Fuzzy ART for Relatively Fast Unsupervised Image Color Quantization

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1 Fuzzy ART fo Relatively Fast Unsuevised Image Colo Quantization Nicholas Shote and Takis Kasais Univesity of Cental Floida School of Electical Engineeing and Comute Science, Olando, FL, USA Abstact. The use of Fuzzy Adative Resonance Theoy (FA) is exloed fo the unsuevised colo quantization of a colo image. The ed, geen and blue colo comonent values of a given colo image ae assed as inut instances into FA which then gous simila colos into the same class. The aveage of all of the colos in a given class then elaces the ixel values whose oiginal colos belonged to that class. The FA unsuevised clusteing is caable of ealizing colo quantization with cometitive accuacy and aguably low comutation time. Keywods: Image Colo Quantization, Fuzzy ART, Clusteing, Unsuevised. 1 Intoduction Colo quantization (CQ) is the ocess of educing the numbe of colos in a given image by elacing a gou of simila colos with a single colo. The elacement should be such that the colo quantized image looks as simila as ossible to the oiginal image. Colo quantization is often used as a eocesso to vaious digital image ocessing alications such as segmentation. The JSEG [1] and B-JSEG [2] algoithms both ely on image CQ eocessing. We intend to segment high esolution aeial images (4000x000 ixels at 15cm ixel esolution) and theefoe needed a comutationally efficient colo quantizing technique. Because of Fuzzy Adative Resonance Theoy s (Fuzzy ART o FA) [] ability to ealize one shot fast leaning and because of its cometitive unsuevised leaning efomance; it is chosen as the clusteing algoithm to gou the oiginal image s colo alette into a smalle subset of colos. 2 Poblem Fomulation 2.1 Fuzzy ART Paametes Caente et al s FA has thee use defined aametes: the vigilance aame-. Fo te ρ [ 0,1], the leaning ate [ 0,1] β, and the choice aamete α > 0 fast leaning, the choice aamete is chosen close to 0 (actually set to α = ) N. da Vitoa Lobo et al. (Eds.): SSPR&SPR 2008, LNCS 542, , Singe-Velag Belin Heidelbeg 2008

2 148 N. Shote and T. Kasais and the leaning ate is set to 1 ( β = 1). These settings theefoe coesond to one shot stable leaning whee no weight change o seach occus afte each item of an inut set is esented just once. The vigilance aamete contols how esilient FA is to acceting new inut instances to existing atten temlates. Theefoe, the highe (o close to 1) the vigilance aamete is set, the moe categoies (o quantized colos) ae ceated. The lowe (o close to 0) the vigilance aamete is set, the less categoies (o quantized colos) ae ceated. 2.2 Inut Reesentation and Peocessing An inut image is defined as matix in I, i { 1,2,..., R}, j { 1, 2,..., C}, with R C elements. Each element in the matix I in is a dimensional vecto coesonding to a given ixel s colo RGB comonents: { } I in( i, j) = Iin, ( i, j), Iin, g ( i, j) Iin, b( i, j) (1) Assume the inut image is 24 bits e ixel. The inuts ae then esented to FA as an aay of colos: Iay ( k) = I in( i j) whee k { 1,2,..., R C} = (2) The inut image is ead into memoy via the cimg heade file [4]. In [5], Mooe descibes a oblem in FA whee a lage numbe of inuts eode the nom of weight vectos causing FA to ceate moe categoies than necessay to encode a given atten (known as categoy olifeation). Caente et. al in [] note that nomalizing the inut by comliment encoding it can enable FA to avoid the categoy olifeation oblem. Assuming 8 bits, each colo comonent inut is scaled to exist between 0 and 1: a( k) = Iay, ( k) 255 () Equation () shows how the ed colo comonent is scaled. Simila equations can be deived fo the blue and geen colo scalings. The comlement of a is calculated as follows: c a ( k) = 1 a( k) (4) The comliment encoding of the inut then is fomulated as follows: {,,, c, c, c g b g b } X ( k) = a ( k) a ( k) a ( k) a ( k) a ( k) a ( k) (5) 2. Fuzzy ART Classification The inuts (aay of colos) ae esented to FA which then clustes simila colos into the same class. Theefoe gous of ixels ae now assigned membeshi to a given class:

3 X Fuzzy ART fo Relatively Fast Unsuevised Image Colo Quantization 149 {,,, },,, c, c, c g b g b ( k) = a ( k) a ( k) a ( k) a ( k) a ( k) a ( k) { C} 1,2,..., l whee C l is the total numbe of classes detected by FA and ae the ixel class labels. The colo comonents of all of the ixels belonging to a single class ae aveaged. Then fo all of those ixels in that class, thei colos ae elaced by the class colo comonent aveage: 1 A I i j (, ) = in, (7) N Equation (7) is the class aveage fo the ed colo comonent. Simila equations can be deived fo the blue and geen colo class aveages. Note that N is the total numbe of oints in a given class. The outut, O(i,j), fo a given ixel located at (i,j) and belonging to class is theefoe the ed, blue and geen class colo comonent aveages: ( i, j) = { A,, Ag Ab } O (8) (6) Exeiments.1 Pefomance Metics Seveal metics wee devised to evaluate the efomance of FA s ability to colo quantize a given inut image. The algoithm s execution time was noted. Execution time is obviously elative to the machine which uns the algoithm. Ou tests wee benchmaked on an AMD 700+ Sandiego Single Coe Pocesso 2.2 GHz. The comute also had 2 gigabytes of DDR400 (PC200) am with timings. In ode to evaluate the quality of the colo quantized image, the oot mean squaed eo (RMSE) between the oiginal colo image and the colo quantized image is used. The RMSE is also used in both [6] and [7] whee they have benchmaked thei algoithms on the Lenna and Mandill images as well. The MSE fo the ed colo comonent is calculated as follows: R C 1 MSE = O i j I i j ( (, ) ) 2, (, ) in RC i = 1 j = 1 (9) Substituting the Outut tem fo the colo class aveage: 1 MSE = A I i j R C ( ) 2, (, ) (10) in RC i = 1 j = 1

4 150 N. Shote and T. Kasais Simila equations can be witten fo the MSE s fo geen and blue colos (denoted MSE and MSE esectively). The total RMSE then is the aveage of the thee g b colo comonent mean squae eos: 1 RMSE = ( MSE + MSEg + MSEb) (11).2 Exeimental Results Seveal exeiments wee efomed to benchmak FA s clusteing ability. Fist, thee sets of 9 exeiments wee conducted fo the entie ange of the vigilance aamete. In the fist set, β = 1 and α = fo one shot stable leaning. Fuzzy ART was only allowed to execute fo a single eoch (the Inut was only esented to FA once). In the second set, the leaning ate was emiically set to β = 0.8 and the choice aamete set to α = 0.1. These values wee found to give the best esults fo when FA was allowed to execute fo thee eochs. Finally, the leaning ate was set to β = 0.8 and the choice aamete set to α = Now FA was allowed to un fo as many eochs until it conveged. Convegence fo FA was defined when two conditions wee met. The fist condition was that no new categoies wee ceated duing the cuent and last eoch. The second condition was that the magnitudes of the weight changes fo all dimensions wee less than a edefined value: l, w w (12) new old l, l, whee l is the dimension index and its ange is as follows: l { 1, 2,...,6}. The is the class index and its ange is as follows: { C}. These thee sets of 9 1,2,..., l exeiments wee conducted on two images: the Lenna image (labeled as '-L') and the Mandill image (labeled as '-M') with esults shown fo both images in [Fig. 1]. Then, six additional sets of exeiments (testing methods on both the Lenna and Mandill images) wee efomed using the one shot stable leaning aametes. The vigilance aametes wee fine tuned so that the colo quantized image had the following numbe of colos: 16, 2, 64, 128, 256 and 512. Fist, as descibed in section 2., all R,G,B colo comonents in a single class (as clusteed by FA) wee aveaged and then the aveage elaced the ixels belonging to that class in the outut image. The vigilance aametes (Vig.), the RMSE, and the execution time in seconds (Sec) fo the coesonding numbe of colos (Col) fo both the Lenna image and the Mandill image ae esented in [Table 1] (shown on age 8). Second, instead of using the aveage, we exeimented with eesenting a given class in the outut image with the median of the class' sectal comonents. We soted

5 Fuzzy ART fo Relatively Fast Unsuevised Image Colo Quantization RMSE vs. Vig. fo Eochs fo Lenna and Mandill Single Eoch-L Thee Eochs-L Convegence-L Single Eoch-M Thee Eochs-M Convegence-M Root Mean Squae Eo Vigilance Paamete Fig. 1. RMSE fo Diffeent Eochs fo Lenna each of the R, G and B sectal comonents of ixels in a single class in ascending ode and then found the median of each of those ixel sectal comonents fo each class: { ( )} in,, M Median i j = I (1) whee M is the median of all the ed comonents of all ixels in class. Simila equations can be witten fo the median of all blue and geen ixels in class. Then all ixels belonging to class ae elaced in the outut image with the median of thei class sectal comonents: ( i, j) = { M,, Mg Mb } O (14) Thid, a timmed mean median was used to eesent the ixels of a given class in the outut image. Fist, all of the sectal comonents (RGB colos) fo all ixels in a given class wee soted: ( in ( )),, S = sot I i j (15) Then, the mean was calculated fo only samles fom colo comonent in a given class : N to 2 N fo each

6 152 N. Shote and T. Kasais ( N ) 2 N 1 TA = S k k= N ( ) Equations simila to (15) and (16) can be witten fo the geen and blue colos. Each outut ixel is then elaced by the class' timmed mean median: ( i, j) = { TA,, TAg TAb} (16) O (17) All of the diffeent outut methods (aveage, median of each colo, and timmed aveage) ae all comaed in tems of thei RMSE fo the vaious colos tested fo the Lenna ('-L')and Mandill ('-M') images in [Fig. 2] RMSE Fo Diff. Methods fo Lenna and Mandil Avg-L MedCol-L TimAvg-L Avg-M MedCol-M TimAvg-M Root Mean Squae Eo # of Colos in Outut Image Fig. 2. RSME fo Diff. Methods fo Lenna Fig.. Lenna as 64 Colos with FA CQ Fig. 4. Mandill as 64 Colos with CQ

7 Fuzzy ART fo Relatively Fast Unsuevised Image Colo Quantization 15 The 64 colo quantized Lenna and Mandill images (RMSE listed in [Table 1]) ae esented as [Fig. ] and [Fig. 4] esectively. Both of these figues wee geneated by substituting the outut ixel with the class aveage. The oiginal vesion of the standad Lenna image exists as a 512x512 ixel image with 148,279 colos while the Mandill image is 512x512 ixels with 20,427 colos. Finally, two additional sets of exeiments wee conducted whee the FA colo quantization algoithm was tested on two eal aeial images deicting commecial and esidential buildings existent in Faifield, Austalia. The aeial images have a 15 cm/ixel sacing esolution. Fo these two images, only the aveage of each of the R,G,B sectal comonents of inuts belonging to the same class, as descibed in section 2., wee used to eesent the colo clustes in the outut image. The esults fo Scene 1 (atially deicted in [Fig. 5]) and Scene 2 (atially deicted in [Fig. 6]) ae shown in [Table 2]. Table 1. Results fo Aveage Each Colo Table 2. Results fo Scenes 1 and 2 Aveage of Each Colo in Class Aveage of Each Colo in Class Col Vig RMSE Sec Vig Scene 1 Mandil Im age Lenna Im age RMSE Sec Col Vig Scene 2 RMSE Sec Vig RMSE Sec Fig. 5. Scene 1 Coed 7.95 Fig. 6. Scene 2 Coed

8 154 N. Shote and T. Kasais Fig. 7. Scene 1 Quantized to 64 Colos Coed Fig. 8. Scene 2 Quantized to 64 Colos Coed Coed sections of the 64 colo quantized Scene 1 (1510x197 ixels with 698,84 colos) and Scene 2 (1595x1878 with 519,51 colos) ae shown in [Fig. 7] and [Fig. 8] esectively. Both ictues wee about 8.5 megabytes in size. The coed sections shown wee selected whee thee existed a athe lage amount of colos so that the colo quantization eos wee somewhat visible. This algoithm s executable is ublicly available at [8]. 4 Conclusions Because the oiginal Mandill image has about 80,000 moe colos than the Lenna image, it makes sense that in [Fig. 1] all of the RMSE values ae highe fo the Mandill than that of the Lenna image fo the listed, quantized colos. Obseve how letting FA execute fo 1 to as many eochs as needed fo convegence fo both the Lenna and the Mandill does not esult in a significant eduction in the RMSE fo vigilance values of 0.5 (which in most cases coesond 16 o moe esultant quantized colos) and highe (see [Fig. 1]). Theefoe, we ecommend the use of one shot stable leaning ( β = 1, α = , and only one eoch o inut list esentation) fo FA inut aametes as the execution time is significantly lowe and the RMSE is on a with that of letting the algoithm un fo additional eochs. We have exeimented with seveal diffeent methods fo eesenting a given cluste of colos existent in the inut image in the outut image. These methods ae as follows: aveaging all the colos in a given class; taking the median of the colos in a

9 Fuzzy ART fo Relatively Fast Unsuevised Image Colo Quantization 155 given class; and taking a timmed mean median of all the colos in a given class. It was obseved that simly aveaging all the colos in a given class and elacing the ixels in that class with those colo aveages woked the best, yielding not only the lowest RMSE, fo the tested Lenna and Mandill images as shown in [Fig. 2] but the lowest execution time as well (as oosed to using the othe two median based outut eesentation methods). In addition to testing the algoithm on the Lenna and Mandill images, tests fo vaious colo quantizations wee also done on eal images catued ove Faifield, Austalia. The oosed imlementation oduced accetable colo quantized vesions of the images with an aguably low execution time and RMSE. The use of FA theefoe has seveal advantages fo CQ. One of which is its low execution time (on the ode of less than a minute fo both the Lenna and Mandill images when the numbe of colos desied is 512 o less, see [Table 1]) and on the ode of minutes fo the lage aeial images (see [Table 2]). The RMSE yielded between the oiginal and colo quantized image, as shown in [Fig. 2], is cometitive with that of othe methods in the liteatue [6], [7]. One unfotunate disadvantage of using FA fo CQ is one cannot diectly secify the numbe of colos that esults in the colo quantized image. 5 Futue Wok One ossible futue diection in which to take this wok would be to exloe the use of additional featues to enable the FA clusteing to account fo the ixel s location and context in the image in addition to the ixel s RGB value. This could ossibly futhe imove the algoithm s efomance. Acknowledgments This eseach effot has been funded by a gant fom Hais Cooation. The authos would like to thank the anonymous eviewes fo thei citiques. The authos would also like to thank D. Simone Clode and D. Fanz Rottensteine fo donating the Faifield, Dataset which was used in this eseach effot. Refeences 1. Deng, Y., Manjunath, B.S.: Unsuevised segmentation of colo-textue egions in images and video. IEEE Tansactions on Patten Analysis and Machine Intelligence 2(8), (2001) 2. Wang, Y.-G., Yang, J., Chang, Y.-C.: Colo-textue image segmentation by integating diectional oeatos into JSEG method. Patten Recognition Lettes 27 (2006). Caente, G.A., Gossbeg, S., Rosen, D.B.: Fuzzy ART: Fast Stable Leaning and Categoization of Analog Pattens by an Adative Resonance System (1991) 4. CImg Heade File, htt://cimg.soucefoge.net/ 5. Mooe, B.: ART1 and Patten Clusteing. In: Poceedings of the 1998 Connectionist Models Summe School, Mogan Kaufmann Publishes, San Mateo (1989)

10 156 N. Shote and T. Kasais 6. Ashutosh, D., Bose, N.S.C., Kandula, P., Kala, P.K.: Modified Fowad Only Counteoogation Netwok (MFOCPN) fo Imoved Colo Quantization by Entoy based Subclusteing. In: Intenational Joint Confeence on Neual Netwoks, IJCNN 2007, (2007) 7. El-Mihoub, T., Nolle, L., Schaefe, G., Nakashima, T., Hogood, A.: A Self-Adative Hybid Genetic Algoithm fo Colo Clusteing. In: IEEE Intenational Confeence on Systems, Man and Cybenetics, SMC 2006, Octobe 8-11, vol. 4, (2006) 8. Nicholas Shote s eseach site, htt://