Deterministic and Statistical Properties of Multi-Resolution 3D Modeling

Multi-resolution schemes for 3D modeling from an input video sequence are becoming very popular. However, multi-resolution techniques may not be the best solution strategy in many scenarios and it is important to understand the characteristics of such algorithms. In this paper, we present a multiresolution structure from motion algorithm, using monocular video as input, that exploits the bilinear relationship between depth and translation parameters to propagate the estimates through a coarseto-fine reconstruction framework. We present a detailed analysis of the deterministic and statistical properties of such an algorithm. We show that our optimization procedure is guaranteed to converge to the local minimum at each resolution. We also derive analytical expressions for the error covariances of depth and motion estimates, obtained using a multi-resolution structure from motion reconstruction algorithm, as a function of the error covariance of the feature tracks in the the input video. The method is based on using the implicit function theorem for deriving the error covariance at each resolution, and propagating the statistics from coarse to fine resolution, again taking advantage of the bilinear nature of the problem. The statistical calculations do not require the assumptions of Gaussianity of the error distributions and are valid for dense depth reconstruction estimates. Simulations have been carried out using real-life video sequences. It is shown how multi-resolution techniques can either succeed or fail in reconstructing the same scene depending on the quality of the input video of that scene, thus justifying the need for a theoretical analysis of error propagation in the 3D modeling from video.