A bundle adjustment approach with inner constraints for the scaled orthographic projection

Bundle adjustment is a method for simultaneously calculating both the interior and exterior orientation parameters of a set of images, and the object-space coordinates of the observed points. In the case of long focal length lenses and narrow field-of-view (FOV) imaging situations, collinearity based (perspective projection) algorithms may result in linear dependencies between parameters that cause solution instability. The use of a scaled orthographic projection model based on linear algebraic formulations was therefore adopted to reduce this risk. Using quaternions, a new mathematical model is derived that includes the partial derivatives as well as the inner constraint equations for a scaled orthographic bundle adjustment. The model was then tested using two image sets of a single, small vessel (about 6 m length) with a cube target of known dimensions at two distinct ranges; perspective solutions were also calculated for comparison. RMS residual errors of 0.74–0.78 pixels associated with the new method compare favorably to a residual error range of 0.59–0.74 pixels using a perspective bundle adjustment of the same target points. Relative precisions (as a ratio of target size) of between 1:1650 and 1:750 have been achieved at ranges of 375 m and 850 m, respectively, given comparisons with the known cube dimensions. A third image dataset consisting of a network of 16 images was solved with a 1:2200 relative precision showing the new method can successfully handle high redundancy. For the experiments that were conducted, the new method was found to produce less precise results than the perspective bundle solution for a FOV of 0.50–0.65° where the object fills 5–8% of the image. However, it was found to match the precision of the perspective model (with an uncalibrated camera) for a FOV of 0.20–0.30° where the object of interest fills only 1–2% of the full image.