Analytical results of error sensitivity of motion estimation from two views

Fundamental instabilities have been observed in the performance of the majority of the algorithms for three dimensional motion estimation from two views. Many geometric and intuitive interpretations have been offered to explain the error sensitivity of the estimated parameters. In this paper, we address the importance of the form of the error norm to be minimized with respect to the motion parameters. We describe the error norms used by the existing algorithms in a unifying notation and give a geometric interpretation of them. We then explicitly prove that the minimization of the objective function leading to an eigenvector solution suffers from a crucial instability. The analyticity of our results allows us to examine the error sensitivity in terms of the translation direction, the viewing angle and the distance of the moving object from the camera. We propose a norm possessing a reasonable geometric interpretation in the image plane and we show by analytical means that a simplificati on of this norm leading to a closed form solution has undesirable properties.