Efficient Homotopy Method for Total Variation Image Registration

The so-called image registration is to find an optimal spatial transformation such that the transformed template image becomes similar to the reference image as much as possible. Several partial differential equations (PDEs) based variational methods can be used for deformable image registration, mainly differing in how regularization for deformation fields[1]. Regularization techniques based on total variation (TV) preserving discontinuities of the deformation field are useful to a class of problems where smoothness is less a concern. A previous study by C. Frohn-Schauf, S. Henn [2, 3] considered multigrid method and the sequential quadratic approximation on total variation based image registration. On one hand, although the approximation solutions obtained from C. Frohn-Schauf, S. Henn [2, 3] are visually pleasing, they may not fulfill the necessary condition for being a minimiser of the variational problem (4), we can refer to [12, P₆₆₀]. On the other hand, when the smoothing parameter β is very small, the corresponding Euler-Lagrange equation (EL)is very difficult to solve. In this paper, we propose a homotopy method to solve the resulting TV based EL equation and consider using curve tracking to select smoothing parameter β adaptively. Numerical experiments confirms that our proposed method can effectively find a highly accurate solution and produce excellent image registration results in terms of image quality.