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THE EXPLICIT SOLUTIONS OF LINEAR LEFT-INVARIANT SECOND ORDER STOCHASTIC EVOLUTION EQUATIONS ON THE 2D EUCLIDEAN MOTION GROUP

Quarterly of applied mathematics, 2008-01, Vol.66 (1), p.27-67 [Peer Reviewed Journal]

2008 Brown University ;2008 INIST-CNRS ;ISSN: 0033-569X ;EISSN: 1552-4485 ;DOI: 10.1090/s0033-569x-07-01066-0 ;CODEN: QAMAAY

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Title:
THE EXPLICIT SOLUTIONS OF LINEAR LEFT-INVARIANT SECOND ORDER STOCHASTIC EVOLUTION EQUATIONS ON THE 2D EUCLIDEAN MOTION GROUP
Author: DUITS, REMCO ; VAN ALMSICK, MARKUS
Subjects: Applied sciences ; Approximation ; Artificial intelligence ; Computer science; control theory; systems ; Eigenvalues ; Evolution equations ; Exact sciences and technology ; Fourier transformations ; Greens function ; Image analysis ; Mathematical functions ; Mathieu function ; Pattern recognition. Digital image processing. Computational geometry ; Sine function ; Stochastic processes
Is Part Of: Quarterly of applied mathematics, 2008-01, Vol.66 (1), p.27-67
Description: We provide the solutions of linear, left-invariant, second order stochastic evolution equations on the 2D Euclidean motion group. These solutions are given by group-convolution with the corresponding Green's functions which we derive in explicit form. 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 basis 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, second order stochastic evolution equations.
Publisher: Providence, RI: Brown University
Language: English
Identifier: ISSN: 0033-569X
EISSN: 1552-4485
DOI: 10.1090/s0033-569x-07-01066-0
CODEN: QAMAAY
Source: American Mathematical Society Publications (Freely Accessible)

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THE EXPLICIT SOLUTIONS OF LINEAR LEFT-INVARIANT SECOND ORDER STOCHASTIC EVOLUTION EQUATIONS ON THE 2D EUCLIDEAN MOTION GROUP

2008 Brown University ;2008 INIST-CNRS ;ISSN: 0033-569X ;EISSN: 1552-4485 ;DOI: 10.1090/s0033-569x-07-01066-0 ;CODEN: QAMAAY

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