DENOISING OF SPHERE- AND SO(3)-VALUED DATA BY RELAXED TIKHONOV REGULARIZATION

. Manifold-valued signal- and image processing has received attention due to modern image acquisition techniques. Recently, a convex relaxation of the otherwise non-convex Tikhonov-regularization for denoising circle-valued data has been proposed by Condat (2022). The circle constraints were encoded in a series of low-dimensional, positive semi-definite matrices. Using Schur complement arguments, we showed 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 the special orthogonal group SO(3), where we relied 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 used the alternating direction method of multipliers to demonstrate the denoising behavior of the proposed method. In a series of experiments, we demonstrated the numerical convergence of the signal- or image values to the underlying manifold.