Regularization in Image Non-Rigid Registration: I. Trade-off between Smoothness and Intensity Similarity

In this report, we first propose a new classification of non-rigid registratio- n algorithms into three main categories: in one hand, the geometric algorithms- , and in the other hand, intensity based methods that we split here into standard intensity-based (SIB) and pair-and-smooth (P&S) algorithms. We then focus on the subset of SIB and P&S algorithms that are competitive, i.e. that use a regularization energy which is minimized together with the intensity similarity energy. In SIB algorithms, these two energies are combined in a weighted sum, and thus the trade-off between them is direct. P&S algorithms alternates their respective minimization, leading to the characteristic two steps: pairing of points, and smoothing. We theoretically compare the behavior of SIB and P&S algorithms, and more precisely, we explain why in practice the smoothness of the transforms estimated by SIB algorithms is non-uniform, thus difficult to control, while P&S algorithms estimate a motion that is more uniformly smooth. We give an example illustrating this behavior. Very few P&S algorithms minimize a global energy. We therefore propose a new image registration energy whose minimization leads to a \PAS algorithm. This energy is general, and can use any existing similarity or regularization energy. Its behavior is also compared to the previous SIB and \PAS algorithms. This new energy allows uniformly smooth solutions, as for our previous P&S algorithm, while preventing registration of non-informative, noisy areas, as for SIB algorithms.