Journal of Sgnal and Informaton Processng, 2013, 4, 351-358 Publshed Onlne ovember 2013 (http://www.scrp.org/journal/jsp) http://dx.do.org/10.4236/jsp.2013.44044 351 Mult-Level Halftonng by IGS Quantzaton adahko Kmoto Faculty of Scence and Engneerng, oyo Unversty, Kawagoe, Japan. Emal: kmoto@toyo.jp Receved August 19 th, 2013; revsed September 15 th, 2013; accepted September 24 th, 2013 Copyrght 2013 adahko Kmoto. hs s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. ABSRAC Improved gray-scale (IGS) quantzaton s a known method for re-quantzng dgtal gray-scale mages for data compresson whle producng halftones by addng a level of randomness to mprove vsual qualty of the resultant mages. In ths paper, frst, analyzng the IGS quantzng operatons reveals the capablty of conservng a DC sgnal level of a source mage through the quantzaton. hen, a complete procedure for producng a mult-level halftone mage by IGS quantzaton that can acheve the DC conservaton s presented. Also, the procedure uses the scannng of source pxels n an order such that geometrc patterns can be prevented from occurrng n the resultng halftone mage. ext, the performance of the mult-level IGS halftonng s evaluated by experments conducted on 8-bt gray-scale test mages n comparson wth the halftonng by error dffuson. he expermental result demonstrates that a sgnal level to be quantzed n the IGS halftonng vares more randomly than that n the error dffuson halftonng, but not entrely randomly. Also, vsual qualty of the resultng halftone mages was measured by subjectve evaluatons of vewers. he result ndcates that for 3 or more-bt, n other words, 8 or more-level halftones, the IGS halftonng acheves mage qualty comparable to that by the error dffuson. Keywords: Dgtal Halftone; Mult-Level Halftone; Improved Gray-Scale Quantzaton; Error Dffuson; Subjectve estng 1. Introducton Dgtal halftonng s a technque to re-quantze a dgtal mage to fewer bts whle preventng the mage appearance from beng corrupted by producng contnuouslookng tones, that s, so-called halftones [1]. For nstance, for beng prnted wth a black-and-whte prnter, an 8-bt gray-scale mage s to be re-quantzed to one bt or two possble levels. hus, halftonng wth two levels s stll ndspensable n prntng. Also, halftonng wth more than two levels s useful for output devces that can produce many levels, such as electrostatc prnters and compact LED dsplays. Generally n a halftonng method, some knd of sgnal s added to each pxel of a source mage and then, the resultng sgnal s re-quantzed to fewer bts. hese sgnals determne the resultng mage appearance. A varety of halftonng methods to generate the addtonal sgnals can yeld good halftone qualty n the resultng mages that have been proposed. Error dffuson s a well-known halftonng method that can produce halftone mages of good qualty. In the error dffuson, the addtonal sgnals are generated by dstrbutng error sgnals, whch are the dfference of a source pxel level and the re-quantzed pxel level, by a dgtal flter. hus, the resultng halftone qualty depends on the dgtal flter. he mproved gray-scale (IGS) quantzaton [2] s regarded as a halftonng method where the addtonal sgnals are generated from the low-order bts of neghborng pxels. he low-order bts n a natural scene mage are generally a random varable dependng on the mage characterstcs, and also, they are more random as the bt orders are lower. Consequently, the addtonal sgnals become a random varable wth these propertes. In ths paper, we consder the use of the IGS quantzaton for mult-level halftonng. By evaluatng the performance of mult-level quantzaton and the vsual qualty of the resultng halftone mages, a manner of usng the IGS halftonng nstead of the error dffuson halftonng s nvestgated. he rest of ths paper s organzed as follows: Secton 2 defnes the IGS halftonng. Frst, the IGS quantzaton s formulated on the bass of the lterature. From an analyss of the recurson formula, t s proven that the DC
352 Mult-Level Halftonng by IGS Quantzaton component of a source mage can be conserved through the quantzaton. hen, a complete algorthm to apply the quantzaton to gray-scale mages whle achevng the DC conservaton s presented. In Secton 3, stochastc performance of IGS halftonng s evaluated through an experment conducted on 8-bt test mages. he extent of randomness of the addtonal sgnals s measured and compared between three IGS methods and two error dffuson methods. In Secton 4, vsual qualty of IGS halftone mages s evaluated through a subjectve testng conducted to measure mage qualty by the subjectve evaluatons of human observers. From the result, a number of halftone bts can be yelded as good mage appearance as that acheved by error dffuson that s estmated. Secton 5 concludes the paper. 2. IGS Halftonng 2.1. Prncple of IGS Quantzaton he procedure of the IGS quantzaton from M-bt sgnals to -bt ones, 1 M, s expressed as follows. Here, we suppose that source pxels are of M bts wthout statng t clearly n the rest of ths paper. Let Ln v denote the n least sgnfcant bts of the bnary expresson of v. Also let Un v denote a value that conssts of the n most sgnfcant bts same as those of v and the other bts of zero. hen, an M-bt value v can be expressed as v U v L v. (1) M For an M-bt pxel value p n a scannng order, 1, 2,,, where s the number of pxels n the mage, an M-bt value S s defned by the recurson relaton S p L S (2) 1 M where S 0 s supposed to be 0. o ensure that S 2 M, f M p 2 1 2, we use nstead of Equaton (2) S p. (3) he -bt result of quantzng p, denoted by Q p, s gven by the most sgnfcant bts of S. In other words, Q p U S. (4) 2 M Fgure 1 shows the relatonshp between source levels and output levels of the IGS quantzer. In summary, Q p s gven by the relaton ether k or k1, M M f k2 p k12 for k n the range 0,1,, 2 2 k, M Q p f p k2 (5) for k n the range 0,1,,2 1 2 1, M M f 2 12 p 2. Fgure 1. Input-output level relatonshp of IGS quantzer. M M 2 1 2 and 0 2 has two possble values, whch both depend on In ths relaton, for k p k k 2, Q p L S M 1. 2.2. Mathematcal Propertes A sum of the quantzed pxels can be descrbed as follows: Expressng S 1 n the form of Equaton (1), then, substtutng LM S 1 nto Equaton (2) and usng U S yeld the relaton Equaton (4) for 1 S Q p S p. (6) 1 2 M 1 he summaton of each sde of Equaton (6) for from 1 to yelds 1 M 2 p (7) S Q p 1 1 Snce S0 0 and hence, Q p0 0. Substtutng the form of Equaton (1) for S yelds the fnal relaton: M 2 p. (8) L S Q p M 1 1 Dvdng both sdes of Equaton (8) by gves LM S M Q p2 p (9) where x represents the average of a sequence of x. M Because LM S takes a value of at most 2 1, the frst term of the left-hand sde of Equaton (9) can be neglected for large s; thus, we obtan 2 M Q p p. (10) hs relaton means that a level of drect current (DC) of the source mage s left almost unchanged n the quantzed mage. Also, because Equaton (8) holds for any, M the DC level changes by at the most 2 1 at any pxel durng the quantzng of the mage. ote that ' Ss gven by Equaton (3) are excluded n dervng Equaton (8).
Mult-Level Halftonng by IGS Quantzaton 353 2.3. Implementng of IGS Halftonng 1) Source sgnal range he DC conservaton descrbed above s acheved under the condton that all of Ss ' are gven by Equaton (2). hs condton can be satsfed by lmtng the nput sgnals of the quantzer n the range assocated wth Equaton (2). o lmt the source pxel levels n the range, we apply a level transformaton that maps the whole M- M bt range 0, 2 1 onto the range 0, 2 1 2 M to a source mage before carryng out the IGS quantzaton. For smplcty of mplementaton, a lnear mappng s used n the transformaton n ths paper. On the other hand, the level dstrbuton n an mage s lkely to be dstorted due to both the above level transformaton and the followng IGS quantzaton. On the assumpton both that all the pxel levels n the range M 0, 2 1 occur unformly n an M-bt source mage and that LM S 1 of Equaton (2) takes on equally lkely random values, t s derved from stochastc analyss that n the resultng -bt mage, both the occurrence frequency of level 0 and that of level 2 1 are 1 2 2 1, and those of the other levels are 1 2 1. Image areas of pxel levels n the range M M 2 12,2 1 are panted over n the sold level 2 1 after the IGS quantzaton. Such areas posterzed n the hghest level may degrade the resultng halftone qualty. he above level transformaton prevents the posterzaton from occurrng whle ensurng that the source level 2 M 1 s always quantzed to 2 1. 2) Scannng order he scannng order n an mage durng the IGS quantzaton affects the results accordng to the recurson relaton of Equaton (2). Such processng dependent on the scannng drecton s lkely to produce vsble artfacts n the resultng mage. Scannng n a more complcated order than the raster order can reduce a relaton between the scannng result and the mage content [3]. Such scannng s mplemented by, for example, the Hlbert path. Fgure 2 llustrates an example of Hlbert path through a squared mage wth one sde of nteger-power-of-2 pxels, where the path starts at the upper-left pxel and ends at the lower-left pxel. Fgure 3 shows the system dagram of IGS halftonng process. hs dagram nvolves scannng the whole of a source mage once and works n pont processng: For every source pxel extracted n the scannng order, frst, the level transformaton descrbed n the precedng secton s carred out and then, the resultant pxel value s quantzed by the IGS manner. 3. Stochastc Performance of IGS Quantzaton 3.1. Measurng Randomness of Addtonal Sgnals In ths secton, we evaluate the propertes of the addtonal sgnals, that s, LM S 1 of Equaton (2) as a random varable by an experment wth real mages. For a gven, let s represent a varable that produces LM S for 1, 2,, and also, let s be regarded M as a random varable n the range 0, 2 1. As measures of randomness of s, the followng two quanttes are used: 1) he memory-less entropy of s1, s2,, s, denoted by, gven by M 2 1 s0 P s log P s (11) Fgure 2. An example of Hlbert scannng path. Fgure 3. Dagram of IGS halftonng.
354 Mult-Level Halftonng by IGS Quantzaton Ps s the probablty of s. 2) he Entropy of s1, s2,, s p p p where condtonal on px-,,,, denoted by, gven by els 1 2 M M 2 1 2 1 P p P s p P s p log p0 s0 (12) Ps p s the condtonal probablty that, gven where that a pxel level s p, a sgnal of level s s added to t, and P p represents the occurrence probablty of level p. hese quanttes are compared between the followng fve methods: 1) Method IGS-RAS mplements the IGS quantzaton by raster scannng. 2) Method IGS-HIL mplements the IGS quantzaton by the Hlbert scannng descrbed n Secton 2.3 (2). 3) Method IGS-RD uses an M -bt unform random number, whch s actually generated by a pseudo-random number generator wth a computer, nstead of LM S 1 n Equaton (2). 4) Method EDF-FS mplements the error dffuson scheme wth the error flter proposed by Floyd and Stenberg [4]. 5) Method EDF-JJ mplements the error dffuson scheme wth the error flter proposed by Jarvs, Judce and nke [5]. In EDF-FS and EDF-JJ, the error dffuson scheme works to quantze M-bt pxels to bts, 1 M. Because the addtonal sgnal that plays the role of LM S 1 of Equaton (2) takes on fractonal values n the error dffuson, ts to be dgtzed to M-bt levels n calculatng and. 3.2. Expermental Results Experments of the above methods were conducted on 256-gray scale (that s, M 8 ) mages wth varous values of. he source mages used n the experments nclude vertcal and horzontal ramp mages shown n Fgures 4 and, respectvely, and two natural scene mages Barbara and Lena shown n Fgures 4 and (d), respectvely. ote that the mage Barbara ncludes strped regons easy to see, and Lena ncludes wde smooth regons. For each gven, the source mages were transformed by the level transformaton descrbed n Secton 2.3 (1) pror to undergong any of the halftonng methods. Fgure 5 shows the measurements of the entropy that were taken n performng each method by varyng for three of the source mages. As ths fgure shows, the entropes n IGS-HIL are almost the same as those n IGS-RD, and consequently, the addtonal sgnals n IGS-HIL preserve a nearly unform randomness for any as far as the memory-less entropy s concerned. Fgure 4. Examples of 8-bt source mages of 256 by 256 pxels used n the experments: Vertcal ramp wth 1- level per pxel gradaton; Horzontal ramp wth 1-level per pxel gradaton; Barbara and (d) Lena. Fgure 6 shows the measurements of the entropy for the three source mages. he result ndcates that n the three IGS methods, the addtonal sgnals n IGS-HIL have more randomness for source pxels than those n IGS-RAS and less randomness than those n IGS-RD, and also, more randomness than those n the two error dffuson methods. In addton, we can observe a dfference between the two error dffuson methods due to the error flters; the addtonal sgnals are more constraned n EDF-JJ than n EDF-FS. As regards IGS-RAS, a large dfference between the measurements of the entropy of the vertcal ramp mage and the entropy of the horzontal ramp mage s observed from Fgure 6. hs result demonstrates that the propertes of the addtonal sgnals n IGS-RAS are apt to depend on the scannng drecton. Fgure 7 reveals a vsual dfference between the IGS results of raster scannng and those of Hlbert scannng. We observe that the IGS quantzaton yelds geometrcal patterns from regons of a unform level gradaton through the raster scannng, and the generated patterns depend on the relaton between the drecton of raster scannng and that of gradaton. On the contrary, the Hlbert scannng s effectve n nhbtng such patterns from beng produced regardless of the drecton of gradaton. 4. Vsual Qualty of Halftones 4.1. Subjectve Evaluaton In ths secton, we consder appearances of mult-level (d)
Mult-Level Halftonng by IGS Quantzaton 355 Fgure 5. Entropes Barbara. of addtonal sgnals n halftonng. he vertcal ramp mage; he horzontal ramp mage; and Fgure 6. Entropes of addtonal sgnalsn halftonng. he vertcal ramp mage; he horzontal ramp mage; and Barbara.
356 Mult-Level Halftonng by IGS Quantzaton Fgure 7. A comparson between the scannng orders n the IGS quantzaton: Results from the vertcal ramp mage by Raster scannng and by Hlbert scannng; results from the horzontal ramp mage by Raster scannng and by (d) Hlbert scannng. All the mages are dsplayed at a resoluton of 200 pxels per nch (PPI). halftones acheved by the IGS halftonng. Assumng halftone mages produced by EDF-FS to be of standard qualty, we compare the IGS halftone mages wth the EDF-FS ones, and then, evaluate a halftone qualty of the IGS produced mages from the amount of perceptble dfference. hus, a subjectve qualty of mult-level halftones produced by each IGS method wll be evaluated. Also, the prntng resoluton s to be taken nto account as a condton of evaluaton. For measurng the above mage dfference by subjecttve evaluatons of human observers, the followng experment was conducted by usng a category-judgment method [6]: In preparaton, the halftone mages each produced from the same source mage by EDF-FS and the three IGS methods wth a gven were prnted on the same photographc paper wth a photo prnter at a prntng resoluton of 400 dots per nch. Comparng wth the EDF-FS halftone mage on the paper, the observers evaluated each IGS halftone mage from the degree of perceptble dfference between them on the scale lsted n able 1. Each mage was evaluated by 21 observers. All the observers, who were n ther twentes, were unfamlar wth any of the methods. he result of evaluatng each IGS mage s presented by computng a mean value, whch s generally referred to as a Mean Opnon Score (MOS), from the collected values. (d) 4.2. Expermental Result and Dscusson Fgures 8 and show the varatons of MOS wth for the 200-PPI halftone mages of the source mage Barbara and Lena, respectvely, for each IGS method separately. he error bars n the fgures llustrate the 0.95-confdence ntervals of the respectve MOS values. Fgure 9 shows those for the 100-PPI halftone mages. In addton, Fgure 10 presents a comparson between EDF-FS halftone mages and IGS-HIL halftone mages of a resoluton of 200 PPI for each. hese fgures demonstrate the propertes of -bt halftones by the IGS quantzaton. he values of MOS ncrease wth ncreasng n any IGS method, and ths means that the IGS halftone mage looks closer to the EDF-FS halftone mage, and consequently, to the source mage as more levels are used n halftonng. We can observe such effect of level multplcty n Fgure 10. Also, the methods IGS-RAS and IGS-HIL have almost the same ncreasng propertes wth. However, the curves of IGS-RAS show more fluctuaton wth than those of Value able 1. Scale used n subjectve evaluatons. Descrpton 3 Dfference s hardly perceptble. 2 Dfference s perceptble but neglgble. 1 Dfference s defntely observed. Fgure 8. Measurements of subjectve qualty of -bt halftone mages prnted at 200 PPI: Result from he source mage Barbara; Lena.
Mult-Level Halftonng by IGS Quantzaton 357 Fgure 9. Measurements of subjectve qualty of -bt halftone mages prnted at 100 PPI: Result from the source mage Barbara; Lena. IGS-HIL. he fluctuaton s probably due to the dependence of IGS-RAS on scannng drecton. Let us suppose that an IGS halftone mage wth MOS over 2 can be used nstead of the correspondng EDF-FS halftone mage. hen, as Fgures 8 and 9 show, for IGS- HIL, the halftones of 3 or more bts 3 acheve the MOS values larger than 2 ncludng those n the 0.95-confdence nterval. As for IGS-RD, at least 4 bts are necessary to acheve such MOS values. In addton, the vsual qualty of mult-level halftone mages ncludng those by EDF-FS was evaluated by subjectve testng. In the experment, the observers chose one or more mages that look closest to the source mage n the fve -bt halftone mages produced by the respectve methods of IGS-RAS, IGS-HIL, IGS-RD, EDF-FS and EDF-JJ. By usng four dfferent set of a prnted source mage and the fve prnted halftone mages, the ratos that the halftone mages by each method were chosen n all the sets were calculated. Fgure 11 shows the varaton of the rato of each method wth. hs expermental result ndcates that EDF-FS acheves the best vsual qualty n the fve methods for any -bt mult-level halftones and also that both IGS-RAS and IGS-HIL methods ncrease the capablty of achevng the best -bt mult-level halftone qualty wth an ncrease n. 5. Conclusons In halftonng by the IGS quantzaton, the resultng half (d) Fgure 10. -bt Halftone mages of the source mage Barbara by EDF-FS (left column) and by IGS-HIL (rght column), whch are zoomed parts of the respectve mages: = 1; = 2; = 3; (d) = 4. tones are subject to the scannng order n a source mage. Accordngly, geometrc patterns are lkely to be produced n areas of smooth gray levels. Such artfacts degrade the mage qualty, n partcular n b-level 1 halftones. Such a drectonal relaton between a level gradent and the scannng path can be reduced by usng a complcated path such as the Hlbert path. he expermental
358 Mult-Level Halftonng by IGS Quantzaton represent a 10-gray scale pxel. he IGS quantzng s one of what we call pont processng. Hence, the operaton of the quantzaton s obvously smpler than that of the convoluton of flterng n the error dffuson. Although the IGS quantzaton may be an old-fashoned method, the operatonal smplcty as well as the above qualty comparablty to EDF-FS makes the IGS mult-level halftonng useful n recent mage output devces. Fgure 11. Expermentally measured ratos of each -bt halftone mage lookng closest to the source mage between those generated by the fve methods. result demonstrated that the Hlbert scannng makes the addtonal sgnals for the IGS quantzaton ndependent of level gradents, and also, ncreases ther randomness. Consequently, the IGS halftone mage by Hlbert scannng looks a lttle graner wth hgh-frequency sgnal components somewhat degraded than that by smple raster scannng. B-level halftone mages by IGS-HIL probably look worse than those by the error dffuson scheme. Usng mult-level halftones mproves vsual qualty of IGS-HIL halftone mages. By usng three or more bts, that s, 8 or more levels for a halftone level, the IGS-HIL mages look comparable to the EDF-FS mages. Such 8-level halftone mages are supposed to be prnted at a resoluton of one pxel per dot wth a prnter of a capablty of eght levels per dot. Otherwse, they can be prnted wth a b-level prnter, by expressng one 8-level halftone pxel by a set of b-level dots by usng another halftonng method such as clustered-dot ordered dtherng [7]. For nstance, a set of 3 3 b-level dots can REFERECES [1] R. Ulchney, Dgtal Halftonng, MI Press, Cambrdge, 1993. [2] R. C. Gonzalez and R. E. Woods, Dgtal Image Processng, Addson-Wesley, Boston, 1993. [3]. D. Venkata, B. L. Evans and V. Monga, Color Error Dffuson Halftonng, IEEE Sgnal Processng Magazne, Vol. 20, o. 4, 2003, pp. 51-58. http://dx.do.org/10.1109/msp.2003.1215231 [4] R. W. Floyd and L. Stenberg, An Adaptve Algorthm for Spatal Grayscale, Proceedngs of the Socety of Informaton Dsplay, Vol. 17, o. 2, 1976, pp. 75-77. [5] J. F. Jarvs, C.. Judce and W. H. nke, A Survey of echnques for the Dsplay of Contnuous-one Pctures on Blevel Dsplays, Computer Graphcs and Image Processng, Vol. 5, o. 1, 1976, pp. 13-40. http://dx.do.org/10.1016/s0146-664x(76)80003-2 [6] A.. etraval and B. G. Haskell, Dgtal Pctures, Plenum, 1988. http://dx.do.org/10.1007/978-1-4684-1294-9 [7] P. G. Roetlng and R. P. Loce, Dgtal Halftonng, In: E. R. Dougherty, Ed., Dgtal Image Processng Methods, Marcel Dekker, ew York, 1994, pp. 363-413.