TY - JOUR

T1 - The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D-Euclidean motion group

AU - Duits, R.

AU - Almsick, van, M.A.

PY - 2008

Y1 - 2008

N2 - We provide the solutions of linear, left-invariant, 2nd-order stochastic
evolution equations on the 2D-Euclidean motion group. These solutions
are given by group-convolution with the corresponding Green’s functions
that we derive in explicit form in Fourier space. A particular case coincides
with the hitherto unsolved forward Kolmogorov equation of the so-called direction
process, the exact solution of which is required in the field of image
analysis for modeling the propagation of lines and contours. By approximating
the left-invariant base elements of the generators by left-invariant generators
of a Heisenberg-type group, we derive simple, analytic approximations of the
Green’s functions. We provide the explicit connection and a comparison between
these approximations and the exact solutions. Finally, we explain the
connection between the exact solutions and previous numerical implementations,
which we generalize to cope with all linear, left-invariant, 2nd-order
stochastic evolution equations.

AB - We provide the solutions of linear, left-invariant, 2nd-order stochastic
evolution equations on the 2D-Euclidean motion group. These solutions
are given by group-convolution with the corresponding Green’s functions
that we derive in explicit form in Fourier space. A particular case coincides
with the hitherto unsolved forward Kolmogorov equation of the so-called direction
process, the exact solution of which is required in the field of image
analysis for modeling the propagation of lines and contours. By approximating
the left-invariant base elements of the generators by left-invariant generators
of a Heisenberg-type group, we derive simple, analytic approximations of the
Green’s functions. We provide the explicit connection and a comparison between
these approximations and the exact solutions. Finally, we explain the
connection between the exact solutions and previous numerical implementations,
which we generalize to cope with all linear, left-invariant, 2nd-order
stochastic evolution equations.

M3 - Article

VL - 66

SP - 27

EP - 67

JO - Quarterly of Applied Mathematics

JF - Quarterly of Applied Mathematics

SN - 0033-569X

IS - 1

ER -