Discretisation of 3D deformation fields: Implications for establishing correspondence between 2D X-ray mammographic projections

The registration of temporal 2D X-ray mammograms is a challenging task which is difficult in part due to the superimposition of structures found in the breast together with the complex 3D deformation that the breast undergoes during compression. This paper presents an investigation into a method for registration of 2D mammographic images which accounts for varying displacements through the breast by allowing additional degrees of freedom for the transformation at different depths. We use simulated compressions of MR derived volumes to determine what is the maximum sampling interval of the deformation field that yields a deformation field that agrees with the simulation within a certain tolerance. We found that a subsampling of the deformation field by a factor of 17 in-plane and 10 out-of-plane increased error in the estimate of displacement of tissue projected onto the imaging plane by less than 1mm for all points, excluding outliers. This has important ramification for the search for the appropriate 2D non-diffeomorphic transformation between 2 2D X-ray mammograms. This work supports the development of more accurate registration algorithms by taking into account the realistic 3D movement of tissue as well as reducing the complexity of the problem.