2,514 research outputs found
Anisotropic Fast-Marching on cartesian grids using Lattice Basis Reduction
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 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 prove
the consistence of the algorithm, and illustrate its efficiency by numerical
experiments. The algorithm relies on the computation at each grid point of a
special system of coordinates: a reduced basis of the cartesian grid, with
respect to the symmetric positive definite matrix encoding the desired
anisotropy at this point.Comment: 28 pages, 12 figure
Automatic segmentation of the left ventricle cavity and myocardium in MRI data
A novel approach for the automatic segmentation has been developed to extract the epi-cardium and endo-cardium boundaries of the left ventricle (lv) of the heart. The developed segmentation scheme takes multi-slice and multi-phase magnetic resonance (MR) images of the heart, transversing the short-axis length from the base to the apex. Each image is taken at one instance in the heart's phase. The images are segmented using a diffusion-based filter followed by an unsupervised clustering technique and the resulting labels are checked to locate the (lv) cavity. From cardiac anatomy, the closest pool of blood to the lv cavity is the right ventricle cavity. The wall between these two blood-pools (interventricular septum) is measured to give an approximate thickness for the myocardium. This value is used when a radial search is performed on a gradient image to find appropriate robust segments of the epi-cardium boundary. The robust edge segments are then joined using a normal spline curve. Experimental results are presented with very encouraging qualitative and quantitative results and a comparison is made against the state-of-the art level-sets method
Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D
We propose an efficient approach for the grouping of local orientations
(points on vessels) via nilpotent approximations of sub-Riemannian distances in
the 2D and 3D roto-translation groups and . In our distance
approximations we consider homogeneous norms on nilpotent groups that locally
approximate , and which are obtained via the exponential and logarithmic
map on . In a qualitative validation we show that the norms provide
accurate approximations of the true sub-Riemannian distances, and we discuss
their relations to the fundamental solution of the sub-Laplacian on .
The quantitative experiments further confirm the accuracy of the
approximations. Quantitative results are obtained by evaluating perceptual
grouping performance of retinal blood vessels in 2D images and curves in
challenging 3D synthetic volumes. The results show that 1) sub-Riemannian
geometry is essential in achieving top performance and 2) that grouping via the
fast analytic approximations performs almost equally, or better, than
data-adaptive fast marching approaches on and .Comment: 18 pages, 9 figures, 3 tables, in review at JMI
Can local single-pass methods solve any stationary Hamilton-Jacobi-Bellman equation?
The use of local single-pass methods (like, e.g., the Fast Marching method)
has become popular in the solution of some Hamilton-Jacobi equations. The
prototype of these equations is the eikonal equation, for which the methods can
be applied saving CPU time and possibly memory allocation. Then, some natural
questions arise: can local single-pass methods solve any Hamilton-Jacobi
equation? If not, where the limit should be set? This paper tries to answer
these questions. In order to give a complete picture, we present an overview of
some fast methods available in literature and we briefly analyze their main
features. We also introduce some numerical tools and provide several numerical
tests which are intended to exhibit the limitations of the methods. We show
that the construction of a local single-pass method for general Hamilton-Jacobi
equations is very hard, if not impossible. Nevertheless, some special classes
of problems can be actually solved, making local single-pass methods very
useful from the practical point of view.Comment: 19 page
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