We consider the Cauchy problem for the one-dimensional Perona-Malik equation

in the interval , with homogeneous Neumann boundary conditions.

We prove that the set of initial data for which this equation has a local-in-time classical solution is dense in . Here ``classical solution'' means that , , and are continuous functions in .

This paper addresses the problem of feature enhancement in noisy images when the feature is known to be constrained to a manifold. As an example, we approach the direction denoising problem in a general dimension via the geometric Beltrami framework for image processing. The spatial-direction space is a fiber bundle in which the spatial part is the base manifold and the direction space is the fiber. The feature (direction) field is represented accordingly as a section of the spatial-feature fiber bundle. The resulting Beltrami flow is a selective smoothing process that respects the bundle's structure, i.e., the feature constraint. Direction diffusion is treated as a canonical example of a non-Euclidean feature space. The structures of the fiber spaces of interest in this paper are the unit circle S¹, the unit sphere S², and the unit hypersphere Sⁿ. Applications to color analysis are discussed, and numerical experiments demonstrate again the benefits of the Beltrami framework in comparison to other feature enhancement schemes for nontrivial geometries in image processing.