Functions of bounded deformation (BD) are widely used in the theory of elastoplasticity to describe the possibly discontinuous displacement fields inside elastoplastic bodies. BD functions have been proved suitable for deformable image registration, the goal of which is to find the displacement field between a moving image and a fixed image. Recently BD functions have been generalized to symmetric tensor fields of bounded generalized variation. In this paper, we focus on the first-order symmetric tensor fields, i.e., vector-valued functions, of bounded generalized variation. We specify these functions as functions of bounded generalized deformation (BGD) since BGD functions are natural generalizations of BD functions. We propose a BGD model for deformable image registration problems by regarding concerned displacement fields as BGD functions. BGD model employs not only the first-order but also higher-order coupling information of components of the displacement field. It turns out that BGD model allows for jump discontinuities of displacements while, in contrast to BD model, at the same time is able to employ higher-order derivatives of displacements in smooth regions. As a result, BGD model tends to capture possible discontinuities of displacements appeared around edges of the target objects while keep the smoothness of displacements inside the target objects as well. This characteristic enables BGD model to obtain better registration results than BD model and other variational models. To our knowledge, it is the first time in literature to use BGD functions for image registration. A first-order adaptive primal–dual algorithm is adopted to solve the proposed BGD model. Numerical experiments on 2D and 3D images show both effectiveness and advantages of BGD model.
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The authors would like to thank the multidisciplinary team of liver, billiary and pancreatic tumors in Nanjing Drum Tower Hospital, China for providing CT liver images used in the second 2D numerical experiments. The first author would like to thank Dong WANG for his help with formatting the Latex code of this manuscript.
Conflict of interest
The authors declare that they have no conflict of interest.
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This work is supported by National Natural Science Foundation of China (Grant Nos. 11971229, 12090023) and China’s Ministry of Science and Technology (Grant No. SQ2020YFA070208)
Communicated by Xavier Pennec.
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Nie, Z., Li, C., Liu, H. et al. Deformable Image Registration Based on Functions of Bounded Generalized Deformation. Int J Comput Vis (2021). https://doi.org/10.1007/s11263-021-01439-x
- Functions of bounded generalized deformation
- Deformable image registration
- Functions of bounded deformation
- Primal–dual algorithm