Total Variation Constrained Non-Negative Matrix Factorization for Medical Image Registration Chengcai Leng, Hai Zhang, Guorong Cai, Zhen Chen, and Anup Basu, Senior Member, IEEE Abstract—This paper presents a novel medical image registration algorithm named total variation constrained graph- regularization for non-negative matrix factorization (TV-GNMF). The method utilizes non-negative matrix factorization by total variation constraint and graph regularization. The main contributions of our work are the following. First, total variation is incorporated into NMF to control the diffusion speed. The purpose is to denoise in smooth regions and preserve features or details of the data in edge regions by using a diffusion coefficient based on gradient information. Second, we add graph regularization into NMF to reveal intrinsic geometry and structure information of features to enhance the discrimination power. Third, the multiplicative update rules and proof of convergence of the TV-GNMF algorithm are given. Experiments conducted on datasets show that the proposed TV-GNMF method outperforms other state-of-the-art algorithms. Index Terms—Data clustering, dimension reduction, image registration, non-negative matrix factorization (NMF), total variation (TV).    I. Introduction I MAGE registration is an important research topic for aligning two or more images of the same scene taken at different times, viewpoints, or sensors [1]. Registration is widely used in computer vision and medical image processing, including multimodal image fusion, medical image reconstruction, and the monitoring of tumors. For example, the fusion of multimodal information can be realized by registering two images, which provides better visualization of anatomical structures and functional changes to facilitate diagnosis and treatment [2]. Area-based registration methods [3] mainly uses gray level information to optimize the maximum similarity measure, including mutual information (MI), by adapting optimization algorithms for registration [4]. Gong et al. [5] proposed a novel image registration method including the pre-registration and a fine-tuning process based on scale-invariant feature transform (SIFT) and MI. Woo et al. [6] presented a novel registration method based on MI by incorporating geometric and spatial context to compute the MI cost function in large spatial variation regions for multimodal image registration. However, these methods are very sensitive to intensity variations and suffer from noise interference. Feature-based methods for image registration directly detect salient features and construct feature descriptors, which are robust and invariant to noise, illumination, and distortion. SIFT [7] is one of the most popular methods invariant to rotation, scale, translation, and illumination changes. Rister et al. [8] extended SIFT to arbitrary dimensions by adjusting the orientation assignment and gradient histogram of key points. We can often treat the feature matching problem as a graph matching problem in image registration, since spectral graph theory [9] is widely used for image segmentation [10], [11], graph matching [12]–[15], and image registration [16]–[21]. In order to make many algorithms practical in several real-life applications, dimensionality reduction is necessary. In order to avoid the curse of dimensionality, some dimensionality reduction matching or registration methods have been introduced [22]–[24]. Xu et al. [24] proposed such a method for high-dimensional data sets using the Cramer-Rao lower bounds to estimate the transformation parameters and achieve data set registration. In addition, some manifold learning methods [25] have also been presented, such as ISOMAP [26], locally linear embedding (LLE) [27], and Laplacian Eigenmap [28]. However, many of these algorithms have high computational complexity, and deal poorly with large data sets [29]. Liu et al. [30] proposed the text detection method based on morphological component analysis and Laplacian dictionary, which can reduce the adverse effects of complex backgrounds and improve the discrimination power of dictionaries. Recently, some low-rank matrix factorization methods have been introduced in data representation [31]. Among these Manuscript received June 2, 2020; revised July 2, 2020, July 29, 2020; accepted August 6, 2020. This work was supported by the National Natural Science Foundation of China (61702251, 41971424, 61701191, U1605254), the Natural Science Basic Research Plan in Shaanxi Province of China (2018JM6030), the Key Technical Project of Fujian Province (2017H6015), the Science and Technology Project of Xiamen (3502Z20183032), the Doctor Scientific Research Starting Foundation of Northwest University (338050050), Youth Academic Talent Support Program of Northwest University (360051900151), and the Natural Sciences and Engineering Research Council of Canada, Canada. Recommended by Associate Editor Xin Luo. (Corresponding author: Chengcai Leng.) Citation: C. C. Leng, H. Zhang, G. R. Cai, Z. Chen, and A. Basu, “Total variation constrained non-negative matrix factorization for medical image Registration,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 5, pp. 1025–1037, May 2021. C. C. Leng is with the School of Mathematics, Northwest University, Xi’an 710127, and with the Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China, and also with the Department of Computing Science, University of Alberta, Edmonton, AB T6G 2E8, Canada (e-mail: ccleng@ nwu.edu.cn). H. Zhang is with the School of Mathematics, Northwest University, Xi’an 710127, China (e-mail: zhanghai@nwu.edu.cn). G. R. Cai is with the College of Computer Engineering, Jimei University, Xiamen 361021, China (e-mail: guorongcai.jmu@gmail.com). Z. Chen is the School of Measuring and Optical Engineering, Nanchang Hangkong University, Nanchang 330063, China (e-mail: chenzhen@nchu. edu.cn). A. Basu is with the Department of Computing Science, University of Alberta, Edmonton, AB T6G 2E8, Canada (e-mail: basu@ualberta.ca). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JAS.2021.1003979 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 8, NO. 5, MAY 2021 1025