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- 1. 1070 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 3, MARCH 2012 Image Reduction Using Means on Discrete Product Lattices Gleb Beliakov, Senior Member, IEEE, Humbeto Bustince, Member, IEEE, and Daniel Paternain AbstractWe investigate the problem of averaging values on lat- over a product lattice are, in general, different from the Carte-tices and, in particular, on discrete product lattices. This problem sian products of the averages. This has an implication over thearises in image processing when several color values given in RGB, methods of color image reduction.HSL, or another coding scheme need to be combined. We showhow the arithmetic mean and the median can be constructed by We recall the problem of image reduction for grayscale im-minimizing appropriate penalties, and we discuss which of them ages, and we justify the importance of penalty functions. Wecoincide with the Cartesian product of the standard mean and the prove that, when we reconstruct a reduced image, the error withmedian. We apply these functions in image processing. We present respect to the original image may be determined by the reduc-three algorithms for color image reduction based on minimizing tion method that has been employed.penalty functions on discrete product lattices. We present three new color image reduction algorithms that Index TermsAggregation operators, image reduction, mean, are based on minimizing a penalty function dened over productmedian, penalty functions. lattices. We carry out an experimental study in which we com- pare the proposed algorithms with the alternative methods that can be found in the literature, and we analyze the stability of the I. INTRODUCTION algorithms with respect to noise in the images. The structure of this paper is as follows. In Section II, we pro- vide preliminary denitions. In Section III, we give the deni-T HE NEED TO aggregate several inputs into a single rep- tions of aggregation functions based on penalties, i.e., dened on resentative output frequently arises in many practical ap- product lattices, and we present the problem of image reductionplications. In image processing, it is often necessary to average algorithms. We discuss solutions to resulting optimization prob-the values of several neighboring pixels (to reduce the image lems in Section IV. In Section V, we present the color image http://ieeexploreprojects.blogspot.comsize or apply a lter) or to average pixel values in two different reduction algorithms, and we present an experimental study inbut related images (e.g., in stereovision [1]). When the images Section VI. Conclusions are presented in Section VII.are in color, i.e., typically coded as discrete RGB, CMY, or HSLvalues, then it is customary to average the values in the respec-tive channels. It is not immediately clear that this is appropriate II. PRELIMINARIESand what are the other ways to average color values. In this paper, we study averaging on product lattices (RGB or A. Aggregation Functionsanother color coding scheme is an example of a product lattice).We note previous works related to triangular norms on posets The research effort concerning aggregation functions, theirand lattices [2], [3] and on discrete chains [4]. Our setting is behavior, and properties has been disseminated throughout var-different as we do not deal with associativity of aggregation op- ious elds including decision making, knowledge-based sys-erations but, in contrast, require averaging behavior. tems, articial intelligence, and image processing. Recent works We focus on a large class of averages based on minimizing providing a comprehensive overview include [9][13].a penalty function [5][8]. We show that, with an appropriately Denition 1: Function is called anchosen class of penalties, the resulting penalty-based functions aggregation function if it is monotonically nondecreasing inare monotone and idempotent. We also show that the averages each variable and satises and , with and , respectively. Denition 2: The aggregation function is called averaging Manuscript received September 15, 2010; revised July 17, 2011; accepted if it is bounded by the minimum and the maximum of itsSeptember 04, 2011. Date of publication September 15, 2011; date of currentversion February 17, 2012. This work was supported in part by the Govern- argumentsment of Spain under Grant TIN2010-15055. The associate editor coordinatingthe review of this manuscript and approving it for publication was Dr. Rick P.Millane. G. Beliakov is with the School of Information Technology, Deakin University,Burwood 3125, Australia (e-mail: [email protected]). H. Bustince and D. Paternain are with the Department of Automatics andComputation, Public University of Nevarra, 31006 Pamplona, Spain (e-mail: It is immediate that averaging aggregation functions [email protected]; [email protected]). idempotent (i.e., ) and (because Color versions of one or more of the gures in this paper are available onlineat http://ieeexplore.ieee.org. of monotonicity) vice versa. Then clearly, the boundary condi- Digital Object Identier 10.1109/TIP.2011.2168412 tions and are satised. 1057-7149/$26.00 2011 IEEE
- 2. BELIAKOV et al.: IMAGE REDUCTION USING MEANS ON DISCRETE PRODUCT LATTICES 1071 Well-known examples of averaging functions are the arith- We will deal with Cartesian products of nite chains , whichmetic mean and the median. It is known that the arithmetic mean are precisely the type of a product lattice representing colors inand the median are solutions to simple optimization problems, image processing, with the length of each chain typically beingin which a measure of disagreement between the inputs is min- 256. We note that all nite chains of the same length are isomor-imized (see [5][7], [10], [14]). The main motivation is the fol- phic to each other; hence, we can represent them as nonnegativelowing: Let be the inputs and be the output. If all the inputs integers and elements of product lattices as tuplescoincide , then the output is , and we and .have a unanimous vote. If some input , then we impose Denition 7: Let and be two aggregation functions de-a penalty for this disagreement. The larger the disagreement ned on sets and , respectively. The Cartesian product ofand the more inputs disagree with the output, the larger (in gen- aggregation functions iseral) is the penalty. We look for an aggregated value that mini- dened bymizes the penalty. Thus, we need to dene a suitable measure of disagreementor dissimilarity. Denition 3: Let be a penalty functionwith the properties: C. Image Reduction 1 for all , ; 2 if all ; Image reduction consists in reducing the dimension of the 3 is quasi-convex in for any . image while keeping as much information as possible. Image The penalty-based function is reduction can be used to accelerate computations on an image or just to reduce the cost of its storage or transmission. There exist several methods for image reduction in the litera- ture. Some of them consider the image to be reduced in a globalif is the unique minimizer and if the set of way [15][17] or in a transform domain [18]. Other widely usedminimizers is the interval . methods act locally over pieces (blocks) of the image [19], [20]. Remark 1: is quasi-convex if The division of the image in blocks of small size allows one to for all and all , within its design simple reduction algorithms.domain.