Stabilizing deep tomographic reconstruction: Part B. Convergence analysis and adversarial attacks
Weiwen Wu,
Dianlin Hu,
Wenxiang Cong,
Hongming Shan,
Shaoyu Wang,
Chuang Niu,
Pingkun Yan,
Hengyong Yu,
Varut Vardhanabhuti,
Ge Wang
Affiliations
Weiwen Wu
Biomedical Imaging Center, Center for Biotechnology and Interdisciplinary Studies, Department of Biomedical Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA; School of Biomedical Engineering, Sun Yat-sen University, Shenzhen, Guangdong, China; Department of Diagnostic Radiology, Li Ka Shing Faculty of Medicine, University of Hong Kong, Hong Kong SAR, China
Dianlin Hu
The Laboratory of Image Science and Technology, Southeast University, Nanjing, China
Wenxiang Cong
Biomedical Imaging Center, Center for Biotechnology and Interdisciplinary Studies, Department of Biomedical Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA
Hongming Shan
Biomedical Imaging Center, Center for Biotechnology and Interdisciplinary Studies, Department of Biomedical Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA; Institute of Science and Technology for Brain-inspired Intelligence, Fudan University, Shanghai, China
Shaoyu Wang
Department of Electrical & Computer Engineering, University of Massachusetts Lowell, Lowell, MA, USA
Chuang Niu
Biomedical Imaging Center, Center for Biotechnology and Interdisciplinary Studies, Department of Biomedical Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA
Pingkun Yan
Biomedical Imaging Center, Center for Biotechnology and Interdisciplinary Studies, Department of Biomedical Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA
Hengyong Yu
Department of Electrical & Computer Engineering, University of Massachusetts Lowell, Lowell, MA, USA; Corresponding author
Varut Vardhanabhuti
Department of Diagnostic Radiology, Li Ka Shing Faculty of Medicine, University of Hong Kong, Hong Kong SAR, China; Corresponding author
Ge Wang
Biomedical Imaging Center, Center for Biotechnology and Interdisciplinary Studies, Department of Biomedical Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA; Corresponding author
Summary: Due to lack of the kernel awareness, some popular deep image reconstruction networks are unstable. To address this problem, here we introduce the bounded relative error norm (BREN) property, which is a special case of the Lipschitz continuity. Then, we perform a convergence study consisting of two parts: (1) a heuristic analysis on the convergence of the analytic compressed iterative deep (ACID) scheme (with the simplification that the CS module achieves a perfect sparsification), and (2) a mathematically denser analysis (with the two approximations: [1] AT is viewed as an inverse A-1 in the perspective of an iterative reconstruction procedure and [2] a pseudo-inverse is used for a total variation operator H). Also, we present adversarial attack algorithms to perturb the selected reconstruction networks respectively and, more importantly, to attack the ACID workflow as a whole. Finally, we show the numerical convergence of the ACID iteration in terms of the Lipschitz constant and the local stability against noise. The bigger picture: For deep tomographic reconstruction to realize its full potential in practice, it is critically important to address the instabilities of deep reconstruction networks, which were identified in a recent PNAS paper. Our analytic compressed iterative deep (ACID) framework has provided an effective solution to address this challenge by synergizing deep learning and compressed sensing through iterative refinement. Here, we provide an initial convergence analysis, describe an algorithm to attack the entire ACID workflow, and establish not only its capability of stabilizing an unstable deep reconstruction network but also its stability against adversarial attacks dedicated to ACID as a whole. Although our theoretical results are under approximations, they shed light on the converging mechanism of ACID, serving as a basis for further investigation.
