Optical Flow - Estimation of The Optical Flow

Estimation of The Optical Flow

Sequences of ordered images allow the estimation of motion as either instantaneous image velocities or discrete image displacements. Fleet and Weiss provide a tutorial introduction to gradient based optical flow . John L. Barron, David J. Fleet, and Steven Beauchemin provide a performance analysis of a number of optical flow techniques. It emphasizes the accuracy and density of measurements.

The optical flow methods try to calculate the motion between two image frames which are taken at times t and at every voxel position. These methods are called differential since they are based on local Taylor series approximations of the image signal; that is, they use partial derivatives with respect to the spatial and temporal coordinates.

For a 2D+t dimensional case (3D or n-D cases are similar) a voxel at location with intensity will have moved by, and between the two image frames, and the following image constraint equation can be given:

Assuming the movement to be small, the image constraint at with Taylor series can be developed to get:

H.O.T.

From these equations it follows that:

or

which results in

where are the and components of the velocity or optical flow of and, and are the derivatives of the image at in the corresponding directions., and can be written for the derivatives in the following.

Thus:

or

This is an equation in two unknowns and cannot be solved as such. This is known as the aperture problem of the optical flow algorithms. To find the optical flow another set of equations is needed, given by some additional constraint. All optical flow methods introduce additional conditions for estimating the actual flow.