New Approximation of a Scale Space Kernel on SE(3) and Applications in Neuroimaging

Portegies, JM Jorg; Sanguinetti, GR Gonzalo; Meesters, Spl Stephan; Duits, Remco

doi:10.1007/978-3-319-18461-6_4

Cited by 8 publications

(17 citation statements)

References 19 publications

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“…Moreover, this choice for estimating the kernels in the group is natural, as it provides the weakest upper bound kernel since by direct computation one has α = −γ ⇒ c 6 = 0. Finally, this choice indeed provides us the correct symmetry for the Gaussian approximation of K 1 t (y, n) as stated in the following theorem, that is proven in [42].…”

Section: Analytical Approximations Using Logarithm On Se(3)mentioning

confidence: 78%

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New exact and numerical solutions of the (convection–)diffusion kernels on SE(3)

Portegies

Duits

2017

Differential Geometry and its Applications

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We consider hypo-elliptic diffusion and convection-diffusion on R 3 S 2 , the quotient of the Lie group of rigid body motions SE(3) in which group elements are equivalent if they are equal up to a rotation around the reference axis. We show that we can derive expressions for the convolution kernels in terms of eigenfunctions of the PDE, by extending the approach for the SE(2) case. This goes via application of the Fourier transform of the PDE in the spatial variables, yielding a second order differential operator. We show that the eigenfunctions of this operator can be expressed as (generalized) spheroidal wave functions. The same exact formulas are derived via the Fourier transform on SE(3). We solve both the evolution itself, as well as the time-integrated process that corresponds to the resolvent operator. Furthermore, we have extended a standard numerical procedure from SE(2) to SE(3) for the computation of the solution kernels that is directly related to the exact solutions. Finally, we provide a novel analytic approximation of the kernels that we briefly compare to the exact kernels.

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Section: Analytical Approximations Using Logarithm On Se(3)mentioning

confidence: 78%

“…The proof of this lemma was given in [42]. Now we consider the special case where K represents the solution kernel K 1 t of the diffusion process with generator Q 1 .…”

Section: Analytical Approximations Using Logarithm On Se(3)mentioning

confidence: 99%

New exact and numerical solutions of the (convection–)diffusion kernels on SE(3)

Portegies

Duits

2017

Differential Geometry and its Applications

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“…Next we employ PDE-formulation (10) on the group G = SE(3) to summarize the symmetries for the probability kernels K α t : R 3 S 2 → R + . For details see [78] and [37].…”

Section: Symmetries Of the Pdes Of Interestmentioning

confidence: 99%

“…There also exist Gaussian estimates for the heat kernel K α=1 t that use a weighted modulus on the logarithm on G, [83]. Such Gaussian estimates can account for the quotient structure G/H [78], and can be reasonably close [84,Fig.4.4] to the exact solutions for practical parameter settings in applications [41,85].…”

Section: Theoremmentioning

confidence: 99%

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Fourier Transform on the Homogeneous Space of 3D Positions and Orientations for Exact Solutions to PDEs

Duits¹,

Bekkers²,

Mashtakov³

2018

Preprint

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Fokker-Planck PDEs (incl. diffusions) for stable Lévy processes (incl. Wiener processes) on the joint space of positions and orientations play a major role in mechanics, robotics, image analysis, directional statistics and probability theory. Exact analytic designs and solutions are known in the 2D case, where they have been obtained using Fourier transform on $SE(2)$. Here we extend these approaches to 3D using Fourier transform on the Lie group $SE(3)$ of rigid body motions. More precisely, we define the homogeneous space of 3D positions and orientations $\mathbb{R}^{3}\rtimes S^{2}:=SE(3)/(\{\mathbf{0}\} \times SO(2))$ as the quotient in $SE(3)$. In our construction, two group elements are equivalent if they are equal up to a rotation around the reference axis. On this quotient we design a specific Fourier transform. We apply this Fourier transform to derive new exact solutions to Fokker-Planck PDEs of $\alpha$-stable Lévy processes on $\mathbb{R}^{3}\rtimes S^{2}$. This reduces classical analysis computations and provides an explicit algebraic spectral decomposition of the solutions. We compare the exact probability kernel for $\alpha = 1$ (the diffusion kernel) to the kernel for $\alpha=\frac12$ (the Poisson kernel). We set up SDEs for the Lévy processes on the quotient and derive corresponding Monte-Carlo methods. We verify that the exact probability kernels arise as the limit of the Monte-Carlo approximations.

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Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D

2018

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We propose an efficient approach for the grouping of local orientations (points on vessels) via nilpotent approximations of sub-Riemannian distances in the 2D and 3D roto-translation groups SE(2) and SE(3). In our distance approximations we consider homogeneous norms on nilpotent groups that locally approximate SE(n), and which are obtained via the exponential and logarithmic map on SE(n). In a qualitative validation we show that the norms provide accurate approximations of the true sub-Riemannian distances, and we discuss their relations to the fundamental solution of the sub-Laplacian on SE(n). The quantitative experiments further confirm the accuracy of the approximations. Quantitative results are obtained by evaluating perceptual grouping performance of retinal blood vessels in 2D images and curves in challenging 3D synthetic volumes. The results show that (1) sub-Riemannian geometry is essential in achieving top performance and (2) grouping via the fast analytic approximations performs almost equally, or better, than data-adaptive fast marching approaches on R n and SE(n).

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New Approximation of a Scale Space Kernel on SE(3) and Applications in Neuroimaging

Cited by 8 publications

References 19 publications

New exact and numerical solutions of the (convection–)diffusion kernels on SE(3)

New exact and numerical solutions of the (convection–)diffusion kernels on SE(3)

Fourier Transform on the Homogeneous Space of 3D Positions and Orientations for Exact Solutions to PDEs

Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D

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