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I am trying to derive numerical scheme for optical flow

In Euler-Lagrange equations i obtain following term

$div(\psi(x, y)\nabla(u(x, y)))$ where $\psi, u : R^2 \rightarrow R$, $div(u)$ denotes divergence of $u$

I want to obtain system of linear equation from my Euler-Lagrange equations

In Horn-Schunck work term $div(\nabla(u(x, y)))$ was transformed into $A(u(x, y)) - u(x, y)$ where $A(u(x, y))$ - weighted sum of neighbours of $u(x)$

Is there similiar transformation of $div(\psi(x, y)\nabla(u(x, y)))$ ?

PS Sorry for my poor English and math skills

Daiver
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1 Answers1

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I've found very similar operator in work "From pixels to regions: partial differential equations in image analysis" by Thomas Brox

Page 15 (29 on PDF Document) "2.1.3 Numerics"

If i've not mistaken following formula is good discretization of term from question.

$ \partial_t u = \partial_x {(g(|\nabla u^2|)u_x)} + \partial_y {(g(|\nabla u^2|)u_y)} $

$ \frac{\partial u_{ij}}{\partial t} = \frac{g_{i+1,j} + g_{i, j}}{2}(u_{i+1,j} - u_{i,j}) + \frac{g_{i-1,j} + g_{i, j}}{2}(u_{i-1,j} - u_{i,j}) + \frac{g_{i,j+1} + g_{i, j}}{2}(u_{i,j+1} - u_{i,j}) + \frac{g_{i-1,j} + g_{i, j}}{2}(u_{i-1,j} - u_{i,j}) $

It seems to work in my implementation

Daiver
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