IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 5, MAY 2000 1379 Locally Monotonic Diffusion Scott T. Acton, Senior Member, IEEE Abstract—Anisotropic diffusion affords an efficient, adaptive signal smoothing technique that can be used for signal enhance- ment, 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 sig- nals, 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 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. Index Terms—Anisotropic diffusion, partial differential equa- tions, scale space, signal enhancement. I. INTRODUCTION D IFFUSION processes may be used to model a special class of nonlinear adaptive signal filters. The same approach used to solve heat diffusion and bacteria migration problems [8] can be applied to the adaptive processing of a digital signal for the purposes of signal enhancement or feature extraction, for example. The localized diffusion operation can be modeled by a system of partial differential equations (PDE’s) that depend only on limited signal neighborhoods. The PDE-based signal diffusion methods are attractive for several reasons. Important signal features such as edges can be preserved using the diffusion approach; an anisotropic diffu- sion algorithm [1], [9] can be designed that enacts intraregion smoothing as opposed to interregion smoothing. The diffusion PDE’s generate a signal scale space [14]. The resultant family of signals that vary from coarse to fine may be used in several mul- tiscale signal analyses such as hierarchical searches, segmenta- tion, and coding. Unlike scale spaces generated using linear fil- ters (see Fig. 1), anisotropic diffusion can generate a scale-space Manuscript received May 28, 1998; revised November 18, 1999. This work was supported in part by the U.S. Army Research Office under Grant DAAH04-95-1-0255 and by NASA under EPSCoR Grant NCC5-171. The associate editor coordinating the review of this paper and approving it for publication was Prof. Arnab K. Shaw. The author is with the School of Electrical and Computer Engineering, Oklahoma State University, Stillwater, OK 74078-0321 USA (e-mail: sacton@ceat.okstate.edu). Publisher Item Identifier S 1053-587X(00)03302-X. Fig. 1. Scale space generated by the Gaussian filter. From back to front: original noisy input signal ( ); results of successive convolution with a Gaussian filter (standard deviation ). where edge localization is preserved (see Fig. 2). From an im- plementation perspective, the simple PDE’s can be efficiently executed on a locally connected parallel processor. Caselles et al. provide additional motivation for exploring PDE-based signal processing [4]. The PDE methods use simple models in the continuous domain where discrete filters become PDE’s as the sample spacing goes to zero. The simplified for- malism allows various properties to be proved or disproved in- cluding stability, locality, and causality. In addition, the analysis allows discussion of the existence and uniqueness of solutions. High degrees of accuracy and stability are possible through the mature results of numerical analysis. Finally, the PDE approach facilitates the combination of algorithms through the weighted sum of PDE’s. In a typical PDE approach, we have some knowledge of the dynamics of a physical process, and we model the dynamics with a suitable PDE. However, in the case pursued in this paper, we know the physical characteristics of the solution, and we de- sign PDE’s that converge to signals with the appropriate prop- erties. With the basic premise that we wish to preserve step edges (abrupt signal transitions), ramp edges, and smooth re- gions while eliminating impulses (noise), we can describe the prototypical signal with the property of local monotonicity. Previous diffusion algorithms have been shown to diverge, to converge to constant signals, or to converge to piecewise con- stant signals [3], [16]. Although these results are important to the understanding the current diffusion algorithms, the resultant signals may not be desired in applications where preservation of ramp edges is important. Diffusion algorithms that diverge [9] or converge to signals of zero mean curvature [15] must be halted in progress to obtain a nontrivial signal. The stop- ping conditions are difficult to estimate and do not provide a guaranteed description of the final signal characteristics. Piece- wise constant signals, on the other hand, may be of interest in 1053-587X/00$10.00 © 2000 IEEE