constituting the family F 2 . For each image, the white object represents a discretized compact set and the black pixels represents the background. The binary images are numerated from one to seventy-eight. The image number is mapped to its proper point in the shape diagram D 8 (Figure 4b). The color of the number is related to the convexity parameter value c ∈ [0, 1] of the associated set (dark red for a high c, dark blue for a low c). This convexity range will enable to analyze the convexity discrimination within the shape diagram.

A wider family F 3 of 1370 binary images from the Kimia database [START_REF] Sharvit | Symmetry-based indexing of image databases[END_REF] is also considered and analyzed (Figure 4c). For the two families of sets, the colors are rather irregularly distributed according to the color range. Blue colors (resp. red colors) appear inside (resp. outside) the convex domain. In addition, the artifacts of the discretization can interfer with the results. In conclusion, the shape diagram D 8 shows a moderate discrimination of convex and non convex sets.

B. Discrimination of similar 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.

Let be two families F 4 and F 5 of binary images representing triangles (Figure 5) and disks (Figure 6), respectively, that have undergone minor transformations, modifications, deformations. The morphological parameters are computed, and the images are located by a point in the shape diagram D 8 . The color of the number is related to the convexity parameter value c, and the convex domain boundary is illustrated with black lines. The results show that the spatial distribution of the F 5 sets is dense. Each set is located around the same point. On the contrary, the locations of the F 4 sets are more sparse (the shape roundness of the different sets varies a lot!).

IV. APPLICATION

Let be a binary image with three classes of seeds (Figure 7): circular seeds, small and elongated seeds, and almond-shaped seeds. The connected component corresponding to the seeds are numbered. These discretized compact sets are mapped onto the shape diagram D 8 . The color of the set number represents the group to which the associated seed belongs: red for the circular seeds, blue for the small and elongated seeds, and green for the almond-shaped seeds.

In the shape diagram D 8 , the red locations form a well distinct cluster from the blue and green locations. It is usual as green and blue locations are a little confused. The associated seed shapes have similar shapes. They differ mainly in size, but remember that the shape diagrams axes are morphological functionals that do not depend on the global size of the set. This application shows that the shape diagram can be used for shape classification.

Let be four families of binary images representing apples, camels, sea-snakes and watches that have undergone minor transformations, modifications, deformations. These binary images are not illustrated here. They are extracted from the Kimia database and they are similar enough to the images 39, 13, 14 and 55 respectively of the figure 4. The morphological parameters are computed, and the images are located by a point in the shape diagram D 8 (Figure 8). The color of the point is related to the group to which the set representing in the image belongs: red for a set of 'watch' shape, blue for a set of 'apple' shape, green for a set of 'sea-snake' shape, and black for a set of 'camel' shape.

The results show that the spatial distribution of each family form a well distinct cluster from each other. Thus, the shape diagram can be used for shape classification. The set locations are points whose the color is related to the group to which the set belongs: red for the sets of 'watch' shape, blue for the sets of 'apple' shape, green for the sets of 'sea-snake' shape, and black for the sets of 'camel' shape.

V. CONCLUSION AND PERSPECTIVES

This paper has dealed with shape diagrams of 2D compact sets built from geometrical functionals and inequalities. Each set is represented by a point within a shape diagram whose coordinates are morphological functionals defined as normalized ratios of geometrical functionals. Such a diagram enables to define a shape representation for 2D sets in the 2D Euclidean plane (bounded region).

A particular shape diagram for which mathematical properties have been well-defined has been studied for shape representation and analysis of several families of 2D compact sets. First, it was shown that the shape diagram allowed a moderate discrimination of convex and non convex sets. Secondly, the discrimination of similar sets was not always relevant in relation with the studied family of sets. Lastly, an application related to this study is offered.

Pattern recognition and classification could be investigated using the proposed shape diagrams defining a morphological signature of sets. Also, a more detailed study has to be performed in order to evaluate the quality of these signatures in relation with other standard ones using shape databases.

Currently, the authors are studying the shape discrimination of the twenty-one other possible shape diagrams. A comparative study is being performed [START_REF] Rivollier | A comparative study of shape diagrams for 2D compact sets -Part I: analytic convex sets[END_REF], [START_REF] Rivollier | A comparative study of shape diagrams for 2D compact sets -Part II: analytic simply connected sets[END_REF], [START_REF] Rivollier | A comparative study of shape diagrams for 2D compact sets -Part III: convexity discrimination for analytic and discretized simply connected sets[END_REF].

Fig. 1 .

 1 Fig. 1. Geometrical functionals of a compact set: radii of inscribed (r) and circumscribed (R) circles, minimum (ω) and maximum (d) Feret diameters.

Fig. 2 .

 2 Fig. 2. Family F 1 of 2D compact sets.

8 (D 8 Fig. 3 .

 883 Fig. 3. Shape diagram D 8 (b) for the compact sets (a) of the family F 1 with the boundaries of the convex domain (c).

Fig. 4 .

 4 Fig. 4. Family F 2 of seventy-eight 2D compact sets, represented in white on binary images. The color of each pattern number is related to the convexity parameter value using non-linear color tones (a). Convexity analysis for compact sets of the families F 2 (b) and F 3 (c) mapped onto the shape diagram D 8 .

  Fig. 5. Family F 4 of twenty 2D compact sets, represented in white on binary images. All these sets have a 'triangle' shape. The color of the number is related to the convexity parameter value (a). Discrimination of similar sets for this family F 4 (b) mapped onto the shape diagram D 8 .

  Fig. 6. Family F 5 of twenty 2D compact sets, represented in white on binary images. All these sets have a 'disk' shape. The color of the number is related to the convexity parameter value (a). Discrimination of similar sets for this family F 5 (b) mapped onto the shape diagram D 8 .

Fig. 7 .

 7 Fig. 7. Seeds represented in white on the binary image (a). The seeds are numbered and the color of a number is related to the group to which the seed belongs: red for the circular seeds, blue for the small and elongated seeds, and green for the almond-shaped seeds. Discrimination of seeds (b) mapped onto the shape diagram D 8 .

Fig. 8 .

 8 Fig.8. Discrimination of sets mapped onto the shape diagram D 8 . The set locations are points whose the color is related to the group to which the set belongs: red for the sets of 'watch' shape, blue for the sets of 'apple' shape, green for the sets of 'sea-snake' shape, and black for the sets of 'camel' shape.

Table 1 .

 1 r / d and ω /2 R;• diameter constance: π ω / P and ω / d; • thinness: 2 d / P and 4 R / P. Shape functionals for compact 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.

	The morphological functional	√	3 R / d expresses both the
	equilateral triangularity and the diameter constance. The ratios
	2 r / ω and d /2 R do not have concrete meaning, they are
	equal to one for many different compact sets.
	Geometrical	Geometric		Morphological	Extremal
	functionals	inequalities		functionals	sets
	r, R	r

  2" semi-rings

	Regular pentagrams Regular pentagrams Regular pentagrams Regular pentagrams Regular pentagrams Regular pentagrams Regular pentagrams Regular pentagrams Regular pentagrams Regular pentagrams Regular pentagrams
	Regular pentagons Regular pentagons Regular pentagons Regular pentagons Regular pentagons Regular pentagons Regular pentagons Regular pentagons Regular pentagons Regular pentagons Regular pentagons
	Regular hexagrams Regular hexagrams Regular hexagrams Regular hexagrams Regular hexagrams Regular hexagrams Regular hexagrams Regular hexagrams Regular hexagrams Regular hexagrams Regular hexagrams
	Regular hexagons Regular hexagons Regular hexagons Regular hexagons Regular hexagons Regular hexagons Regular hexagons Regular hexagons Regular hexagons Regular hexagons Regular hexagons
	Reuleaux triangles Reuleaux triangles Reuleaux triangles Reuleaux triangles Reuleaux triangles Reuleaux triangles Reuleaux triangles Reuleaux triangles Reuleaux triangles Reuleaux triangles Reuleaux triangles
	Regular crosses Regular crosses Regular crosses Regular crosses Regular crosses Regular crosses Regular crosses Regular crosses Regular crosses Regular crosses Regular crosses
	Line segments Line segments Line segments Line segments Line segments Line segments Line segments Line segments Line segments Line segments Line segments