Vessel Tracking via Sub-Riemannian Geodesics on the Projective Line Bundle

07/11/2017
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We study a data-driven sub-Riemannian (SR) curve optimization model for connecting local orientations in orientation lifts of images. Our model lives on the projective line bundle R2 x P1, with P1 = S1/~ with identification of antipodal points. It extends previous cortical models for contour perception on R2 x P1 to the data-driven case. We provide a complete (mainly numerical) analysis of the dynamics of the 1st Maxwell-set with growing radii of SR-spheres, revealing the cutlocus. Furthermore, a comparison of the cusp-surface in R2 x Pto its counterpart in R2 x S1 of a previous model, reveals a general and strong reduction of cusps in spatial projections of geodesics. Numerical solutions of the model are obtained by a single wavefront propagation method relying on a simple extension of existing anisotropic fast-marching or iterative morphological scale space methods. Experiments show that the projective line bundle structure greatly reduces the presence of cusps. Another advantage of including R2 x P1 instead of R2 x Sin the wavefront propagation is reduction of computational time.

Vessel Tracking via Sub-Riemannian Geodesics on the Projective Line Bundle

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application/pdf Vessel Tracking via Sub-Riemannian Geodesics on the Projective Line Bundle Erik Bekkers, Remco Duits, Alexey Mashtakov, Yuri Sachkov
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We study a data-driven sub-Riemannian (SR) curve optimization model for connecting local orientations in orientation lifts of images. Our model lives on the projective line bundle R2 x P1, with P1 = S1/~ with identification of antipodal points. It extends previous cortical models for contour perception on R2 x P1 to the data-driven case. We provide a complete (mainly numerical) analysis of the dynamics of the 1st Maxwell-set with growing radii of SR-spheres, revealing the cutlocus. Furthermore, a comparison of the cusp-surface in R2 x Pto its counterpart in R2 x S1 of a previous model, reveals a general and strong reduction of cusps in spatial projections of geodesics. Numerical solutions of the model are obtained by a single wavefront propagation method relying on a simple extension of existing anisotropic fast-marching or iterative morphological scale space methods. Experiments show that the projective line bundle structure greatly reduces the presence of cusps. Another advantage of including R2 x P1 instead of R2 x Sin the wavefront propagation is reduction of computational time.
Vessel Tracking via Sub-Riemannian Geodesics on the Projective Line Bundle
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Vessel Tracking via Sub-Riemannian Geodesics on the Projective Line Bundle E.J. Bekkers1∗ , R. Duits1∗ , A. Mashtakov2∗ , and Yu. Sachkov2? 1 Eindhoven University of Technology, The Netherlands, Department of Mathematics and Computer Science, 2 Program Systems Institute of RAS, Russia, Control Processes Research Center {E.J.Bekkers, R.Duits}@tue.nl, {alexey.mashtakov, yusachkov}@gmail.com Abstract. We study a data-driven sub-Riemannian (SR) curve opti- mization model for connecting local orientations in orientation lifts of images. Our model lives on the projective line bundle R2 × P1 , with P1 = S1 /∼ with identification of antipodal points. It extends previous cortical models for contour perception on R2 ×P1 to the data-driven case. We provide a complete (mainly numerical) analysis of the dynamics of the 1st Maxwell-set with growing radii of SR-spheres, revealing the cut- locus. Furthermore, a comparison of the cusp-surface in R2 × P1 to its counterpart in R2 × S1 of a previous model, reveals a general and strong reduction of cusps in spatial projections of geodesics. Numerical solutions of the model are obtained by a single wavefront propagation method re- lying on a simple extension of existing anisotropic fast-marching or it- erative morphological scale space methods. Experiments show that the projective line bundle structure greatly reduces the presence of cusps. Another advantage of including R2 × P1 instead of R2 × S1 in the wave- front propagation is reduction of computational time. Keywords: Sub-Riemannian geodesic, tracking, projective line bundle 1 Introduction In image analysis extraction of salient curves such as blood vessels, is often tackled by first lifting the image data to a new representation defined on the higher dimensional space of positions and directions, followed by a geodesic tracking [1–3] in this lifted space [4–6]. Benefits of such approaches are that one can generically deal with complex structures such as crossings [4, 6, 7], bifur- cations [8], and low-contrast [5, 6, 9], while accounting for contextual alignment ? Joint main authors. The ERC is gratefully acknowledged for financial support (ERC- StG nr. 335555). Sections 1, 2 of the paper are written by R. Duits and A. Mashtakov, Section 3 is written by A. Mashtakov, Yu. Sachkov and R. Duits, Sections 4, 6 are written by R. Duits and A. Mashtakov, and Section 5 is written by E.J. Bekkers. The work of A. Mashtakov and Yu. Sachkov is supported by the Russian Science Foundation under grant 17-11-01387 and performed in Ailamazyan Program Systems Institute of Russian Academy of Sciences. of local orientations [5, 6]. The latter can be done in the same way as in cortical models of visual perception of lines [10–13], namely via sub-Riemannian (SR) geometry on the combined space of positions and orientations. In these corti- cal models, it is sometimes stressed [12] that one should work in a projective line bundle R2 × P1 with a partition of equivalence classes P1 := S1 /∼ with n1 ∼ n2 ⇔ n1 = ±n2. Furthermore, in the statistics of line co-occurrences in retinal images the same projective line bundle structure is crucial [14]. Also, for many image analysis applications the orientation of an elongated structure is a well defined characteristic of a salient curve in an image, in contrast to an artificially imposed direction. At first sight the effect of the identification of antipodal points might seem minor as the minimizing SR geodesic between two elements in R2 ×P1 is obtained by the minimum of the two minimizing SR geodesics in R2 ×S1 that arise (twice) by flipping the directions of the boundary conditions. However, this appearance is deceptive, it has a rather serious impact on geometric notions such as 1) the 1st Maxwell set (where two distinct geodesics with equal length meet for the first positive time), 2) the cut-locus (where a geodesic looses optimality), 3) the cusp-surface (where spatial projections of SR geodesics show a cusp). Besides an analysis of the geometric consequences in Sect. 2, 3, 4, we show that the projective line bundle provides a better tracking with much less cusps in Sect. 5. 2 The Projective Line Bundle Model The projective line bundle PT(R2 ) is a quotient of Lie group SE(2), and one can define a sub-Riemannian structure (SR) on it. The group SE(2) = R2 o SO(2) of planar roto-translations is identified with the coupled space of positions and orientations R2 × S1 , and for each g = (x, y, θ) ∈ R2 × S1 ∼ = SE(2) one has Lgg0 = g g0 = (x0 cos θ + y0 sin θ + x, −x0 sin θ + y0 cos θ + y, θ0 + θ). (1) Via the push-forward (Lg)∗ one gets the left-invariant frame {A1, A2, A3} from the Lie-algebra basis {A1, A2, A3} = {∂x|e , ∂θ|e , ∂y|e} at the unity e = (0, 0, 0): A1 = cos θ ∂x + sin θ ∂y, A2 = ∂θ, A3 = − sin θ ∂x + cos θ ∂y. Let C : SE(2) → R+ denote a smooth cost function strictly bounded from below. The SR-problem on SE(2) is to find a Lipschizian curve γ : [0, T] → SE(2), s.t. γ̇(t) = u1 (t) A1|γ(t) + u2 (t) A2|γ(t), γ(0) = g0, γ(T) = g1, l(γ(·)) := T R 0 C(γ(t)) p ξ2|u1(t)|2 + |u2(t)|2 dt → min, (2) with controls u1 , u2 : [0, T] → R are in L∞ [0, T], boundary points g0, g1 are given, ξ > 0 is constant, and terminal time T > 0 is free. Thanks to reparametrization invariance the SR distance can be defined as d(g0, g1) = min γ ∈ Lip([0, 1], SE(2)), γ̇ ∈ ∆|γ , γ(0) = g0, γ(1) = g1 Z 1 0 q Gγ(τ)(γ̇(τ), γ̇(τ)) dτ, (3) with Gγ(τ)(γ̇(τ), γ̇(τ)) = C2 (γ(τ)) ξ2 |u1 (τT)|2 + |u2 (τT)|2  , τ = t T ∈ [0, 1], and ∆ := span{A1, A2} with dual ∆∗ = span{cos θ dx + sin θ dy, dθ}. The projec- tive line bundle PT(R2 ) is a quotient PT(R2 ) = SE(2) /∼ with identification (x, y, θ) ∼ (x, y, θ+π). The SR distance in PT(R2 ) ∼ = R2 × P1 = R2 × R/{πZ} is d(q0, q1) := min{d(g0, g1) , d(g0 (0, 0, π), g1 (0, 0, π)), d(g0, g1 (0, 0, π)) , d(g0 (0, 0, π), g1)} = min { d(g0, g1) , d(g0 (0, 0, π), g1)} (4) for all qi = (xi, yi, θi) ∈ PT(R2 ), gi