TY - JOUR T1 - Statistical bias in 3-D reconstruction from a monocular video JF - Image Processing, IEEE Transactions on Y1 - 2005 A1 - Roy-Chowdhury, A.K. A1 - Chellapa, Rama KW - 3D face models KW - 3D video reconstruction KW - algorithms KW - artifacts KW - Artificial intelligence KW - Automated;Signal Processing KW - bias compensation KW - bundle adjustment KW - camera motion estimation KW - Computer simulation KW - Computer-Assisted;Imaging KW - Computer-Assisted;Subtraction Technique;Video Recording; KW - depth estimate KW - error covariance estimation KW - error statistics KW - generalized Cramer-Rao lower bound KW - Image Enhancement KW - Image Interpretation KW - Image reconstruction KW - initialization procedures KW - least squares approximations KW - linear least-squares framework KW - monocular video KW - motion compensation KW - Motion estimation KW - statistical bias KW - Statistical;Pattern Recognition KW - structure from motion algorithms KW - Three-Dimensional;Information Storage and Retrieval;Models KW - video signal processing AB - The present state-of-the-art in computing the error statistics in three-dimensional (3-D) reconstruction from video concentrates on estimating the error covariance. A different source of error which has not received much attention is the fact that the reconstruction estimates are often significantly statistically biased. In this paper, we derive a precise expression for the bias in the depth estimate, based on the continuous (differentiable) version of structure from motion (SfM). Many SfM algorithms, or certain portions of them, can be posed in a linear least-squares (LS) framework Ax=b. Examples include initialization procedures for bundle adjustment or algorithms that alternately estimate depth and camera motion. It is a well-known fact that the LS estimate is biased if the system matrix A is noisy. In SfM, the matrix A contains point correspondences, which are always difficult to obtain precisely; thus, it is expected that the structure and motion estimates in such a formulation of the problem would be biased. Existing results on the minimum achievable variance of the SfM estimator are extended by deriving a generalized Cramer-Rao lower bound. A detailed analysis of the effect of various camera motion parameters on the bias is presented. We conclude by presenting the effect of bias compensation on reconstructing 3-D face models from rendered images. VL - 14 SN - 1057-7149 CP - 8 M3 - 10.1109/TIP.2005.849775 ER -