What can be seen in a noisy optical flow field projected by a moving planar patch in 3D space?

Abstract

In this paper, we would like to propose a brand new interpretation to the so-called “structure-from-motion” (SFM) problem. The optical flow field projected by a moving rigid planar patch in 3D space is our main consideration. Instead of just obtaining an explicit 3D motion/pose solution like the old approaches did before, we focus our attention on analyzing its error sensitivity, uncertainty, and ambiguity from another point of view. Our new method can handle the above error analysis easily. As known well before, the optical flow field projected by a 3D moving planar patch can be completely expressed by eight coefficients (two for second-order, four for first-order, and two for zeroth-order). Based on these flow coefficients easily determined by a linear regression method or other similar approaches, the error sensitivity of 3D estimates can be analyzed quantitatively and qualitatively in a coarse-to-fine way. The concepts of camera fixation and singular value decomposition (SVD) play important roles in our analysis. There are three goals for our experiments: (1) To prove the correctness of the algorithms (simulated image). (2) To show the tendency of error sensitivity when the 3D poses of the target planar patch are varied in a controlled manner (simulated image). (3) To show that our analysis is workable in the real-world application (real-world image).