Convex optimisation methods in image orientation (2016)

Motivation:

In photogrammetry one basic task is to derive three-dimensional object information from a set of two-dimensional overlapping images. Most approaches in image orientation are based on a simultaneous estimation of object information and the orientation parameters of the imagery. The functional model of the parameter estimation is nonlinear and hence dependent on initial values. In the determination of initial values it is rather common to proceed in a sequential manner: It starts with a certain minimum number of images (often only two or three) and additional images are added iteratively. In this process errors are accumulated and hence the final result may suffer from only locally optimised parameters and vary by changing that initial configuration.

Goals:

This project aims at formulating different approaches for the orientation of images that obey characteristics of convex optimization, which means that they yield the global minimum solution with respect to a proper metric and under relatively mild constraints.

One approach is based on a two-step procedure and is able to estimate the globally optimal solution out of a set of multiple and redundantly overlapping images with given homologous point tuples. The first step of the approach consists of the estimation of global rotation parameters by averaging relative rotation estimates for image pairs (these are determined from the homologous points via the essential matrix in a pre-processing step). For the averaging we make use of algebraic group theory in which rotations, as part of the special orthogonal group SO(3), form a Lie group with a Riemannian manifold structure. This allows for a mapping to the local Euclidean tangent space of SO(3), the Lie algebra. In this space the redundancy of relative orientations is used to compute an average of the absolute rotation for each image and furthermore to detect and eliminate outliers. In the second step translation parameters and the object coordinates of the homologous points are estimated within a convex L∞ optimisation, in which the rotation parameters are kept fixed. As an optional third step the results can be used as initial values for a final bundle adjustment that does not suffer from bad initialisation and quickly converges to a globally optimal solution.

In future work this approach can be further extended to be robust against gross errors in relative orientations or homologous points. Also different approaches based on convex relaxation have to be investigated, for instance by searching for rotations not part of SO(3) but rather part of its convex hull.