In this paper we study a variational problem in the space of functions of bounded Hessian. Our model constitutes a straightforward higher-order extension of the well known ROF functional (total variation minimisation) to which we add a non-smooth second order regulariser. It combines convex functions of the total variation and the total variation of the first derivatives. In what follows, we prove existence and uniqueness of minimisers of the combined model and present the numerical solution of the corresponding discretised problem by employing the split Bregman method. The paper is furnished with applications of our model to image denoising, deblurring as well as image inpainting. The obtained numerical results are compared with results obtained from total generalised variation (TGV), infimal convolution and Euler's elastica, three other state of the art higher-order models. The numerical discussion confirms that the proposed higher-order model competes with models of its kind in avoiding the creation of undesirable artifacts and blocky-like structures in the reconstructed images -- a known disadvantage of the ROF model -- while being simple and efficiently numerically solvable.
A Nonlinear Variational Approach to Motion-Corrected Reconstruction of Density Images
University of Münster（明斯特大学）；Institute for Computational and Applied Mathematics, University of Münster（明斯特大学计算与应用数学研究所）；Cells in Motion Cluster of Excellence, University of Münster（明斯特大学运动细胞卓越集群）；University of Lübeck（吕贝克大学）；Institute of Mathematics and Image Computing, University of Lübeck（吕贝克大学数学与图像计算研究所）；Fraunhofer MEVIS Project Group Image Registration（弗劳恩霍夫 MEVIS 图像配准项目组）
A Variational Reconstruction Method for Undersampled Dynamic X-ray Tomography based on Physical Motion Models
Department of Computer Science, University College London（伦敦大学学院计算机科学系）；Department of Mathematics and Statistics, University of Helsinki（赫尔辛基大学数学与统计学系）
A New Variational Model for Joint Image Reconstruction and Motion Estimation in Spatiotemporal Imaging
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（LSEC，ICMSEC，数学系统科学研究所，中国科学院）；Department of Mathematics, KTH–Royal Institute of Technology（数学系，皇家理工学院）
Position-Blind Ptychography: Viability of image reconstruction via data-driven variational inference
Department of Informatics, University of Hamburg（汉堡大学信息学院）；Center for Free-Electron Laser Science CFEL, Deutsches Elektronen-Synchrotron DESY（自由电子激光科学中心 CFEL，德意志电子同步辐射实验室）；Department of Mathematics, Bundesstr. 55, University of Hamburg（汉堡大学数学系）；CIT School, Technical University of Munich（慕尼黑技术大学 CIT 学院）；Munich Center for Machine Learning, München（慕尼黑机器学习中心）；Massachusetts Institute of Technology (MIT)（麻省理工学院）；The NSF AI Institute for Artificial Intelligence and Fundamental Interactions（国家科学基金会人工智能与基本相互作用研究院）；Helmholtz Imaging, Deutsches Elektronen-Synchrotron DESY（海德堡成像，德意志电子同步辐射实验室）
Reference-free logged energy-oracle recovery for neural approximations of symmetric coercive variational problems: conforming Riesz reconstruction and archive-level selection
Laboratory for Analysis and Modeling of Systems and Decision Support (LAMSAD), Hassan 1st University of Settat（塞塔特哈桑一世大学系统与决策支持分析建模实验室）；Al Akhawayn University in Ifrane（伊夫兰阿赫拉万大学）
