The problem of motion estimation, in general, is made difficult by large illumination variations and by motion discontinuities. In recent papers, we and others have proposed global approaches to deal with both problems simultaneously within the regularization framework. A major drawback of such global methods is that several regularization parameters responsible for the integration of the illumination and motion components need to be determined in advance. This has reduced the applicability of global methods. In this paper, a parameter-free local approach, which solves a linear regression problem using a simple parametric model, is presented. To achieve robustness for the linear regression problem, we introduce a modified version of the least median of squares algorithm. We show quantitative error comparisons between the results obtained by our local approach and those produced by several global methods. Our results show that our local method is comparable to the best results obtained by the global approaches yet does not require any manual selection of parameters.