Towards a theory of a solution space for the biplane imaging geometry problem

Biplane angiographic imaging is a primary method for visual and quantitative assessment of the vasculature. In order to reliably reconstruct the three-dimensional (3D) position, orientation, and shape of the vessel structure, a key problem is to determine the rotation matrix R and the translation vector t which relate the two coordinate systems. This so-called Imaging Geometry Determination problem is well studied in the medical imaging and computer vision communities and a number of interesting approaches have been reported. Each such technique determines a solution which yields 3D vasculature reconstructions with errors comparable to other techniques. From the literature, we see that different techniques with different optimization strategies yield reconstructions with equivalent errors. We have investigated this behavior, and it appears that the error in the input data leads to this equivalence effectively yielding what we call the solution space of feasible geometries, i.e., geometries which could be solutions given the error or uncertainty in the input image data. In this paper, we lay the theoretical framework for this concept of a solution space of feasible geometries using simple schematic constructions, deriving the underlying mathematical relationships, presenting implementation details, and discussing implications and applications of the proposed idea. Because the solution space of feasible geometries encompasses equivalent solutions given the input error, the solution space approach can be used to evaluate the precision of calculated geometries or 3D data based on known or estimated uncertainties in the input image data. We also use the solution space approach to calculate an imaging geometry, i.e., a solution.