Left-invariant parabolic evolutions on $SE(2)$ and contour enhancement via invertible orientation scores Part I: Linear left-invariant diffusion equations on $SE(2)$

Abstract

Abstract. We provide the explicit solutions of linear, left-invariant, diffusion equations and the corresponding resolvent equations on the 2D-Euclidean motion group SE(2) = R 2 T. These parabolic equations are forward Kolmogorov equations for well-known stochastic processes for contour enhancement and contour completion. The solutions are given by group convolution with the corresponding Green's functions. In earlier work we have solved the forward Kolmogorov equations (or Fokker-Planck equations) for stochastic processes on contour completion. Here we mainly focus on the forward Kolmogorov equations for contour enhancement processes which do not include convection. We derive explicit formulas for the Green's functions (i.e., the heat kernels on SE(2)) of the left-invariant partial differential equations related to the contour enhancement process. By applying a contraction we approximate the left-invariant vector fields on SE(2) by left-invariant generators of a Heisenberg group, and we derive suitable approximations of the Green's functions. The exact Green's functions are used in so-called collision distributions on SE(2), which are the product of two left-invariant resolvent diffusions given an initial distribution on SE(2). We use the left-invariant evolution processes for automated contour enhancement in noisy medical image data using a so-called orientation score, which is obtained from a grey-value image by means of a special type of unitary wavelet transformation. Here the real part of the (invertible) orientation score serves as an initial condition in the collision distribution.

Left Invariant Evolution Equations on Gabor Transforms

“…We adapt the group theoretical approach developed for the Euclidean motion groups in the recent works [9,18,12,13,15,11], thus illustrating the scope of the methods devised for general Lie groups in [10] in signal and image processing. Reassignment will be seen to be a special case of left-invariant convection.…”

“…. , a 2d ) T and/or conductivity matrix D. We will use ideas similar to our previous work on adaptive diffusions on invertible orientation scores [17], [12], [11], [13] (where we employed evolution equations for the 2D-Euclidean motion group). We use the absolute value to adapt the diffusion and convection to avoid oscillations.…”

“…Moreover, the omission does not affect the smoothness and uniqueness of the solutions of (13), since the initial condition is infinitely differentiable (if ψ is a Schwarz function) and the Hörmander condition [23], [10] is by (12) still satisfied. The removal of the ∂ s direction from the tangent space does not imply that one can entirely ignore the ∂ s -axis in the domain of a (processed) Gabor transform.…”

“…Simply, because [∂ p , ∂ q ] = 0 whereas we should have (12). For further differential geometrical details see the appendices of [14], analogous to the differential geometry on orientation scores, [13] …”

Riemann-Finsler Geometry for Diffusion Weighted Magnetic Resonance Imaging

“…Clinical use of dwMRI is hampered by the fact that dwMRI analysis requires radically new approaches, based on abstract representations, a development still in its infancy. Examples are rank-2 symmetric positive-definite tensor representations in diffusion tensor imaging (DTI), pioneered by Basser, Mattiello and Le Bihan et al [1,2] and explored by many others [3,4,5,6,7,8,9,10,11,12,13,14], higher order symmetric positive-definite tensor representations [15,16,17,18,19,20,21], spherical harmonic representations in high angular resolution diffusion imaging (HARDI) [22,23,24,25,26], and SE(3) Lie group representations [27,28,29]. The latter type of representation, developed by Duits et al, appears to bear a particularly close relationship to the theory outlined below.…”