52 research outputs found
Automatic differentiation of non-holonomic fast marching for computing most threatening trajectories under sensors surveillance
We consider a two player game, where a first player has to install a
surveillance system within an admissible region. The second player needs to
enter the the monitored area, visit a target region, and then leave the area,
while minimizing his overall probability of detection. Both players know the
target region, and the second player knows the surveillance installation
details.Optimal trajectories for the second player are computed using a
recently developed variant of the fast marching algorithm, which takes into
account curvature constraints modeling the second player vehicle
maneuverability. The surveillance system optimization leverages a reverse-mode
semi-automatic differentiation procedure, estimating the gradient of the value
function related to the sensor location in time N log N
Geodesic Tracking via New Data-driven Connections of Cartan Type for Vascular Tree Tracking
We introduce a data-driven version of the plus Cartan connection on the
homogeneous space of 2D positions and orientations. We formulate
a theorem that describes all shortest and straight curves (parallel velocity
and parallel momentum, respectively) with respect to this new data-driven
connection and corresponding Riemannian manifold. Then we use these shortest
curves for geodesic tracking of complex vasculature in multi-orientation image
representations defined on . The data-driven Cartan connection
characterizes the Hamiltonian flow of all geodesics. It also allows for
improved adaptation to curvature and misalignment of the (lifted) vessel
structure that we track via globally optimal geodesics. We compute these
geodesics numerically via steepest descent on distance maps on
that we compute by a new modified anisotropic fast-marching method.
Our experiments range from tracking single blood vessels with fixed endpoints
to tracking complete vascular trees in retinal images. Single vessel tracking
is performed in a single run in the multi-orientation image representation,
where we project the resulting geodesics back onto the underlying image. The
complete vascular tree tracking requires only two runs and avoids prior
segmentation, placement of extra anchor points, and dynamic switching between
geodesic models.
Altogether we provide a geodesic tracking method using a single, flexible,
transparent, data-driven geodesic model providing globally optimal curves which
correctly follow highly complex vascular structures in retinal images.
All experiments in this article can be reproduced via documented Mathematica
notebooks available at GitHub
(https://github.com/NickyvdBerg/DataDrivenTracking)
Generalized fast marching method for computing highest threatening trajectories with curvature constraints and detection ambiguities in distance and radial speed
Work presented at the 9th Conference on Curves and Surfaces, 2018, ArcachonWe present a recent numerical method devoted to computing curves that globally minimize an energy featuring both a data driven term, and a second order curvature penalizing term. Applications to image segmentation are discussed. We then describe in detail recent progress on radar network configuration, in which the optimal curves represent an opponent's trajectories
Nonholonomic Motion Planning as Efficient as Piano Mover's
We present an algorithm for non-holonomic motion planning (or 'parking a
car') that is as computationally efficient as a simple approach to solving the
famous Piano-mover's problem, where the non-holonomic constraints are ignored.
The core of the approach is a graph-discretization of the problem. The
graph-discretization is provably accurate in modeling the non-holonomic
constraints, and yet is nearly as small as the straightforward regular grid
discretization of the Piano-mover's problem into a 3D volume of 2D position
plus angular orientation. Where the Piano mover's graph has one vertex and
edges to six neighbors each, we have three vertices with a total of ten edges,
increasing the graph size by less than a factor of two, and this factor does
not depend on spatial or angular resolution. The local edge connections are
organized so that they represent globally consistent turn and straight
segments. The graph can be used with Dijkstra's algorithm, A*, value iteration
or any other graph algorithm. Furthermore, the graph has a structure that lends
itself to processing with deterministic massive parallelism. The turn and
straight curves divide the configuration space into many parallel groups. We
use this to develop a customized 'kernel-style' graph processing method. It
results in an N-turn planner that requires no heuristics or load balancing and
is as efficient as a simple solution to the Piano mover's problem even in
sequential form. In parallel form it is many times faster than the sequential
processing of the graph, and can run many times a second on a consumer grade
GPU while exploring a configuration space pose grid with very high spatial and
angular resolution. We prove approximation quality and computational complexity
and demonstrate that it is a flexible, practical, reliable, and efficient
component for a production solution.Comment: 34 pages, 37 figures, 9 tables, 4 graphs, 8 insert
ChebLieNet: Invariant Spectral Graph NNs Turned Equivariant by Riemannian Geometry on Lie Groups
We introduce ChebLieNet, a group-equivariant method on (anisotropic)
manifolds. Surfing on the success of graph- and group-based neural networks, we
take advantage of the recent developments in the geometric deep learning field
to derive a new approach to exploit any anisotropies in data. Via discrete
approximations of Lie groups, we develop a graph neural network made of
anisotropic convolutional layers (Chebyshev convolutions), spatial pooling and
unpooling layers, and global pooling layers. Group equivariance is achieved via
equivariant and invariant operators on graphs with anisotropic left-invariant
Riemannian distance-based affinities encoded on the edges. Thanks to its simple
form, the Riemannian metric can model any anisotropies, both in the spatial and
orientation domains. This control on anisotropies of the Riemannian metrics
allows to balance equivariance (anisotropic metric) against invariance
(isotropic metric) of the graph convolution layers. Hence we open the doors to
a better understanding of anisotropic properties. Furthermore, we empirically
prove the existence of (data-dependent) sweet spots for anisotropic parameters
on CIFAR10. This crucial result is evidence of the benefice we could get by
exploiting anisotropic properties in data. We also evaluate the scalability of
this approach on STL10 (image data) and ClimateNet (spherical data), showing
its remarkable adaptability to diverse tasks.Comment: submitted to NeurIPS'21, https://openreview.net/forum?id=WsfXFxqZXR
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Models for Human Navigation and Optimal Path Planning Using Level Set Methods and Hamilton-Jacobi Equations
We present several models for different physical scenarios which are centered around human movement or optimal path planning, and use partial differential equations and concepts from control theory. The first model is a game-theoretic model for environmental crime which tracks criminals' movement using the level set method, and improves upon previous continuous models by removing overly restrictive assumptions of symmetry. Next, we design a method for determining optimal hiking paths in mountainous regions using an anisotropic level set equation. After this, we present a model for optimal human navigation with uncertainty which is rooted in dynamic programming and stochastic optimal control theory. Lastly, we consider optimal path planning for simple, self-driving cars in the Hamilton-Jacobi formulation. We improve upon previous models which simplify the car to a point mass, and present a reasonably general upwind, sweeping scheme to solve the relevant Hamilton-Jacobi equation
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
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