Stochastic Uncertainty Models for the Luminance Consistency Assumption

In this paper, a stochastic formulation of the brightness consistency used in many computer vision problems involving dynamic scenes (for instance, motion estimation or point tracking) is proposed. Usually, this model, which assumes that the luminance of a point is constant along its trajectory, is expressed in a differential form through the total derivative of the luminance function. This differential equation linearly links the point velocity to the spatial and temporal gradients of the luminance function. However, when dealing with images, the available information only holds at discrete time and on a discrete grid. In this paper, we formalize the image luminance as a continuous function transported by a flow known only up to some uncertainties related to such a discretization process. Relying on stochastic calculus, we define a formulation of the luminance function preservation in which these uncertainties are taken into account. From such a framework, it can be shown that the usual deterministic optical flow constraint equation corresponds to our stochastic evolution under some strong constraints. These constraints can be relaxed by imposing a weaker temporal assumption on the luminance function and also in introducing anisotropic intensity-based uncertainties. We also show that these uncertainties can be computed at each point of the image grid from the image data and hence provide meaningful information on the reliability of the motion estimates. To demonstrate the benefit of such a stochastic formulation of the brightness consistency assumption, we have considered a local least-squares motion estimator relying on this new constraint. This new motion estimator significantly improves the quality of the results.