On the Curve Reconstruction in Riemannian Manifolds

Abstract

In this article we extend the computational geometric curve reconstruction approach to the curves embedded in the Riemannian manifold. We prove that the minimal spanning tree, given a sufficiently dense sample, correctly reconstructs the smooth arcs which can be used to reconstruct closed and simple curves in Riemannian manifolds. The proof is based on the behavior of the curve segment inside the tubular neighborhood of the curve. To take care of the local topological changes of the manifold, the tubular neighborhood is constructed in consideration with the injectivity radius of the underlying Riemannian manifold. We also present examples of successfully reconstructed curves and show applications of curve reconstruction to ordering motion frames.

Appendix: Exponential and Logarithmic Maps

A1

Given [ω]∈so(3),

$$ \exp[\omega]=I + \frac{\sin\Vert \omega \Vert}{\Vert \omega \Vert}\cdot[\omega]+ \frac{1-\cos\Vert \omega \Vert}{\Vert \omega \Vert^2}\cdot [ \omega]^2 $$

(19)

A2

Let (ω,v)∈se(3). Then

$$ \exp \left [\begin{array}{c@{\quad}c} [\omega] & v\\ 0 & 0 \end{array} \right ] = \left [\begin{array}{c@{\quad}c} \exp[\omega] & Av\\ 0 & 1 \end{array} \right ] $$

(20)

where

$$A = I+\frac{1-\cos\Vert\omega\Vert}{\Vert\omega\Vert^2}\cdot [\omega]+ \frac{\Vert\omega\Vert-\sin\Vert\omega\Vert}{\Vert\omega\Vert^3}\cdot[ \omega]^2 $$

A3

Given θ∈SO(3) such that $\operatorname{Tr}(\theta)\neq -1$. Then

$$ \log(\theta) = \frac{\phi}{2 \sin\phi}\bigl(\theta-\theta^T\bigr) $$

(21)

where ϕ satisfies $1+2 \cos\phi = \operatorname{Tr}(\theta)$, |ϕ|<π. Further more, ∥logθ∥²=ϕ ².

A4

Suppose θ∈SO(3) such that $\operatorname{Tr}(\theta)\neq -1$, and let b∈ℝ³. Then

$$ \log \left [\begin{array}{c@{\quad}c} \theta & b\\ 0 & 1 \end{array} \right ] = \left [\begin{array}{c@{\quad}c} [\omega] & A^{-1}b\\ 0 & 0 \end{array} \right ] $$

(22)

where [ω]=logθ, and

$$A^{-1}= I-\frac{1}{2}\cdot[\omega]+\frac{2\sin\Vert\omega\Vert-\Vert\omega\Vert(1+\cos\Vert\omega\Vert)}{ 2\Vert\omega\Vert^2\sin\Vert\omega\Vert}\cdot[ \omega]^2 $$

A5

Let θ ₁,θ ₂∈SO(3). Then the distance L=d(θ ₁,θ ₂) induced by the standard bi-invariant metric on SO(3) is

$$ d(\theta_1,\theta_2)=\bigl\Vert \log\bigl( \theta_1^{-1} \theta_2\bigr)\bigr\Vert $$

(23)

where ∥⋅∥ denotes the standard Euclidean norm.

A6

Let X ₁=(θ ₁,b ₁) and X ₂=(θ ₂,b ₂) be two points in SE(3). Then the distance L=d(X ₁,X ₂) induced by the scale dependent left-invariant metric on SE(3) is

$$ d(X_1,X_2)=\sqrt{c\bigl\Vert \log\bigl( \theta_1^{-1} \theta_2\bigr) \bigr\Vert^2+d\Vert b_2-b_1 \Vert^2} $$

(24)

where ∥⋅∥ denotes the Euclidean norm.