Riemannian Fast-Marching on Cartesian Grids, Using Voronoi's First Reduction of Quadratic Forms

Mirebeau, Jean-Marie

doi:10.1137/17m1127466

Cited by 33 publications

(74 citation statements)

References 71 publications

(199 reference statements)

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“…(40) and (41), we can construct the tensor filed M g and the vector field ω g respectively via Eqs. (29) and (30). Indeed, one has ψ f (x) ≈ ψ b (x) ≈ 1 for the points x located in the homogeneous region of the image I where ρ(x) ≈ 0.…”

Section: Tissot's Indicatrixmentioning

confidence: 98%

Fast Asymmetric Fronts Propagation for Image Segmentation

Chen

Cohen

2017

J Math Imaging Vis

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In this paper, we introduce a generalized asymmetric fronts propagation model based on the geodesic distance maps and the Eikonal partial differential equations. One of the key ingredients for the computation of the geodesic distance map is the geodesic metric, which can govern the action of the geodesic distance level set propagation. We consider a Finsler metric with the Randers form, through which the asymmetry and anisotropy enhancements can be taken into account to prevent the fronts leaking problem during the fronts propagation. These enhancements can be derived from the image edge-dependent vector field such as the gradient vector flow. The numerical implementations are carried out by the Finsler variant of the fast marching method, leading to very efficient interactive segmentation schemes. We apply the proposed Finsler fronts propagation model to image segmentation applications. Specifically, the foreground and background segmentation is implemented by the Voronoi index map. In addition, for the application of tubularity segmentation, we exploit the level set lines of the geodesic distance map associated to the proposed Finsler metric providing that a thresholding value is given.

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Section: Tissot's Indicatrixmentioning

confidence: 98%

Fast Asymmetric Fronts Propagation for Image Segmentation

Chen

Cohen

2017

J Math Imaging Vis

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“…new models, that can be used to enhance these methods. However, the present paper is primarily focused on the algorithmic aspects of minimal path computation, similarly to [Mir14b,Mir14a,Mir17], and our experiments in §5 thus only involve synthetic data. Applications to real data will be published elsewhere in collaboration with experts in the field, as they previously were [SBD + 15, CMC17, CMC16b, DMMP16].…”

Section: Applications To Image Processingmentioning

confidence: 99%

“…They are designed using the following result, whose proof presented in §4 relies on Voronoi's first reduction, a tool from discrete geometry characterizing the interaction of a positive quadratic form with an additive lattice [Sch09]. Similar techniques are used for anisotropic diffusion PDEs in [FM14], for Monge-Ampere equations in [Mir16], and for eikonal PDEs associated to Riemannian, sub-Riemannian and Rander metrics in [Mir17].…”

Section: Discretizationmentioning

confidence: 99%

“…Their qualitative features differ widely: depending on the model, minimal paths may or may not be smooth, and the associated distance may or may not be continuous. For that purpose, we discretize generalized eikonal equations, also called first order static Hamilton-Jacobi-Bellman PDEs, with a unified approach relying on a tool from lattice geometry named Voronoi's first reduction of quadratic forms, also considered in [Mir17]. Our discretizations are monotone and causal, hence can be solved using the fast marching algorithm/dynamic programming principle, with complexity O(N ln N ) where N is the number of points in the discrete domain.…”

Section: Introductionmentioning

confidence: 99%

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Fast-Marching Methods for Curvature Penalized Shortest Paths

Mirebeau

2017

J Math Imaging Vis

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118

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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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“…We design weights c ξ (x, y), x, y ∈ X such that for any tangent vectorẋ at x one has Figure 1. Their construction exploits the additive structure of the discretization grid X and relies on techniques from lattice geometry [14], see [6,10,11] for details. The generalized eikonal PDE H ξ (x, ∇ x u ξ (x)) = 1/2 is discretized as…”

Section: Discretization Of Generalized Eikonal Equationsmentioning

confidence: 99%

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

Mirebeau

Dréo²

2017

Lecture Notes in Computer Science

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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 O(N ln N ).

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Riemannian Fast-Marching on Cartesian Grids, Using Voronoi's First Reduction of Quadratic Forms

Cited by 33 publications

References 71 publications

Fast Asymmetric Fronts Propagation for Image Segmentation

Fast Asymmetric Fronts Propagation for Image Segmentation

Fast-Marching Methods for Curvature Penalized Shortest Paths

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

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