Curve denoising by multiscale singularity detection and geometric shrinkage

We propose a method for denoising piecewise smooth curves, given a number of noisy sample points. Using geometric variants of wavelet shrinkage methods, our algorithm preserves corners while enforcing that the smoothed arcs lie in an L2 Sobolev space Hα of order α chosen by the operator. The reconstruction is scale-invariant when using the Sobolev space H3/2, adapts to the local noise level, and is essentially free of tuning parameters. In particular, our noise-adaptivity ensures that there is no arbitrarily-chosen “diffusion time” parameter in the denoising. Further, in cases where the distinction between signal and noise is unclear, we show how statistics gathered from the curve can be used to identify a finite number of “good” choices for the denoising.