The Horn–Schunck method of estimating optical flow is a global method which introduces a global constraint of smoothness to solve the aperture problem (see Optical Flow for further description).
The Horn-Schunck algorithm assumes smoothness in the flow over the whole image.
Thus, it tries to minimize distortions in flow and prefers solutions which show more smoothness.
The flow is formulated as a global energy functional which is then sought to be minimized.
are the derivatives of the image intensity values along the x, y and time dimensions respectively,
is the optical flow vector (which is to be solved for), and the parameter
This functional can be minimized by solving the associated multi-dimensional Euler–Lagrange equations.
is the integrand of the energy expression, giving where subscripts again denote partial differentiation and
In practice the Laplacian is approximated numerically using finite differences, and may be written
Using this notation the above equation system may be written which is linear in
The following iterative scheme is derived using Cramer's rule: where the superscript k+1 denotes the next iteration, which is to be calculated and k is the last calculated result.
This is in essence a Matrix splitting method, similar to the Jacobi method, applied to the large, sparse system arising when solving for all pixels simultaneously[citation needed].
Advantages of the Horn–Schunck algorithm include that it yields a high density of flow vectors, i.e. the flow information missing in inner parts of homogeneous objects is filled in from the motion boundaries.
On the negative side, it is more sensitive to noise than local methods.