Comparison of Bundle Adjustment Formulations

Bundle adjustment (BA) is an optimization process refining the estimates of extrinsic camera parameters (position and orientation, or pose) and the three dimensional (3D) positions of features using overlapping images from multiple views. The optimization in BA may be formulated many ways. We compare four alternatives, including the standard formulation, from the perspectives of the region of convergence for errors in the initial estimates of pose and the complexity of the formulation. We consider cost functions formed by the sum of squared elements and minimized using a Levenberg-Marquardt (LM) algorithm. The customary elements of the cost function for BA are found using corresponding features in the images. In the standard formulation (SBA) the cost elements are the difference in the measured image coordinates and the image coordinates found by projecting the current estimate of the 3D position of the features into the camera’s focal plane, and the camera poses and the feature positions are varied as explicit parameters in the minimization. In another formulation, implicit BA (IBA), the cost function is the same but the current feature position estimates are implicit parameters not directly varied. They are functions of the current camera pose estimates. This reduces the number of parameters used in the minimization search and therefore reduces the size of the Jacobian matrix, but complicates the calculation of the Jacobian elements. The third formulation, reduced BA (RBA), is the same as the second but uses a simplified approximation of the Jacobian elements. The fourth and last formulation is Alternate Cost BA (ABA). As the name suggests, this formulation is the only one to use a completely different cost function. Again it only uses camera poses as search parameters, but uses a cost function formulated in 3D space rather than image space. The conclusions demonstrate that IBA is inferior to SBA in region of convergence despite the reduced number of parameters and that RBA and ABA are similar to SBA in convergence.