Beyond the Epipolar Constraint: Integrating 3D Motion and Structure Estimation

This paper develops a novel solution to the problem of recovering the structure of a scene given an uncalibrated video sequence depicting the scene. The essence of the technique lies in a method for recovering the rigid transformation between the different views in the image sequence. Knowledge of this 3D motion allows for self-calibration and for subsequent recovery of 3D structure. The introduced method breaks away from applying only the traditionally used epipolar constraint and introduces a new constraint based on the interaction between 3D motion and shape. Up to now, structure from motion algorithms proceeded in two well defined steps, where the first and most important step is recovering the rigid transformation between two views, and the subsequent step is using this transformation to compute the structure of the scene in view. Here both aforementioned steps are accomplished in a synergistic manner. Existing approaches to 3D motion estimation are mostly based on the use of optic flow which however poses a problem at the locations of depth discontinuities. If we knew where depth discontinuities were, we could (using a multitude of approaches based on smoothness constraints) estimate accurately flow values for image patches corresponding to smooth scene patches; but to know the discontinuities requires solving the structure from motion problem first. In the past this dilemma has been addressed by improving the estimation of flow through sophisticated optimization techniques, whose performance often depends on the scene in view. In this paper the main idea is based on the interaction between 3D motion and shape which allows us to estimate the 3D motion while at the same time segmenting the scene. If we use a wrong 3D motion estimate to compute depth, then we obtain a distorted version of the depth function. The distortion, however, is such that the worse the motion estimate, the more likely we are to obtain depth estimates that are locally unsmooth, i.e., they vary more than the correct ones. Since local variability of depth is due either to the existence of a discontinuity or to a wrong 3D motion estimate, being able to differentiate between these two cases provides the correct motion, which yields the “smoothest” estimated depth as well as the image locations of scene discontinuities. Although no optic flow values are computed, we show that our algorithm is very much related to minimizing the epipolar constraint when the scene in view is smooth. When however the imaged scene is not smooth, the introduced constraint has in general different properties from the epipolar constraint and we present experimental results with real sequences where it performs better.