Optimality in homography in 3D reconstruction from canonical stereo setup

This paper presents an experimental study and investigation of errors in 3D reconstruction from views in a canonical stereo camera setup with point correspondences of a grid known apriori. The objective of investigation is to find the estimate of P by Direct Linear Transform (DLT) solution i.e by minimizing the algebraic error (||AP||) subjected to normalizing constraint ||P||=1 and then further minimizing the total cost function (C) involving reprojection error and 3D geometric error through iterative Levenberg-Marquardt technique, to see whether the second level of minimization of C brings any remarkable improvement in solution or not. Here A is the measured value matrix with measurement error in image and space point coordinates and P is the estimate of camera projection matrix. The problem of reconstructed from stereo pair is very much documented and well researched for both precisely calibrated and uncalibrated cameras. But much literature is not available in investigating the optimality of the homography by minimizing the total cost function considering both geometric error in the image pair and 3d geometric error term. We focus our investigation in studying the results of optimizing the total cost function including 3D geometric error term and see its effects in homography and in reprojection.