The problem of determining the velocity field is investigated. This problem is considered by many authors in various formulations. The most well-known statement of the problem is proposed with the use of a concept of optical flow of constant distribution density function (brightness of images) along trajectories of the system under consideration. In addition, besides the common grey value constancy assumption, also, gradient constancy, as well as the constancy of the Hessian and the Laplacian are considered. In this statement functionals of quality are constructed, that also require the smoothness of the considered velocity field. The minimization of the constructed functional usually reduces to solving the Euler-Lagrange equations by numerical methods.

In this paper a new formulation of the problem is proposed. The density along the trajectories is assumed to vary. The velocity field is defined as a function depending on the vector of unknown parameters. In this paper an optimization approach to constructing the velocity field is proposed, which is based on the study of the integral functional on trajectories ensembles. The variation of integral functional is represented in an analytical form, which makes it possible to use gradient methods to find the required parameters.

The proposed approach can be used in the analysis of various images, in particular, of radionuclide images.