Summary: A new morphological multiscale method in three-dimensional (3D) image processing is presented which combines the image processing methodology based on nonlinear diffusion equations and the theory of geometric evolution problems. Its aim is to smooth the level sets of a 3D image while simultaneously preserving geometric features such as edges and corners on the level sets. This is obtained by an anisotropic curvature evolution, where time serves as the multiscale parameter. Thereby the diffusion tensor depends on a regularized shape operator of the evolving level sets. As one suitable regularization, local \(L^{2}\) projection onto quadratic polynomials is considered. The method is compared to a related parametric surface approach, and a geometric interpretation of the evolution and its invariance properties is given. A spatial finite element discretization on hexahedral meshes and a semi-implicit, regularized backward Euler discretization in time are the building blocks of the easy-to-code algorithm. Different applications underline the efficiency and flexibility of the presented image processing tool.