Lucie Barque

Alexis Nasr

From the Definitions of the Trésor de la Langue Française to a Semantic Database of the French Language

Introduction

The Definiens project aims at building a database of French lexical semantics that is formal and structured enough to allow for a fine-grained semantic access to the French lexicon-for such tasks as automatic extraction and computation. To achieve this in a relatively short time, we process the definitions of the Trésor de la Langue Française informatisé [START_REF] Dendien | Le Trésor de la Langue Française informatisé: un exemple d'informatisation d'un dictionnaire de langue de référence[END_REF][START_REF] Tlfi | Trésor de la Langue Française informatisé[END_REF])-hereafter TLFi-, enriching TLFi's definitions with an XML tagging that makes explicit their internal organization. Definiens builds on the results of the BDéf project [START_REF] Altman | La BDéf : base de définitions dérivée du Dictionnaire explicatif et combinatoire[END_REF], that derived a small database of fully formalized definitions from the Explanatory Combinatorial Dictionary of Contemporary French [START_REF] Mel | Dictionnaire explicatif et combinatoire du français contemporain: recherches lexico-sémantiques I-IV[END_REF][START_REF] Mel | Dictionnaire explicatif et combinatoire du français contemporain: recherches lexico-sémantiques I-IV[END_REF][START_REF] Mel | Dictionnaire explicatif et combinatoire du français contemporain: recherches lexico-sémantiques I-IV[END_REF][START_REF] Mel | Dictionnaire explicatif et combinatoire du français contemporain: recherches lexico-sémantiques I-IV[END_REF]. There is, to our knowledge, no existing broad coverage database for the French lexicon that offers to researchers and NLP developers a structured decomposition of the meaning of lexical units. Definiens is an ongoing research that will hopefully fill this gap in the near future. In Section 2, we explain the problem at hand. Then, in Section 3, we detail the first stage of definition structuring, namely the segmentation into definitional components, and explain how this operation can be achieved by automatic procedures. In Section 4, we detail the second stage of definition structuring, which consists of enhancing TLFi's definitions with semantic labels.

Context and aims of Definiens

Using analytical definitions

There are two main reasons why we decided to start from the Trésor de la Langue Française, rather than from any other dictionary, in order to build a French semantic database. First of all, there is the obvious reason that the TLF was available to us for research purposes in electronic form, as TLFi. All other large-scale descriptions of the French lexicon are commercial products or are embedded in commercial products, and therefore not available for this kind of research. Second, TLF's lexicographers have adopted what we view as good practice in structuring their definitions: most of TLF's definitions belong to a specific class of lexicographic definitions that have been termed analytic definitions in Polguère (2008:182-188). Analytic definitions correspond closely to so-called "definitions by genus and diffentiae", which makes them compatible with the type of formal structuring we are targeting. Analytic definitions possess the two following properties:

1. they are analytic paraphrases of the lexical unit under description-the term analytic implies that they decompose the meaning of this unit in terms of simpler meanings (they are not lists of synonyms or quasi-synonyms); 2. they are made up of two main parts: a) the genus-or generic/central component of the definition-, that stands in a hyperonymic relation with the defined unit, and b) the differentiae-or peripheral components-, that specify the unit's meaning relative to the genus and to other semantically related lexical units.

This type of definition is the norm in most reference dictionaries, especially in learners' dictionaries, sometimes with some "pedagogical" adaptation, such as in the Cobuild (Sinclair, 1990;[START_REF] Rundell | More Than One Way to Skin a Cat: Why Full-Sentence Definitions Have Not Been Universally Adopted[END_REF]. Analytical definitions are also used in theoretical dictionaries, such as [START_REF] Mel | Dictionnaire explicatif et combinatoire du français contemporain: recherches lexico-sémantiques I-IV[END_REF][START_REF] Mel | Dictionnaire explicatif et combinatoire du français contemporain: recherches lexico-sémantiques I-IV[END_REF][START_REF] Mel | Dictionnaire explicatif et combinatoire du français contemporain: recherches lexico-sémantiques I-IV[END_REF][START_REF] Mel | Dictionnaire explicatif et combinatoire du français contemporain: recherches lexico-sémantiques I-IV[END_REF] and [START_REF] Wierzbicka | English Speech Act Verbs. A Semantic Dictionary[END_REF], and have proved very useful for the study of the semantics of natural languages-for instance, in the formal study of French polysemy [START_REF] Martin | La polysémie verbale, esquisse d'une typologie formelle[END_REF][START_REF] Barque | Description et formalisation de la polysémie régulière du français[END_REF].

Advantage of making explicit the internal structure of definitions

We believe analytical definitions to be the most straightforward and natural way of describing lexical meanings. Though it is Aristotle-cf. Topics (Aristotle, 1939 translation)-who is credited for their first theoretical conceptualization, they are probably as old as natural languages themselves as they correspond to the most spontaneous way of accounting for meanings in a metalinguistic fashion (Q: -What does "X" mean? A: -It means ...). From a natural language processing or, more generally, formal point of view, however, analytical definitions present one major drawback: they are "texts" rather than fully-formalized structures-like typed feature structures proposed by the Generative Lexicon [START_REF] Pustejovsky | The Generative Lexicon[END_REF], for instance. One way to remedy, at least partially, this problem is to make explicit the internal structure of text definitions by means of an XML tagging that will identify:

1. the central and peripheral components of the definition; 2. the normalized content of the central component-encoded by means of semantic labels [START_REF] Polguère | Étiquetage sémantique des lexies dans la base de données DiCo[END_REF]; 3. the normalized semantic role played by each peripheral component relative to the central component-encoded by means of a small subset of semantic labels.

This tagging will superimpose on the definitional text a fully-formalized semantic skeleton that will allow for non-trivial automatic processing of the resulting database. Though the target of research in formal semantics should be a complete formalization of semantic descriptions, Definiens will represent a significant advance in the field by offering access to the semantics of French that will go far beyond what is now available (which is simple processing of definitional texts as strings of characters).

A heterogeneous corpus of definitions

Identifying the exact list of lexical units described in the TLFi in order to tag its definitions is not a straightforward task. The dictionary contains 54,281 main entries (i.e. polysemic units), in which are embedded 18,095 entries that correspond to morphological derivatives (e.g. FRUITARISME under FRUIT). Main entries also include 59,168 defined phrases that are of two types: some correspond to idioms (e.g. FRUIT DEFENDU under FRUIT), others correspond to collocations (e.g vol domestique under VOL). By collocations, we mean semi-idiomatic expressions, also called semi-phrasemes in Explanatory Combinatorial Lexicology [START_REF] Mel | Introduction à la lexicologie explicative et combinatoire[END_REF]. For the project, we have automatically extracted from the TLFi a total of 271,164 definitions that describe the meaning of these three types of lexical entitieslexemes, idioms and collocations. To date, about 5 percent of our corpus (that is, of 271,164 definitions) has been manually segmented. The manually segmented data are currently used to evaluate the automatic segmentation of the TLFi definitions and to describe the grammar of the TLFi metalanguage.

Segmentation into definitional components

During their training period, the annotators work in pairs for the segmentation, and then alone.

The agreement rate between annotators has not yet been evaluated. Annotators work on files (extracted from TLFi files and edited in the oXygen XML Editor) that contain only the relevant information for Definiens, that is, the name of the defined units and their definitions. The TLFi's structuration of the different meanings of the polysemic units (vocable) is preserved, as illustrated in figure 1 below. Segmenting and tagging the TLFi definitions is a time consuming and tedious task that can be partly automated using NLP techniques. We have devised a series of tools called MACAON that realizes the segmentation.

The segmentation process is decomposed into four consecutive sub-processes. The first one decomposes the definitional text into elementary elements called tokens, according to typographic rules. The second sub-process, called lexer, performs a search in a wide coverage inflected forms lexicon. During this step, tokens that have been identified by the tokenizer can be put together to form multi-word expressions. For example, the sequence en dessous de ('below') is recognized as a complex preposition. The output of the lexer is then passed on to the third sub-process, a part of speech tagger, whose output is passed on to the final subprocess, a chunker. The chunker groups POS-tagged words together to form complex units called chunks among which the chunks that are relevant for the segmentation work: the Central Component (CC), the Peripherial Components (CP) and the paraphrase (PARAPH).

The drawback of this sequential organization of the sub-processes is that most of them can, and generally do, make mistakes. An error made at one stage of the process usually provokes errors in the following stages. To address this problem, every module can take as input several solutions produced by the preceding module, and can produce several outputs. The chunker, for example, accepts several possible part-of-speech tag assignments to words and can produce several possible groupings of the tag sequences into chunks.

Ambiguous input and output are represented by way of weighted finite state automata and most modules process their input through standard operation on automata [START_REF] Mohri | Finite-State Transducers in Language and Speech Processing[END_REF].

Each module also associates a score to every solution produced, and the scores of the different modules accumulate as long as the input sentence goes through the different modules. At the end of the process, every solution produced has a score which is a linear combination of the score given by each module that the solution passed through. The association of a score to each solution built by a module allows the researcher to limit, at any stage of the process, the ambiguity to the n highest score solutions. A scoring technique such as this has been shown by [START_REF] Nasr | Integrating a Part of Speech Tagger and a Chunker Implemented as Weighted Finite-State Machines[END_REF] to improve the quality of a pipeline architecture. The highest score solution produced by the process is then reviewed by a human annotator.

First results

The automatic processing of the definitions is under development. The most delicate part of the development is the creation of the chunk grammar. Such a grammar is very ambiguous since there are generally many possible (syntactic) segmentations of a given definition. The ambiguity is further increased by the fact that several part of speech taggings of the definition are given as input to the chunker. We devised some simple heuristics in order to rank the possible segmentations and select the segmentation which achieves the highest score with respect to these heuristics. The current state of the chunk grammar allows us to parse 70.42% of a sample of the definitions. The quality of a segmentation H (for Hypothesis) of a definition D selected by the heuristics can be compared to the correct (manual) segmentation of D, which we call R (for Reference). H and R can be compared by means of recall and precision. The precision of H is the proportion of components (either central component or peripheral component) in H that are in R while the recall is the proportion of components of R that are found in H. The current state of the grammar achieves a precision of 0.74 and a recall of 0.58. In other words, 74% of the components automatically built are correct and 58% of the expected components have been built.

It is important to keep in mind that the aim of the automatic processing is to speed up the manual segmentation of the definition. In order to evaluate the efficacy of the automatic processing, a small experiment has been conducted. The annotators were asked to both annotate plain definitions and to correct automatically their annotated counterparts (and vice versa with another set of definitions). The time to perform both tasks was compared. The results are quite disappointing-at this stage, there is no noticeable gain in time: correcting the automatic segmentation is not faster than a manual segmentation from scratch. It is however most likely due to the quality of the automatic segmentation at this stage of the project; an automatic segmentation of better quality would surely yield some gains.

Several improvements are under study in order to increase the quality and coverage of the automatic segmentation and, hopefully, speed up the manual segmentation. The first one is the improvement of the chunk grammar and the heuristics used to rank the competing segmentations. The second one is more ambitious-it aims at segmenting all the definitions in parallel and not in an iterative way as is currently done. The idea is that a central component, for example, cannot be identified only on syntactic criteria, as is the case in the current system. Refer to our previous example, the definition of BROUETTE. In order to decide whether the central component of BROUETTE is véhicule or véhicule à une roue, we need to look up other definitions which have véhicule à une roue as a central component. The idea is to introduce this comparative search during the automatic segmentation stage: for each definition, all its potential central components are built based on syntactic criteria, but the choice of the best one is made after a comparison with the candidates of all the other definitions in the database. In our example, the candidate véhicule à une roue will not be selected since it does not appear as a potential central component of any other definition. The definition of the semantic label guides us in the second step of the semantic tagging, namely the assignment of a definitional role to each peripheral component of a given definition. Indeed, the definition of VÉHICULE given above indicates that a vehicle has some "characteristic parts" (constitué d'un châssis muni de roues), is characterized by a "type of motion" (à traction animale ou autopropulsé) and, finally, has a "function" (servant au transport routier ou ferroviaire). This tells us that these three basic roles will likely be found in the definition of lexical units labeled with véhicule (or a more specific label). An examination of the definitions of the set of lexical units labeled véhicule reveals indeed other roles like "speed", "appearance", etc. The definition of BROUETTE will hence be semantically tagged as follows:

Semantic enhancement

BROUETTE (sense B.1): <PARAPH><CC=véhicule>Véhicule</CC> <CP=parties charactéristiques>à une roue et à deux brancards</CP> <CP=fonction>servant au transport des matériaux</CP></PARAPH> 4.2 A closer look into the hierarchy of semantic labels

The hierarchy of semantic labels, developed during the construction of the DiCo database [START_REF] Polguère | Étiquetage sémantique des lexies dans la base de données DiCo[END_REF], currently contains 790 labels that have been defined with the relevant definition from the TLFi, as illustrated by véhicule in the previous section. As these labels have been created for the description of a limited but representative set of lexical units, we do not as yet know to what extent the labels will cover the TLFi word list.

An important lexicographic work has to be done to enhance the label hierarchy with roles. As previously mentioned, each label is associated a list of possible roles for predicted peripheral components, as illustrated below with véhicule:

véhicule: {parties caractéristiques, fonction, mode de fonctionnement, vitesse, apparence} This kind of information will be very useful for the continuation of our project, which will consists of creating a new lexicographic database made of fully formalized definitions. Indeed, the set of roles associated with each label allows for a semantic control of the definitions. For example, a semantically well-formed definition of a lexical unit labelled véhicule has to contain peripheral components whose roles are associated to this label in the hierarchy.

Conclusion

In this paper, we have presented an ongoing lexicographic project which aims at providing an explicit and formalized structuring for the definitions of the French electronic dictionary Le Trésor de la Langue Française informatisé (TLFi). The project is ambitious and will take significant time to be completed. In the meantime, the following intermediary results can be achieved. First of all, the basic segmentation of the definitions into central and peripheral components will allow for a better exploitation of the dictionary. For instance, one will be able to request the set of lexical units that denote a vehicle with two wheels, or the set of lexical units that denote an "intense" feeling. This type of request is very useful for a language-related task which requires well specified semantic clusters of lexical units (for example, information extraction, etc.). The work in progress will also lead to a precise characterization of the TLFi metalangage and will thus allow for the description of a definition format in order to develop an NLP lexicon derived from the TLFi. This could be done using the Lexical Markup Framework (LMF), for example (Francopoulo, 2005:19-21).

Finally, the semantic label hierarchy, which will probably be completed before the end of the project, will offer a valuable resource to lexical semantics researchers.

Figure 1 :

 1 Figure 1 : Edition in oXygen for the segmentation task

  4.1 Aims of semantic taggingSemantic tagging first consists of assigning the central component a semantic label, i.e. a normalized expression that accounts for the semantic value of the central component. Every semantic label is defined by its appropriate definition in the TLFi, as illustrated below with the semantic label that will tag the central component of the definition of BROUETTE: véhicule (from VÉHICULE sense II.A): engin constitué d'un châssis muni de roues, à traction animale ou autopropulsé, servant au transport routier ou ferroviaire 'machine made up of a chassis equipped with wheels, pulled by animals or selfpropelled, used for road or rail transport'

Acknowledgements

This research is supported by a grant from the Social Sciences and Humanities Research Council (SSHRC) of Canada and by an internal grant from the ATILF CNRS (Nancy) research lab. We wish to thank our invaluable Canadian and French team of annotators: Évelyne Arbour-Barbeau, Jean-Luc Benoit, Claudia Fecteau, Anaïs Ferte, Anne-Laure Jousse, Caroline Poudrier, Mélissa Samson-Fauteux and Olivier Taïs. Finally, we are deeply grateful to Helen Lim for proofreading the final version of this paper.


M Fernandes
email: fernandes@emse.fr

Y Gavet
email: gavet@emse.fr

J C Pinoli

A FEATURE-BASED DENSE LOCAL REGISTRATION OF PAIRS OF RETINAL IMAGES

Keywords: Feature-based registration, retinal images, opthalmology, local transformation, dense transformation

A method for spatial registering pairs of digital images of the retina is presented, using intrinsic feature points (landmarks) and dense local transformation. First, landmarks, i.e. blood vessel bifurcations, are extracted from both retinal images using filtering followed by thinning and branch point analysis. Correspondances are found by topological and structural comparisons between both retinal networks. From this set of matching points, a displacement field is computed and, finally, one of the two images is transformed. Due to complex retinal registration problem, the presented transformation is dense, local and adaptive. Expermimental results established the effectiveness and the interest of the dense registration method.

INTRODUCTION

The problem of image registration is fundamental to many applications of computer vision. Solving this problem requires estimating transformation(s) between images and applying them in order to place the data in a common coordinate system [START_REF] Zitova | Image registration methods: a survey[END_REF]. In retinal imaging (Figure 1), disease diagnosis and treatment planning are facilitated by multiple spatial images acquired from the same patient. Spatial registration techniques allow to integrate informations into a comprehensive single image. They are typically classified as feature-based or area-based.

In area-based techniques, a similarity measure quantifies the matching between images under an assumption of global transformation and is generally optimized with global search algorithms [START_REF] Ritter | Registration of stereo and temporal images of the retina[END_REF]. There are many factors that may degrade the performance of area-based methods: large textureless regions, nonconsistent contrast and nonuniform illumination within images.

Feature-based methods focus on aligning extracted features of the images, i.e bifurcations of the retinal vasculature. From landmarks between both images and with the assumption of correspondances, a global transformation is estimated. These methods are usually more reliable and faster in the case of sufficient and accurate landmarks but a global transformation is always applied. Indeed, image distorsions and aberrations come from different sources:

• changes in head posture and eye movements,

• projection of the retinal surface on the camera plane,

• optical systems (camera, cornea and crystalline),

• eye deformations due to defects and diseases.

In order to obtain an accurate registration, a global transformation must consider all this considerations with a limited overlapping regions between images. This is why this problem requires intuitively an adaptive local dense approach.

First, the developed feature-based technique using a dense local transformation will be described. Next, we will discuss the results obtained. Finally, conclusions and possible extensions will be given.

FEATURE-BASED TECHNIQUE

A feature-based registration method is made of three main steps: landmarks extraction, matching between landmarks and image transformation.

Landmarks extraction

Traditionnally, bifurcations are extracted automatically by a retinal vessels segmentation, followed by thinning and branch point analysis. For example, Zana and Klein [START_REF] Zana | A multimodal registration algorithm of eye fundus images using vessels detection and hough transform[END_REF] enhanced vessels with a sum of top-hats with linear revolving structuring elements and detected bifurcations using a supremum of openings with revolving structuring elements with T shape. In [START_REF] Becker | Image processing algorithms for retinal montage synthesis, mapping, and real-time location determination[END_REF], the boundaries of retinal vessels are detected using standard Sobel filter and the vasculature is thickened using a minimum filter.

Proposed methods. Retinal vessels can be approximated by a succession of linear segments (of length L) at different orientations. Afterwards, all used parameter values are experimetal. First, retinal vessels are emphasized using a minimum of morphological contrasts with linear (L = 9 pixels) and revolving (30˚increments) structuring elements. A median filter smoothed the result image (Figure 2.b). Second, retinal vessels, whose cross section can be approximated by a Gaussian shaped curve (standard deviation σ), are detected by matched filters with 6 orientations, and with L = 9 pixels and σ = 2 [START_REF] Chaudhuri | Detection of blood vessels in retinal images using two-dimensional matched filters[END_REF]. Next, the thresholded image is cleaned: small objects (≤ 200 pixels) and small holes (≤ 15 pixels) are deleted (Figure 2.c). Third, the centreline of the vascular tree is obtained with a thinning operation and is pruned so as to eliminate small branches (≤ 15 pixels) (Figure 2.d). Fourth, bifurcations are extracted as skeleton pixels with at least six binary transitions between adjacent pixels of V8 or V16 neighbourhoods. Finally, adjacent and closer landmarks (≤ 10 pixels) are joined: the new landmark corresponds to the centre of mass of the system with equal weights and may not belong to the skeleton (Figure 2.e).

Landmarks matching

After extraction, pairs of matching landmarks need to be determined between both images. [START_REF] Can | A feature-based, robust, hierarchical algorithm for registering pairs of images of the curved human retina[END_REF] and [START_REF] Zana | A multimodal registration algorithm of eye fundus images using vessels detection and hough transform[END_REF] suggested similarity measures between bifurcations depending on surrounding vessels angles. Due to nonuniform illuminations, a similarity measure may not be robust. [START_REF] Becker | Image processing algorithms for retinal montage synthesis, mapping, and real-time location determination[END_REF] and [START_REF] Ryan | Registration of digital retinal images using landmark correspondence by expectation maximization[END_REF] computed simple transformation parameters from all possible combinations of landmarks. From this data set, matched landmark pairs form a tight cluster which is unfortunately demarcated with difficulty.

Proposed methods. Let I p and I q denote both images called arbitrarily reference and transformed image respectively and with P and Q extracted landmark sets respectively. (u, v) and (u ′ , v ′ ) are the coordinates systems of I p and I q respectively. In this paper, the matching technique proceeds in two steps.

The first step is a similarity measure between landmark signatures of both images and results in an initial couples set S . For a landmark p, the signature is the number of surrounding vessels n p and the angles between them θ p 1 , • • • , θ p np obtained by computing the intersection between the pruned skeleton and a circle of fixed diameter o 24 pixels centred on it.

For each (p, q) belonging to P × Q , (p, q) belongs to S if and only if n = n p = n q ≤ 5 and θ q i -α ≤ θ p i ≤ θ q i + α for i = 1, • • • , n and with α = 10˚. This step restricts landmarks sets P and Q before the second step which is more time consuming.

The second step consists in estimating, for each initial couple, the spatial agencement of landmarks between the two images (Figure 3). For an initial verified in I q with a given tolerance distance δ. Initial couple, pairs 1, 2, 4 and 5 are preserved and landmarks p 3 and q 6 are eliminated. For the displacement vector computation, 2 and 5 are the two nearest preserved pairs of the pixel of I q .

couple (p S j , q S j ) belonging to S , whose locations (u S j , v S j ) and (u ′ S j , v ′ S j ) constitute now images origins, landmarks from the two images that have the same locations with a given tolerance value δ are preserved :

C j =    (p, q) ∈ P × Q | q = (u ′ , v ′ ) ∈ [u ′ S j + ∆u -δ ; u ′ S j + ∆u + δ] ×[v ′ S j + ∆v -δ ; v ′ S j + ∆v + δ]    , (1)
with ∆u = uu S j , ∆v = vv S j , p = (u, v) and δ = 8 pixels. The final matching set C corresponds to the best spatial agencement similarity set:

C = C argmax j=1,••• ,cardS card(C j ) .
(2)

This step increases reliable matching and robustness.

Image transformation

Having established landmark matchings, the next task is to identify a suitable transformation. All cited references employed global linear or not transformations. According to [START_REF] Becker | Image processing algorithms for retinal montage synthesis, mapping, and real-time location determination[END_REF], the affine model is appropriate because the retina is roughly planar over small regions and, consequently, parallel with the camera plane. In [START_REF] Can | A feature-based, robust, hierarchical algorithm for registering pairs of images of the curved human retina[END_REF], the retina is modelled by a quadratic surface, the camera movements by rigid transformations and the projections on the camera plane by weak perspective projections. The resulting quadratic transformation have 12 parameters. However, the difficult problem of spatial retinal registration leads to produce local adaptive deformations and motivates the following transformation.

Proposed methods. To match I p , a dense displacement vector field, function of C , locally deforms I q . For one pixel i of I q , first, the two nearest couples belonging to C are sought:

(p 1 , q 1 ) = argmin (p C ,q C )∈C d(i, q C ) (3) (p 2 , q 2 ) = argmin (p C ,q C )∈C -{(p 1 ,q 1 )} d(i, q C ) , (4)
with d the Euclidean distance defined on the spatial support of I q (Figure 3). Then, the distinct displacement vector of i is:

- → T i =    1/d(i,q 1 ).(u 1 -u ′ 1 )+1/d(i,q 2 ).(u 2 -u ′ 2 ) 1/d(i,q 1 )+1/d(i,q 2 ) 1/d(i,q 1 ).(v 1 -v ′ 1 )+1/d(i,q 2 ).(v 2 -v ′ 2 ) 1/d(i,q 1 )+1/d(i,q 2 )    , (5)
with

p 1 = (u 1 , v 1 ), q 1 = (u ′ 1 , v ′ 1 ), p 2 = (u 2 , v 2 ) and q 2 = (u ′ 2 , v ′ 2 )
. When all pixels of I q are computed, a dense displacement vector field is obtained which is regularized using a Gaussian filter. Finally, I q is transformed: the mapped positions of pixels is calculated as the sum of their original locations and their corresponding displacement vectors.

RESULTS

All visible bifurcations are globally extracted except ones belonging to very narrow vessels due to the unscalability of used matched filters. The matching process is able to match landmarks with large coordinates differences between images (typically the case of a spatial registration) and, intrinsically, to obtain correspondances which are very rarely incorrect.

In this paper, visual assessment on the fused image between reference and transformed images is adopted. With limited overlapping regions (typically the case of spatial registration), i.e with few matching landmarks, the dense local registration outperforms the quadratic model estimated using linear regression [START_REF] Laliberte | Registration and fusion of retinal images -an evaluation study[END_REF] [START_REF] Ryan | Registration of digital retinal images using landmark correspondence by expectation maximization[END_REF] (Figure 4). The quadratic model estimated using linear regression needs larger number of matching landmarks in order to obtain conveniently optimal parameters estimation and to achieve an accurate result. However, both registration methods are affected by the lack of matching informations in some areas. Therefore, in the case of the presented dense local registration, vessel misalignments appear in some peripheral areas.

CONCLUSIONS

In this paper, we have described a feature-based, dense, local registration for eye fundus images. This method is efficient and avoids iterations or heavy calculations. It allows to cope with complex recognitions of global transformation. On the one hand, it avoids high-order global transformation due to complex retinal registration problem. On the other hand, obtaining conveniently optimal parameters with few matching landmarks is a difficult task due to limited overlapping regions of spatial registration.

In order to minimize misalignments due to the lack of intrinsic landmarks uniformity, we are currently investigating the use of additional regional landmarks like the whole vessels. We are also developing a protocol so as to quantitatively compare different registration processes.

Figure 1 :

 1 Figure 1: Pair of retinal images of size 924 × 912 pixels.

Figure 2 :

 2 Figure 2: (a) Part of a retinal image. (b) Result of morphological contrast and median filter. (c) Result of matched filters, thresholding and cleaning. (d) Thinned and pruned binary image. (e) Detected landmarks (white dots).

Figure 3 :

 3 Figure 3: Illustrating the second point matching step and the displacement vector computation. Red dots indicate one initial couple belonging to S . Blue dots indicate the others landmarks belonging to P or to Q . All landmarks locations of I p are

Figure 4 :

 4 Figure 4: Overlapping regions of registration for a pair of images (Figure 1) and with 15 extracted matching couples.

ACKNOWLEDGEMENT

The authors wish to thank Pr. Gain from University Hospital Centre, Saint-Etienne, France for supporting this work and for providing pictures.


Séverine Rivollier
email: rivollier@emse.fr

Johan Debayle
email: debayle@emse.fr

Jean-Charles Pinoli
email: pinoli@emse.fr

Integral Geometry and General Adaptive Neighborhoods for Multiscale Image Analysis

Keywords: GAN-based Minkowski maps and histograms, General Adative Neighborhoods, Integral Geometry, Minkowski functionals, Multiscale image analysis

In quantitative image analysis, Minkowski functionals are becoming standard parameters for topological and geometrical measurements. Nevertheless, they are limited to binary images or to sections of gray-tone images and are achieved in a global and monoscale way. The use of General Adaptive Neighborhoods (GANs) enables to overcome these limitations. The GANs are spatial neighborhoods defined around each point of the spatial support of a gray-tone image, according to three (GAN) axiomatic criteria: a criterion function (luminance, contrast, . . . ), an homogeneity tolerance with respect to this criterion, and an algebraic model for the image space. Thus, the GANs are simultaneously adaptive with the analyzing scales, the spatial structures and the image intensities. This paper aims to introduce the GAN-based Minkowski functionals, which allow a gray-tone image analysis to be realized in a local, adaptive and multiscale way. The Minkowski functionals are computed on the GAN of each point of the spatial support of a gray-tone image, enabling to define the so-called Minkowski maps by assigning the Minkowski functional value to each point. The histograms of these maps provide a statistical distribution of the topology and geometry of the gray-tone image structures, and not only of the image intensities. The impact of the GAN characteristics, as well as the impact of multiscale transformations, are analyzed in a qualitative global and local way through these GAN-based Minkowski maps and histograms. This multiscale image analysis is illustrated on the test image 'Lena' and also applied in both the biomedical and materials areas.

NOMENCLATURE

This section presents a list of all the used symboles and their meaning. This paper aims to introduce a novel approach for analyzing gray-tone images in a local, adaptive and multiscale way. A segmentation process, generally used before quantitative image analysis, is not here required. The quantitative description is directly applied on the raw graytone images. Geometrical and topological measurements, through Minkowski functionals, see [START_REF] Michielsen | Integral-geometry morphological image analysis[END_REF], are performed on spatial neighborhoods associated to each point of the gray-tone image support. These specific neighborhoods, named General Adaptive Neighborhoods (GANs), see Debayle andPinoli (2005, 2006), are simultaneously adaptive with the spatial structures, the image intensities and the analyzing scales. They are based on three axiomatic criteria: an analyzing criterion, an homogeneity tolerance and an algebraic model. These GAN enable to define the so-called GAN-based Minkowski maps which assign a measurement (based on the local Minkowski functionals) to each point of the spatial support of the gray-tone image to be studied.

A
First, the second and third sections recall the notions of Minkowski functionals and of general adaptive neighborhoods, respectively. Then, the fourth and fifth sections introduce the GAN-based Minkowski maps and histograms and illustrate them for various analyzing critera and algebraic models. Thereafter, the impact of the homogeneity tolerance and of a multiscale transformation are analyzed through these GAN-based Minkowski maps. This multiscale image analysis is illustrated on the test image 'Lena' and also applied in both the biomedical and materials areas.

MINKOWSKI FUNCTIONALS AND DENSITIES

In quantitative image analysis, Minkowski functionals, see [START_REF] Minkowski | Volumen und Oberfläche[END_REF], are becoming standard parameters for topological and geometrical measurements, see [START_REF] Coster | Précis d'analyse d'image[END_REF] and [START_REF] Michielsen | Integral-geometry morphological image analysis[END_REF]. A description of the shape of a two-dimensional spatial pattern is provided, using the three 2D Minkowski functionals (up to a constant): area, perimeter, and Euler number, denoted A, P and χ, respectively.

These functionals are defined on the class of nonempty compact convex sets (convex bodies) of R 2 , and satisfy five properties (increasing, invariance under rigid motions, homogeneity, additivity, continuity vs. the Hausdorff metric). They have been extended (excluding the properties of increasing and continuity) to the convex ring, see [START_REF] Mecke | Statistical physics and spatial statistics[END_REF], i.e. the set of all finite unions of convex bodies of R 2 , which may be considered as a realistic Euclidean model for digital planar images.

In this paper, the densities instead of the intensities of these functionals, i.e. the ratio of the Minkowski intensity functionals by the area of the spatial support, will be used. These Minkowski density functionals are called the specific area, the specific perimeter, and the specific Euler number, and are denoted, A A , P A and χ A , respectively.

GENERAL ADAPTIVE NEIGHBORHOODS

The GANIP (General Adaptive Neighborhood Image Processing) approach, see [START_REF] Debayle | Spatially adaptive morphological image filtering using intrinsic structuring elements[END_REF][START_REF] Rivollier | Analyse spectrale morphologique adaptative d'image[END_REF], 2009), provides a general framework for multiscale, local and adaptive processing and analysis of gray-tone images.

It is based on an image representation by means of specific spatial neighborhoods, named General Adaptive Neighborhoods (GANs). Indeed, GANs are simultaneously adaptive with:

• the spatial structures: the size and the shape of the neighborhoods are adapted to the local context of the image, • the analyzing scales: the scales are given by the image itself, and not a priori fixed, • the intensity values: the neighborhoods are defined according to the GLIP (General Linear Image Processing) algebraic mathematical framework, see [START_REF] Oppenheim | Generalized Superposition[END_REF] and Pinoli (1997), enabling to involve the physical and/or psychophysical image formation laws.

For a gray-tone image f , a GAN of a point x belonging to the spatial support D ⊆ R 2 , denoted V h m (x), is a connected subset of D. It is homogeneous with respect to an analyzing criterion h (such as luminance, contrast, thickness, . . . ) using an homogeneity tolerance, denoted m . The symbol stands for the algebraic model for the space of gray-tone images. The space of gray-tone images (resp. analyzing criteria), defined on the spatial support and valued in a real number interval Ẽ (resp. E ) is denoted I (resp. C). The homogeneity tolerance m belongs to the positive intensity value range E + := {t ∈ E |t ≥ 0 }, where 0 denotes the neutral element for vector addition, see Debayle and Pinoli (2006). The GAN of a point x is mathematically defined as follows:

V h m (x) := C h -1 ([h(x) -m ;h(x) + m ]) (x) (1)
where C X (x) denotes the path-connected component (with the usual Euclidean topology on D) of X ⊆ D holding x. The criterion function h is represented in a GLIP (General Linear Image Processing) model, see [START_REF] Oppenheim | Generalized Superposition[END_REF] and Pinoli (1997), i.e. a vector space with its vector addition + and its scalar multiplication × .

For instance, the operations + and × of the usual CLIP (Classical Linear Image Processing) framework correspond to the usual operations between images, + and × respectively. The vector addition and subtraction and the scalar multiplication of the LIP (Logarithmic Image Processing) framework, see Jourlin andPinoli (1988, 2001), denoted + △ , -△ and × △ respectively, are defined as follows:

f + △ g = f + g - f g M (2) f - △ g = M f -g M -g (3) α × △ f = M -M M -f M α (4)
where f and g are gray-tone images, α ∈ R is a scalar, and M ∈ R + denotes the upper bound of the range where intensity images are valued. The LIP framework has been proved to be consistent with the transmittance and the multiplicative reflectance/transmittance image formation model, and with several laws and characteristics of human brightness perception, see Pinoli (1997). The GANIP approach in the special case where the LIP framework is used has been studied and applied in [START_REF] Pinoli | Logarithmic Adaptive Neighborhood Image Processing (LANIP): introduction, connections to human brightness perception and application issues[END_REF].

The GAN satisfy several properties as stated in the following (proved in Debayle and Pinoli (2006)). For all

(h, m , m 1 , m 2 , x, x 1 , x 2 ) ∈ C × (E + ) 3 × D 3 ,
• reflexivity:

x ∈ V h m (x) (5)
• increasing with respect to m :

m 1 ≤ m 2 ⇒ V h m 1 (x) ⊆ V h m 2 (x) (6)
• equality between iso-valued points:

x 1 ∈ V h m (x 2 ) h(x 1 ) = h(x 2 ) ⇒ V h m (x 1 ) = V h m (x 2 ) (7)
• + -translation invariance:

∀c ∈ E : V h + c m (x) = V h m (x) (8)
• × -multiplication compatibility:

∀α ∈ R * + : V α × h m (x) = V h 1 α × m (x) (9)
Figure 1 gives a visual impression, on a 1D example, of the computation of a GAN in the LIP framework (i.e. with the + △ vector addition (Eq. 2) and the -△ vector subtraction (Eq. 3)).

Figure 2 illustrates the GANs of two points computed with the luminance criterion in the CLIP (Classical Linear Image Processing) framework (the operations + and × correspond to the usual operations between images, + and × respectively) on a human retina image.

The General Adaptive Neighborhoods are intrinsically defined by the local image structures. Thus, the GANs {V h m (.)} m allow a new multiscale representation of gray-tone images to be defined. Defined on the spatial support D of an image (Fig. 3), the GANs are simultaneously adaptive with the spatial structures, the analyzing scales and the intensity values. On the contrary, the shape and size of the classical neighborhoods {B r (.)} r (centered homothetic isotropic discs of radius r), generally used as analyzing windows for image transforms, are a priori fixed.

GAN-BASED MINKOWSKI MAPS

This novel multiscale representation of gray-tone images enables to define GANs measurements, allowing a graytone image quantitative analysis, in a local, adaptive and

point line measurement value h(x) x [h(x) - △ m △ , h(x) + △ m △ ] V h m △ (x) Fig. 1. One-dimensional computation of a GAN set V h m △ (x)
in the LIP framework. For a point x, a range of tolerance m △ is first computed around h(x). Secondly, the inverse mapping of this interval gives a subset of the 1D spatial support. Finally, the pathconnected component holding x provides the GAN V h m △ (x).

D

x 1 B r (x 1 )

x 2 B r (x 2 ) x 3 V h m (x 3 ) x 4 V h m (x 4 ) r, m Fig. 3. Example of adaptive V h m
and non-adaptive B r analyzing windows with three values both for the homogeneity tolerance parameter m , and for the disks radius r. The shape of B r (x 1 ) and B r (x 2 ) are identical and {B r (x)} r is a family of homothetic sets for each point x ∈ D. On the contrary, the shapes of V h m (x 3 ) and V h m (x 4 ) are dissimilar and {V h m (x)} m is not a family of homothetic sets. multiscale way. Indeed, for each point x of the gray-tone image f , various measurements, such as area, orientation, concavities number, . . . see [START_REF] Coster | Précis d'analyse d'image[END_REF] and [START_REF] Rivollier | Analyse spectrale morphologique adaptative d'image[END_REF], of the GAN V h m (x), can be computed.

In this paper, the considered measurements are the Minkowski density functionals. These functionals can be estimated in an efficient way, see [START_REF] Nagel | An integralgeometric approach for the Euler-Poincaré characteristic of spatial images[END_REF] and [START_REF] Osher | Statistical Analysis of Microstructures in Materials Science[END_REF].

Definition

A GAN-based Minkowski map is defined by assigning a value for each point x which represents a GAN Minkowski density functional of V h m (x). In a more explicit way, the GAN-based Minkowski map µ h m of a gray-tone image f with respect to the Minkowski density functional µ (area: µ ≡ A A , perimeter: µ ≡ P A , Euler number: µ ≡ χ A ) is defined by:

µ h m : D → R x → µ V h m (x) ( 10
)
where V h m (x) is the GAN of the point x with respect to the analyzing criterion h using the homogeneity tolerance m in a GLIP framework. The GANs depend on three axiomatic criteria (Eq. 1): an analyzing criterion, an algebraic GLIP model for the criterion mapping space, and an homogeneity tolerance with respect to this criterion. The two following subsections show, from a visual point of view, the impact of the two firsts characteristics on the Minkowski maps (Eq. 10). The impact of the homogeneity tolerance will be studied in the next section.

Impact of the analyzing criterion

The GANs are homogeneous with respect to a criterion function h (such as luminance, contrast, thickness, . . . ). For instance, the luminance criterion is defined by h ≡ f where f is the original gray-tone image. A contrast criterion can also be used. For instance, the contrast image, denoted c, associated to the gray-tone image, can be defined in a discrete way in the context of the CLIP framework as:

c :    D → R + x → 1 #N (x) y∈N (x) |f (x) -f (y)| ( 11
)
where N (x) is a neighborhood of the point x (for instance, points in the window 3 × 3 centered on x). The results show that the extracted zones are not the same for the two criteria.

Impact of the mathematical framework

H µ h m (t) = dL 2 x ∈ D : µ h m (x) < t L 2 (D)dt (12)
where dL 2 (X)/dt denotes the derivative according to t of the Lebesgue measure L 2 of a subset X ⊆ D. These GAN-based Minkowski histograms define a topological and geometrical signature of a gray-tone image. They provide a statistical distribution of the topology and geometry of the gray-tone image structures, and not only of the image intensities, like the classical gray-tone histograms. Thanks to this signature, similarity measures between gray-tone images could be determined (distances between histograms). When comparing gray-tone images, the GAN-based Minkowski maps can supply major additional informations: images that cannot be differentiated by their gray-tone histograms (two images visually different can have similar gray-tone histograms) can be differentiated by the histograms of their GAN-based Minkowski histograms. Consequently, application issues to image classification could be adressed, based on this geometrical and topological image characterization.

The next section focuses on the definition of GAN-based Minkowski functions, providing a multiscale analysis of a gray-tone image.

MULTISCALE ANALYSIS

When studying the variation of the homogeneity tolerance on the GAN-based Minkowski maps, as it has been done previously for the two others GAN axiomatic criteria, a multiscale analysis of gray-tone images can be performed. This leads to study the impact of a multiscale transformation on these maps. GAN-based Minkowski functions are thus defined.

Impact of the homogeneity tolerance

The GANs are homogeneous regions with respect to an analyzing criterion h using a tolerance m within a GLIP framework. For one point of the spatial support of a graytone image, an increasing family of GANs is obtained with the variation of the homogeneity tolerance m (Fig. 3). The variation of this parameter allows to built GAN-based Minkowski multiscale maps. These multiscale maps define a GAN-based Minkowski function according to the homogeneity tolerance m . Mathematically, it is thus defined by:

µ f (.) : m → µ f m (.) (13)
Figure 9 shows the cumulative histograms of the GANbased Minkowski maps according to the homogeneity tolerance m . A 2D representation is obtained where each row is a cumulative histogram. The statistical distribution is given with the use of a color scale. (a) x increases, contrary to (P A ) f (.) and (χ A ) f (.). This property can be generalized to the others GLIP frameworks and criterions: the function (A A ) f (.) increases, however (P A ) f (.) and (χ A ) f (.) are not monotonous. This is due to the boundary tortuosity and the number of holes that can occur in the GANs. The increasing of (A A ) f (.) is explained by the increasing of the GANs with respect to m (eq. 6).

(b) V f 10 (x) (c) V f 20 (x) (d) V f 40 (x) (e) (χ A ) f (x), (P A ) f (x) et (A A ) f (x)
Thanks to the variation of the homogeneity tolerance m , a GAN-based Minkowski function is defined, providing a multiscale analysis of gray-tone images. The next section is focused on the analysis of the impact of a multiscale transformation on the GAN-based Minkowski maps.

Impact of a morphological transformation

Mathematical morphology, see Soille (2003), is an important and nowadays a traditional theory in image processing, particularly used for geometrical image analysis

µ T (f ) m (.) : r → µ Tr(f ) m (.) ( 14
)
where r is the radius of the structuring element used for the morphological operator T .

Figure 12 shows the cumulative histograms of the GANbased Minkowski maps according to the radius r of the structuring element. A 2D representation is obtained where each line is a cumulative histogram. The statistical distribution is given with the use of a color scale. Currently, the authors are working on the analysis of the GAN-based Minkowski functions (particularly the discontinuities of the curves) allowing to obtain shape and/or size informations of the structures in a gray-tone image.

  (a) original retina image with two (white) points x and y (b) GANs V h 20 (x) and V h 20 (y) respectively

Fig. 2 .

 2 Fig. 2. The GANs of the two points x and y of the original image (a) are connected and homogeneous (b) with respect to the luminance criterion using the tolerance value m = 20 within the CLIP framework. The GANs are adapted to the image structures.

Figure 4

 4 Figure 4 illustrates some GAN-based Minkowski maps of the image 'Lena' f . The GANs are homogeneous with respect to the luminance criterion (h ≡ f ) using the tolerance value m = 40 in the context of the CLIP framework. Therefore, the value µ h m (x) of each point x of the Minkowski map corresponds to the density µ of the GAN V f 40 (x).The GANs depend on three axiomatic criteria (Eq. 1): an analyzing criterion, an algebraic GLIP model for the criterion mapping space, and an homogeneity tolerance with respect to this criterion. The two following subsections show, from a visual point of view, the impact of the two firsts characteristics on the Minkowski maps (Eq. 10). The impact of the homogeneity tolerance will be studied in the next section.

Figure 5

 5 Figure 5 illustrates the GAN-based Minkowski maps of a fibronectin image f with respect to the luminance criterion

Figure 6

 6 Figure 6 illustrates the GAN-based Minkowski maps of a zinc sulfide image f acquired by scanning electron microscopy imaging with respect to the luminance criterion f using the homogeneity tolerance value 50 in the context of the CLIP and LIP mathematical frameworks, respectively. These GAN-based Minkowski maps can be used for image segmentation. A usual intensity thresholding does not always extract the desired structures of a gray-tone image. Using appropriate GAN axiomatic criteria (analyzing criterion, mathematical framework, homogeneity tolerance), the GAN-based Minkowski maps highlight these structures and the desired segmentation is obtained by thresholding these GAN-based Minkowski maps. The next section focuses on the histograms of these GANbased Minkowski maps.

  lower and upper bound values

Fig. 4 .

 4 Fig. 4. GAN-based Minkowski maps (b-d) of the image 'Lena' (a) with respect to the luminance criterion h ≡ f , using the homogeneity tolerance m = 40 in the context of the CLIP mathematical framework. Grayscale bound values gives the extrema of the Minkowski density functionals values.

  (a) original image f (b) criterion map c (c) (χ A ) f 20 (d) (χ A ) c 20 (e) (P A ) f 20 (f) (P A ) c 20 (g) (A A ) f 20 (h) (A A ) c 20 Gray-scale lower and upper bound values

Fig. 5 .

 5 Fig. 5. GAN-based Minkowski maps (c-h) of a fibronectin image (a) with respect to the luminance criterion f (c,e,g) and the contrast criterion c (d,f,h), respectively, using the homogeneity tolerance m = 20 in the context of the CLIP framework.

  lower and upper bound values

Fig. 6 .

 6 Fig. 6. GAN-based Minkowski maps (b-g) of a zinc sulfide image (a) with respect to the luminance criterion f , using the homogeneity tolerance m = 50 in the context of the CLIP (b,d,f) and LIP (c,e,g) framework, respectively.

Figure 7

 7 Figure 7 illustrates some GAN-based Minkowski histograms of the image 'Lena' f . The GANs are homogeneous with respect to the luminance criterion (h ≡ f ) using the tolerance value m = 40 in the context of the CLIP framework.

Fig. 7 .

 7 Fig. 7. GAN-based Minkowski maps (b-d) and histograms (e-g) of the image 'Lena' (a) with respect to the luminance criterion h ≡ f , using the homogeneity tolerance m = 40 in the context of the CLIP framework.

Figure 8

 8 Figure 8 illustrates the GAN-based Minkowski maps and histograms of a brain MR image f with respect to the luminance criterion f using various homogeneity tolerance values: m = 20, m = 30 and m = 40 in the context of the CLIP framework.

Figure 10

 10 Figure 10 illustrates these GAN-based Minkowski functions with respect to the luminance criterion, for one point in the brain MR image. This figure shows that in the context of the CLIP framework the function (A A ) f (.)

  Fig. 9. GAN-based Minkowski cumulative histograms, with respect to the luminance criterion f in the context of the CLIP framework, according to the homogeneity tolerance m (a-c). A 2D representation is obtained where each row is a cumulative histogram. Abscissa and ordinate coordinates indicate respectively the Minkowski density functionals values and the homogeneity tolerance. The distribution is given with the use of a color scale (d).

Fig. 10 .

 10 Fig. 10. Original image of brain marked with a point x (a), and marked with the GAN of x for the tolerance values m = 10, 20, 40 (b-d). Representation of the GAN-based Minkowski functions according to m (e) for this point x. The dot lines represent the results for the Euler number, perimeter, and area of the GAN (b-d) of the point x with m = 10, 20, 40.

  . The elementary morphological operators of dilation and erosion (and thus the combined operators of closing and opening) act on image intensities with the use of an operational window named structuring element. The GAN-based Minkowski maps can be computed on gray-tone images transformed by morphological operators. For instance, figure 11 first illustrates the morphological transforms T r (f ) (T r denotes the opening (resp. closing) if r < 0 (resp. r > 0)) using a disk of radius |r| ∈ R + as structuring element of a gray-tone image f of corneal endothelium cells. The GANs V Tr(f ) m are determinated with respect to this transformation T r (f ) using the homogeneity tolerance m = 20 in the context of the CLIP framework. Figure 11 shows the associated GAN-based Minkowski maps and histograms. The variation of the radius r allows to built GANbased Minkowski multiscale maps. It defines a GAN-based Minkowski function according to the radius r of the structuring element used for the morphological transformation. Mathematically, it is thus defined by:

Figure 13

 13 Figure 13 illustrates these GAN-based Minkowski functions, for one point in the image f of corneal endothelium cells. This figure shows that in the context of the CLIP framework the function (A A ) T (f ) m (.) is not monotonous, contrary to T (f )(.). Thus, whatever the a priori fixed algebraic model and the homogeneity tolerance, the function µ T (f ) m (.) is not monotonous. A more detailed analysis of the GAN-based Minkowski functions (particularly on the discontinuities of the curves) should allow to obtain shape and/or size informations of

  Fig. 11. GAN-based Minkowski maps and histograms (dl) of an image of corneal endothelium cells (b) with respect to the morphological transformation T r (f ), using the homogeneity tolerance values m = 20, in the context of the CLIP framework. T r denotes the morphological opening (r < 0) and closing (r > 0) with a disk of radius r.

  Fig. 12. GAN-based Minkowski cumulative histograms, with respect to the morphological transformation T r in the context of the CLIP framework, according to the radius r (a-c). T r denotes the morphological opening (r < 0) and closing (r > 0) with a disk of radius r. A 2D representation is obtained where each row is a cumulative histogram. Abscissa and ordinate coordinates indicate respectively the Minkowski density functionals values and the radius. The distribution is given with the use of a color scale (d).

Fig. 13 .

 13 Fig. 13. Original image of cornea cells marked with a point x (a), and marked with the GAN of x for the tolerance value m = 20 on transformed images T r (f ) by opening (r < 0) and closing (r > 0) with disks of radii r = -4, 0, 4 (b-d). Representation of the GANbased Minkowski functions according to r (e) for this point x. The dot lines represent the results for the Minkowski density functionals of the GAN (b-d) of the point x with r = -4, 0, 4.

ACKNOWLEDGEMENTS

The authors wish to thank the University Hospital Center (ophtalmology and radiology) of Saint-Etienne in France, the UMR CNRS 5148, and the INSERM U890, who have kindly provided the original images.

 Mathematics in 1983, 1987and 1992, respectively. From 1985to 1989, he was member of the optoelectronics department of the Angenieux (Thales) company, Saint-Héand, France, where he pioneered researches in the field of digital imaging and artificial vision. In 1990, he joined the Corporate Research Center of Pechiney Company, Voreppe, France, and was member of the computational technologies department in charge of the imaging activities. Since 2001, he is full professor at the 'Ecole Nationale Supérieure des Mines de Saint-Etienne'. He leads the Image Processing and Pattern Analysis Laboratory within the CIS Center and the LPMG, UMR CNRS 5148. His research interests and teaching include Image Processing, Image analysis, Mathematical Morphology and Computer Vision.


Hervé Mohrbach

Albert Johner

Igor M Kulić
email: kulic@unistra.fr

Polymorphic Dynamics of Microtubules

Keywords: numbers: 87.16.Ka, 82.35.Pq, 87.15.-v

Starting from the hypothesis that the tubulin dimer is a conformationally bistable moleculefluctuating between a curved and a straight configuration at room temperature -we develop a model for polymorphic dynamics of the microtubule lattice. We show that tubulin bistability consistently explains unusual dynamic fluctuations, the apparent length-stiffness relation of grafted microtubules and the curved-helical appearance of microtubules in general. Analyzing experimental data we conclude that taxol stabilized microtubules exist in highly cooperative yet strongly fluctuating helical states. When clamped by the end the microtubule undergoes an unusual zero energy motion -in its effect reminiscent of a limited rotational hinge.

Microtubules are the stiffest cytoskeletal component and play versatile and indispensable roles in living cells. They act as cellular bones, transport roads [START_REF] Howard | Mechanics of Motor Proteins and the Cytoskeleton[END_REF] and cytoplasmic stirring rods [START_REF] Kulic | [END_REF]. Microtubules consist of elementary building blocks -the tubulin dimers -that polymerize head to tail into linear protofilaments (PFs). PFs themselves associate side by side to form the hollow tube structure known as the microtubule (MT). Despite a long history of their biophysical study a deeper understanding of MT's elastic and dynamic properties remains elusive to this date. Besides the unusual polymerization related non-equilibrium features like "treadmilling" and the dynamic instability there are a number of other experimental mysteries -in thermal equilibriumthat presently defy coherent explanations, most notably: (i) The presence of high "intrinsic curvature" [3]- [6] of unclear origin, also identified as a long wave-length helicity [3]. (ii) In various active bending [7] or thermal fluctuation experiments [4][5] [8] MTs display length dependent, even non-monotonic apparent stiffness [5]. (iii) They exhibit unusually slow thermal dynamics in comparison with standard semiflexible filaments [5] [6].

The most bizarre and controversial feature (ii) has been the subject of much debate and some theoretical explanation attempts based on low shear stiffness modulus have been put forward [9]. However a careful reanalysis of clamped MT experiments, Figs 2, 3 reveals two features not captured by these initial models: the lateral end-fluctuations scale as ∼ L 2 while the relaxation times scale as ∼ L 3 . This exotic behavior naively suggests the presence of a limited angular hinge at the MT clamping point. On the other hand artifacts that could trivially lead to a "hinged behavior" (like loose MT attachment and punctual MT damage) were specifically excluded in experiments [4] dimer is a conformationally multistable entity and fluctuates between at least 2 states on experimental time scales. (II) There is a nearest-neighbor cooperative interaction of tubulin states along the PF axis. We are lead to assumptions I-II from several independent directions: First, the experimentally observed MT helicity [3] implies that there is a symmetry breaking mechanism of individual PF's conformational properties. In analogy to the classic case of bacterial flagellum the existence of helices in azimuthaly symmetric bundles also necessitates a cooperative longitudinal interaction along protofilaments [11] [12]. Second, investigations of single protofilament conformations by Elie-Caille et al [13] reveal that a single taxol-PF can coexist in at least 2 states with comparable free energy: a straight state κ P F ≈ 0 and a weakly curved state with intrinsic curvature κ P F ≈ 1/250nm. These authors also point out the apparent cooperative nature of straight to curved transition within single PFs. Third, when mechanically buckled by AFM tips tubulin dimers occasionally switch back to the initial straight conformation [14]. Fourth, tubulin multistability was inferred from the formation of stable circular MT arcs in kinesin driven gliding assays by Amos & Amos [15]. Unfortunately their clear, seminal observations were subsequently forgotten for decades leading to much of the confusion about MTs we are witnessing today.

Polymorphic MT Model. Starting from assumptions I -II we model the tubulin dimer state by a two state variable σ n (s) = 0, 1 (the tubulin dimer in the "straight"/"curved" state, cf.

e el = Y 2 (ε -ε pol ) 2 rdrdα ( 1
)
e trans = - ∆G b N n=1 σ n (s), (2)
e inter = - J b N n=1 (2σ n (s) -1) (2σ n (s + b) -1) (3)
where the integration in e el goes over the annular MT cross-section with r 1 ≈ 7.5nm, r 2 ≈ 11.5nm the inner and outer MT radii, with ∆G > 0 the energy difference between the 0 and 1 state and b ≈ 8nm the monomer length, J the "Ising" cooperative coupling term along the PF contour and with the polymorphism induced prestrain ε pol ∝ ε P F σ n (s) [START_REF]r) •I [ 2π N n+q 0 s, 2π N (n+1)+q 0 s] (α) where I[END_REF] where ε P F is the strain generated in the curved state. The latter can be estimated from the switched PF curvature κ P F ≈ (250nm) -1 [13] to be ε P F = d P F κ P F /2 ≈ 10 -2 . For an isotropic Euler-Kirchhoff beam, the actual material deformations are related to the centerline curvature via ε = -κ • r with r the radial vector in the cross-section.

Upon inspection it becomes clear that the phase behavior (straight or curved state stability) is contained in the interplay of the first two terms e el and e trans while the thermal dynamics is governed by the 3rd e inter which rules over defect behavior (cf. Fig. 1d). To understand the basic behavior we first consider a short MT section along which the PFs are in a uniform state σ n (s) = σ n (s + b) (e inter =const. can be dropped). Furthermore we resort to the single block ansatz, i.e. at each cross-section there is only one continuous block of switched PFs of length p. This ansatz was successfully used by Calladine in modelling bacterial flagellin polymorphic states [11]. In this approximation the energy density becomes

e = B 2 (κ -κ pol (p)) 2 + κ 2 1 γ π N p -sin 2 π N p (4)
with the bending modulus

B = Y π 4 r 4 2 -r 4 1 and the polymorphic curvature κ pol (p) = κ 1 sin π N p with κ 1 = κ F P (r2-r1) 2 π(r 2 1 +r 2 2 )
. The MT phase behavior depends on the polymorphic-elastic competition parameter γ

= κ P F κ 1 - 2N ∆G bBκ 2 1
. Physically, γ measures the ratio between polymorphic energy of tubulin switching and the elastic cost of the transition. For γ < -1 all the PFs are in the (highly prestrained) state σ = 1 while for γ > 1 all them are in the state σ = 0 -both corresponding to a straight MT. For -1 < γ < 1 we have coexistence of 2 locally (meta) stable states : straight (p = 0 or p = N ) and curved state with p > 0. For -γ < γ < γ with γ ≈ 0.72 the curved state is the absolute energy minimum and the straight state is only metastable. Therefore in this regime, the ground state of a microtubule bearing natural lattice twist will be helical (cf. Fig. 1b). Assuming a stable helical state as observed in [3] we have p/N ∈ [1/4, 1/2] giving us an estimate for the radius of curvature κ -1 pol ≈ 9 -14µm. This compares favorably with an estimate of observed helices κ -1 ≈ 11µm from [3]. The helical stability and the magnitude of the protofilament curvature κ P F ≈ 1/250nm [13] with a typical protein Young modulus Y ≈ 1-10GP a, allows us also a simple estimate of the transition energy per monomer ∆G ≈ +1.1 to +11kT. In general, the energy in Eqs.1-3 gives rise to a complex behavior and we focus on basic phenomena. It turns out that a most remarkable deviation from standard wormlike chain behavior comes from the change of polymorphic phase that we consider in the following.

Polymorphic Phase Dynamics.

To better understand the central phenomenon, we define at each MT cross-section the complex polymorphic order parameter P (s) = N n=1 e 2πin/N σ n (s) = |P (s)| e iφ(s) where |P (s)| denotes the "polymorphic modulus" and φ the "polymorphic phase" (cf. Fig. 1b). The polymorphic state can then be described by the local (complex) centerline curvature κpol (s) = κ 0 e iq 0 s P (s) with κ 0 = κ 1 sin π/N and q 0 the natural lattice twist that varies with PF number [START_REF] Chrétien | [END_REF]. This gives rise to a helical MT shape described by the curvature |κ pol | = κ pol and torsion τ ≈ φ +q 0 . For large acting forces both the polymorphic phase φ and amplitude |P | will vary along the contour, however for small (thermal) perturbations the phase fluctuations will be dominant [START_REF]induced by double defects which carry a limited local energy cost ∆E = 2Jn (n being the number of double defects). |P | variations are induced by single defects having in general larger energy cost that grows with their end distance[END_REF]. Based on this and on the observation of stable helical states [3] we will now assume |P | = const, and write the total energy of the MT whose centerline deflection is described by a complex angle θ (s) = θ x (s)+iθ y (s) (deflection angles in x/y direction) as follows:

E tot = E el (θ, φ) + E pol (φ) (5)
The first energy term is the "wormlike-chain" bending contribution E el (θ, φ) = B

= 25mm, λ = 7.5µm, κ -1 0 = 18µm, q 0 l φ >> 1. C φ 2 L 0 φ 2 ds with the polymorphic phase stiffness C φ = k B T N 2 b
8π 2 2 + e 2J/k B T which can be related to the density of double defects with energy 2J (cf. Fig. 1d), giving rise to a new length scale -the polymorphic phase coherence length

l φ = C φ /k B T.
The most unusual property of a polymorphic chain is reflected in the rotational invariance of E pol (φ). The broken cylindrical to helical symmetry of the straight state is restored by the presence of a "Goldstone mode" φ → φ + φ 0 [19] consisting of a rotation of P by an arbitrary angle φ 0 in the material frame (cf. Fig. 1c). This mode that we will call the "wobbling mode" is a fundamental property of a helically polymorphic filament. The wobbling mode leads to dramatic effects on chain's fluctuations and is the clue to the resolution of mysteries (i)-(iii). To see this we will first investigate the static properties resulting in length dependent variations of the persistence length.

Persistence Length Anomalies. Among several definitions of the persistence length [START_REF]) -θ (s )) exhibiting a similarly rich behavior as l * p (yet a distinct functional form) will be discussed elsewhere[END_REF] we consider for direct comparison with experiments [4][5], the lateral fluctuation persistence length l * p (s) = (2/3) s 3 / |ρ (s)| 2 with ρ (s) = x(s) + iy(s) the transverse displacement at position s of a MT clamped at s = 0 and .. the statistical average. It is easy to see from Eq. 5 that for small deflections, ρ decouples into independent elastic and polymorphic displacements ρ (s

) = ρ el + ρ pol , such that l * p = l * -1 pol + l -1 B -1 with l * pol = (2/3) s 3 / |ρ pol | 2 where ρ pol = κ 0 s 0 s 1 0 e iq0 s+iφ(s) d sds 1 .
The coherent helix nature of the MT observed in [3] and the absence of a plateau in l * p imply [21] that l φ >> λ = 2πq -1 0 (the helix wave length). In that limit we ob-

tain |ρ pol | 2 ≈ κ 2 0 q -2 0 [ 2 q 2 0 + s 2 + s 3 3l φ -2 q 2 0 e -s 2l φ ((1 + s 2l φ ) cos q 0 s + sq 0 sin q 0 s)].
Whereas for an ideal WLC l * p = l B ≡ B/kT is position and definition independent, the polymorphic fluctuations induce a strong position / distance dependence -a behavior that could be interpreted as "length dependent persistence length". Indeed for l φ >> s >> q -1 0 the persistence length displays a non-monotonic oscillatory behavior around a nearly linearly growing average value l * p (s) ≈ 2 3

q 2 0 κ 2 0 s + 4 3 q0 κ 2 0
sin (q 0 s). This oscillation is related to the helical ground state while the linear growth l * p (s) ∝ α 2 s is associated to the conical rotation of the clamped chain (wobbling mode), cf. Fig. 1c, with an angle α = 2κ 0 q -1 0 . For s >> l φ the saturation regime with a renormalized

l * p (∞) = 1/ l -1 pol + l -1 B with l pol = 2l φ q 2 0 κ -2 0
is reached. The theory can now be compared with the experimental data [4][5] (cf. Fig. 2) that reveal several interesting characteristics in agreement with predictions. In particular the mean linear growth of l * p (L) (single parameter fit l * p ∼ L δ gives δ = 1.05) and the non-monotonic l * p (L) dependence [5] are well captured by the theory. The linearly growing experimental spread of l * p with L is likely linked to the spread of q 0 in the MT lattice populations [START_REF] Chrétien | [END_REF]. The large length plateau s >> l φ is not reached even for longest MTs (∼ 50µm) in agreement with coherent helices [3]. Our best comparison between theory and experiments (cf. Fig. 2) gives l B = 25mm corresponding to Y ≈ 9GP a (proteins with Y up to 19GP a exist [START_REF] Kol | [END_REF]) and a helix wave length λ ≈ 7.5µm. This is close to the expected 6µm corresponding to the twist [START_REF] Chrétien | [END_REF] of the predominant 14 PF MTs fraction in the in-vitro MTs preparation of [4] [5]. It turns out that l B is larger than in previous studies l B ∼ 1 -6mm where however polymorphic fluctuations were neglected. The absence of the plateau also allows a lower estimate of the coherence length l φ > 55µm and the coupling constant J > 4k B T .

Polymorphic Phase Dynamics. To describe the MT fluctuation dynamics we consider the total dissipa-tion functional P diss = P ext + P int which is composed of an internal dissipation contribution P int = 1 2 ξ int φ2 ds and an external hydrodynamic dissipation

P ext = 1 2 ξ ⊥ | ρ|
2 ds with ξ ⊥ = 4πη/ (ln (2L/r) -1/2) the lateral friction constant, η the solvent viscosity, r and L the MT radius and length. The time evolution equation of the phase variable φ (s, t) and elastic displacement ρ el (s, t) is given by the coupled Langevin equations δE δφ = -δP diss δ φ + Γ φ and δE δρ el = -δP diss δ ρel + Γ ρ with Γ φ/ρ the thermal noise term. In general this dynamics is highly non-linear however in the experimentally relevant regime where the behavior is dominated by the wobbling mode the equations simplify greatly and we end up with a simple diffusive behavior of the wobbling mode d dt φ 0 (t) = 1 ξ tot L -1 L 0 Γ φ (s, t) ds with a friction constant given by ξ tot = ξ int + ξ ext where ξ ext = 2ξ ⊥ κ 2 0 q -4 0 ((1 + cos Lq 0 ) -4 sin Lq 0 + q 3 0 L 3 /3). For comparison with the experiment we compute the time correlation of the y deflection. A short calculation gives y pol (L, t)y pol (L, t ) ∝ e -|t-t |/τ(L) with the relaxation time τ (L) ≈ Lξ tot /k B T . For small lengths, τ (L) ≈ Lξ int /k B T is dominated by internal dissipation while for large lengths τ (L) ≈ ξ ⊥ 3k B T (κ 0 /q 0 ) 2 L 3 . A careful analysis of the experimental data [5] reveals in fact the latter scaling. An independent single exponent fit gives τ ∝ L α with α = 2.9. Using the value (κ 0 /q 0 ) 2 ≈ 4. 8 × 10 -3 from Fig. 2 and ξ ⊥ ≈ 2η with η = 10 -3 P a • s [23] we find the theoretical value τ th /L 3 = 7.9 × 10 14 s/m 3 that can be compared with the fit of experimental data (Fig. 3) τ f it /L 3 = 6. 25 × 10 14 s/m 3 . The excellent agreement of both the exponent and the prefactor leads us again to the strong conclusion that in these experiments the clamped MT is an almost rigid helical polymorphic rotor whose behavior is dominated by the zero energy ("wobbling") mode and hydrodynamic dissipation. For very short MTs the linearly scaling internal dissipation dominates and we could measure ξ int from the limit value of τ th /L, for L → 0. For the available data L > 2 µm [5] this plateauregime is not yet fully developed and we can only provide an upper estimate from the data ξ int 4 × 10 -17 N s.

Conclusion. The MT fluctuations are well describedboth dynamically and statically -by the bistable tubulin model and the reason for appearance of MT helices becomes obvious. The otherwise mysterious lateral fluctuations reflected in l * p (L) ∼ L 1 and τ p (L) ∼ L 3 scaling are mere consequences of the "wobbling motion" of a polymorphic cooperatively switching helical lattice. We speculate that the implied conformational multistablity of tubulin and the allosteric interaction are not just nature's way to modulate the elastic properties of its most important cytoskeletal mechano-element. It could also be a missing piece in the puzzle of dynamic instability. Another intriguing possibility of using this switch for long range conformational signalling in vivo, could hardly have been overlooked by evolution. I.M.K thanks Francesco Pampaloni for stimulating discussions.

  FIG. 1: Polymorphic Tube Model: (a) The tubulin dimer fluctuates between two states σ = 0, 1 (straight / curved), b) Tubulin switching on one MT side leads to spontaneous breaking of symmetry. Combined with the built-in lattice twist MT forms a polymorphic helix. The polymorphic order parameter P with phase angle φ describes the distribution of tubulin states in the cross section. c) Polymorphic wobbling -the zero-energy motion of the phase angle at each cross-section. It is responsible for the rotation on a cone with opening angle α when clamped by the end. d) Defects in polymorphic order: Single defects (SD) have a cost proportional to their length. Double defects (DD) give only local energy contribution.

  Fig. 1a) at each lattice site with circumferential PF index n = 1, ...N (N = 11 -16 number of PFs) at longitudinal arclength centerline position s. The total elastic + conformational energy can be written as E M T = L 0 (e el + e trans + e inter ) ds with

2 |θ -κpol | 2 FIG. 2 :

 222 FIG.2: Effective persistence length l * p (s) as a function of the position from the attachment point along the MT contour. The experimental and theoretical prediction with lB = 25mm, λ = 7.5µm, κ -1 0 = 18µm, q 0 l φ >> 1.

FIG. 3 :

 3 FIG.3: Longest relaxation time of a microtubule of length L, experimental data from[5] and theoretical prediction from the wobbling mode approximation and the wobbling angle α = 2κ0/q0 ≈ 7. 9 • extracted from static data[4][5], Fig.2.


!" # !$ % 1 MINES ParisTech, CEMEF -Centre de Mise en Forme des Matériaux, CNRS UMR 7635, 1 Rue Claude Daunesse, 06904 Sophia Antipolis Cedex, France & A new experimental procedure, named « Polymer High Cooling -Optics » and whose performance has been improved, is presented. It allows us to monitor the quiescent crystallization of a polymer film under high and constant cooling rates. With such a set/up, two cooling control modes are available resulting into a range of relatively moderate (from 30 to 500 ° C/min) and a range of high (from 500 to 1600 ° C/min) constant cooling rates. The crystallization experiments are observed by polarized light microscopy. Preliminary results are presented. They are obtained on the α/modification of one industrial grade of isotactic polypropylene, under relatively moderate constant cooling rates. Two relevant crystallization parameters are captured, the temperature of crystallization together with the spherulitic growth kinetics cooling rate. Accurate data are obtained. As well, they are in good agreement with comparable literature results compiled by Janeschitz/Kriegl [Macromolecules, 2006]. The obtained results seem to match the Hoffman and Lauritzen theory. These discussions validate our « Polymer High Cooling / Optics » set/up in relatively moderate constant cooling rates. Next step is to demonstrate the suitability of this first generation set/up for higher constant cooling rates.

'

( & Industrial Processes, Liquid to Solid Transition, Isotactic Polypropylene, High Cooling Rates, Optical Monitoring ( Among the large thermo/mechanical constraints that a polymer undergoes during industrial processes [1], high cooling rates are of prime importance. Consequently, the investigation of crystallization phenomena requires acquiring an expertise in constant high cooling rates, i.e., to develop well/controlled cooling procedures equivalent to those typically encountered in processing. This leads to propose new home/made devices and techniques as well as to interpret the new investigated data from existing models and/or from the formulation of new models.

In the present communication, the choice of an experimental framework and the design and performance of a new set/up are presented with the ability to well monitor the crystallization of semi/crystalline polymers under constant and high cooling rates. In a first part, a brief review of original conceptions employed to investigate polymer crystallization under high cooling rates is presented. The review is followed by the description of crystallization mechanism proposed by Hoffman and Lauritzen. In a second part, our own approach is presented. In a last part, results for relatively moderate cooling rates are discussed from a comparison with literature data and from a theoretical point of view.

) * * '

For following the growth rate of spherulitic entities, hot stage microscopy is successfully used, but it requires original conceptions to reach high degrees of supercooling.

) (

Approaches
)         - - - = ∞ ∆ ) ( * exp 0 (1)
is a pre/exponential term which is not strongly dependent on temperature. and ∞ are the Vogel/ Fulcher/Tamman/Hesse (VFTH) parameters.

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 crystallization temperature. is the universal gas constant. ∆ is the degree of undercooling with -= ∆ 0 . is a factor used to correct the temperature dependence of the heat of fusion (i.e., ( )

0 / 2 + =
).

is the nucleation parameter defined by Equation (2). ξ equals 2 for the regime II and 4 for the regime I and III, denotes the crystal layer thickness along the growth direction, σ and σ are the lateral and fold surface free energy, respectively. 0 is the equilibrium melting temperature. is the Boltzmann constant.

0 ∆ is the equilibrium heat of fusion. , relates to the formation of the critical nucleus on the growth surface. Intuitively, from the competing contributions of the transport and nucleation terms, one expects that there should be a maximum in the growth rate data at a temperature between the glass transition and the melting temperature, when plotted as a function of . Equation ( 1) can be rewritten into Equation (3) as given here after.

( ) ( ) ∆ - = - + ∞ 0 ln * ln (3)
It is possible to factor out the contribution of the transport term to the growth rate through the plotting

( ) ∞ - + / * ln ( ) ∆ / 1
. Nucleation parameter can be then deduced.

+ ( (

This part describes both the material used and the newly revisited home/made technique named « Polymer High Cooling / Optics ».

+

One isotactic polypropylene (iPP) grade for injection molding is used. The commercial reference is 3250 MR1 (ATOFINA Company, France) (now named PPH9081, TOTAL PETROCHEMICALS) with a weight average molecular weight of 213000 g/mol and a poly/ dispersity of 5.

+ ) , - + ) !./ 0 1# 23 /.
! !2$ Optical light microscopy from a LEICA/DMRX microscope (Leica/Microsystèmes, France) with a CMOS camera is used to monitor the structural evolution. Optical micrographs are taken between crossed polarizers, +// 90 ° rotated.

+ ) )

!1 3/13 #4 !!. 1# /3 !13 !. A specially designed hot/stage, optically transparent, is used. It is equipped with a flux control system that can heat or cool a gas passing over the sample. The gaseous flux is used as a heating and cooling transfer medium.

The sample is in a shape of film and has a thickness of about 50 Lm in the area of observation. The sample temperature is continuously measured by a K type thermocouple of 0.25 mm in diameter embedded directly in the polymer sample. This eliminates lag time between the sample and the temperature of the surroundings, i.e., air/polymer heat transfer. To optimize the time/polymer temperature following, attention was paid to locate the thermocouple, in contact with the polymer, very near to the observation area. The polymer plus thermocouple set was sandwiched by a pair of clean thin glass slides. The polymer is heated and cooled down by the flowing gaseous heat transfer medium. The sample is heated and cooled down in the same location in the hot/stage chamber, i.e., without any transfer. The heating of the sample was accomplished by passing air through a separate heating unit or throttle chamber. The heat exchanger chamber receives the gaseous flux coming either from the cooling line or from the heating line. When the cooling process begins, the hot air is mixed with cold air. The rate of mixing is done by controlling proportionally the gap of the two lines. A home/made program was especially written under Labview control. It allows us to simultaneously program the cooling rates, the temperature profile of the sample and the monitoring of optical observation. The program proposes two cooling modes: a proportional one using a rotating throttle to control constant cooling rates at relatively moderate values, and an 'all/or/none' one using an 'open/close' valve for cold flux to attain high cooling rates. The experiments were performed with the same polymer film. The protocol is: prior to crystallization, the sample is kept in the molten state at about 215 ° C for about 90 s to erase the thermal history and to avoid its degradation, enabling thus the processing of the sample for many cycles. Then the experiments in the constant non/ isothermal mode consisted of cooling scans between 215 and 50 ° C. The actual operating range of relatively moderate and constant cooling rates is 30/500 ° C/min. The temperature profiles programmed and measured were accordingly as shown in Figure 1. It proves that the sample temperature is well/controlled.

6 (

The growth kinetics resulting from relatively moderate and constant cooling rates are presented together with the understanding of crystallization mechanism by means of the theory of Hoffman and Lauritzen.

( * (

Figure 2 shows an illustration of the optical snapshots indicating the radial development of crystalline entities under constant cooling rates. Open symbols are from Lovinger et al., 1977(rhombs), Padden & Keith, 1959(circles), Von Falkai, 1960(triangles), Olley & Bassett, 1989 (stars) Our results are in good agreement with the comparable ones from the literature. It is evident that, in the high temperature region, i.e., before attaining the maximum at about = 90 ° C, the spherulitic growth rate follows the typical bell/shaped dependence with .

At the end of the study, the spherulitic phase of the investigated iPP sample was verified by X/Ray analysis.

The condition pertains to crystallization of the α/form.

)

To study the regime of crystallization according to the Hoffman and Lauritzen theory, Equation ( 3) is graphically represented in Figure 4. The regime III is observed when crystallization occurs at relatively moderate cooling rates. The nucleation exponent was estimated to 8.01 10 5 K 2 . The value is in good agreement with literature data on iPP [11].

Figure 1 :

 1 Figure 1: Temperature-time evolution of iPP sample at a constant cooling rate of 144 ° C/min.

Figure 2 :

 2 Figure 2: Development of crystalline iPP spherulites at a constant cooling rate of 144 ° C/min.

Figure 3

 3 Figure 3 represents the relationship between the growth rates of iPP α/spherulites in the quiescent melt and resulting from our new investigation. Additionally, the of many authors, compiled by Janeschitz/Kriegl [10], are visible.

Figure 3 :

 3 Figure 3: Compilation of iPP α-spherulites growth rates versus the crystallization temperatures. Blue open diamonds are from the present investigation at relatively moderate cooling rates (from 50 to 500 ° C/min); data at slow cooling rates are from an investigation using a conventional Mettler hot-stage (at 3 and 10 ° C/min).Open symbols are fromLovinger et al., 1977 (rhombs), Padden & Keith, 1959 (circles), Von Falkai, 1960 (triangles), Olley & Bassett, 1989 (stars). Closed symbols

Figure 4 :

 4 Figure 4: Analysis of the spherulitic 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.

  Lm in thickness, under non/isothermal conditions up to 5000 ° C/min. Brucato, Piccarolo and Titomanlio supplied a general experimental route for which thin polymeric samples, 50/100 Lm in thickness, are quenched at very high cooling rates up to 2000 ° C/s by rapidly moving the polymer sample from a hot into a cold section. The experimental procedure involves the recording of the thermal history that the polymer undergoes during crystallization [4/6]. Moreover, it involves simultaneous detection of overall and depolarized light intensities emerging in the course of polymer crystallization[7].

	150 ) )	(	(
	Hoffman & Lauritzen proposed and revisited repeatedly
	a polymer crystal growth mechanism or secondary
	nucleation kinetic theory [8,9]. They classify different
	regimes of lamellar growth: regime I, regime II and
			dealing with the crystallization under
			constant and high cooling rates consist in designing and
			setting up model experiments. Ding & Spruiell [2]
			developed a technique based on the earlier work of
			Magill [3] who introduced the light/depolarizing
			microscopy (LDM) technique as a method of
			investigating the overall kinetics of the crystallization
			process. Their technique allows studies of polymer films,

regime III, respectively. These regime differ by the deposit mechanism of molecular nuclei on the growing lamella or, in other words, by the relative values of the nucleation rate on one substrate and of the substrate completion rate. The regime I is observed for high crystallization temperature where one surface nucleus causes the completion of the entire substrate; the regime II is observed when multiple surface nuclei begin to occur on the substrate because of a larger undercooling; the regime III is similar to the regime II but is entered when distance separation between characteristics of the substrate in the Regime II approaches the value of a molecular stem. In the context of Hoffman & Lauritzen secondary nucleation theory, the linear growth rate of a crystalline entity (e.g., spherulite) is dependent on the degree of undercooling ∆ as presented in Equation (1) below.

7

The design and performances of a newly revisited home/ made optical experimental set/up to monitor structure development with a proper control of high and constant cooling rates are described. Spherulitic growth/ rate was measured at relatively moderate (from 30 to 500 ° C/min) constant cooling rates. Results obtained match the data base done by Janeschitz/Kriegl as well as the Hoffman and Lauritzen theory.

Before going to implement such information into new crystallization models and then into numerical simulation, the effect of higher constant cooling rates (from 500 to 1600 ° C/min) on crystallization event needs further investigations. Moreover, the reliability of the growth kinetics data will be enhanced by comparison with the results of alternative methods, i.e., overall kinetics from constant high cooling rate differential scanning calorimetry.

' (

The authors present their acknowledgements to Alain le Floc'h, Francis Fournier, Thierry Collin, and Lionel Freire for the help during the revisit of the set/up.

[