IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 11, NOVEMBER 2006 4271 Weighted Median Filters for Multichannel Signals Yinbo Li, Gonzalo R. Arce, Fellow, IEEE, and Jan Bacca, Student Member, IEEE Abstract—Weighted medians over multichannel signals are not uniquely defined. Due to its simplicity, Astola et al.’s Vector Median (VM) has received considerable attention particularly in image processing applications. In this paper, we show that the VM and its direct extension the Weighted VM are limited as they do not fully utilize the cross-channel correlation. In fact, VM treats all sub-channel components independent of each other. By revisiting the principles of Maximum Likelihood estimation of location in a multivariate signal space, we propose two new and conceptually simple multichannel weighted median filters which can capture cross-channel information effectively. Their optimal filter derivations are also presented, followed by a series of simula- tions from color image denoising to array signal processing where the advantages of the new filtering structures are illustrated. Index Terms—Multichannel signal processing, nonlinear fil- tering, vector medians, weighted medians. I. INTRODUCTION M ULTICHANNEL signal processing [1], [2] has under- gone rapid developments in the past decade, primarily due to its importance to multispectrum imaging, array pro- cessing, and medical signal processing [3]. Noise in color imaging is often impulsive, and since edges and details are of paramount importance, it is natural to extend weighted medians and order statistic filters [4], [5] from the scalar domain into the multichannel space. The extension of the weighted median (WM) for use with multidimensional (multichannel) signals is, however, not straightforward. Sorting multicomponent (vector) values and selecting the middle value is not as well defined as in the scalar case; see [6]. In consequence, the weighted median filtering operation of a multidimensional signal can be achieved in a number of ways, among which the most well known are marginal medians of orthogonal coordinates in [7], -norm median from [8] and [9] that minimizes the sum of distances to all samples, the half-plane median from [10] that minimizes the maximum number of samples on a half-plane convex hull median from [6] and [11] that results from the continuous “peeling” off pairs of extreme samples. For historical reviews and comprehensive introduction on multivariate medians, see [12] and [13]. Other approaches can be found in [14]–[17]. Manuscript received July 22, 2005; revised January 31, 2006. This work was prepared through collaborative participation in the Communications and Networks Consortium supported by the U.S. Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD19-01-2-0011. This work was also supported in part by the National Science Foundation under Grant 0312851. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Ioan Tabus. The authors are with the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716 USA (e-mail: yli@eecis.udel.edu; arce@eecis.udel.edu; baccarod@eecis.udel.edu). Digital Object Identifier 10.1109/TSP.2006.881208 A problem with many definitions of multivariate medians is that they have more conceptual meaning than practical use be- cause of their high computational complexities. The algorithms used to compute the median often involve gradient tech- niques or iterations that can only provide numerical solutions, as shown by [18], and even the fastest algorithm up-to-date for the Oja median [19] is about in time; see [13]. More- over, they usually have difficulties with their extension to struc- tures admitting weights and the optimal design of these weights. The so-called vector median (VM), proposed by Astola et al. [1] in 1990, has since received considerable attention in signal- processing research. Relaxed on its mathematical rigorousness but focused on its practical use, the basic idea of VM is to con- fine the filter output to be one of the vector-valued input sam- ples that minimizes the sum of distances from this output to all other samples in the observation window. Denote as the sample set. Astola’s VM is defined as (1) where denotes norm, where . An otherwise full multivariate space search is thus replaced by at most calculations of the cost function. The computational complexity is thus greatly reduced. It is easy to see from the definition that the vector median is a selection type 1 of robust filter, a property also possessed by univariate median filters. This property is particularly useful in image processing; since no new colors will be created, the chromaticity consistency of the fil- tered image is thus guaranteed. Compared to marginal vector processing methods that filter image channels independently, VM’s selection type filtering outputs also preserve the corre- lation that exists between the image channels. Combining the fact that VMs have good edge and detail preservation capabil- ities, just as their univariate counterparts, their application in color imaging has proven effective [1], [4]. In order to expand the capabilities of the vector median, the weighted vector me- dian (WVM) was introduced as a direct extension [20] (2) where a weighted cost function is defined such that the dis- tances are first weighted by nonnegative scalars before they are summed together. Although the WVM finds its immediate applications in color imaging, and has several optimization algorithms in existence [21]–[24], the principles of parameter estimation reveal that the very structure of weighted vector medians is 1 Selection type refers to the property that the filter output value is always equal to that of one of the input samples. 1053-587X/$20.00 © 2006 IEEE