Identifying degrees of freedom in pushbroom bundle adjustment

An analytical method of detecting and enumerating degrees of freedom (DOF) in bundle adjustment (BA) is developed, and used to analyze several scene geometries for pushbroom cameras. While it is well recognized in the photogrammetric community that BA can be poorly constrained, especially with pushbroom imagery, the literature is absent of techniques to study the DOF in BA except through numerical analysis, which does not delineate them except through the suggestion of their existence in degraded results. The analytical methods presented here provide insight into the DOF. The method is based on finding the singular values, and the associated singular directions, of a Hessian matrix from the minimization process in BA. These directions are scrutinized as possible DOF along which the minimal cost of BA will not change from the true solution. To demonstrate the use of the methods six different scene geometries are analyzed, some of which are commonly employed by current remote sensing satellites. Several of the scene geometries are shown to have multiple DOF beyond the well known scaling and absolute position and orientation that are not recoverable from stereo imagery without exterior orientation control of the cameras or ground control points. As a simple example, the effects of these DOF images from the HiRISE camera on the Mars Reconnaissance Orbiter are used along with the associated ephemeris data to demonstrate possible distortions in the results of a terrain reconstruction problem.