Statistical Error Propagation in 3D Modeling From Monocular Video

A significant portion of recent research in computer vision has focused on issues related to sensitivity and robustness of existing techniques. In this paper, we study the classical structure from motion problem and analyze how the statistics representing the quality of the input video propagates through the reconstruction algorithm and affects the quality of the output reconstruction. Specifically, we show that it is possible to derive analytical expressions of the first and second order statistics (bias and error covariance) of the solution as a function of the statistics of the input. We concentrate on the case of reconstruction from a monocular video, where the small baseline makes any algorithm very susceptible to noise in the motion estimates from the video sequence. We derive an expression relating the error covariance of the reconstruction to the error covariance of the feature tracks in the input video. This is done using the implicit function theorem of real analysis and does not require strong statistical assumptions. Next, we prove that the 3D reconstruction is statistically biased, derive an expression for it and show that it is numerically significant. Combining these two results, we also establish a new bound on the minimum error in the depth reconstruction. We present the numerical significance of these analytical results on real video data.