다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

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Convex Analysisの二,三の進展について

This is a short presentation of the freefem++ software. In Section 1, we recall most of the characteristics of the software, In Section 2, we recall how to to build the weak form of a partial differential equation (PDE) from the strong form. In the 3 last sections, we present different examples and tools to illustrated the power of the software. First we deal with mesh adaptation for problems in two and three dimension, second, we solve numerically a problem with phase change and natural convection, and the finally to show the possibilities for HPC we solve a Laplace equation by a Schwarz domain decomposition problem on parallel computer.

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New development in freefem

Computational geometry

Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions, two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a "global" cost to every such transport, using the "local" consideration of how much it costs to move a grain of sand from one place to another. Recent years have witnessed the spread of OT in several fields, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.

Computational Optimal Transport

We introduce a modification of the Fast Marching Algorithm, which solves the generalized eikonal equation associated to an arbitrary continuous riemannian metric, on a two or three dimensional box domain. The algorithm has a logarithmic complexity in the maximum anisotropy ratio of the riemannian metric, which allows to handle extreme anisotropies for a reduced numerical cost. We establish that the output of the algorithm converges towards the viscosity solution of continuous problem, as the discretization step tends to zero. The algorithm is based on the computation at each grid point of a reduced basis of the unit lattice, with respect to the symmetric positive definite matrix encoding the desired anisotropy at this point.

https://hal.archives-ouvertes.fr/hal-00657608/document

Anisotropic Fast-Marching on cartesian grids using Lattice Basis Reduction

We introduce a novel discretization of the Monge-Ampere operator, simultaneously consistent and degenerate elliptic, hence accurate and robust in applications. These properties are achieved by exploiting the arithmetic structure of the discrete domain, assumed to be a two dimensional cartesian grid. The construction of our scheme is simple, but its analysis relies on original tools seldom encountered in numerical analysis, such as the geometry of two dimensional lattices, and an arithmetic structure called the Stern-Brocot tree. Numerical experiments illustrate the method's efficiency.

https://hal.archives-ouvertes.fr/hal-01067540/document

Monotone and consistent discretization of the Monge-Ampère operator

We study the discretization of the escape time problem: find the length of the shortest path joining an arbitrary point $$z$$ z of a domain $$\Omega $$ Ω , to the boundary $$\partial \Omega $$ ? Ω . Path length is measured locally via a Finsler metric, potentially asymmetric and strongly anisotropic. This optimal control problem can be reformulated as a static Hamilton---Jacobi partial differential equation, or as a front propagation model. It has numerous applications, ranging from motion planning to image segmentation. We introduce a new algorithm, fast marching using anisotropic stencil refinement (FM-ASR), which addresses this problem on a two dimensional domain discretized on a cartesian grid. The local stencils used in our discretization are produced by arithmetic means, like in the FM-LBR (Mirebeau in Anisotropic fast Marching on Cartesian grids, using lattice basis reduction, preprint 2012), a method previously introduced by the author in the special case of Riemannian metrics. The complexity of the FM-ASR, in an average sense over all grid orientations, only depends (poly-)logarithmically on the anisotropy ratio of the metric, while most alternative approaches have a polynomial dependence. Numerical experiments show, in several occasions, that the accuracy/complexity compromise is improved by an order of magnitude or more.

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Efficient fast marching with Finsler metrics

In this paper, we propose a novel curvature penalized minimal path model via an orientation-lifted Finsler metric and the Euler elastica curve. The original minimal path model computes the globally minimal geodesic by solving an Eikonal partial differential equation (PDE). Essentially, this first-order model is unable to penalize curvature which is related to the path rigidity property in the classical active contour models. To solve this problem, we present an Eikonal PDE-based Finsler elastica minimal path approach to address the curvature-penalized geodesic energy minimization problem. We were successful at adding the curvature penalization to the classical geodesic energy (Caselles et al. in Int J Comput Vis 22(1):61---79, 1997; Cohen and Kimmel in Int J Comput Vis 24(1):57---78, 1997). The basic idea of this work is to interpret the Euler elastica bending energy via a novel Finsler elastica metric that embeds a curvature penalty. This metric is non-Riemannian, anisotropic and asymmetric, and is defined over an orientation-lifted space by adding to the image domain the orientation as an extra space dimension. Based on this orientation lifting, the proposed minimal path model can benefit from both the curvature and orientation of the paths. Thanks to the fast marching method, the global minimum of the curvature-penalized geodesic energy can be computed efficiently. We introduce two anisotropic image data-driven speed functions that are computed by steerable filters. Based on these orientation-dependent speed functions, we can apply the proposed Finsler elastica minimal path model to the applications of closed contour detection, perceptual grouping and tubular structure extraction. Numerical experiments on both synthetic and real images show that these applications of the proposed model indeed obtain promising results.

Global Minimum for a Finsler Elastica Minimal Path Approach

We present a PDE-based approach for finding optimal paths for the Reeds–Shepp car. In our model we minimize a (data-driven) functional involving both curvature and length penalization, with several generalizations. Our approach encompasses the two- and three-dimensional variants of this model, state-dependent costs, and moreover, the possibility of removing the reverse gear of the vehicle. We prove both global and local controllability results of the models. Via eikonal equations on the manifold $$\mathbb {R}^d \times {\mathbb {S}}^{d-1}$$
 we compute distance maps w.r.t. highly anisotropic Finsler metrics, which approximate the singular (quasi)-distances underlying the model. This is achieved using a fast-marching (FM) method, building on Mirebeau (Numer Math 126(3):515–557, 2013; SIAM J Numer Anal 52(4):1573–1599, 2014). The FM method is based on specific discretization stencils which are adapted to the preferred directions of the Finsler metric and obey a generalized acuteness property. The shortest paths can be found with a gradient descent method on the distance map, which we formalize in a theorem. We justify the use of our approximating metrics by proving convergence results. Our curve optimization model in $$\mathbb {R}^{d} \times \mathbb {S}^{d-1}$$
 with data-driven cost allows to extract complex tubular structures from medical images, e.g., crossings, and incomplete data due to occlusions or low contrast. Our work extends the results of Sanguinetti et al. (Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications LNCS 9423, 2015) on numerical sub-Riemannian eikonal equations and the Reeds–Shepp car to 3D, with comparisons to exact solutions by Duits et al. (J Dyn Control Syst 22(4):771–805, 2016). Numerical experiments show the high potential of our method in two applications: vessel tracking in retinal images for the case $$d=2$$
 and brain connectivity measures from diffusion-weighted MRI data for the case $$d=3$$
 , extending the work of Bekkers et al. (SIAM J Imaging Sci 8(4):2740–2770, 2015). We demonstrate how the new model without reverse gear better handles bifurcations.

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Jean-Marie Mirebeau

Papers

Anisotropic Fast-Marching on cartesian grids using Lattice Basis Reduction

Monotone and consistent discretization of the Monge-Ampère operator

Efficient fast marching with Finsler metrics

Global Minimum for a Finsler Elastica Minimal Path Approach

Optimal Paths for Variants of the 2D and 3D Reeds–Shepp Car with Applications in Image Analysis