An augmented lagrangian method for solving a new variational model based on gradients similarity measures and high order regulariation for multimodality registration

Abstract

In this work we propose a variational model for multi-modal image registration. It minimizes a new functional based on using reformulated normalized gradients of the images as the fidelity term and higher-order derivatives as the regularizer. We first present a theoretical analysis of the proposed model. Then, to solve the model numerically, we use an augmented Lagrangian method (ALM) to reformulate it to a few more amenable subproblems (each giving rise to an Euler-Lagrange equation that is discretized by finite difference methods) and solve iteratively the main linear systems by the fast Fourier transform; a multilevel technique is employed to speed up the initialisation and avoid likely local minima of the underlying functional. Finally we show the convergence of the ALM solver and give numerical results of the new approach. Comparisons with some existing methods are presented to illustrate its effectiveness and advantages.

Diffeomorphic unsupervised deep learning model for mono- and multi-modality registration

“…Various similarity terms can be used in the registration of multi-modal images such as mutual information Pluim et al., 21 normalised gradient fields Ruhaak et al. 22 normalised gradients fitting Theljani and Chen 6 ; Zhang et al. 23 In this work, we use parallel level sets similarity measure which is well suitable for measuring alignment between the gradients of two images.…”

“…There exist various deformable variational models for image registration where the unknown displacement field u is sought in a properly chosen functional space. [6][7][8][9][10][11] Generally speaking, the variational problem consists in solving the optimisation problem…”

“…In Jaderberg, 12 a spatial transformer network is developed to learn transformations for 2D images; however only affine and thin plate spline transformations were used. More general nonparametric transformations were considered in Haskins et al 13 ; Li and Fan 14 ; Theljani and Chen 6,25 for monomodal images. In these approaches, the transformations are controlled by penalising the derivatives of the deformation u in the loss function, which promotes smoothness of predicted transformations.…”

“…There exist various deformable variational models for image registration where the unknown displacement field u is sought in a properly chosen functional space. 6–11 Generally speaking, the variational problem consists in solving the optimisation problem where φ(x)=x+u(x). In (1), S(u) is a regularisation term which controls the smoothness of u and reflects our expectations by penalising unlikely transformations.…”

“…More general non-parametric transformations were considered in Haskins et al. 13 ; Li and Fan 14 ; Theljani and Chen 6,25 for mono-modal images. In these approaches, the transformations are controlled by penalising the derivatives of the deformation u in the loss function, which promotes smoothness of predicted transformations.…”