A Fokker–Planck Model for Optical Flow Estimation and Image Registration

Tudor Barbu (), Costică Moroşanu and Silviu-Dumitru Pavăl
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Tudor Barbu: Institute of Computer Science of the Romanian Academy—Iasi Branch, Bd. Carol I, No. 8, 700506 Iaşi, Romania
Costică Moroşanu: Faculty of Mathematics, “Al. I. Cuza” University, Bd. Carol I, No. 11, 700506 Iaşi, Romania
Silviu-Dumitru Pavăl: Faculty of Automatic Control and Computer Engineering, Technical University “Gheorghe Asachi” of Iasi, Str. Prof. dr. doc. Dimitrie Mangeron, nr. 27, 700050 Iaşi, Romania

Mathematics, 2025, vol. 13, issue 17, 1-17

Abstract: The optical flow problem and image registration problem are treated as optimal control problems associated with Fokker–Planck equations with controller u in the drift term. The payoff is of the form 1 2 | y ( T ) − y 1 | 2 + α ∫ 0 T | u ( t ) | 4 4 d t , where y 1 is the observed final state and y = y u is the solution to the state control system. Here, we prove the existence of a solution and obtain also the Euler–Lagrange optimality conditions which generate a gradient type algorithm for the above optimal control problem. A conceptual algorithm to compute the approximating optimal control and numerical implementation of this algorithm is discussed.

Keywords: Fokker–Planck equation; gradient flow; semigroup; optical flow; optimal control; optimality conditions; difference method; stochastic equations; tensor metric (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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