Locally monotonic diffusion

Anisotropic diffusion affords an efficient, adaptive signal smoothing technique that can be used for signal enhancement, signal segmentation, and signal scale-space creation. This paper introduces a novel partial differential equation (PDE)-based diffusion method for generating locally monotonic signals. Unlike previous diffusion techniques that diverge or converge to trivial signals, locally monotonic (LOMO) diffusion converges rapidly to well-defined LOMO signals of the desired degree. The property of local monotonicity allows both slow and rapid signal transitions (ramp and step edges) while excluding outliers due to noise. In contrast with other diffusion methods, LOMO diffusion does not require an additional regularization step to process a noisy signal and uses no ad hoc thresholds or parameters. In the paper, we develop the LOMO diffusion technique and provide several salient properties, including stability and a characterization of the root signals. The convergence of the algorithm is well behaved (nonoscillatory) and is independent of the signal length, in contrast with the median filter. A special case of LOMO diffusion is identical to the optimal solution achieved via regression. Experimental results validate the claim that LOMO diffusion can produce denoised LOMO signals with low error using less computation than the median-order statistic approach.