33 research outputs found
Sub-Riemannian Fast Marching in SE(2)
We propose a Fast Marching based implementation for computing sub-Riemanninan
(SR) geodesics in the roto-translation group SE(2), with a metric depending on
a cost induced by the image data. The key ingredient is a Riemannian
approximation of the SR-metric. Then, a state of the art Fast Marching solver
that is able to deal with extreme anisotropies is used to compute a SR-distance
map as the solution of a corresponding eikonal equation. Subsequent
backtracking on the distance map gives the geodesics. To validate the method,
we consider the uniform cost case in which exact formulas for SR-geodesics are
known and we show remarkable accuracy of the numerically computed SR-spheres.
We also show a dramatic decrease in computational time with respect to a
previous PDE-based iterative approach. Regarding image analysis applications,
we show the potential of considering these data adaptive geodesics for a fully
automated retinal vessel tree segmentation.Comment: CIARP 201
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
A PDE Approach to Data-driven Sub-Riemannian Geodesics in SE(2)
We present a new flexible wavefront propagation algorithm for the boundary
value problem for sub-Riemannian (SR) geodesics in the roto-translation group
with a metric tensor depending on a smooth
external cost , , computed from
image data. The method consists of a first step where a SR-distance map is
computed as a viscosity solution of a Hamilton-Jacobi-Bellman (HJB) system
derived via Pontryagin's Maximum Principle (PMP). Subsequent backward
integration, again relying on PMP, gives the SR-geodesics. For
we show that our method produces the global minimizers. Comparison with exact
solutions shows a remarkable accuracy of the SR-spheres and the SR-geodesics.
We present numerical computations of Maxwell points and cusp points, which we
again verify for the uniform cost case . Regarding image
analysis applications, tracking of elongated structures in retinal and
synthetic images show that our line tracking generically deals with crossings.
We show the benefits of including the sub-Riemannian geometry.Comment: Extended version of SSVM 2015 conference article "Data-driven
Sub-Riemannian Geodesics in SE(2)
Cortical Functional architectures as contact and sub-Riemannian geometry
In a joint paper, Jean Petitot together with the authors of the present paper
described the functional geometry of the visual cortex as the symplectization
of a contact form to describe the family of cells sensitive to position,
orientation and scale. In the present paper, as a "homage" to the enormous
contribution of Jean Petitot to neurogeometry, we will extend this approach to
more complex functional architectures built as a sequence of contactization or
a symplectization process, able to extend the dimension of the space. We will
also outline a few examples where a sub-Riemannian lifting is needed