Automatic Differentiation of Non-holonomic Fast Marching for Computing Most Threatening Trajectories Under Sensors Surveillance

Mirebeau, Jean-Marie; Dréo, Johann

doi:10.1007/978-3-319-68445-1_91

Cited by 9 publications

(14 citation statements)

References 16 publications

(30 reference statements)

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“…Hard constraints can also be accounted for, such as vehicles with a bounded turning radius. In the proof of concept work [41], the HFM library is used to optimize a surveillance system for detecting an enemy vehicle, subject to non-holonomic constraints, in a worst case scenario.…”

Section: Applications To Image Processing and Motion Planningmentioning

confidence: 99%

“…Our contribution is thus two fold: we generalize these previous works to anisotropic fast marching, and we implement reverse sensitivity analysis for the first time in this context. The algorithms presented in this subsection are used in [41] to solve two player zero-sum games, where the first player places a surveillance system and the second player wants to visit a target undetected.…”

Section: Sensitivity Analysismentioning

confidence: 99%

“…(ρ, π) = J T · φ with its adjoint), where ξ, ζ and φ are arbitrary. In the context of the fast marching algorithm, the first reported uses of forward and backward differentiation respectively appear in [5] and [41]. For simplicity, these functionalities are illustrated here in the context of isotropic metrics, but they are applicable to all the minimal path models implemented in the HFM library, including those featuring dimension lifting and curvature penalization techniques [41].…”

Section: Automatic Differentiationmentioning

confidence: 99%

“…We next present an optimization problem in which one aims to maximize the distance between two points p 0 and p 1 , by adjusting the cost function c : Ω →]0, ∞[ of an isotropic metric. See [6,41] for problems of similar nature, some of them more complex. The cost function is constrained by upper and lower bounds, and subject to an integral penalty term, all determined by positive constants α, β, γ.…”

Section: Automatic Differentiationmentioning

confidence: 99%

“…See [10] and references therein on the topic of pedestrian crowd motion models based on eikonal equations. Optimal paths make most sense when the environnement is well known and path length (measured with the appropriate metric) is the dominant criterion for success: for instance if one is playing a race, or trying to avoid a surveillance system [41].…”

Section: Motion Planningmentioning

confidence: 99%

See 4 more Smart Citations

Hamiltonian Fast Marching: A Numerical Solver for Anisotropic and Non-Holonomic Eikonal PDEs

Mirebeau¹,

Portegies²

2019

Image Processing on Line

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We introduce a generalized Fast-Marching algorithm, able to compute paths globally minimizing a measure of length, defined with respect to a variety of metrics in dimension two to five. Our method applies in particular to arbitrary Riemannian metrics, and implements features such as second order accuracy, sensitivity analysis, and various stopping criteria. We also address the singular metrics associated with several non-holonomic control models, related with curvature penalization, such as the Reeds-Shepp's car with or without reverse gear, the Euler-Mumford elastica curves, and the Dubins car. Applications to image processing and to motion planning are demonstrated. Source CodeThis paper is related to the HamiltonFastMarching code, designed to solve various classes of eikonal equations, and extract the related minimal paths. The software is written in C++, but comes with interfaces for the scripting languages Python, Matlab R and Mathematica R . The codes are available at the web page of the article 1 . Supplementary MaterialA series of notebooks 2 written in the Python language, present a hands on usage of the code provided in this paper, and allow to reproduce the numerical experiments. Additional notebooks 3 by the second author are written in the Mathematica R language.

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Section: Applications To Image Processing and Motion Planningmentioning

confidence: 99%

Section: Sensitivity Analysismentioning

confidence: 99%

Section: Automatic Differentiationmentioning

confidence: 99%

Section: Automatic Differentiationmentioning

confidence: 99%

Section: Motion Planningmentioning

confidence: 99%

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Hamiltonian Fast Marching: A Numerical Solver for Anisotropic and Non-Holonomic Eikonal PDEs

Mirebeau¹,

Portegies²

2019

Image Processing on Line

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Massively parallel computation of globally optimal shortest paths with curvature penalization

Mirebeau

Gayraud

Barrère

et al. 2022

Concurrency and Computation

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We address the computation of paths globally minimizing an energy involving their curvature, with given endpoints and tangents at these endpoints, according to models known as the Reeds-Shepp car (reversible and forward variants), the Euler-Mumford elasticae, and the Dubins car. For that purpose, we numerically solve degenerate variants of the eikonal equation, on a three-dimensional domain, in a massively parallel manner on a graphical processing unit. Due to the high anisotropy and nonlinearity of the addressed Partial Differential Equation, the discretization stencil is rather wide, has numerous elements, and is costly to generate, which leads to subtle compromises between computational cost, memory usage, and cache coherency. Accelerations by a factor 30 to 120 are obtained w.r.t a sequential implementation. The efficiency and the robustness of the method is illustrated in various contexts, ranging from motion planning to vessel segmentation and radar configuration.

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Fast-Marching Methods for Curvature Penalized Shortest Paths

Mirebeau

2017

J Math Imaging Vis

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124

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We introduce numerical schemes for computing distances and shortest paths with respect to several planar paths models, featuring curvature penalization and data-driven velocity: the Dubins car, the Euler/Mumford elastica, and a two variants of the Reeds-Shepp car. For that purpose, we design monotone and causal discretizations of the associated Hamilton-Jacobi-Bellman PDEs, posed on the three dimensional domain R 2 × S 1. Our discretizations involve sparse, adaptive and anisotropic stencils on a cartesian grid, built using techniques from lattice geometry. A convergence proof is provided, in the setting of discontinuous viscosity solutions. The discretized problems are solvable in a single pass using a variant of the Fast-Marching algorithm. Numerical experiments illustrate the applications of our schemes in motion planning and image segmentation.

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Automatic Differentiation of Non-holonomic Fast Marching for Computing Most Threatening Trajectories Under Sensors Surveillance

Cited by 9 publications

References 16 publications

Hamiltonian Fast Marching: A Numerical Solver for Anisotropic and Non-Holonomic Eikonal PDEs

Hamiltonian Fast Marching: A Numerical Solver for Anisotropic and Non-Holonomic Eikonal PDEs

Massively parallel computation of globally optimal shortest paths with curvature penalization

Fast-Marching Methods for Curvature Penalized Shortest Paths

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