2023/07/20 by Robert Beinert, Beinert, Robert, Jonas Bresch +3 · 1 citation
Computer Science · Mathematics · Medicine · #65J22 #90C22 #90C25 #94A08 #94A12 #Algorithm #Applied mathematics #Artificial intelligence #Computer science #Convex function #Convex optimization #FOS: Mathematics #Geometry #Image and Signal Denoising Methods #Inverse problem #Mathematical analysis #Mathematics #Medical Imaging Techniques and Applications #Noise reduction #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Optimization and Control (math.OC) #Quaternion #Regular polygon #Regularization (linguistics) #Signal processing #Tikhonov regularization
paper · pdf · doi:10.48550/arxiv.2307.10980
openalex publication_date 2023/07/20 · openalex created_date 2023/07/22 · openalex updated_date 2026/07/28
Manifold-valued signal- and image processing has received attention due to modern image acquisition techniques. Recently, a convex relaxation of the otherwise nonconvex Tikhonov-regularization for denoising circle-valued data has been proposed by Condat (2022). The circle constraints are here encoded in a series of low-dimensional, positive semi-definite matrices. Using Schur complement arguments, we show that the resulting variational model can be simplified while leading to the same solution. The simplified model can be generalized to higher dimensional spheres and to SO(3)-valued data, where we rely on the quaternion representation of the latter. Standard algorithms from convex analysis can be applied to solve the resulting convex minimization problem. As proof-of-the-concept, we use the alternating direction method of multipliers to demonstrate the denoising behavior of the proposed method. In a series of experiments, we demonstrate the numerical convergence of the signal- or image values to the underlying manifold.