Segmentation of 3D pore space from CT images using curvilinear skeleton: application to numerical simulation of microbial decomposition

Recent advances in 3D X-ray Computed Tomographic (CT) sensors have stimulated research efforts to unveil the extremely complex micro-scale processes that control the activity of soil microorganisms. Voxel-based description (up to hundreds millions voxels) of the pore space can be extracted, from grey level 3D CT scanner images, by means of simple image processing tools. Classical methods for numerical simulation of biological dynamics using mesh of voxels, such as Lattice Boltzmann Model (LBM), are too much time consuming. Thus, the use of more compact and reliable geometrical representations of pore space can drastically decrease the computational cost of the simulations. Several recent works propose basic analytic volume primitives (e.g. spheres, generalized cylinders, ellipsoids) to define a piece-wise approximation of pore space for numerical simulation of draining, diffusion and microbial decomposition. Such approaches work well but the drawback is that it generates approximation errors. In the present work, we study another alternative where pore space is described by means of geometrically relevant connected subsets of voxels (regions) computed from the curvilinear skeleton. Indeed, many works use the curvilinear skeleton (3D medial axis) for analyzing and partitioning 3D shapes within various domains (medicine, material sciences, petroleum engineering, etc.) but only a few ones in soil sciences. Within the context of soil sciences, most studies dealing with 3D medial axis focus on the determination of pore throats. Here, we segment pore space using curvilinear skeleton in order to achieve numerical simulation of microbial decomposition (including diffusion processes). We validate simulation outputs by comparison with other methods using different pore space geometrical representations (balls, voxels).Comment: preprint, submitted to Computers & Geosciences 202

Blending pieces of Dupin cyclides for 3D modeling and reconstruction : study in the space of spheres

La thèse porte sur le raccordement de surfaces canal en modélisation géométriques en utilisant des morceaux de cyclides de Dupin. Elle tente de répondre à un problème de reconstruction de pièces controlées et usinées par le CEA de Valduc. En se plaçant dans l'espace adéquat, l'espace des sphères, dans lequel nous pouvons manipuler à la fois les points, les sphères et les surfaces canal, nous simplifions considérablement certains problèmes. Cet espace est représenté par une quadrique de dimension 4 dans un espace de dimension 5, muni de la forme de Lorentz : l'espace de Lorentz. Dans l'espace des sphères, les problèmes de recollements de surfaces canal par des morceaux de cyclides de Dupin se simplifient en problèmes linéaires. Nous donnons les algorithmes permettant de réaliser ce type de jointures en utilisant l'espace des sphères puis nous revenons dans l'espace à 3 dimensions usuel. Ces jointures se font toujours le long de cercles caractéristiques des surfaces considérées. En résolvant le problème dit des trois conditions de contact, nous mettons en évidence une autre courbe particulière, sur une famille à un paramètre de cyclides, que nous appellons courbe de contact qui permettrait d'effectuer des jointures le long d'autres courbesThe thesis deals with the blending of canal surfaces in geometric modeling using pieces of Dupin Cyclides. We try to solve a problem of reconstructing real parts manufactured and controlled by the CEA of Valduc. Using the space of spheres in which we can manipulate both points, spheres and canal surfaces, we simplify some problems. This space is represented by a 4-dimensional quadric in a 5-dimensional space, equipped with the Lorentz form, it is the Lorentz space. In the space of spheres, problems of blending canal surfaces by pieces of Dupin cyclides are simplified in linear problems. We give algorithms to make such blends using the space of spheres and after we come back to 3 dimensions to draw the result. These blends are always made along characteristics circles of the considered surfaces. By solving the problem of three contact conditions, we highlight another particular curve, on a one parameter familly of cyclides, that we call contact curve along which we could also make this kind of blend

Recollements de morceaux de cyclides de Dupin pour la modélisation et la reconstruction 3D : étude dans l'espace des sphères

The thesis deals with the blending of canal surfaces in geometric modeling using pieces of Dupin Cyclides. We try to solve a problem of reconstructing real parts manufactured and controlled by the CEA of Valduc. Using the space of spheres in which we can manipulate both points, spheres and canal surfaces, we simplify some problems. This space is represented by a 4-dimensional quadric in a 5-dimensional space, equipped with the Lorentz form, it is the Lorentz space. In the space of spheres, problems of blending canal surfaces by pieces of Dupin cyclides are simplified in linear problems. We give algorithms to make such blends using the space of spheres and after we come back to 3 dimensions to draw the result. These blends are always made along characteristics circles of the considered surfaces. By solving the problem of three contact conditions, we highlight another particular curve, on a one parameter familly of cyclides, that we call contact curve along which we could also make this kind of blendsLa thèse porte sur le raccordement de surfaces canal en modélisation géométriques en utilisant des morceaux de cyclides de Dupin. Elle tente de répondre à un problème de reconstruction de pièces controlées et usinées par le CEA de Valduc. En se plaçant dans l'espace adéquat, l'espace des sphères, dans lequel nous pouvons manipuler à la fois les points, les sphères et les surfaces canal, nous simplifions considérablement certains problèmes. Cet espace est représenté par une quadrique de dimension 4 dans un espace de dimension 5, muni de la forme de Lorentz : l'espace de Lorentz. Dans l'espace des sphères, les problèmes de recollements de surfaces canal par des morceaux de cyclides de Dupin se simplifient en problèmes linéaires. Nous donnons les algorithmes permettant de réaliser ce type de jointures en utilisant l'espace des sphères puis nous revenons dans l'espace à 3 dimensions usuel. Ces jointures se font toujours le long de cercles caractéristiques des surfaces considérées. En résolvant le problème dit des trois conditions de contact, nous mettons en évidence une autre courbe particulière, sur une famille à un paramètre de cyclides, que nous appellons courbe de contact qui permettrait d'effectuer des jointures le long d'autres courbe

Blending canal surfaces along given circles using Dupin cyclides

International audienc

Points massiques, hyperbole et hyperboloïde à une nappe

National audienceLes courbes de Bézier rationnelles quadratiques jouent un rôle fondamental pour la modélisation d'arcs de coniques propre. Cependant, lorsque les deux points extrémaux de l'arc ne sont pas sur la même branche d'une hyperbole, l'utilisation des courbes de Bézier classiques est impossible. Il suffit de considérer les points massiques, à la place des points pondérés, pour remédier à ce problème. De plus, nous gardons la structure (pseudo)-métrique du plan dans lequel nous nous trouvons et il possible de modéliser une branche d'hyperbole dont les extrémités sont deux vecteurs, non colinéaires, de même norme, définis par les directions des asymptotes. Nous donnons comme application le tracé d'une branche d'hyperbole sur un hyperboloïde à une nappe. Afin de simplifier ce travail, nous utilisons une forme quadratique de signature (2; 1) permettant de manipuler cette quadrique comme une sphère unité : ainsi, les courbes seront manipulables comme des cercles. It is well known that proper conic arcs can be modeled by rational quadratic Bézier curves but we can not use Bézier curves when the two endpoints of the arc do not belong to the same branch of a hyperbola. A solution is the use of massic points and moreover, it is easy to model a branch of hyperbole : the bounds are vectors having the same direction than the asymptote of the hyperbola and we can determine hyperbola parameters. The use of the massic points does not depend on the Euclidean structure : we can define a quadratic form such as the hyperboloid of revolution of one sheet is seen as an unit sphere. Then, the hyperbolae on this quadric have almost the properties of a circle (with two asympototes). Mots-clés : Hyperbole, points massiques, courbe de Bé-zier, forme quadratique, hyperboloïde à une nappe, cercle

Iterative constructions of central conic arcs using non-stationary IFS

Poster à Solid ModelingSeveral methods of subdivision exist to build parabola arcs or circle arcs in the usual Euclidean affine plane. Using a compass and a ruler, it is possible to construct, from three weighted points, circles arcs in the affine space without projective considerations. This construction is based on rational quadratic Bézier curve properties. However, when the conic is an ellipse or a hyperbola, the weight computation is relatively hard. As the equation of a conic is \qaff(x,y)=1, where \qaff is a quadratic form, one can use the pseudo-metric associed to \qaff in the affine plane and then, the conic geometry is also handled as an Euclidean circle. At each step of the iterative algorithm, the constructed point belongs to a principal perpendicular bissector of the control polyhedron and then, our construction is regular. Moreover, we can pass through the point at infinity when the bounds do not belong to the same branch of the hyperbola, using massic points defined by J.C. Fiorot: we compute two subdivisions with two collinear direction vectors of the same asymptote such that these two vectors have opposite senses. Moreover, at each step, we know the tangent lines to the conic at the built vertex. At each step, two weights are equal to $1$ or $0$, we just have to find an induction relation to compute the third weight and so, it is possible to run our algorithms using a non-stationary I.F.S

Iterative construction of Dupin cyclides characteristic circles using non-stationary Iterated Function Systems (IFS)

International audienceA Dupin cyclide can be defined, in two different ways, as the envelope of an one-parameter family of oriented spheres. Each family of spheres can be seen as a conic in the space of spheres. In this paper, we propose an algorithm to compute a characteristic circle of a Dupin cyclide from a point and the tangent at this point in the space of spheres. Then, we propose iterative algorithms (in the space of spheres) to compute (in 3D space) some characteristic circles of a Dupin cyclide which blends two particular canal surfaces. As a singular point of a Dupin cyclide is a point at infinity in the space of spheres, we use the massic points defined by J.C. Fiorot. As we subdivide conic arcs, these algorithms are better than the previous algorithms developed by Garnier and Gentil

A Survey of De Casteljau Algorithms and Regular Iterative Constructions of Bézier Curves with Control Mass Points

International audienceDrawing a curve on a computer actually involves approximating it by a set of segments. The De Casteljau algorithm allows to construct these piecewise linear curves which approximate polynomial Béziercurves using convex combinations. However, for rational Bézier curves,the construction no longer admits regular sampling. To solve thisproblem, we propose a generalization of the De Casteljau algorithmthat addresses this issue and is applicable to Bézier curves withmass points (a weighted point or a vector) as control points and using a homographic parameter change dividing the interval $\begin{bmatrix}0,1\end{bmatrix}$into two equal-length intervals $\begin{bmatrix}0,\frac{1}{2}\end{bmatrix}$ and $\begin{bmatrix}\frac{1}{2},1\end{bmatrix}$. If the initial Bézier curve is in standard form, we obtain two curves in standard form, unless the mass endpoint of the curve is a vector. This homographic parameter change also allows transforming curves defined over an interval $\begin{bmatrix}\alpha,+\infty \end{bmatrix}$, \alpha\in \r, into Bézier curves, which then enables the use of the De Casteljau algorithm. Some examples are given: three-quart of circle, semicircle and a branch of a hyperbola (degree $2$), cubic curve on $\begin{bmatrix}0;+\infty \end{bmatrix}$ and loop of a Descartes Folium (degree $3$) and a loop of a Bernouilli Lemniscate (degree $4$)

Recollements de morceaux de cyclides de Dupin pour la modélisation et la reconstruction 3D (étude dans l'espace des sphères)

La thèse porte sur le raccordement de surfaces canal en modélisation géométriques en utilisant des morceaux de cyclides de Dupin. Elle tente de répondre à un problème de reconstruction de pièces controlées et usinées par le CEA de Valduc. En se plaçant dans l'espace adéquat, l'espace des sphères, dans lequel nous pouvons manipuler à la fois les points, les sphères et les surfaces canal, nous simplifions considérablement certains problèmes. Cet espace est représenté par une quadrique de dimension 4 dans un espace de dimension 5, muni de la forme de Lorentz : l'espace de Lorentz. Dans l'espace des sphères, les problèmes de recollements de surfaces canal par des morceaux de cyclides de Dupin se simplifient en problèmes linéaires. Nous donnons les algorithmes permettant de réaliser ce type de jointures en utilisant l'espace des sphères puis nous revenons dans l'espace à 3 dimensions usuel. Ces jointures se font toujours le long de cercles caractéristiques des surfaces considérées. En résolvant le problème dit des trois conditions de contact, nous mettons en évidence une autre courbe particulière, sur une famille à un paramètre de cyclides, que nous appellons courbe de contact qui permettrait d'effectuer des jointures le long d'autres courbesThe thesis deals with the blending of canal surfaces in geometric modeling using pieces of Dupin Cyclides. We try to solve a problem of reconstructing real parts manufactured and controlled by the CEA of Valduc. Using the space of spheres in which we can manipulate both points, spheres and canal surfaces, we simplify some problems. This space is represented by a 4-dimensional quadric in a 5-dimensional space, equipped with the Lorentz form, it is the Lorentz space. In the space of spheres, problems of blending canal surfaces by pieces of Dupin cyclides are simplified in linear problems. We give algorithms to make such blends using the space of spheres and after we come back to 3 dimensions to draw the result. These blends are always made along characteristics circles of the considered surfaces. By solving the problem of three contact conditions, we highlight another particular curve, on a one parameter familly of cyclides, that we call contact curve along which we could also make this kind of blendsDIJON-BU Doc.électronique (212319901) / SudocSudocFranceF

Approximation of pore space with ellipsoids: a comparison of a geometrical method with a statistical one.

International audienceWe work with tomographic images of pore space in soil. The images have large dimensions and so in order to speed-up biological simulations (as drainage or diffusion process in soil), we want to describe the pore space with a number of geometrical primitives significantly smaller than the number of voxels in pore space. In this paper, we use the curve skeleton of a volume to segment it into some regions. We describe the method to compute the curve skeleton and to segment it with a simple segment approximation. We approximate each obtained region with an ellipsoid. The set of final ellipsoids represents the geometry of pore space and will be used in future simulations. We compare this method which we call geometrical method with the one described in the paper [8], which we name statistical method (using k-means algorithm)