图像配准中方向场正则化模型的适定性和收敛性

郑晓俊, 郇中丹, 刘君

数学学报 ›› 2021, Vol. 64 ›› Issue (3) : 385-404.

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数学学报 ›› 2021, Vol. 64 ›› Issue (3) : 385-404. DOI: 10.12386/A2021sxxb0033
论文

图像配准中方向场正则化模型的适定性和收敛性

    郑晓俊1, 郇中丹2, 刘君2
作者信息 +

Well-posedness and Convergence of the Vector Field Regularization Model in Image Registration

    Xiao Jun ZHENG1, Zhong Dan HUAN2, Jun LIU2
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文章历史 +

摘要

图像配准是图像处理的一个重要方面.方向场正则化模型是现有配准方法中效果相对突出的模型.然而它仍然无法正确对齐所有感兴趣的区域.因此,本文从理论角度研究方向场正则化模型,希望寻找模型设计可能存在的问题.由于模型中有一个直接变量和一个由直接变量通过常微初值问题确定的间接变量,所以它是数学上的一类新颖的正则化模型.方向场正则化模型定义为minv{α||v||H2+ρTyvτ)),S)},其中TS分别是模板图像和参考图像,yvτ):x?yvτ;0,x)是由初值问题dy/ds=vs,y),y(0)=x的解yvs;0,x)定义的变换,ρ是相似性泛函,α>0是正则化参数,H是希尔伯特空间.本文首先证明了方向场正则化模型有稳定解,然后证明了其收敛性.结合yvτ)与v的收敛关系和正则化问题的经典理论可得上述结论.然而,在现有理论下,ρST需满足较强的条件.本文通过充分利用yvτ)的性质,提出了关于ρST的相对弱的条件.此外,我们还验证了配准常用的3个相似性泛函都满足所提条件.

Abstract

Image registration is fundamental to image processing. The vector field regularization model performs relatively well among a large number of registration methods. However, it still can't correspond to all interested regions across images correctly. Therefore, we hope to study the theory of the vector field regularization model to see whether there are some problems with the design of the model. Moreover, as there are two unknowns which are related by an initial value problem in the regularization model, it is novel in mathematics. The vector field regularization model takes the form minv {α||v||H2 + ρ(T (yv(τ)), S)}, where T is a template image, S is a reference image, yv(τ):x ? yv(τ;0, x) is a transformation determined by the solution yv(s;0, x) of the initial value problem dy/ds=v(s, y), y(0)=x, ρ is a similarity functional, α> 0 is a regularization parameter and H is a Hilbert space. In this paper, we firstly show the vector field regularization model has stable solutions and then demonstrate its convergence. The above results can be obtained by the standard arguments of regularized problems together with the convergence relation of yv(τ) and v. However, the requirements for ρ, S and T are relatively strong under the existing regularization theory. We give relatively weak conditions for ρ, S and T by taking full advantage of the good properties of yv(τ). In addition, we verify that three commonly used similarity functionals in image registration satisfy the given conditions.

关键词

方向场正则化模型 / 存在性 / 稳定性 / 收敛性 / 图像配准

Key words

vector field regularization model / existence / stability / convergence / image registration

引用本文

导出引用
郑晓俊, 郇中丹, 刘君. 图像配准中方向场正则化模型的适定性和收敛性. 数学学报, 2021, 64(3): 385-404 https://doi.org/10.12386/A2021sxxb0033
Xiao Jun ZHENG, Zhong Dan HUAN, Jun LIU. Well-posedness and Convergence of the Vector Field Regularization Model in Image Registration. Acta Mathematica Sinica, Chinese Series, 2021, 64(3): 385-404 https://doi.org/10.12386/A2021sxxb0033

参考文献

[1] Arnold V. I., Ordinary Differerntial Equations, MIT Press, Cambridge, 1973.
[2] Avants B. B., Epstein C. L., Grossman M., et al., Symmetric diffeomorphic image registration with crosscorrelation:evaluating automated labeling of elderly and neurodegenerative brain, Med. Image Anal., 2008, 12:26-41.
[3] Avants B. B., Tustison N. J., Song G., et al., Ants:Advanced Open-source Normalization Tools for Neuroanatomy, Penn Image Computing and Science Laboratory, Philadelphia, 2009.
[4] Avants B. B., Tustison N. J., Song G., et al., A reproducible evaluation of ANTs similarity metric performance in brain image registration, NeuroImage, 2011, 54:2033-2044.
[5] Beg M. F., Miller M. I., Trouvé A., et al., Computing large deformation metric mappings via geodesic flows of diffeomorphisms, Int. J. Comput. Vision, 2005, 61:139-157.
[6] Brown L. G., A survey of image registration techniques, ACM Comput. Surv., 1992, 24:325-376.
[7] Burger M., Osher S., Convergence rates of convex variational regularization, Inverse Probl., 2004, 20:1411-1421.
[8] Christensen G. E., Johnson H. J., Consistent image registration, IEEE T. Med. Imaging, 2001, 20:568-582.
[9] Chung A. C., Wells W. M., Norbash A., et al., Multi-modal image registration by minimising Kullback-Leibler distance, The 5th Int. Conf. on Medical Image Computing and Computer-Assisted Intervention (Lecture Notes in Computer Science), Vol. 2489, Springer-Verlag, Berlin, 2002:523-532.
[10] Cover T. M., Thomas J. A., Elements of Information Theory, John Wiley & Sons, New York, 2012.
[11] Crum W. R., Griffin L. D., Hill D. L., et al., Zen and the art of medical image registration:correspondence, homology, and quality, NeuroImage, 2003, 20:1425-1437.
[12] Droske M., Rumpf M., A variational approach to nonrigid morphological image registration, SIAM J. Appl. Math., 2004, 64:668-687.
[13] Du J., Younes L., Qiu A., Whole brain diffeomorphic metric mapping via integration of sulcal and gyral curves, cortical surfaces, and images, NeuroImage, 2011, 56:162-173.
[14] Dupuis P., Grenander U., Miller M. I., Variational problems on flows of diffeomorphisms for image matching, Q. Appl. Math., 1998, 56:587-600.
[15] Engl H. W., Kunisch K., Neubauer A., Convergence rates for Tikhonov regularisation of non-linear ill-posed problems, Inverse Probl., 1989, 5:523-540.
[16] Evants L. C., Partial Differential Equations, American Mathematical Society, Providence, 2010.
[17] Fiez J. A., Damasio H., Grabowski T. J., Lesion segmentation and manual warping to a reference brain:intra-and interobserver reliability, Hum. Brain Mapp., 2000, 9:192-211.
[18] Flemming J., Theory and examples of variational regularization with non-metric fitting functionals, J. Inverse Ill-Pose Probl., 2010, 18:677-699.
[19] Flemming J., Generalized Tikhonov Regularization:Basic Theory and Comprehensive Results on Convergence Rates, Ph.D. thesis, Chemnitz University of Technology Chemnitz, 2011.
[20] Hill D. L., Batchelor P. G., Holden M., et al., Medical image registration, Phys. Med. Biol., 2001, 46:R1-R45.
[21] Hofmann B., Kaltenbacher B., Poeschl C., et al., A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Probl., 2007, 23:987-1010.
[22] Jensen J. R., Introductory Digital Image Processing:A Remote Sensing Perspective, University of South Carolina, Columbia, 1986.
[23] Kasturi R., Jain R. C., Computer Vision:Principles, IEEE Computer Society Press, Los Alamitos, 1991.
[24] Kim M., Wu G., Yap P. T., et al., A general fast registration framework by learning deformation-appearance correlation, IEEE T. Image P., 2012, 21:1823-1833.
[25] Klein A., Anderson J., Ardekani B. A., et al., Evaluation of 14 nonlinear deformation algorithms applied to human brain MRI registration, NeuroImage, 2009, 46:786-802.
[26] Klein S., Staring M., Pluim J. P., Evaluation of optimization methods for nonrigid medical image registration using mutual information and B-splines, IEEE T. Image P., 2007, 16:2879-2890.
[27] Maintz J. B., Viergever M. A., A survey of medical image registration, Med. Image Anal., 1998, 2:1-36.
[28] Modersitzki J., Numerical Methods for Image Registration, Oxford University Press, New York, 2003.
[29] Reducindo I., Arce-Santana E. R., Campos-Delgado D. U., et al., Non-rigid multimodal medical image registration based on the conditional statistics of the joint intensity distribution, Procedia Tech., 2013, 7:126-133.
[30] Rohlfing T., Maurer C. R., Bluemke D. A., et al., Volume-preserving nonrigid registration of MR breast images using free-form deformation with an incompressibility constraint, IEEE T. Med. Imaging, 2003, 22:730-741.
[31] Sotiras A., Davatzikos C., Paragios N., Deformable medical image registration:a survey, IEEE T. Med. Imaging, 2013, 32:1153-1190.
[32] Studholme C., Cardenas V., Blumenfeld R., et al., Deformation tensor morphometry of semantic dementia with quantitative validation, NeuroImage, 2004, 21:1387-1398.
[33] Stytz M. R., Frieder G., Three-dimensional medical imaging:algorithms and computer systems, ACM Comput. Surv. (CSUR), 1991, 23:421-499.
[34] Vogel C. R., Computational Methods for Inverse Problems, Society for Industrial and Applied Mathematics, Philadelphia, 2002.
[35] Wang H., Suh J. W., Das S. R., et al., Multi-atlas segmentation with joint label fusion, IEEE T. Pattern Anal. Mach. Intell., 2013, 35:611-623.
[36] Yosida K., Functional Analysis, Springer-Verlag, Berlin, 1965.
[37] Yushkevich P. A., Avants B. B., Pluta J., et al., A high-resolution computational atlas of the human hippocampus from postmortem magnetic resonance imaging at 9.4 T, NeuroImage, 2009, 44:385-398.

基金

海南省自然科学基金资助项目(118QN023)

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