# A PDE approach to data-driven sub-Riemannian geodesics in SE(2)

E.J. Bekkers, R. Duits, A. Mashtakov, G.R. Sanguinetti

22 Citaties (Scopus)

### Uittreksel

We present a new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group $SE(2) = \mathbb{R}^2 \rtimes S^1$ with a metric tensor depending on a smooth external cost $\mathcal{C}:SE(2) \to [\delta,1]$, $\delta>0$, computed from image data. The method consists of a first step where an SR-distance map is computed as a viscosity solution of a Hamilton--Jacobi--Bellman system derived via Pontryagin's maximum principle (PMP). Subsequent backward integration, again relying on PMP, gives the SR-geodesics. For $\mathcal{C}=1$ 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 $\mathcal{C}=1$. Regarding image analysis applications, trackings of elongated structures in retinal and synthetic images show that our line tracking generically deals with crossings. We show the benefits of including the SR-geometry.
Taal Engels 2740-2770 31 SIAM Journal on Imaging Sciences 8 4 10.1137/15M1018460 Gepubliceerd - 1 dec 2015

### Vingerafdruk

Maximum principle
Data-driven
Geodesic
Pontryagin Maximum Principle
Wavefronts
Sub-Riemannian Geometry
Image analysis
Boundary value problems
Tensors
Costs
Global Minimizer
Hamilton-Jacobi
Viscosity Solutions
Viscosity
Cusp
Image Analysis
Wave Front
Numerical Computation
Geometry
Tensor

### Citeer dit

Bekkers, E.J. ; Duits, R. ; Mashtakov, A. ; Sanguinetti, G.R./ A PDE approach to data-driven sub-Riemannian geodesics in SE(2). In: SIAM Journal on Imaging Sciences. 2015 ; Vol. 8, Nr. 4. blz. 2740-2770
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abstract = "We present a new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group $SE(2) = \mathbb{R}^2 \rtimes S^1$ with a metric tensor depending on a smooth external cost $\mathcal{C}:SE(2) \to [\delta,1]$, $\delta>0$, computed from image data. The method consists of a first step where an SR-distance map is computed as a viscosity solution of a Hamilton--Jacobi--Bellman system derived via Pontryagin's maximum principle (PMP). Subsequent backward integration, again relying on PMP, gives the SR-geodesics. For $\mathcal{C}=1$ 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 $\mathcal{C}=1$. Regarding image analysis applications, trackings of elongated structures in retinal and synthetic images show that our line tracking generically deals with crossings. We show the benefits of including the SR-geometry.",
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A PDE approach to data-driven sub-Riemannian geodesics in SE(2). / Bekkers, E.J.; Duits, R.; Mashtakov, A.; Sanguinetti, G.R.

In: SIAM Journal on Imaging Sciences, Vol. 8, Nr. 4, 01.12.2015, blz. 2740-2770.

TY - JOUR

T1 - A PDE approach to data-driven sub-Riemannian geodesics in SE(2)

AU - Bekkers,E.J.

AU - Duits,R.

AU - Mashtakov,A.

AU - Sanguinetti,G.R.

PY - 2015/12/1

Y1 - 2015/12/1

N2 - We present a new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group $SE(2) = \mathbb{R}^2 \rtimes S^1$ with a metric tensor depending on a smooth external cost $\mathcal{C}:SE(2) \to [\delta,1]$, $\delta>0$, computed from image data. The method consists of a first step where an SR-distance map is computed as a viscosity solution of a Hamilton--Jacobi--Bellman system derived via Pontryagin's maximum principle (PMP). Subsequent backward integration, again relying on PMP, gives the SR-geodesics. For $\mathcal{C}=1$ 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 $\mathcal{C}=1$. Regarding image analysis applications, trackings of elongated structures in retinal and synthetic images show that our line tracking generically deals with crossings. We show the benefits of including the SR-geometry.

AB - We present a new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group $SE(2) = \mathbb{R}^2 \rtimes S^1$ with a metric tensor depending on a smooth external cost $\mathcal{C}:SE(2) \to [\delta,1]$, $\delta>0$, computed from image data. The method consists of a first step where an SR-distance map is computed as a viscosity solution of a Hamilton--Jacobi--Bellman system derived via Pontryagin's maximum principle (PMP). Subsequent backward integration, again relying on PMP, gives the SR-geodesics. For $\mathcal{C}=1$ 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 $\mathcal{C}=1$. Regarding image analysis applications, trackings of elongated structures in retinal and synthetic images show that our line tracking generically deals with crossings. We show the benefits of including the SR-geometry.

KW - roto-translation group

KW - Hamilton-Jacobi equations

KW - vessel tracking

KW - sub-Riemannian geometry

KW - morphological scale spaces

U2 - 10.1137/15M1018460

DO - 10.1137/15M1018460

M3 - Article

VL - 8

SP - 2740

EP - 2770

JO - SIAM Journal on Imaging Sciences

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SN - 1936-4954

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