. .. Imagerie-médicale-et-géométrie,

. .. , Forces et limitations des réseaux convolutifs, p.203

, Travailler avec des surfaces, courbes et nuages de points, p.206

, Doter un espace de formes d'une structure géométrique, p.210

. .. , Une introduction accessible au domaine, p.215

, Développer des méthodes géométriques robustes, p.216

B. , Imagerie médicale et géométrie

L. Médicale, Au cours des quarante dernières années, dans nos sociétés occidentales, un facteur essentiel pour l'amélioration des standards de santé a été le perfectionnement constant des matériels d'imagerie. Bénéficiant du fruit de décennies de recherche, un médecin peut aujourd'hui inspecter l'intérieur de ses patients en quelques minutes. L'industrie médicale produit plusieurs milliers de scanners IRM

, Au fur et à mesure, les techniques modernes d'imagerie deviennent donc accessibles à toujours plus de patients. Malheureusement, la formation de radiologues qualifiés pour interpréter ce volume croissant d'images ne peut pas suivre la même cadence : le manque de ressources humaines est donc rapidement devenu le principal facteur limitant l'accès aux soins dans nos campagnes

, transformer un signal physique brut en une donnée utile est un processus complexe que nous détaillons Figure B.1. Dans cette thèse, nous nous concentrons sur un maillon spécifique de cette chaîne : l'analyse de données anatomiques. Partant d'images 3D produites par nos collègues en amont, nous tâchons d'extraire une information géométrique pertinente pour des analyses ultérieures. En deux mots, notre métier est de fournir une représentation de haut niveau, fiable et interprétable de l'anatomie d'un patient qui puisse être utilisée simplement par les médecins et statisticiens. Comme on le voit Figure B.2, cette question peut être découpée en trois grands types de problèmes : 1, Un mathématicien peut-il être utile? L'automatisation partielle d'examens cliniques est un problème difficile : de l'acquisition d'un scan IRM à l'estimation de tendances globales dans une population

, L'analyse de formes : quantifier les variations anatomiques d'un organe. 3. La simulation biomécanique : utiliser notre connaissance fine du corps humain pour extraire une information physiologique de simples images

L. , Cette thèse est consacrée au problème "intermédiaire" de l'analyse de formes, illustré Figure B.2.b. Nous aborderons parfois des questions liées à des domaines de recherche voisins ; mais en fin de compte, nos efforts se concentreront toujours sur des cas d'utilisation pertinents pour le traitement de données médicales

, Les médecins interviennent directement dans nos algorithmes en annotant les données. 2. Les statisticiens choisissent les meilleurs paramètres d'un algorithme

, Les développeurs implémentent des codes efficaces sur carte graphique pour tirer parti des dernières avancées matérielles

, Les mathématiciens et informaticiens encodent a priori et hypothèses dans l'architecture de leurs programmes

, L'architecture des réseaux convolutifs est partiellement inspirée par la structure du cortex visuel (Hubel and Wiesel, 1962, 1968) et de nombreux chercheurs rêvent de pouvoir un jour simuler de véritables cerveaux humains sur ordinateur. Il faut toutefois nous garder d'attribuer de trop nombreuses qualités à des algorithmes qui ne sont, après tout, Vers l'intelligence artificielle ? Le vocabulaire pseudo-biologique qui prévaut dans notre domaine découle des liens historiques entre recherches sur les "réseaux de neurones" et véritables neurosciences, 1980.

D. Skodras, Mais il serait déraisonnable d'espérer l'émergence de comportements de haut niveau dans ces algorithmes rudimentaires : empiler des filtres de convolution les uns sur les autres n'a pas, les réseaux convolutifs sont les proches cousins d'algorithmes classiques comme la transformée en ondelettes rapide qui sous-tend le standard de compression JPEG-2000 pour le cinéma numérique, 2001.

. Nowak, Néanmoins, même après une coûteuse phase d'entraînement, les réseaux convolutifs restent toujours fortement biaisés vers l'analyse de texture et la détection de motifs, Limitations des algorithmes convolutionnels. Le traitement d'images et la vision par ordinateur ont considérablement progressé depuis les premiers travaux sur les pyramides de Laplace (Burt and Adelson, 1983) ou les descripteurs SIFT, 1999.

, 2 L'analyse de données géométriques, p.207

, 2 L'analyse de données géométriques

, En venant des sciences des données ou des statistiques, la spécificité notable de l'analyse de formes est l'absence des opérations algébriques "+" et "×". Calculer la somme de deux cerveaux ne fait guère de sens, et aucun modèle de coeur canonique ne pourra vraiment remplacer l, Les formes ne sont pas des vecteurs

, Heureusement, des distances entre formes peuvent toujours être définies : dire que deux crânes sont "proches" ou "éloignés" l'un de l'autre peut être tout à fait légitime. Un problème d'intérêt en anatomie computationnelle est donc de définir des structures métriques sur des espaces de formes qui soient : 1. Anatomiquement pertinentes et raisonnables du point de vue médical, Une théorie géométrique pour les données géométriques

, Algorithmiquement peu coûteuses pour être déployables sur des données cliniques

;. .. L'analyse-procrustéenne, N ) dans R N×D sont deux nuages de points non dégénérés (i.e. pas réduits à un seul point), on dira que x et y ont la même forme au sens de Procruste s'il existe un vecteur de translation ? ? R D , une matrice de rotation R ? O(D) ? R D×D, La plus simple de toutes ces métriques est celle que l'on peut définir sur l'espace des polygones indexés, définis à similitude près

. Géodésiques, En allant plus loin, on définit les courbes géodésiques comme des chemins continus ? : t ? [0, 1] ? ?(t) ? S qui minimisent la longueur localement -mais pas nécessairement entre leurs extrémités, pour cause de courbure et non-unicité, En un mot : les géodésiques sont des lignes droites généralisées

(. Si and . Est-une-variété-riemannienne--i.e, si elle est localement équivalente à un espace euclidien, comme une sphère est localement équivalente à ses plans tangents -on peut montrer que toute géodésique obéit à une équation différentielle d'ordre 2 (Lee, 2006)

. Fletcher, Le segment géodésique ? i entre une moyenne x * et un sujet x i est paramétré par sa vitesse initiale v i = d dt ? i (t = 0). Cela nous permet de réaliser des analyses statistiques comme l'ACP dans l'espace tangent T x * S à l'espace de formes S au point x *, Régression géodésique, modèles longitudinaux. En identifiant les géodésiques aux lignes droites, on peut généraliser la régression linéaire aux espaces de formes, 2004.

. Durrleman, Dans de nombreux cas d'application cliniques, on cherche à comparer la trajectoire d'un patient à la tendance globale de son groupe, Comme illustré Figure B, vol.18, 2013.

. Montagnat, Si D = 2 ou 3 et si ? : R D ? R D est une application qui envoie une forme source A sur une cible ?(A) = B, l'écart entre ? et la fonction identité Id : x ? R D ? x ? R D peut être utilisé pour définir une distance d(A, B). Réciproquement, une déformation plausible ? peut être comprise comme une géodésique qui transforme petit à petit Id(A) = A en ?(A) = B en suivant, pour une certaine métrique, une trajectoire de moindre effort dans un espace de déformations, Analyse de formes et recalage. Au cours des vingt dernières années, une littérature abondante s'est structurée autour de l'idée que des métriques entre formes peuvent être définies à partir d'algorithmes de recalage, 2001.

. Von-tycowicz, Un manuel de référence sur le sujet a récemment été publié, avec de nombreuses applications aux statistiques, ou via des déformations SVF (Arsigny et al., 2006) et LDDMM (Beg et, 1999.

. Ollier, il nous faut donc résoudre les problèmes suivants : 1. Vitesse d'exécution. Recaler finement deux volumes 3D à l'aide d'algorithmes itératifs prend au mieux une poignée de secondes, En 2020, la communauté semble avoir atteint les limites de ce qui peut raisonnablement être réalisé à l'aide d'équations explicites et de codes Matlab ou C++, 2005.

. Pertinence, Aujourd'hui, bien peu de modèles peuvent vraiment extrapoler de manière vraisemblable en dehors de leurs bases d'entraînement : les modèles à noyaux, SVF ou LDDMM reposent sur des hypothèses de régularité qui ne font pas beaucoup de sens d'un point de vue médical

. Durrleman, Pour apporter une réponse à ces questions, les chercheurs se sont principalement concentré sur des implémentations multi-échelles, ou basses fréquences (Zhang and Fletcher, 2019) d'algorithmes standards, sur des hypothèses de régularité implicites, 2014.

, Dans des cadres favorables comme la neuro-anatomie, il semble maintenant possible de calculer des déformations vraisemblables en quelques fractions de seconde. Toutefois, aucune réponse pleinement satisfaisante n'a encore pu être fournie à la question de la pertinence anatomique : le recherche de métriques robustes et inspirées par les données

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