Estimation of 3-D Motion and Structure Based on a Temporally-Oriented Approach With the Method of Regression

In this paper we argue that the 3-D velocity of a single point up to a scalar factor can be recovered from its 2-D trajectory under the perspective projection. We then extend the idea to the recovery of 3-D motion of rigid objects. In both cases measurements are collected through temporal axis first, while keeping the amount of measurements in each frame minimal. We may use multiple features to get a more accurate estimate if they are available. This approach called temporally oriented approach requires us to introduce the explicit model for the evolution of 3-D motion. Our analysis is based on the assumption that the 3-D motion is smooth so that its 3-D velocity can be approximated as a truncated Taylor series. Regression relations between unknown motion parameters and measurements for a single point and rigid body are derived. The method of Maximum Likelihood is used to estimate the motion. The uniqueness of determining the 3-D motion of a single point is discussed. Experimental results obtained from simulated data and real images are given to illustrate the robustness of this approach.