Pascale Elbaz

From Intercultural communication to Translation ... and back

• Reading based on a similar network regardless of the target language, with only minor variations in the degree of involvement of the different nodes of the network • according to the graphic complexity and consistency • According to the linguistic grain size that predominates in a given writing system

Writing system and reading ability in French and Chinese children Xiaoxia Feng & al. (2020) eLife 9:e54591. 2020work and its modulation by writing system and reading ability in French and Chineschildren Xiaoxia Feng & al. 2019 Are the brain mechanisms of reading acquisition similar across writing systems? And do similar brain anomalies underlie reading difficulties in alphabetic and ideographic reading systems? In a cross-cultural paradigm, we measured the fMRI responses to words, faces, and houses in 96 Chinese and French 10-year-old children, half of whom were struggling with reading. We observed a reading circuit which was strikingly similar across languages and consisting of the left fusiform gyrus, superior temporal gyrus/sulcus, precentral and middle frontal gyri. Activations in some of these areas were modulated either by language or by reading ability, but without interaction between those factors. In various regions previously associated with dyslexia, reading difficulty affected activation similarly in Chinese and French readers, including the middle frontal gyrus, a region previously described as specifically altered in Chinese. Our analyses reveal a large degree of cross-cultural invariance in the neural correlates of reading acquisition and reading impairment.

The return of Unicode CJC ideograms as seen from Japan

Nadine Lucas 2002

The standard commonly known as Unicode, which is based on the ISO/CEI 10646 standard, provides a unified way of handling doublebyte coded scripts, in particular ideographic scripts, which were scorned for a while. Unihan, which is based on ideograms, and the UTF conversion code enable information to be exchanged between countries in the Chinese-speaking world, as well as with other cultural areas. (...) The integration of Unicode allows access to documentary resources of any origin, regardless of coding, as well as their processing and computer storage.

I. Exploration

The diversity of types of communication

Conceptual communication between two cultures

Concept. Notion. Conceptualization.

Socrates and Confucius: an encounter

The Socratic method= a technique of philosophical discussion

Ideas that are difficult to convey

The importance of statuary Nudity in art

Idea #1 The statue as a materialisation of power Shigu 石鼓 (The stone drums)

• Ten stones in the shape of drums (hence the name).

• Two chi 尺 high, one chi 尺 in diameter.

• Thick base, narrow rounded top

Visual communication between two cultures

Idea #2 The nude is beautiful

The nude is not a naked body

The male nude

• Practice of live model drawing

• An essential source of inspiration

• A touchstone of an artist's talent

• Embodies the ideals of beauty and truth

• Reference to the nudes of ancient Greece

• Sculpt the interaction of skin, muscle and bone

• Representation of the gods (of God) in human form

• Intellectual and moral life is conceived in physical terms

The doryphoros by

Polycletus, 440 BC

The female nude

• Cult of Aphrodite

• Art collection: the statue from public space to private homes (Rome)

• Christianity (5th-15th century )=No nude=sinful, shameful

• XVI e s= female nudes in painting

• Mythological context (Giorgione, Sleeping Venus; Titian, The Venus from Urbino)

• Humanist reflection on the human body (truth to be reached through the contemplation of beauty) The Salle des États in 1931

摹文艺复兴以还名画 拙己约同在此研究艺术之同志十余 人,每日下午赴鲁佛尔宫分别摹文艺 复兴以还名画。拙目下所摹者,为法 国十八世纪大师浪漫主义巨头 德拉 克洛瓦(Delacroix)之代表作《但丁与 维齐尔》,长二丈,画幅至 巨,取 材于但丁《神曲》,表现人生之苦闷 与肉体之壮美,极其致者也!
Every afternoon I go to the Louvre with a group of ten or so of my fellow painters who study art to make copies of famous paintings from the Renaissance onwards. The painting I am copying is a masterpiece by Delacroix, one of the great French Romantics of the eighteenth century, Dante and Virgil, which is two feet long and the largest of its kind. The painting is based on Dante's Divine Comedy and is a masterful expression of the misery of life and the beauty of the body.

Terminological communication between two cultures

General and specialised languages -individual culture (13 translators) -generational culture (junior/senior translators) -postural culture (translators, an author) -national culture (French, Chinese) -the culture of French readers 67 Zhang Dainian: "All opposing beings can unite into one and every entity necessarily includes its opposite. Two opposing beings repel each other by attracting each other, confront each other by seeking each other out. Hence change.

"The first is the interdependence between two opposing beings: if there is one, there is the other, i.e. each exists because it depends on the other; the second is mutual inclusion, where the two beings are contained in each other, i.e. each being is preserved in the other; the third is their permutation, where each being can become the other, i.e. can come into being as a result of the mutual transformation ; the fourth is the completeness of the two beings, where all opposition is eliminated, i.e. where the difference between the two poles is annihilated; the fifth is the unity of the two beings, where they are unified by a larger unity, and become one. "

From

  temples to public places From public to private Michael Bird, 100 Ideas That Changed Art, London, Laurence King Publishing, 2012 Artistic, religious, political idea: the statue • A statue (stare, to stand): telling a story, displaying movement, expressing emotion • Exhibited in the public space=religious or sociopolitical role • Materialisation of power= religious power (offering, ex-voto), political power • Shapes, attitudes, techniques transmitted from Greece • Do statues have some magical power? • An allegory, an icon 1504 place de la Signoria, Florence 15 -A character from the Bible -An allegory of the Republic of Florence -A symbol of the Renaissance -An advertising logo Winged Victory of Samothrace 220-185 BC • Xiong, Bingming 熊秉明 (1922-2002), Zhongguo shufa lilun tixi 中國書法理論體 系, Xianggang : Shangwu yinshuguan Xianggang fenguan, 1984

•

  Bear inscriptions relating to a court hunt (hence its other name: liejie "猎碣" (Hunting Tablet))。 • Dating difficult: Duke Mu of the Qin (秦穆公)? Late Spring and Autumn (770-476) or early Warring Kingdoms (475-221) • Of around five hundred characters, only three hundred and twenty-one are legible today. • Form: poem in four characters • Shiguwen writing style: intermediate style between the Big Seal (8th century BC) and the Small Seal (221 BC). an album of stampings known as "後勁本" from the Song period Wu Changshuo 吴昌硕 (1844-1927)

• « Mais il y a aussi la partie immergée de la langue, sa partie basse... Une langue est riche lorsqu'elle intègre des corpus particuliers. »• "But there is also the immersed part of the language, its lower part... A language is rich when it integrates particular corpora. »

Bridging the gap

("The Dao that can be said is not the constant Dao", Daodejing, 1 )

Microsoft Translator and classical Chinese (1)

Oranges grow in the south of the Huaihe River is oranges, growth in the north of the Huaihe River becomes a dragonfly. From the "Tweezers make Chu" "Zizi avoids the carpet to the dragonfly: baby smell, orange born Huainan is orange, born in Huaibei is a dragonfly, leaf is similar, in fact, taste different. And then? Water and soil are also different. The livelihood of people today is longer than Qi do not fly, in Chu is flying, no chu water and land so that people good flight yes?" ( 54

II. Didactic scope

Intercultural communication in the classroom

Collaborative translation as a forum for intercultural communication

Project creation and shared management In conclusion


De l'Ecosse au Japon

Depuis les années 1850 et l'intervention militaire de l'amiral américain Perry en 1853 dans la baie de Tokyo, le Japon, très peu ouvert au monde occidental tout au long de l'ère Tokugawa, s'est lancé dans une politique de développement-rattrapage intensif pour échapper au destin des pays voisins colonisés par les puissance occidentales (guerre de l'opium gagnée par les Britanniques contre la Chine en 1842). Ce rattrapage fera du Japon un pays « moderne sans être occidental » selon le titre de l'ouvrage de Pierre-François Souyri [START_REF] Souyri | Moderne sans être occidental, aux origines du Japon d'aujourd'hui[END_REF]

  Clin d'oeil pour Entreprise & Histoire, numéro « transfert technologique », D. Barjot ed. De l'Ecosse au Japon et retour : les transferts innovants de technologies d'enseignement par Henry Dyer (1850-1890). Les questions de transfert technologique portent fréquemment sur le dilemme entre transférer une technologie existante ou transférer la capacité à (re)concevoir la technologie pour mieux l'adapter à des contextes d'usage singuliers. On sait aujourd'hui que ce second type a été intense, notamment par le jeu des imitations et des importations des modèles de formation des concepteurs eux-mêmes. Ainsi l'École des Mines de Paris à sa création (1783) s'est inspirée de la formation-modèle délivrée à l'École des Mines de Freiberg ; dans les années 1820 l'Institut Polytechnique de Karlsruhe s'est inspiré du modèle de formation de l'Ecole Polytechnique ; et fin 19 ème les écoles d'arts appliquées allemandes qui préfiguraient le

	formations
	d'ingénieurs à l'Anderson College de Glasgow (qui deviendra plus tard la prestigieuse université
	technique de Strathclyde) et au Finsbury Technical College de Londres et à la Central Institution
	of the City and Guilds of London Institute (qui deviendront plus tard partie intégrante
	d'Imperial College).
	La connaissance de ce cas est relativement récente : ce n'est que dans les années 1980 que
	son principal protagoniste, Henry Dyer (1848-1918), a été redécouvert grâce aux travaux de
	Nobuhiro Miyoshi, travaux traduits en anglais près de quinze plus tard (Miyoshi 2004; Cortazzi
	2004), repris et étendus grâce aux travaux d'Erich Pauer et de ses collègues au Centre
	Européen d'Etudes Japonaises (Pauer 2022, 2010) (Pauer and Mathias 2022; Pauer and Meade
	2020). Ce sont les sources de cet article 1 .

Bauhaus se sont inspiré des écoles d'arts appliqués anglaises. Ces transferts sont d'autant plus intrigants que, comparés à un transfert de technologie, ils peuvent sembler tout à la fois plus 'simples' (moins capitalistique, moins matériels…) et plus 'complexes' (car survenant sur dans les écoles, un des lieux emblématiques de la transmission de la culture).

Voici un curieux cas de « double transfert » de « technologies de formation des concepteurs » entre Japon et Royaume Uni dans les années 1870-1890. Il a conduit d'une part à la création réussie de l'Imperial College of Engineering à Tokyo (qui deviendra plus tard la prestigieuse formation d'ingénieur de l'Université de Tokyo et qui servira aussi de modèle à de nombreuses formations d'ingénieur privées au Japon) et d'autre part à la création de nouvelles

  avec la restauration de Meiji (à partir de 1868) : le gouvernement a l'ambition de créer un nouveau système de formation technique. En 1871, Ito Hirobumi et Yozo Yamao, hauts fonctionnaires du nouveau ministère des Travaux Publics, formés au contact de l'occident grâce à leur long séjour secret en Ecosse à la fin de l'ère Tokugawa, créent l'Imperial College of Engineering (ICE) (1871). Puis Ito participe à la mission Iwakura, un voyage d'études rassemblant des hommes politiques du plus haut rang, des scientifiques et des ingénieurs qui, de 1871 à 1873 visitent les grands pays occidentaux et, sur les conseils de James Matheson, membre de la maison de commerce Jardine, Matheson & Co, io recrute un diplômé de l'Université de Glasgow de 24 ans, Henry Dyer pour diriger l'ICE.C'est ainsi qu'en 1873 Dyer part pour le Japon. Il y restera neuf ans et va y créer un enseignement original, qui sera intégré ensuite à l'Université de Tokyo (non sans de lourdes transformations) et essaimera largement dans l'ensemble du tissu des écoles techniques japonaises[START_REF] Pauer | Der industrielle Aufstieg Japans und die Rolle des Imperial College of Engineering (1873-1885) für die Humankapitalbildung im technischen Bereich[END_REF].Ce premier transfert, de l'Ecosse vers le Japon, est en fait un curieux hybride : Dyer n'est pas complètement satisfait par l'enseignement très pratique de l'Université de Glasgow ou des cours du soir qu'il a suivis à l'Anderson College voisin 2 . Il veut hybrider ces formations pratiques avec les formes d'enseignements qui font alors référence : ceux de Karlsruhe ou de Zurich, dont les rapports officiels de l'époque décrivent la qualité de l'enseignement théorique. En outre Dyer cherche aussi à développer un cursus adapté au contexte socio-technique japonais -un pays avec peu de technologies et une élite de fonctionnaires, les samouraïs, formée dans une logique d'imitation et de répétition et très réticente aux activités manuelles. Il développe ainsi un nouvel hybride entre théorie et pratique. Il est significatif que le cursus soit jugé « trop théorique » par le grand ingénieur anglais Richard Henry Brunton de passage au Japon mais ) cette large couverture conduit à développer trois types de savoirs : i/ un savoir scientifique lié à la profession, suivant le modèle des professions libérales (avocat, médecin)('learned professions'), avec à la fois des savoirs généraux (maths, physique) et des savoirs plus spécialisés (eg minéralogie, thermodynamique,…) caractéristiques de chaque branche ; ii/ un savoir pratique vu comme une façon de produire des idées utiles, appliquées, personnelles et originales et iii/ un savoir général qui se veut 'non-professionnel' et inclut littérature, art et philosophie pour permettre aux étudiants 'd'échapper à l'étroitesse d'esprit, aux préjugés et aux passions qui assaillent les hommes du métier' (Dyer, cité parMiyoshi, p. 36). Cette tripartion dépasse l'habituelle dichotomie théorie / pratique et renvoie plutôt à une triple tension entre l'existant et le nouveau dans les professions libérales : i/ le savoir professionnel permet d'améliorer chaque profession ; ii/ le savoir pratique permet d'innover dans la profession à partir des cas d'application singuliers ; iii/ le savoir général permet d'éviter l'enfermement dans une profession et de mieux penser sa régénération. c) Dyer et les autres enseignants qui le rejoignent (eux aussi venant de Grande Bretagne, tels Ayrton et Perry) développent des ateliers de très hauts niveaux, en lien avec les entreprises publiques qui se développent au cours des premières années de l'ère Meiji. Les ateliers de construction de machine d'Akabane (largement développés par Dyer) constituent une référence pour l'enseignement comme pour l'industrie ; Ayrton développe un atelier pour une formation à la télégraphie ; Ayrton et Perry développent ensemble des ateliers de génie électrique qui sont aussi des centres de recherche et qui sont d'un tel niveau que Maxwell dira que le centre d'excellence du génie électrique a basculé de l'Angleterre au Japon (propos rapporté par[START_REF] Miyoshi | Pioneer of Education in Japan[END_REF].Mais l'histoire ne s'arrête pas là. Miyoshi parle d'un étonnant effet boomerang. En 1882, à 34 ans, Dyer rentre en Ecosse pour postuler à un poste de professeur à l'université de Glasgow. Ses candidatures malheureuses le conduisent à développer son enseignement dans une autre institution, l'Anderson College. Comme Dyer l'expliquera lui-même à plusieurs reprises (notamment dans son papier de 1904, cité par Miyoshi, p. 71), cet enseignement est largement inspiré de son expérience japonaise. Il se caractérise par un 'sandwich system' combinant à différents niveaux, théorie et pratiques : au début du cursus, on trouve une alternance entre des cours théoriques généraux et des visites, puis des cours plus spécialisés pour chaque branche, avec une combinaison de théorie et d'exercices en ateliers et, enfin, en dénominateur commun jusque dans le plus petit exercice, l'usage de méthodes graphiques et expérimentales qui permettent de combiner systématiquement approches générales et expérience pratique spécifique. Ces enseignements continueront de s'épanouir dans ce qui deviendra la très réputée université de Strathclyde. En parallèle, deux professeurs ayant travaillé avec Dyer au ICE vont s'illustrer eux-aussi à leur retour en Grande Bretagne : à Londres Ayrton et Perry développent des enseignements originaux en electrical engineering au Finsbury Technical College de Londres et à la Central Institution of the City and Guilds of London Institute. Leurs méthodes de formation au laboratoire vont faire l'admiration des contemporains. Ces enseignements deviendront des références au sein de ce qui deviendra Imperial College.Dyer et ses collègues illustrent ainsi une forme de transfert de technologies éducatives pour concevoir des technologies ! Mais ce transfert a moins été une exportation de modèles figés qu'un moment privilégié pour faire émerger de nouveaux modèles à la croisée de trois mondes : l'enseignement pratique écossais, les formations d'Europe continentales considérées comme plus théoriques et les spécificités du contexte socio-économique japonais. Dyer et ses collèges ont su s'emparer d'héritages passés pour en faire une création originale. Le résultat a été cet étonnant mélange de théorie et de pratique qui permettait d'affirmer que les ingénieurs étaient une forme de profession libérale, une learned profession à laquelle il n'était pas possible d'appartenir sans maîtriser des savoirs spécifiques -à la manière des médecins et des avocats ; mais de considérer aussi qu'un ingénieur doit être capable d'innover

	Un effet boomerang
	Conclusion

. Dans le domaine de la technologie, une première phase est fondée sur les traductions d'ouvrages et d'articles et sur les échanges avec des experts de passage (1850'-1870) ; une seconde phase s'ouvre « trop pratique » par les élites japonaises descendants des samourais ! En fait ce « transfert » est plutôt une forme d'invention originale à partir de plusieurs héritages passés, comme le soulignent trois traits singuliers pour l'époque : a) l'ICE couvre 7 branches d'enseignements techniques (génie civil, construction de machines, télégraphie, architecture, chimie appliquée, métallurgie, et mines 3 ), une étendue plus large que ce qui est pratiqué ailleurs. Un si large éventail permet de dégager des années initiales d'enseignements généraux partagés et des années ultérieures de 'spécialisation' ; il permet aussi d'avoir des principes généraux communs à toutes les branches (combinaisons de savoir théorique et pratique tout au long de la scolarité) tout en ayant une forte différentiation sur les objets traités. b

Nous remercions le Professeur Erich Pauer pour nous avoir fourni très obligeamment un très riche ensemble de texte très utiles pour ce présent clin d'oeil.

L'Anderson college est une institution qui délivre des cours du soir pour les techniciens de Glasgow, fondée en 1796 grâce au leg de John Anderson, professeur de philosophie naturelle de l'université de Glasgow (1726-1796) qui avait encouragé James Watt dans ses travaux sur la machine à vapeur.

3 Seule la construction navale n'est pas intégrée, elle est enseignée dans une autre institution dirigée par François Léonce Verny, sur le modèle de l'Ecole Polytechnique.

dans la profession voire d'innover pour transformer la profession, et cela grâce à une curiosité pour des savoirs « non-professionnels » qui les rendraient plus créatif.

Ces transferts inventifs répétés nous montrent aussi qu'en matière de formation à la conception, les héritages culturels sont des socles sur lesquels les nouvelles formations s'appuient pour mieux les régénérer. L'ICE, tout comme les formations créées en Grande Bretagne par Dyer, Ayrton et Perry, entrent en résonance avec l'héritage intellectuel et technologique passé pour mieux l'enrichir. De sorte qu'on peut aussi bien considérer qu'il y a des traditions nationales de formation d'ingénierie et considérer que ces traditions sont aussi le fruit de leurs capacités d'hybridation et de régénération. Et il faut peut-être saluer les capacités des inventeurs comme des institutions qui les ont accueillis à oser rendre plus créatifs les patrimoines dont ils avaient hérité pour faire face aux enjeux de leur temps. Une leçon qui peut avoir son importance à l'heure où les défis contemporains conduisent à régénérer la façon dont on forme les concepteurs pour mieux les préparer à concevoir pour les transitions à mener.


Paul Escande
email: paul.escande@univ-amu.fr

F R Éd

Éric J P Richard
email: frederic.richard@univ-amu.f

©xxxx Taras

FULL INFERENCE FOR THE ANISOTROPIC FRACTIONAL BROWNIAN FIELD

Keywords: 2020 Mathematics Subject Classification. Primary 62M40; Secondary 78M50 inference from random field, anisotropic fractional Brownian fields, image texture analysis, image texture synthesis

The anisotropic fractional Brownian field (AFBF) is a non-stationary Gaussian random field which has been used for the modeling of textured images. In this paper, we address the open issue of estimating the functional parameters of this field, namely the topothesy and Hurst functions. We propose an original method which fits the empirical semi-variogram of an image to the semi-variogram of a turning-band field that approximates the AFBF. Expressing the fitting criterion in terms of a separable non-linear least square criterion, we design a minimization algorithm inspired from the variable projection approach. This algorithm also includes a coarse-to-fine multigrid strategy based on approximations of functional parameters. Compared to existing methods, the new method enables to estimate both functional parameters on their whole definition domain. On simulated textures, we show that it has a low estimation error, even when the parameters are approximated with a high precision. We also apply the method to characterize mammograms and sample images with synthetic parenchymal patterns.

Introduction

The anisotropic fractional Brownian field (AFBF) is a non-stationary Gaussian random field with stationary increments; see [START_REF] Bonami | Anisotropic analysis of some Gaussian models[END_REF] and Section 2 for details. When the field is defined in two dimensions, the finite-dimensional probability distributions of an AFBF Z are completely determined by a semi-variogram of the form

∀ h ∈ R 2 , v 0 (h; τ, β) = 1 2 E (Z(x + h) -Z(x)) 2 , = 1 2 R 2
1 -e i⟨h,w⟩ 2 τ (arg(w))|w| -2β(arg(w))-2 dw.

(1.1)

In this expression, the functions τ and β are the parameters of the AFBF. They are called the topothesy and Hurst functions, respectively. They are non-negative π-periodic functions whose values only depend on the direction arg(w) of w in R 2 . The Hurst function β further ranges in (0, 1). In this paper, we propose an original method to estimate these functional parameters from a single realization of a two dimensional AFBF.

The AFBF has been used for the modeling of image textures [START_REF] Biermé | Analysis of texture anisotropy based on some Gaussian fields with spectral density[END_REF][START_REF] Biermé | Anisotropic texture modeling and applications to medical image analysis, Mathematical Methods for Imaging and Inverse Problems[END_REF][START_REF] Richard | Anisotropy of Hölder Gaussian random field: characterization, estimation and application to image textures[END_REF][START_REF]Some anisotropy indices for the characterization of Brownian textures and their application to breast images[END_REF][START_REF] Richard | Analysis of anisotropic Brownian textures and application to lesion detection in mammograms[END_REF][START_REF]Tests of isotropy for rough textures of trended images[END_REF][START_REF] Richard | Statistical tests of anisotropy for fractional Brownian textures. Application to full-field digital mammography[END_REF]. Figure 1 shows some textures together with the Hurst and topothesy functions of the AFBF they are generated from. On the first column, constant topothesy and Hurst functions are associated to an isotropic texture, i.e. a texture whose properties Figure 1. Textures generated from AFBF. First row: the realization of the AFBF, second and third rows: representations of their Hurst and topothesy functions, respectively. The textures were generated using the PyAFBF package [START_REF] Biermé | A turning-band method for the simulation of anisotropic fractional Brownian field[END_REF][START_REF]PyAFBF: a Python library for sampling image textures from the anisotropic fractional Brownian field[END_REF]. are invariant to any image rotation. By contrast, as observed on other columns, nonconstant topothesy or Hurst functions result in anisotropic textures whose properties vary according to the direction. Let us also notice that the texture anisotropy manifests differently depending on the shape of the topothesy and Hurst functions.

Actually, these two functional parameters are critical to characterize the texture. Such a characterization may serve for the classification of textures. In [START_REF] Richard | Anisotropy of Hölder Gaussian random field: characterization, estimation and application to image textures[END_REF], a feature was derived from the topothesy function to measure a degree of texture anisotropy. This feature was used in conjunction with a measure of texture regularity to classify photographic films with respect to their paper type. In [START_REF]Some anisotropy indices for the characterization of Brownian textures and their application to breast images[END_REF][START_REF] Richard | Analysis of anisotropic Brownian textures and application to lesion detection in mammograms[END_REF], another measure of anisotropy depending both on the topothesy and the Hurst functions was designed and applied to detect or diagnose lesions in mammograms. The AFBF can be further extended to an heterogeneous field where the topothesy and Hurst function both vary in space [START_REF] Vu | Statistical tests of heterogeneity for anisotropic multifractional Brownian fields[END_REF][START_REF] Polisano | Texture modeling by Gaussian fields with prescribed local orientation[END_REF][START_REF] Benassi | Elliptic Gaussian random processes[END_REF]. Due to this model, we can also consider characterizing locally the texture through the topothesy and Hurst functions. Such a characterization can be used to segment images, i.e. partition the image into several regions of homogeneous textures. In [START_REF]Tests of isotropy for rough textures of trended images[END_REF], biological microscopic images were segmented using local features related to the topothesy and Hurst functions. The work presented in this paper is mainly motivated by these applications of AFBF to the classification or the segmentation of image textures. By proposing a method to estimate topothesy and Hurst functions, we enable the computation of texture descriptors that are critical to these two image-processing tasks. For instance, descriptors may include variances of the topothesy and Hurst functions that would measure a degree of anisotropy (the variance of the function is defined as the square of the L 2 -norm of the difference between the function and its average value over the interval [-π, π].).

Another important motivation concerns the simulation of textures. In [START_REF] Biermé | A turning-band method for the simulation of anisotropic fractional Brownian field[END_REF], AFBF were simulated with a method which was later implemented in a Python package [START_REF]PyAFBF: a Python library for sampling image textures from the anisotropic fractional Brownian field[END_REF]. Currently, this package enables to generate textures from AFBF whose parameters are randomly sampled. One of our goals would be to extend this package to the simulation of realistic textures. Such a simulation could be obtained from AFBF whose parameters were previously estimated from one or several real examples.

However, the realistic simulation, as well as the texture classification, require a complete estimation of the Hurst and topothesy functions. Methods proposed in the literature only give a partial solution to that estimation issue. In [START_REF] Biermé | Estimation of anisotropic Gaussian fields through Radon transform[END_REF][START_REF] Richard | Statistical tests of anisotropy for fractional Brownian textures. Application to full-field digital mammography[END_REF], a method was developed for the estimation of the Hurst function. It relies upon a theoretical relationship that exists between an AFBF and its Radon projection in a direction orthogonal to θ [START_REF] Bonami | Anisotropic analysis of some Gaussian models[END_REF]: in dimension d = 2, this projection is locally asymptotically self-similar of order H = β(θ) + 1 2 . Hence, by estimating the self-similarity order of the projections, it is theoretically possible to recover the Hurst function of the field. However, in practice, such an approach is only feasible in the vertical and horizontal directions where the Radon transform can be discretized.

In [START_REF] Richard | Anisotropy of Hölder Gaussian random field: characterization, estimation and application to image textures[END_REF], another method was designed for the estimation of the so-called asymptotic topothesy. This function is equal to the topothesy function in directions where the Hurst function is minimal, and vanishes elsewhere. The estimation approach is based on quadratic variations (averages of squared increments) computed from the image at different scales and in different directions. Asymptotically, the logarithm of these variations is linearly related to the Hurst index of the field (minimal value of the Hurst function). This linear relationship includes some intercepts that can be estimated by linear regression. These intercepts can further be expressed as a convolution of the asymptotic topothesy with some known functions. Due to this expression, the asymptotic topothesy can be recovered from the intercepts by solving an inverse problem.

The construction of our estimation method takes its root in the work in [START_REF] Biermé | A turning-band method for the simulation of anisotropic fractional Brownian field[END_REF]. Following [START_REF] Biermé | A turning-band method for the simulation of anisotropic fractional Brownian field[END_REF], the semi-variogram of an AFBF can be expressed in a polar form as

(1.2) v 0 (h; τ, β) = 1 2 π 2 -π 2 τ (θ)|⟨h, u(θ)⟩| 2β(θ) dθ,
where u(θ) = (cos θ, sin θ) and τ (θ) = τ (θ)

+∞ 0

1 -e iρ 2 ρ -2β(θ)-1 dρ. Given some ordered angles (θ m ) M m=1 in [-π 2 , π 2 ] and appropriate positive weights (λ m ) M m=1 (see Section 3 of the supplementary materials of [START_REF] Biermé | A turning-band method for the simulation of anisotropic fractional Brownian field[END_REF] for details), it can be further approximated by

(1.3) v(h; τ , β) = 1 2 M m=1 λ m τ (θ m )|⟨h, u(θ m )⟩| 2β(θm) .
The function v corresponds to the semi-variogram of a turning band-field defined as

Z M (x) = M m=1 λ m τ (θ m )Y m (⟨x, u(θ m )⟩),
where the random processes Y m are independent standard fractional Brownian motions of Hurst index β(θ m ). In [START_REF] Biermé | A turning-band method for the simulation of anisotropic fractional Brownian field[END_REF], it was shown that v converges to v 0 and Z M to Z as the maximal spacing between successive angles θ m tends to 0. This result justifies using the turning-band field Z M for simulating the AFBF Z and the semi-variogram v for approximating v 0 . In this work, we thus propose to estimate the topothesy and Hurst functions of an AFBF by fitting the semi-variogram v of its approximating turning-band field to the empirical semi-variogram of a textured image.

Preliminary considerations about the AFBF

A random field is a collection Z = {Z(x), x ∈ R d } of random variables Z(x) indexed by a position x of the space R d of dimension d. In dimension d = 2 or d = 3, such a field can represent an image, Z(x) being in this case the grey-level value of the image at a pixel x.

A random field is Gaussian if any finite linear combination i λ i Z(x i ) of its variables is a Gaussian random variable. The finite-dimensional probability distributions of a Gaussian random field can be characterized by two functions: an expectation function x → E(Z(x)) giving the mean value of the field at each position x and the auto-covariance function (x, y) → cov(Z(x), Z(y)) describing the interactions of the field variables between pairs (x, y) of positions.

A Gaussian random field is stationary if its expectation function is constant and its auto-covariance function at (x, y) only depends on x -y. Such a field may describe images whose properties are spatially homogeneous. When it is not the case, we can resort to a weaker stationarity assumption which is stated on field increments rather than the field itself. An increment W of a field Z is a field defined for any position

x ∈ R d as W (x) = Z(x + h) -Z(x)
given some fixed lag h ∈ R d . We say that a field has stationary increments whenever its increment fields are stationary. The finitedimensional probability distributions of such a field can be characterized by the semi-

variogram v(h) = 1 2 E((Z(y + h) -Z(y)) 2
) of the field. The AFBF forms a family of random fields with stationary increments which is defined by semi-variograms of the form (1.1).

Problem formulation

The estimation context is the following. We observe the gray-level values

Y [i] of an image Y at points i of a grid [[1, I]] 2 .
We assume that this image can be modeled as the sum

Y [i] = Z i I + W [i]
of a turning-band field Z having a semi-variogram v of the form (1.3) and a noise W formed by independent centered Gaussian variables W [i] of variance τ 0 . It follows that the semi-variogram of the image is given by

(3.1) w(h; τ, β) = τ 0 + v h I ; τ, β .
In this equation, the topothesy and Hurst functions τ and β are unknown. The problem is to estimate these functional parameters.

For that, we compute the empirical semi-variogram ( Section 4 for the choice of these lags):

w n ) N n=1 at some lags (h n ) N n=1 of [[1, I]] 2 (see
(3.2) w n = 1 ν n i (Y [i + h n ] -Y [i]) 2 ,
where ν n is the number of sum terms depending on h n .

We then propose to fit the theoretical semi-variogram w to the empirical one w by minimizing the least-square criterion

(3.3) L(τ, β) = 1 2 N n=1 (w(h n ; τ, β) -w n ) 2 .
To minimize the least-square criterion, the topothesy and Hurst functions are expanded into two finite-dimensional subspaces T and B of the space of square integrable π-periodic functions. Given some bases (T j ) J j=1 and (B k ) K k=1 of T and B, the topothesy and Hurst functions are represented as

τ (θ) = J j=1 τ j T j (θ), β(θ) = K k=1 β k B k (θ). (3.4)
Using these representations, the semi-variogram of the image may be written as

(3.5) w(h n ; τ, β) = τ 0 + 1 2 J j=1 τ j M m=1 T j (θ m )λ m h n I , u(θ m ) 2β(θm) = F (β) n τ,
where τ is a notation abuse for the column-vector (τ j ) J j=0 and F n is a row-vector-valued function whose components are

(3.6) F (β) nj = 1 2 M m=1 T j (θ m )λ m h n I , u(θ m ) 2β(θm) = v(h n ; T j , β), for j = 1, • • • , J and F (β) n0 = 1.
We will denote F (β) the matrix of terms F (β) nj . Consequently, the least-square criterion can be reformulated as

(3.7) L(τ, β) = 1 2 N n=1 (ϵ n (τ, β)) 2 .
where ϵ n (τ, β) are residuals defined by

(3.8) ϵ n (τ, β) = F n (β)τ -w n .
In the optimization literature, such a criterion is known as a separable nonlinear least square (SNLLS) criterion [START_REF] Golub | The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate[END_REF].

Problem resolution

There is a wide literature devoted to the minimization of SNLLS criteria such as the one defined by Equations (3.7) and (3.8); see, for instance, [START_REF] Golub | The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate[END_REF][START_REF] Kaufman | A variable projection method for solving separable nonlinear least squares problems[END_REF][START_REF]Separable nonlinear least squares: the variable projection method and its applications[END_REF][START_REF] Ruhe | Algorithms for separable nonlinear least squares problems[END_REF][START_REF] O'leary | Variable projection for nonlinear least squares problems[END_REF]. The construction of minimization algorithms is based on the variable projection method introduced in [START_REF] Golub | The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate[END_REF]. Let

K(β) = L(τ * (β), β), where τ * (β) minimizes L for a fixed β, τ * (β) ∈ arg min τ L(τ, β).
The variable projection method consists in reducing the minimization of L with respect to the couple (τ, β) to the minimization of K with respect to the single variable β.

The function K is usually minimized using some variants of the Newton algorithm involving approximations of the Hessian of K (e.g. the Gauss-Newton algorithm or the Levenberg-Marquardt algorithm [START_REF] Levenberg | A method for the solution of certain non-linear problems in least squares[END_REF][START_REF] Marquardt | An algorithm for least-squares estimation of nonlinear parameters[END_REF]). The approximation of the Hessian are specifically derived according to the form of the Hessian of a SNLLS [START_REF] Golub | The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate[END_REF][START_REF] Kaufman | A variable projection method for solving separable nonlinear least squares problems[END_REF][START_REF]Separable nonlinear least squares: the variable projection method and its applications[END_REF][START_REF] Ruhe | Algorithms for separable nonlinear least squares problems[END_REF][START_REF] O'leary | Variable projection for nonlinear least squares problems[END_REF]. Here, the Hessian of K can be written as follows (see Sections A and B of the appendix for the computation details). Denote

A 11 (β) = N n=1 F n (β) T F n (β), A 22 (β) = N n=1 DF n (β) T τ * (β)τ * (β) T DF n (β), E 22 (β) = N n=1 ϵ n (τ * (β), β) j=1 ∇ 2 F nj (β)τ * j (β), A 12 (β) = N n=1 F n (β) T τ * (β) T DF n (β), E 12 (β) = N n=1 ϵ n (τ * (β), β) DF n (β). (4.1)
where ϵ n are residuals given by Equation (3.8), F n (β) is the function defined by Equation (3.6), DF n (β) its Jacobian and ∇ 2 F nj (β) the Hessian of its j component F nj . Then, the Hessian of K can be expressed as

(4.2) ∇ 2 K(β) = A 22 (β) + E 22 (β) -(A 12 (β) + E 12 (β)) T (A 11 (β)) -1 (A 12 (β) + E 12 (β)),
Approximations of the Hessian usually exclude terms that depends on the second derivatives of the function (here, the term E 22 ). In our algorithm, we use a simple approximation of K by the matrix A 22 (β) + λI; for λ = 0, this corresponds to the Algorithm III pointed out in [START_REF] Ruhe | Algorithms for separable nonlinear least squares problems[END_REF]. The minimization algorithm was implemented in Python using the method least squares of the optimization toolbox of the Scipy library [START_REF] Virtanen | SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python[END_REF]. At iteration t + 1, the algorithm updates the current value β (t) of β by computing

β (t+1) = β (t) -A 22 (β (t) ) + λI -1 ∇K(β (t) ),
where λ is a non-negative damping factor adjusted at each iteration, and the gradient of K is given by

(4.3) ∇K(β) = N n=1 DF n (β) T τ * (β) ϵ n (τ * (β), β).
Besides, the solution τ * (β) minimizes a linear least square problem, and is characterized as the solution of the linear system:

F (β) T F (β)τ = F (β) T w.
In our implementation, it is found using the method lsq linear of the Scipy library, which is robust to ill-conditioned systems.

To design our algorithm, we set up a coarse-to-fine multigrid strategy: the parameters (τ, β) are successively approximated in a series of embedded subspaces (T s × B s ) S s=0 . In the current implementation of the algorithm, the subspaces T s and B s are both defined as the space of piecewise constant functions on the intervals

- π 2 + m 2 s π, - π 2 + m + 1 2 s π , m = 0, • • • , 2 s -1 .
Moreover, some constraints are added to these spaces to ensure that the topothesy function is non-negative and the Hurst function ranges in (0, 1). This is directely implemented in the least squares and lsq linear methods that use a Trust-Region-Reflective algorithm to deal with these constraints [START_REF] Branch | A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems[END_REF].

Set the algorithm parameters S, T and η (by default, S = 3, T = 10000 and η = 1e-3). Initialization:

Set L 0 = +∞ for h ∈ {0.1, • • • , 0.9} do Set β ∈ B 0 equal to h. Find τ * (β) = arg min τ ∈T0
L(τ, β) using a few iterations of the algorithm least squares (method TRF). until L(τ0,β0)-L(τ,β) L(τ0,β0) < η or iter > T . end for Figure 2. Algorithm for the estimation of the topothesy and Hurst functions.

using lsq linear (method TRF) . if L(τ * (β), β) < L 0 then Set L 0 = L(τ * (β), β), Set τ 0 = τ * (β
The overall algorithm is summarized in Figure 2. In practice, we observed that the choice of the lags where the semi-variograms are evaluated had an influence on the minimization. In particular, we believe that having two orthogonal lags with a same modulus can make the problem ill-posed with permutable solutions. So we arranged the lags to avoid such pathological configuration. Moreover, we spread the lags uniformly in all directions to avoid privileging a few directions. To do so, we picked up lags in a set composed by points of [[0, L]] × [[-L, L]] within a disk of radius L. Points of this set having a given argument φ are in a subset {ρ k (cos φ, sin φ), k = 1, • • • , K φ } defined by some ordered sequence of moduli ρ k . Similarly, points having the orthogonal argument φ + π 2 are in the set {ρ k (cos φ, -sin φ), k = 1, • • • , K φ }. For each pair of orthogonal arguments (φ, φ + π 2 ), we selected lags with different modulus as

{ρ 2k-1 (cos φ, sin φ), k = 1, • • • , K φ } ∪ {ρ 2k (cos φ, -sin φ), k ∈ 1, • • • , K φ }.
A typical set of lags is shown on Figure 3 for L = 40. Besides, we set angles θ m of the turning-band field representation using the dynamic programming algorithm described in [START_REF] Biermé | A turning-band method for the simulation of anisotropic fractional Brownian field[END_REF] with M = 657 and set λ m = θ m -θ m-1 .

Numerical study

In order to assess the performance of the proposed method, we conducted experiments on field realizations generated by the PyAFBF package [START_REF] Biermé | A turning-band method for the simulation of anisotropic fractional Brownian field[END_REF][START_REF]PyAFBF: a Python library for sampling image textures from the anisotropic fractional Brownian field[END_REF]. The topothesy and Hurst functions of the simulated fields were represented as piecewise constant functions on uniformly spaced intervals of ] -π 2 , π 2 ]. In a first series of experiments, we made the number of constants vary so as to study the effect of the model complexity on the estimation precision. We built seven sets of hundred realizations. Realizations were images of size 1024 × 1024. On each set, they were obtained from different fields having a same number of parameters. Depending on the set, the number of parameters was M = 2 m for m = 0, • • • , 6 for the topothesy and Hurst functions. These parameters correspond to the constants of the topothesy and Hurst functions. The higher the number of these constants, the more precise is the directional description of the field. To account for this precision, we defined a so-called radial precision, expressed in percent, as r = (1 -1/M ) * 100, with 0 corresponding to the worst precision and 100 to the best one. The estimation error was evaluated on the Hurst function using a L 1 error, expressed in percent, as

e = 100 π π 2 -π 2 |β(s) -β(s)|ds,
β and β being the true and estimated Hurst functions, respectively. We averaged the estimation errors over the hundred realizations of each set. To obtain the estimates, we applied the method with the coarse-to-fine multigrid strategy (see Section 4), making the size parameter s vary from 0 to m. The maximal number of iterations and the tolerance were set to T = 10000 and η = 1e -3, respectively. All experiments can be reproduced or modified using the PyAFBF package. The results are shown on Figure 4. The estimation error was quite low (below 10 percent), even when the radial precision was high. The method outperformed the one in [START_REF] Biermé | Estimation of anisotropic Gaussian fields through Radon transform[END_REF][START_REF] Richard | Statistical tests of anisotropy for fractional Brownian textures. Application to full-field digital mammography[END_REF] which could only achieved the estimation with a low radial precision of 50 % (4 parameters) and with an error above 10 %. The coarse-to-fine multigrid strategy was critical for the estimation procedure. Without this strategy, the estimation error was above 10 % for a radial precision of 87.5 %.

Besides, the estimation error depended on the error made when estimating the field semi-variogram with the empirical semi-variogram (see Equation (3.2)): after having replaced the empirical semi-variogram by the field semi-variogram in the fitting criterion, the estimation error dropped down below 2 % for a radial precision of 87.5%. The estimation quality of the empirical semi-variogram depends on the size of the field realizations. Hence, to see the effect of this size on the estimation of model parameters, we conducted some other experiments on several sets of hundred realizations of a same size M ×M with a radial precision of 87.5%. On Figure [START_REF] Benassi | Elliptic Gaussian random processes[END_REF], estimation errors are reported as a function of the parameter size M .

We notice that the error significantly increased as the size of the image decreased. For a radial precision of 87.5%, a realization larger than 512 is required to ensure that the estimation error remains below 10 %.

Illustration

To illustrate the interest of our estimation procedure, we applied it to mammograms. Mammograms are projective images of the breast obtained by capturing part of an X-ray beam that passes through the adipose and fibro-glandular tissues organized in hierarchical matrix of fibrous compartments. Radiographic projections of this matrix create the mammographic texture, so called, parenchymal pattern. Mammographic texture has a critical effect on cancer detection, as it can hide early tumors (causing missed cancers) or mimic tumors (causing false positive findings). The statistical nature of the parenchymal pattern was acknowledged by several authors. In [START_REF] Burgess | Human observer detection experiments with mammograms and power-law noise[END_REF][START_REF] Heine | On the statistical nature of mammograms[END_REF][START_REF] Heine | Spectral analysis of full field digital mammography data[END_REF], a power-law noise model was proposed to characterize mammogram textures and study either the lesion detectability or assess the breast cancer risk. This model accounts for the self-similarity of mammographic textures, but it is global and stationary. In [START_REF] Kestener | Wavelet-based multifractal formalism to assist in diagnosis in digitized mammograms[END_REF][START_REF] Arneodo | A wavelet-based multifractal image analysis: from theoretical concepts to experimental applications[END_REF], another model was design using a wavelet-based multifractal formalism to locally characterize mammogram textures. In [START_REF] Richard | Statistical tests of anisotropy for fractional Brownian textures. Application to full-field digital mammography[END_REF], the AFBF were applied to mammograms to account for and analyze their anisotropy.

In this part, we propose to revisit the study of [START_REF] Richard | Statistical tests of anisotropy for fractional Brownian textures. Application to full-field digital mammography[END_REF] by taking the same dataset. This dataset was provided by the Department of Radiology of the University of Pennsylvania. Images of this dataset were acquired in medio-lateral oblique position using a Senographe 2000D (General Electric Medical Systems, Milwaukee, WI), with a spatial resolution of 0.1 mm × 0.1 mm per pixel (image size: 1914 × 2294 pixels). In each image of the database, a region of interest of size 512 × 512 was manually extracted within the densiest region of the breast. Originally, the dataset included a total of 58 cases, each case being composed of the left and right breasts of a woman. For this study, we removed 12 images that contained a part of the background (area outside the breast), causing an estimation failure.

We applied to these images our procedure to estimate the Hurst and topothesy functions represented as piecewise constant functions with 16 parameters each. We then studied several field features extracted from these functional estimates.

The first feature was the minimal value H of the Hurst function (called the Hurst index), that measures a degree of texture regularity. The repartitions of this index over the mammograms of left and right breasts are shown on histograms of Figure 6 (a). These histograms are similar to the ones published in [START_REF] Richard | Statistical tests of anisotropy for fractional Brownian textures. Application to full-field digital mammography[END_REF]: the Hurst index is also distributed in a same way on left and right breasts. The mean value is around 0.3; it is of 0.28 with the standard deviation of 0.11. In the previous distributions of [START_REF] Richard | Statistical tests of anisotropy for fractional Brownian textures. Application to full-field digital mammography[END_REF], there were much fewer index values in the interval (0.3, 0.5) than in the present distributions. Contrarily to the approach proposed in this paper, the model used in [START_REF] Richard | Statistical tests of anisotropy for fractional Brownian textures. Application to full-field digital mammography[END_REF] did not take into account the image noise. This probably led to an underestimation of the Hurst index.

The second feature was the length of the Hurst function range measured by the difference between its maximal and minimal values. In [START_REF] Richard | Statistical tests of anisotropy for fractional Brownian textures. Application to full-field digital mammography[END_REF], such a feature could not be precisely computed due to the fact that the estimation of the Hurst function was limited We can see that the length is more uniformly distributed over the interval (0, 1), and often above 0.3. This length is an indicator of the degree of texture anisotropy: the larger this length, the more anisotropic the texture. Hence, the result obtained in this study shows that the degree of anisotropy of mammograms is even larger than expected in [START_REF] Richard | Statistical tests of anisotropy for fractional Brownian textures. Application to full-field digital mammography[END_REF]. On Figure 7, some mammograms are shown together with their corresponding topothesy and Hurst functions. We can observe a typical form of the Hurst function which is peaked in one direction. The peak location indicates a direction where image variations have lowest frequencies. The peaks are also present in the topothesy functions at a same place as the ones of the Hurst function. The peaks of the Hurst and topothesy functions seem to vary in sharpness and height depending on the image anisotropy.

The Figure 7 also contains a simulation sampled from the estimated AFBF. These illustrations account for the ability of the simulated model to reproduce the mammogram aspect, including its regularity, directionality, anisotropy, trends and details. We observe that some fine structures present for instance in the fifth example could not be reproduced truthfully. Our simulation method is alternative to physic-based methods designed in [START_REF] Bakic | Mammogram synthesis using a 3D simulation. I. breast tissue model and image acquisition simulation[END_REF][START_REF]Mammogram synthesis using a 3D simulation. II. evaluation of synthetic mammogram texture[END_REF][START_REF]Mammogram synthesis using a 3D simulation. III. modeling and evaluation of the breast ductal network[END_REF][START_REF] Carton | Development of a physical 3D anthropomorphic breast phantom[END_REF].

Discussion

We designed a method for the estimation of the functional parameters of the anisotropic fractional Brownian field, namely the topothesy and Hurst functions. This method consisted of fitting the empirical semi-variogram computed from an image to the semivariogram of a turning-band field that approximates the AFBF. The fitting criterion was formulated in terms of a separable non-linear least square criterion, the linear part relating to the topothesy function and the non-linear part to the Hurst function. We then proposed an algorithm relying on a variable projection to minimize this criterion. This algorithm was combined with a coarse-to-fine multigrid strategy to improve its convergence. We presented a numerical study of the algorithm on simulated textures. In this study, the estimation error was evaluated as a function of a radial precision, directly related to the number of parameters appearing in the discrete representations of the Hurst and topothesy functions. This error was below ten percent for the finest radial precision, when the empirical semi-variogram was computed on large images. This error increased as the image size descreased. It reached values above 10 % when the image size was below 400 × 400. For illustration, the estimation method was applied to mammograms of a private dataset studied in a previous work. From the estimated functional parameters, we derived some field features that described the mammogram texture. We also sampled the estimated fields to generate some realistic simulations of mammograms. This work represents a breakthrough concerning the estimation of parameters of AFBF, which had only been partly and inaccurately achieved. Features can now be derived from the estimated field and used to classify textures. However, the actual method is not accurate enough to envisage its application to small-image classification or texture segmentation. Indeed, such an application require estimating model parameters in a local way on small neighborhoods of image pixels. One way to overcome this limitation would be to introduce penalizations of parameters within the fitting criterion. Such an approach was adopted for the estimation of the local Hurst index in [START_REF] Pascal | Automated data-driven selection of the hyperparameters for total-variation-based texture segmentation[END_REF][START_REF] Pascal | Strongly convex optimization for joint fractal feature estimation and texture segmentation[END_REF]. Penalty terms would constrain the model parameters in shape or space. For instance, they could be a TV-norm on topothesy and Hurst functions that would regularize the shape of these functions. As the penalty terms could be non-differentiable, our algorithm would no longer be appropriate for the minimization of the fitting criterion. It could however be adapted following the approach developed in [START_REF] Van Leeuwen | Variable projection for nonsmooth problems[END_REF] for separable non-linear least square criterion or, alternately, specifying generic algorithms for the minimization of non-smooth functions [START_REF] Wang | Global convergence of ADMM in nonconvex nonsmooth optimization[END_REF].

Fields with long range dependencies are non-stationary and have a semi-variogram which tends to infinity as the lag module tends to infinity. State-of-the art texture models [START_REF] Galerne | Random phase textures: Theory and synthesis[END_REF][START_REF] Galerne | A texton for fast and flexible gaussian texture synthesis[END_REF] are based on stationary Gaussian fields, whose variograms are bounded, and are therefore unable to simulate fields with long-range dependencies. Despite the ability of the AFBF model to render long-range dependencies, it is not appropriate for the modeling of fields with bounded variograms. Hence, the AFBF model should be used in complement to state-of-the-art methods for simulating images which do not fulfill the stationary assumption.

Besides, the facility brought by this work to sample realistic textures from AFBF is also of interest for applications. For instance, synthetic textures could be used for data augmentation in machine learning [START_REF] Shorten | A survey on image data augmentation for deep learning[END_REF]: they could supply examples for learning the neural network architectures in domains where there is a lack of data to achieve this task.

Bibliography

  ) and β 0 = β. end if end for Main iterations: Set τ = τ 0 and β = β 0 . Set iter = 0. for s = 0 to S do Project the current functions τ, β into the space T s × B s . Set iter = 0. repeat iter ← iter + 1. Set τ 0 = τ and β 0 = β Fix β and find τ * (β) = arg min τ ∈Ts L(τ, β). using lsq linear (method TRF). Fix τ = τ * (β) and approach arg min β∈Bs K(β).

Figure 3 .

 3 Figure 3. Lags where the semi-variograms are computed.

Figure 4 .

 4 Figure 4. Estimation error on the Hurst function as a function of the radial precision.

Figure 5 .

 5 Figure 5. Estimation error on the Hurst function with respect to the realization size.

Figure 6 .

 6 Figure 6. Histograms of (a) the Hurst index and (a) Hurst function range computed on the mammogram dataset.

HFigure 7 .

 7 Figure 7. Estimation of the topothesy (τ ) and Hurst (β) functions from some mammograms. First column: original image ROI, second and third columns, the graphs of τ and β estimated from the ROI of the same row. Below the graphs, the values of the different field features are also reported: H is Hurst index, H l the range of β. Last column: a synthetic image sampled from the AFBF estimated from the original image.

∇

  τ L(τ, β) = N n=1 ∇ τ ϵ n (τ, β) ϵ n (τ, β), ∇ β L(τ, β) = N n=1 ∇ β ϵ n (τ, β) ϵ n (τ, β).

(A. 6 )

 6 Using Equation (A.1), it follows that∇ τ L(τ, β) = N n=1 F n (β) T ϵ n (τ, β), ∇ β L(τ, β) = N n=1 DF n (β) T τ ϵ n (τ, β).

(A. 7 )F∇ 2 F

 72 The second-order partial derivatives of L are∂ 2 L ∂τ m ∂τ k (β) = N n=1 ∂ϵ n ∂τ m (τ, β) ∂ϵ n ∂τ k ϵ n (τ, β) + ∂ 2 ϵ n ∂τ m ∂τ k (τ, β) ϵ n (τ, β) , k ϵ n (τ, β) + ∂ 2 ϵ n ∂β m ∂β k (τ, β) ϵ n (τ, β) , k ϵ n (τ, β) + ∂ 2 ϵ n ∂τ m ∂β k (τ, β) ϵ n (τ, β) . (A.8)The Hessians of L with respect to the two variables (τ, β) are thus given by∇ 2 τ τ L(τ, β) = N n=1 ∇ τ ϵ n (τ, β)∇ τ ϵ n (τ, β) T + ϵ n (τ, β) ∇ 2 τ τ ϵ n (τ, τ ), ∇ 2 ββ L(τ, β) = N n=1 ∇ β ϵ n (τ, β)∇ β ϵ n (τ, β) T + ϵ n (τ, β) ∇ 2 ββ ϵ n (τ, β), ∇ 2 τ β L(τ, β) = N n=1 ∇ τ ϵ n (τ, β)∇ β ϵ n (τ, β) T + ϵ n (τ, β) ∇ 2 τ β ϵ n (τ, β). n (β) T F n (β), ∇ 2 ββ L(τ, β) = N n=1 DF n (β) T τ τ T DF n (β) + N n=1 ϵ n (τ, β) j=1 nj (β)τ j , ∇ 2 τ β L(τ, β) = N n=1 F n (β) T τ T DF n (β) + N n=1ϵ n (τ, β) DF n (β).

(A. 10 )

 10

Appendix A. Derivatives of L.

Let ϵ n (τ, β) = F n (β)τ -w n be the n-th residual of the function L. Gradients of ϵ n with respect to τ and β are

where DF n (β) is the jacobian matrix of F n (β) whose terms are (A.2)

The Hessians of ϵ n are given by

where ∇ 2 F nj (β) is the Hessian matrix of F nj whose terms are

The first-order derivatives of L are

The corresponding gradients of L with respect to the two variables (τ, β) are Appendix B. Derivatives of K.

Using the chain-rule, the first-order partial derivatives of K are

But, since τ * (β) minimizes L(•, β), we have

Hence, the gradient of K is given by

Using Equation (A.7), the expression of the gradient given in Equation ( 4.3) follows. Furthermore, second-order partial derivatives of K are given by

Hence, the Hessian of K is

, where Dτ * (β) is the Jacobian of τ * at β. Moreover, since u(β) = ∇ τ L(τ * (β), β) = 0, we have ∇u(β) = 0. Hence, using the chain-rule,

Consequently, the Hessian of K can be expressed as

ββ L(τ * (β), β) -∇ 2 βτ L(τ * (β), β)(∇ 2 τ τ L(τ * (β), β)) -1 ∇ 2 τ β L(τ * (β), β). Using Equation (A.10), the expression of the Hessian of K given by Equation (4.2) follows.