Horn-Schunck method is a global method which introduces a global constraint of smoothness to solve the aperture problem in Optical Flow.
The flow is formulated as a global energy functional which is then sought to be minimized.
E(u,v)=∫∫[(Ixu+Iyv+It)2+α2(∣∣∇u∣∣2+∣∣∇v∣∣2)]dxdy E(u,v) = \int \int [(I_x u + I_yv + I_t)^2 + \alpha^2(||\nabla u||^2 + ||\nabla v||^2)] dx dy E(u,v)=∫∫[(Ixu+Iyv+It)2+α2(∣∣∇u∣∣2+∣∣∇v∣∣2)]dxdy
Because
[uv]=f([xy]) \begin{bmatrix} u \\ v \\ \end{bmatrix} = f(\begin{bmatrix} x \\ y \\ \end{bmatrix}) [uv]=f([xy])
By multi-dimensional Euler-Lagrange equation:
∂L∂u−∂∂x∂L∂ux−∂∂y∂L∂uy=0 \frac{\partial L}{\partial u} - \frac{\partial}{\partial x} \frac{\partial L}{\partial u_x} - \frac{\partial}{\partial y} \frac{\partial L}{\partial u_y} = 0 ∂u∂L−∂x∂∂ux∂L−∂y∂∂uy∂L=0
∂L∂v−∂∂x∂L∂vx−∂∂y∂L∂vy=0 \frac{\partial L}{\partial v} - \frac{\partial}{\partial x} \frac{\partial L}{\partial v_x} - \frac{\partial}{\partial y} \frac{\partial L}{\partial v_y} = 0 ∂v
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