On the smoothness constraint in the intensity-based estimation of the parallax field

Abstract

In this contribution, the estimation of the parallax field is considered in a mathematical way. First, a mathematical model for the image formation is developed : This model describes how the parallax field influences the left and the right image. Then, we tackle the inverse problem : We calculate the parallax field given the left and the right image. In fact, the problem of estimating the parallax field is mathematically formulated as the maximum a posteriori probability (MAP) estimation of a Gaussian signal given two (noisy) measured signals, i.e. the left and the right image. Estimation problems like this are considered in [2]. We apply that theory to the problem at hand and prove that the MAP estimate for the parallax field minimizes a certain functional. This functional is compared to the functional of the traditional intensity-based method. Both functionals consist of a term associated with the displaced frame difference and a term associated with the smoothness of the parallax field. Our functional differs from the traditional one in the smoothness term. Next, we consider the searching for the minimum of this functional and develop a new iterative procedure to obtain the MAP estimate in a few iterations. Our procedure is quite different from the Gauss-Seidel or Jacobi schemes [3] that are normally used for this purpose. Finally, some results are given to compare the performance of our smoothness constraint to the traditional one.