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In this paper, we introduce a set of methods for processing and analyzing long time series of 3D images representing embryo evolution. The images are obtained by in vivo scanning using a confocal microscope where one of the channels represents the cell nuclei and the other one the cell membranes. Our image processing chain consists of three steps: image filtering, object counting (center detection) and segmentation. The corresponding methods are based on numerical solution of nonlinear PDEs, namely...
Given a domain of class , , we construct a chart that maps normals to the boundary of the half space to normals to the boundary of in the sense that and that still is of class . As an application we prove the existence of a continuous extension operator for all normal derivatives of order 0 to on domains of class . The construction of this operator is performed in weighted function spaces where the weight function is taken from the class of Muckenhoupt weights.
Soit un opérateur pseudodifférentiel (ou microdifférentiel) tel que soit aussi un opérateur pseudodifférentiel. Alors le symbole de s’ecrit avec un symbole . Pour la réciproque, si est un opérateur à symbole , il existe un opérateur tel que . Tous ces résultats reposent sur la théorie développée dans la Note I de cette série. Comme application, on obtient une condition suffisante d’inversibilité pour les opérateurs pseudodifférentiels d’ordre infini.
Cet article s’intéresse au calcul symbolique des opérateurs microdifférentiels avec symboles exponentiels. On donne la loi de composition des symboles exponentiels. Comme application, on trouve une condition suffisante d’ellipticité pour les opérateurs microdifférentiels d’ordre infini.
In this paper, we describe an efficient method for 3D image segmentation. The method uses a PDE model – the so called generalized subjective surface equation which is an equation of advection-diffusion type. The main goal is to develop an efficient and stable numerical method for solving this problem. The numerical solution is based on semi-implicit time discretization and flux-based level set finite volume space discretization. The space discretization is discussed in details and we introduce three...
In recent years the study of interpolation of Banach spaces has seen some unexpected interactions with other fields. (...) In this paper I shall discuss some more interactions of interpolation theory with the rest of mathematics, beginning with some joint work with Coifman [CS]. Our basic idea was to look for the methods of interpolation that had interesting PDE's arising as examples.
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