Green Noise Digital Halftoning with Multiscale Error Diffusion

Transcription

1 Green Nise Digital Halftning with Multiscale Errr Diffusin Yik-Hing Fung and Yuk-Hee Chan Center f Multimedia Signal Prcessing Department f Electrnic and Infrmatin Engineering The Hng Kng Plytechnic University, Hng Kng ABSTRACT Multiscale errr diffusin (MED) is superir t cnventinal errr diffusin algrithms as it can eliminate directinal hysteresis cmpletely and pssesses a gd blue nise characteristic. Hwever, due t its filter design, it is nt suitable fr systems with pr islated dt generatin and instable dt gain. In this paper, we prpse a multiscale errr diffusin algrithm t prduce halftnes f desirable green nise characteristics. This algrithm allws ne t adjust the desirable cluster size freely thrugh a single parameter and supprts a linear relatinship between the cluster size and the input gray level. With a clse-t-istrpic diffusin filter, the algrithm can effectively remve pattern artifacts, eliminate directinal artifacts and preserve riginal image details. Analysis and simulatin results shw that it prvides better perfrmance in terms f varius aspects including dt distributin, anistrpy and utput image quality as cmpared with ther cnventinal green nise errr diffusin algrithms. EDICS: ELI-PRT Index Terms Multiscale errr diffusin (MED), errr diffusin, halftning, multiscale prcessing, printing, green nise. Crrespnding authr (Tel: (85)766664; Fax: (85)368439; enyhchan@plyu.edu.hk)

2 I. INTRODUCTION Digital halftning is a technique used t turn a gray-level image int a bi-level image and has been widely used in printing applicatins []. Basically, halftning can be accmplished in tw ways. One is amplitude mdulatin(am) [] in which a halftne is prduced by varying the size f printed dts arranged alng a regular grid. The ther ne is frequency mdulatin(fm) [3-5] in which a halftne is prduced by varying the relative dt density f fixed-size printed dts. As cmpared with AM halftning, FM halftning prduces halftnes f higher spatial reslutin and better image quality[4]. Unlike utputs f AM halftning, halftnes prduced by FM halftning are free f miré artifacts as neighbring pixels have been taken int accunt during errr diffusin. Hwever, halftnes prduced by FM halftning are sensitive t the dt gain f a printer, which is the increase in size f the printed dt relative t the intended dt size. In practice, the size and the shape f a printed dt are nt as perfect as they are expected and the printed halftne generally appears darker than expected. When the variatin in dt size and shape frm printed dt t printed dt is small, printing can rely n dt gain cmpensatin technique t minimize this distrtin [6]. Hwever, dt gain cmpensatin des nt wrk effectively when the reliability t prduce islated dts is nt stable. In such a case, ther than using AM halftning, using a halftning methd which prduces clustered dts helps t reduce the dt gain distrtin as a cluster has a lwer perimeter t area rati as cmpared with an islated dt. AM-FM halftning is a hybrid versin f AM and FM halftning which aims at prducing clustered dts. Fr example, Levien [7] prpsed an utput-dependent feedback errr diffusin algrithm t frm dt clusters f adjustable size t increase the printing stability. As a result, the utput halftne image can be tuned t have larger cluster dts in a high dt gain situatin t cater fr different printing systems. Thrugh a systematic study n the halftnes prduced with varius algrithms, Lau fund that a halftne bearing green nise characteristics which cntains nly mid-frequency spectral cmpnents is less susceptible t image degradatin frm nn-ideal printing device [8]. Based n Lau s idea, Damera-Venkata et al. prpsed an adaptive threshld mdulatin framewrk t imprve the halftne quality by ptimizing errr diffusin parameters in the least squares sense and derived an adaptive algrithm t prvide halftnes f green nise characteristics [9]. A blck errr diffusin algrithm which allws a user t determine the dt size and shape directly was later prpsed by the same research grup []. Multiscale errr diffusin (MED) was first prpsed in [] and then extensively imprved in [-5]. It has already been prven t be superir t cnventinal errr diffusin algrithms in terms f varius measuring criteria[4-5]. The cmprehensive empirical analysis prvided in [4] shws that the utputs f MED algrithms are free f directinal hysteresis and pssess gd blue nise characteristics. Its success in prviding a blue-nise halftne mtivates us t explre whether the MED framewrk can als be used in green nise halftning which aims at prviding halftnes f green nise characteristics. In this paper, a MED algrithm is prpsed t prduce halftnes f desirable green nise characteristics. This algrithm allws ne t adjust the desirable cluster size freely thrugh a single parameter and supprts a linear

3 relatinship between the cluster size and the input gray level. With a prpsed clse-t-istrpic diffusin filter, the algrithm effectively remves pattern artifacts, eliminates directinal artifacts and preserves riginal image details. Simulatin results shw that it prvides better perfrmance in terms f varius aspects including dt distributin, anistrpy and visual image quality as cmpared with ther cnventinal green nise errr diffusin algrithms. The rganizatin f this paper is as fllws. In sectin II, we briefly review a MED algrithm and intrduce the mtivatin f ur prpsal. In sectin III, diffusin filters are redesigned fr MED algrithms t achieve green nise halftning. In sectin IV, hw the parameters f the suggested diffusin filter affects the average cluster size f the halftning utput is addressed. Sectin V prvides a detailed empirical analysis n the perfrmance f the prpsed algrithm in terms f varius spatial and spectral statistics. In sectin VI, simulatin results n real images are prvided t evaluate the perfrmance f varius green nise errr diffusin algrithms. The cmplexity issue is addressed in Sectin VII. Finally, a cnclusin is given in sectin VIII. II. REVIEW OF FMED In this sectin, we first briefly review the feature-preserving MED algrithm (FMED) prpsed in [3]. This MED algrithm serves as a typical example f cnventinal MED algrithms and frms the basis f the prpsed algrithm. Then, we explain why cnventinal MED algrithms [-4] d nt prvide halftnes f green nise characteristics and suggest a mdificatin t make it. Withut lss f generality, cnsider we want t halftne an input gray-level image X f size l l, where l is a psitive integer, t btain an utput binary image B. The values f X are within and. Here we assume that the maximum and the minimum intensity values are, respectively, and. Nte that there is n limitatin f the input image size when using MED algrithms. The mentined size is fr easier illustratin nly. FMED is a tw-step iterative algrithm. At the beginning, an errr image E is initialized t be the gray-level input image X. Pixels f B are then picked iteratively t determine their intensity values until the terminatin criterin is satisfied. Fr reference purpse, the intensity values f pixels (m,n) f E and B are, respectively, dented as e m,n and b m,n. In the first step f each iteratin cycle, a pixel in B is selected via the extreme errr intensity guidance based n the mst updated E. Let the crdinates f the selected pixel be (i,j). A dt is then intrduced t pixel (i,j) f B in the secnd step by assigning a crrespnding value ( r ) t b i,j, and the errr image E is updated by diffusing the errr (= b i,j - e i,j ) t pixel (i,j) s neighbrs in E. The value f e i,j is reset t after the diffusin. Figure briefly summarizes hw FMED perates in an iteratin cycle. These tw steps are repeated until the sum f all pixels f E is bunded in abslute value by.5. One may refer t [3] fr the details f FMED. The utput f FMED bears gd blue-nise characteristics [4], which is gd t a stable printing situatin. Hwever, when the dt gain is high and dts cannt be cnsistently reprduced, clustered dts are preferred in the halftning utput t cmpensate fr the dt-gain distrtin. As a matter f fact, cnventinal MED algrithms [-4] are prne t prduce islated dts as they all use small diffusin filters (3 3 in mst cases). After (i,j) is lcated and b i,j is determined by quantizing e i,j, the 3

4 quantizatin errr (b i,j - e i,j ) is diffused t (i,j) s nearest neighbring pixels in E. This encurages the frmatin f islated dts. As an example, when ne puts a black dt t pixel (i,j) by assigning b i,j t, an errr diffusin with a 3 3 diffusin filter increases the intensity values f pixels (i,j±), (i±,j±) and (i±,j) unless e i,j is already befre the diffusin. This intensity increase increases the likeliness f assigning white dts t b i,j±, b i±,j and b i±,j± in the future. Accrdingly, it is mre likely that the black dt at (i,j) will be surrunded by white dts in the final halftning utput. If it is desirable t frm a cluster f dts fr pixel (i,j), the errr shuld nt be diffused t the clsest neighbring pixels with a 3 3 filter. Instead, it shuld be diffused t the pixels that are farther away frm pixel (i,j). By ding s, the intensity values f the clsest neighbring pixels are nt affected by the diffusin and hence it des nt directly affect the chance f assigning a particular type f dts t them after the diffusin. In cntrast, the intensity values f the uter neighbring pixels t which the errr is diffused are increased, which increases these pixels chance f having white dts in the future. In view f energy cnservatin, the number f white dts appeared in the utput shuld equal t the runding result f the ttal energy f the input. As the number f white dts t be assigned t the utput is fixed and nw the uter neighbring pixels are mre likely t be white, the inner neighbring pixels are actually mre likely t be black indeed. Cnsequently, it encurages the frmatin f a dt cluster centered at pixel (i,j). Based n the afrementined bservatins, the diffusin filter used in FMED is redesigned in this paper such that halftnes f green nise characteristics can be prduced with FMED. The details are discussed in Sectin III. III. DIFFUSION FILTERS FOR GREEN-NOISE HALFTONING As mentined in Sectin II, diffusing the quantizatin errr f pixel (i,j) t pixel (i,j) s farther neighbrs instead f its cnnected neighbrs encurages the frmatin f a dt cluster at (i,j). Based n this idea, a straight frward apprach t make FMED prduce dt clusters is t replace the riginal default 3 3 diffusin filter H =[.5,,.5;,, ;.5,,.5]/6 used in FMED with a (k+) (k+) square filter H k defined as ( /( m + n )) (/ S) if m = k r n = k h k ( m, n) = if m < k and n < k fr k=,3 () where h k ( m, n) is the (m,n) th filter cefficient f filter k H and S = ( /( m + n )) m r n = k factr which makes the sum f all filter cefficients be. As an example, we have is a nrmalizatin / 8 / 5 / 4 / 5 / 8 / 5 / 5 H = / 4 / 4 () 3 / 5 / 5 / 8 / 5 / 4 / 5 / 8 4

5 The determinatin f the filter cefficients is based n the idea that, in an istrpic diffusin prcess, the intensity at a particular pint away frm the surce is inversely prprtinal t the square f the distance frm the surce. Filter H k is k dependent. Its size is f (k+) (k+) pixels. The larger the k value, the larger the filter size is and the larger the clusters can be prduced in the halftne utput fr a fixed gray-level input. Our simulatin results verified this truth. Specifically, when FMED wrks with H, the average cluster size is 5 pixels fr a cnstant input f gray-level value.5. Thugh replacing filter H with a filter H k with k> can help FMED t prduce dt clusters, a better diffusin filter can yet be designed t further imprve the halftning perfrmance. As discussed in [3], ne f the strengths f MED algrithms is that they can use a nn-causal filter t diffuse the quantizatin errr in a radially symmetrical way t eliminate directinal hysteresis. T enhance this strength, ne shuld select an istrpic diffusin filter t diffuse the errr t all directins equally. Frm that pint f view, square filters filter. H k are hence nt ideal and a circular ring filter wuld be a better alternative t wrk as a diffusin When H k is used, ne can change the average cluster size fr a particular input gray level by adjusting the value f k. Hwever, the filter size (k+) (k+) and hence the average cluster size is nt cntinuusly adjustable. A filter which allws ne t cntinuusly adjust the cluster size easily wuld be mre preferable frm applicatin s pint f view. A. Ideal circular ring filter in cntinuus spatial dmain Cnsider the ideal case that the image spatial dmain is cntinuus. A circular ring filter F R, R ) fr diffusing the errr at psitin (,) can be defined as ( f ( x, y) f ( x, y) )/ ( R )π ) f ( x, y) = x, y are real numbers (3) R R R ( where R and R are, respectively, the inner and the uter radii f the ring, and f R k ( x, y) = if x + y else R k fr R k = R, R (4) In plar crdinate system, eqn.(3) can be rewritten as ( f ' ( r) f ' ( r) )/ ( R )π ) f '( r) = (5) R R R 5

6 where as = x y and r + if r Rk f ' R k ( r) = fr R k = R, R. The Furier transfrm f f '( r) is then given else ' RJ(πR ρ) R J(πR ρ) F ( ρ) = (6) ρ( R R ) π where ρ is the radial crdinate in the frequency dmain and J ( ) is the Bessel functin f the first kind f rder. The inner radius R f the circular ring filter helps t determine the average cluster size f the clusters frmed when the filter wrks with FMED t prduce a halftne. Rughly speaking, as we shall shw in Sectin IV, when the rati f the uter radius R t the inner radius R clses t, the average cluster size fr a cnstant gray level input can be apprximated t be gr π, where g is the gray level value. Cnceptually, this can be interpreted as that, n average, FMED encircles an area f R π in a cnstant gray level input (=g) and then puts a white cluster (=) f size M g n a black backgrund in that area t emulate the intensity distributin f the area in its utput as shwn in Figure. By energy cnservatin it results in M g = gr π. One can hence make use f parameter R t adjust the desirable cluster size in the halftning utput. When R /R, the ring width f the circular ring filter is small and filter functin (3) apprximates the wavefrnt f the diffusin at radius R frm a pint errr surce. Theretically, the narrwer the ring width, the better the apprximatin t the wavefrnt is. Hwever, there are sme ther factrs t be cnsidered when selecting R and we will discuss it later. B. Apprximated ring filter in discrete spatial dmain In practice, the spatial dmain f a digital image is nt cntinuus and hence the circular ring diffusin filter F has t be adjusted t fit the pixel grid. T achieve this, ne can apprximate filter F, ) with R, R ) ( R R ( F R, R ) ( whse filter cefficients are defined as m+.5 x= m.5 n+.5 f ( m, n) = f ( x, y) dxdy m, n are integers (7) y= n.5 In this arrangement, the filter cefficient fr a pixel which is (m,n) pixels away frm the pint surce is the area cvered by the circular ring R x + y > R in the grid unit assciated with that pixel. By substituting eqn.(3) int eqn. (7), we have A( m, n, R ) A( m, n, R ) m, n) = (8) ( R R )π f ( 6

7 where m+.5 x= m.5 n+.5 A( m, n, R ) = f ( x, y) dxdy fr R R k k =, R (9) k y= n.5 R is the area cvered by circle x + y R in pixel (m,n). k In practice, f (,) must be zer as all errr must be diffused away frm the surce pixel. This can be achieved by making R /. Then we have A (,, R ) = A (,, R ) = and hence f (,) =. The details f the cmputatin f A ( m, n, Rk ) can be fund in the appendix f this paper. With apprpriate A ( m, n, Rk ) values, the filter cefficients f the apprximated ring filter F ( R, R ) can be determined by using eqn. (8). As an example, the apprximated versin f filter F R, R ) fr R =. 8 and R = is given as ( F (.8,.8 ) = () As distrtin is intrduced in the apprximatin, the filter supprt f the errr cannt be diffused t all directins equally. F R, R ) ( is n lnger a circular ring and Figure 3 shws the frequency respnses f filters F, (.8,.8 ) and H ( fr cmparisn. They are all f F.8,.8 ) cmparable filter size. T a certain extent, bth H and are apprximated versins f F. By ( (.8,.8 ) F.8,.8 ) inspecting their frequency respnses, F.8,.8 ) is bviusly a better apprximatin t F as cmpared with ( (.8,.8 ) H. In thery, when a filter is istrpic, in its frequency respnse the magnitude variance f the frequency cmpnents cvered in an annular ring f any radius shuld be zer as lng as the ring width is sufficiently small. Figure 4a plts the crrespnding measures in rings f different radii fr different filters. It als shws that is better than H ( in a way that it can diffuse the errr t all directins mre unifrmly. F.8,.8 ) Figure 4a als shws the perfrmance f F.8,.8.) ( fr shedding sme idea n what we shuld cnsider when selecting R. In practice, the narrwer the ring width f the riginal F R, R ), the mre distrtin is intrduced by the apprximatin prcess gverned by eqn.(7) due t the spatial grid cnstraint and, after the apprximatin, the ( filter supprt f F R, R ) ( deviates mre frm a perfect circular ring. This can be verified by the fact that the curve f filter F.8,.8.) ( deviates mre than that f F.8,.8 ) ( frm zer in Figure 4a. Figure 4b shws the case when a larger R is used (=.8), and that F.8,.8 ) ( is clser than F.8,.8.) ( t be istrpic is even mre bvius. Frm 7

8 that pint f view, a larger value f rati R / R wuld be better fr F R, R ) ( t diffuse the errr t all directins equally. Hwever, a larger filter supprt results in higher cmputatinal effrt and visually mre blurred halftnes. / = In ur prpsal, R R is suggested. With such an arrangement, the area f the filter supprt R ( R π π ) equals t that f the inner regin ( R π ). We will shw in sectin IV why it is suggested. C. Wrking with FMED Theretically, filter F R, R ) ( can wrk with any MED algrithms [-4] t realize green nise halftning. Here, we use FMED as an example t shw hw ne can mdify a MED algrithm t achieve green nise halftning. Fr reference, the mdified FMED is referred t as FMEDg hereafter. Like all MED algrithms, FMEDg is a tw-step iterative algrithm. In each iteratin cycle, it selects a pixel t assign a dt in the st step and then diffuses the errr t update the errr plane E in the nd step. E is initialized t be X at the beginning. The st step f FMEDg is exactly the same as the st step f FMED except that parameters x ff and y ff, the randm shifts added t the starting searching windw, are bunded in set {, ±, ±} instead f {, ±}. Assume that pixel (i,j) is selected and a white(black) dt is assigned t it by making b i,j =() in the st step. In the nd step, the errr between b i,j and e i,j is diffused with a pre-selected default filter F R, R ) ( r, if necessary, its rectified versin. The R value f the default filter is used t select the desirable cluster size fr a particular input gray level. In particular, the errr image E is updated as if ( m, n) = ( i, j) e m, n = () em, n f ( m i, n j) dm, n ( bi, j ei, j ) / s if ( m i, n j) Ω( R, R ) where Ω R, R ) is the filter supprt f ( F R, R ) (, if bm, n has been assigned a value d m, n = and else s = f ( m i, n j) d () ( m i, n j) Ω ( R, R ) m, n In the case when s=, we gradually increase the value f R by.5 t make s and keep the algrithm wrking. 8

9 IV. ANALYSIS OF CLUSTER SIZE The mtivatin fr green nise halftning is t prduce patterns with adjustable carseness that can be tuned t the reliability f a given printer t prduce dts cnsistently. In ideal case, the average size f the resultant clusters fr a particular gray level input shuld be cntinuusly adjustable and the average cluster size shuld be linearly prprtinal t the input gray level. This sectin prvides an empirical analysis n hw parameters R and R f filter F R, R ) ( affect the cluster size in a halftne image when F R, R ) ( wrks with FMED. A simulatin was first perfrmed t study hw FMEDg changes the cluster size accrding t different gray levels. In ur study, F R, R ) ( was used as the default diffusin filter. Fr each particular gray level, a cnstant gray-level image f size 8 8 was generated and halftned with FMEDg t prduce 5 halftnes. The average cluster size Mˆ g in the resultant halftnes was then measured. Figures 5a and 5b, respectively, shw the average and the standard derivatin f the cluster sizes measured in the simulatin fr different cmbinatins f R and g when R = R. Fr reference, Figure 5c shws the surface f a mdel functin frmulated as M g = R πg (3). The small scale f difference ( M ˆ g M g ) shwn in Figure 5d verifies that eqn. (3) is an apprpriate mdel t describe the relatinship amng the invlved items when R = R. Frm the mdel, ne can see that the cluster size is prprtinal t (i) the input gray level g cnsistently fr different R and (ii) the square value f R. Accrding t the prperties f a given printer, ne can adjust R t select the desirable cluster size fr a particular gray level input and allw FMEDg t generate clusters the average size f which is apprximately linearly prprtinal t the input intensity value fr ther input gray levels. The simulatin results presented earlier shws the case when R = R. Figure 6 shws the case f sme ther cmbinatins f R and R. One can see that, when the rati f R t R becmes large, the apprximately linear relatinship between the measured Mˆ g and the input gray level g is n lnger valid fr a large R. The larger the rati, the mre the practical situatin deviates frm the mdel given by (3). By cnsidering that a F ) smaller rati makes the filter supprt f ( R, R mre deviated frm a ring as discussed in Sectin IIIB, R = R is hence suggested in this paper. V. EMPIRICAL PERFORMANCE ANALYSIS The perfrmance f a halftning algrithm can be quantitatively evaluated by measuring varius spatial and spectral statistics f its halftning utputs. An empirical analysis was carried ut t study the perfrmance f 9

10 FMEDg and the results are reprted in this sectin. Fr cmparisn, the perfrmance f sme ther green nise errr diffusin algrithms was als evaluated. In ur analysis, varius green nise errr diffusin algrithms were applied t a set f cnstant gray-level images f size and the dt distributins in their utputs were studied in terms f different statistics. Levien [7], Damera-Venkata [9], Damera-Venkata [] and FMEDg were included in the cmparisn. In the realizatin f [7] and [9], serpentine scanning is used and the hysteresis cnstant H is set t.. The hysteresis filter and the errr filter used in [7] are, respectively, fixed t be [,.6, ;.4,, ] and [,.5, ;.5,, ]. These tw filters are adaptive in [9] and they are, respectively, initialized t be [,.6, ;.4,, ] and [,.5, ;.5,, ] as well. In the realizatin f FMEDg, filter F.8,.8 ) ( is used as the default filter. The settings f [7], [9] and FMEDg are selected t make them prduce clusters f cmparable size ( 5.6 pixels) at the utput when the input gray level is.5 fr fair cmparisns. The dt shape used in simulating [] is a cluster (=[, ;, ]). It is selected because in [] the allwable default cluster size must be an integer. A cluster f dts is f a shape clse t istrpic while its size is clse t 5.6 pixels. Figure 7 shws sme prtins f the halftning results fr g = 33/55, 6/55, 8/55 and 6/55. The selected gray levels represent different ranges f input gray levels. A. Spatial statistics In [8], Lau develped a directinal distributin functin D ( ) t measure the directinal distributin f r, r dts in a dt pattern. It is defined as the expected number f dts per unit area in an angular segment f the ring bunded by an inner radius r and an uter radius r in the spatial dmain. The annular ring is centered at the center pixel f a cluster and the segment is indexed by which specifies the segment s directinal psitin with respect t the center. In ideal case, we have D ( ) fr all s, which indicates an istrpic distributin in r =, r the pattern. T reprt the directinal dt distributin in an intra-cluster regin and that in an inter-cluster regin separately, D ( ) fr ={,π/4, 7π/4} and D ( ) fr ={,π/8, 5π/8} are, respectively, prvided in ur analysis., Δ Δ, λ + Δ g Figure 8 shws cnceptually hw a cluster s neighbrhd is partitined t cmpute D ( ) and D ( )., Δ Δ, λ + Δ Specifically, λ g is the green nise principal wavelength f the input gray-level g and it is the average distance between tw neighbring clusters. In frmulatin, it is defined as g λ = ˆ g (4) g M g / where Mˆ g is the average number f minrity pixels per cluster fr a particular gray level g. In ther wrds, Mˆ g is the average cluster size. is the radius f a circle which is large enugh t encircle a cluster fr any gray level g (,.5]. In ur measure, is selected t be.

11 The reslutin f D ( ) dubles that f D ( ) in terms f number f segments. This is because in a Δ, λ + Δ g, Δ plar system a regin further away frm the rigin can supprt a higher pixel reslutin. Intra-cluster distributin: Figure 9 shws the D ( ) measures f the utputs f varius algrithms fr g = 33/55, 6/55, 8/55 and, Δ 6/55. Only the upper halves f the plts are shwn here as the lwer half f the plt f D ( ) can be, Δ btained with D ( + ) = D ( ). Tw bservatins can be btained frm Figure 9. First, whatever algrithm, Δ π, Δ is used, the intra-cluster dt distributin is generally mre biased t sme directins (i.e. The D ( ) value is, Δ larger fr sme s.) when the input gray-level is lw. Secnd, the distributin f the utput f [7] is the mst uneven amng the evaluated algrithms and it is remarkably biased t the vertical directin except the case f g=6/55. On the cntrary, thugh the D ( ) s f the thers are als nt equal fr all s, there is n verall, Δ directinal bias as their plts are mre r less radially symmetric. T have a cmplete picture f the directinal characteristic f the intra-cluster dt distributin fr all input gray levels, a directinal index functin based n D ( ) is defined as, Δ 8 D Intra ( g) = ( D, Δ ( )) fr.5 g> (5) 8 = In ideal case, D Intra (g) shuld be zer fr all g because an istrpic distributin f dts makes D ( ), Δ = fr all. The larger the value, the mre severe the distributin is directinal fr the specific input gray level. Figure shws the crrespnding perfrmance f the evaluated algrithms and frm it tw bservatins can be btained. First, the intra-cluster dt distributin f the utput f [] is cnsistent fr a wide range f input gray levels. This is due t the fact that the shape f a cluster generated by [] is fixed fr a wide range f gray levels as shwn in Figure 7(iii). This cnsistency makes [] prvide the mst unifrm distributin fr mst input gray levels. Secnd, the perfrmance f FMEDg is cmparable t that f [] when the input gray level g is larger than. and can be even better than [] when g is in the range frm.38 t.48. Inter-cluster distributin: Figure shws the D ( ) plts fr the evaluated algrithms. Again, nly the upper halves f the plts Δ, λ g + Δ are shwn here due t the symmetric prperty f D ( ). Tw bservatins can be btained frm Figure. Δ, λ g + Δ First, thugh [] can prvide a unifrm intra-cluster dt distributin as shwn in Figure due t its fixed predefined cluster characteristic, its inter-cluster dt distributin is severely directinal fr small g (e.g. g=33/55). Secnd, the plt f FMEDg is radially symmetric, which implies there is n directinal hysteresis in their

12 halftning utputs whatever the input gray level is. Mrever, when g is large enugh (e.g. g>.), its D ( ) Δ, λ g + Δ values are all clse t fr all and it is very clse t the ideal situatin. T have a cmplete picture f the directinal characteristic f the inter-cluster dt distributin fr all input gray levels, a directinal index functin based n D ( ) is defined as Δ, λ g + Δ 6 D Inter ( g) = ( DΔ, λ +Δ ( )) fr.5 g> (6) g 6 = Similar t D Intra (g), D Inter (g) shuld be zer fr all g in ideal case and a large D Inter (g) value indicates a severely directinal characteristic f the inter-cluster dt distributin fr the crrespnding input gray level g. Figure shws the crrespnding perfrmance f the evaluated algrithms and frm it tw bservatins can be btained. First, the inter-cluster dt distributin f the utput f [] is highly directinal as cmpared with the ther evaluated algrithms fr mst input gray levels. This implies the distributin f the clusters in the utput f [] is als highly directinal and there is severe directinal hysteresis. Secnd, the perfrmance f FMEDg is remarkably better than that f the ther evaluated algrithms fr all input gray levels. This is expected as a clset-istrpic diffusin filter and a nt-predetermined scanning path is used in FMEDg. B. Spectral statistics Radically averaged pwer spectrum density (RAPSD) and anistrpy are tw measures prpsed by Ulichney t analyze the spectral statistics f a halftne pattern[4]. Bth f them are defined based n the pwer spectrum f a halftne pattern. RAPSD RAPSD is defined as the average pwer f the frequency cmpnents in the annular ring with center radius f p in the spectral dmain as fllws. P ( f p ) = Pˆ( f ) (7) N( R( f )) p f R( f p ) where R ( f p ) is an annular ring f width Δ p partitined frm the spectral dmain and N ( R( f p )) is the number f frequency cmpnents in R f ). P ˆ( f ) is the magnitude square f the Furier transfrm f the utput pattern ( p divided by the sample size. Figure 3 shws the perfrmance f varius algrithms in terms f RAPSD. A gd green nise generatr shuld prduce a result the spectrum f which carries weak lw- and high-frequency spectral cmpnents and has

13 a spectral peak at green nise principal frequency f g. We can see that all the evaluated algrithms have green nise characteristics in terms f RAPSD. Anistrpy Anistrpy is defined as ( Pˆ( f ) P( f p )) A ( f p ) = (8) N( R( f )) P ( f ) p f R( f p ) p It prvides the nise-t-signal rati f frequency samples f P ˆ( f ) in R f ) and is used t measure the strength f directinal artifact. Figure 4 shws the perfrmance f varius algrithms in terms f anistrpy. As mentined in [4], when A ( ) > db happens, directinal cmpnents are cnsidered t be strng r nticeable t human eyes. T f p ( p prvide a reference t study the perfrmance f the algrithms, a surface defined by A ( ) = db is added in f p each f the plts. The plts shw that the prpsed FMEDg is better than the ther algrithms. Its crrespnding anistrpy is well belw db. Green nise halftning is characterized by a distributin f clusters f dts and the distributin shuld be as hmgeneusly as pssible [8]. Clusters distributed in this way create an aperidic and istrpic pattern, which makes the utput visually pleasant as it des nt clash with the structure f an image. FMEDg is better in this aspect. VI. SIMULATION RESULTS T study the perfrmance f the evaluated algrithms in handling real images, sme 8-bit gray-level testing images including seven 5 5 natural images shwn in Figure 5 and their versins were als used in ur simulatins. Halftne visibility metrics prpsed in [6] were used t measure the distrtin bserved by a human viewer between an riginal gray-level image X and its binary halftne B. Figure 6 shws the perfrmance f varius algrithms in terms f MSE v. In particular, MSE v is defined as MSE v = hvs(x, vd, dpi) hvs(b, vd, dpi) (9) N N where hvs is the HVS filter functin defined in[6], vd is the viewing distance in inches and dpi is the printer reslutin. In ur simulatins, the viewing distance changes frm t 8 inches and a printer reslutin f 6dpi was cnsidered. Table shws the perfrmance in terms f Universal Objective Image Quality Index (UQI) [7]. Nte that the value f UQI is bunded in [-,] and a larger value indicates a better perfrmance. One can see that, in terms f bth MSE v and UQI, FMEDg is better than the thers. 3

14 Fr subjective evaluatin, Figure7 shws a prtin f the riginal testing image Barbara and Figure8 shws the crrespnding utputs f the evaluated algrithms. Several bservatins can be btained by cmparing these figures. First, utputs f all evaluated algrithms have green nise characteristics. Secnd, there is severe pattern nise in the cheek and the finger area in Figure 8c. Finally, FMEDg preserves mre feature details f the riginal image than the thers. T verify this, ne can check the scarf cvered the wman s right shulder and left upper arm, and the back f the chair. The prpsed FMEDg allws ne t adjust the cluster size cntinuusly and rughly linearly with a single parameter R. Figure 9 shws the halftning results assciated with different R values. The nise characteristic f Figure 9(a) is actually blue while thse f the thers are green. VII. COMPUTATIONAL COMPLEXITY In this sectin, a cmputatinal cmplexity analysis f the prpsed algrithm is prvided. The analysis is based n an assumptin that the input image is f size N N, where N is a value f t the pwer f a psitive integer. Obviusly, the cmplexity f FMEDg depends n the number f nn-zer cefficients f the diffusin filter, which is R -dependent, and the size f the image. The cnstructin f an energy pyramid at the initializatin stage f FMEDg takes put a dt, N additins. Fr each iteratin cycle, it takes multiplicatins and P additins t diffuse the errr, where P is the number f nn- lg N zer diffusin filter cefficients, and 4 l= 3 lg N cmparisns t lcate a pixel t l P 4P / 3 + lg N additins t update an energy pyramid after the diffusin. Here, we assume that n rectificatin f the diffusin filter is required in the diffusin. Nte that rectificatin generally results in fewer nn-zer filter cefficients. The value f P can be apprximated t be R π, which is the size f the supprt area f the ideal ring filter F ( R, R ). Hwever, this apprximatin always underestimates the true number. Figure shws hw P changes with R in reality. Their relatinship can be apprximated as P = 3.844R + 9.R Fr cmparisn, the cmplexity fr the case using H k, the square filter defined in (), as the default diffusin filter is als presented here. In such a case, we have P=8k. Accrdingly, fr each iteratin cycle, it takes 3 lg N cmparisns t lcate a pixel t put a dt, k+ multiplicatins and 8k additins t diffuse the errr and l lg N = 4 l P 3k / 3 + lg N additins t update an energy pyramid. Nte that the case f k= crrespnds t FMED [3]. Table shws the executin time f the Matlab realizatins f varius green nise errr diffusin algrithms t halftne a image fr cmparisn. Diffusin filter F.8,.8 ) ( is used in the realizatin f FMEDg. The simulatins were carried ut n a.83ghz Intel Cre (TM) PC with 3.5GB RAM. Nte that FMEDg des nt 4

15 need t prcess all N N pixels. As lng as all white dts r black dts are lcated, B is well-defined and FMEDg can be terminated. Accrdingly, the executin time f FMEDg is image dependent. The executin time f FMEDg reprted in the table reflects the cmplexity f its straight-frward Matlab implementatin withut any ptimizatin. An ptimized C realizatin can effectively reduce the executin time. As a matter f fact, the cmplexity f MED algrithms is generally much higher than that f cnventinal errr diffusin algrithms. Hwever, arithmetic peratins can be reduced by recycling the intermediate prcessing results. Besides, the prcessing time can be shrtened by lcalizing the data prcessing peratins t reduce the cmputatin effrt and supprt parallel prcessing. The successful example f cmplexity reductin reprted in [4] shws that ne can achieve blue nise halftning with MED at a cmplexity f 7 arithmetic peratins per pixel fr a image. Similar idea can be used t reduce the cmputatinal cmplexity f FMEDg. In this paper, the fcus is put n the halftne quality and the cluster adjustability. The fast realizatin f FMEDg is t be explred in the future. VIII. CONCLUSIONS MED algrithms are able t prduce halftnes f gd blue nise characteristics withut directinal hysteresis. When the dt gain f a printing prcess is nt stable, this capability may nt be useful as a halftne f green nise characteristics is expected in such a case. In this paper, we prpsed a methd t make FMED [3] prduce halftnes f gd green nise characteristics. Specifically, a diffusin filter is designed based n an istrpic ring filter with their inner and uter radii ( R and R ) as adjustable parameters. FMED can then be mdified t wrk with the prpsed filter t achieve green nise halftning. A detailed empirical analysis was made and we shw that, when R R, the cnnectin f the average / = dt cluster size M g, the inner radius R and the input gray level g can be mdeled with M g = R πg. Cnsequently, the prpsed algrithm allws ne t select any desirable average dt cluster size fr a particular input gray level based n the dt gain nature f a printing prcess easily and freely by adjusting parameter R. This flexible adjustability and the linear relatinship between M g and g are tw advantageus features f the prpsed algrithm t prvide halftnes f desirable green nise characteristics. Empirical perfrmance analysis results shw that the prpsed algrithm can prvide a better perfrmance as cmpared with the ther green nise errr diffusin algrithms in terms f dt distributin, anistrpy, and varius image quality measures including MSE v and UQI. The prpsed algrithm eliminates pattern artifacts and directinal hysteresis cmpletely, and preserves image details better as cmpared with the thers. 5

16 APPENDIX In general, fr any pixel belngs t {(m,n) m,n>}, the cverage f x + y R must be in ne f the k eight scenaris shwn in Figure A. The classificatin f cases 3 t 8 is based n the intersectins f the circle and the grid. The criteria are given as fllws: case : ( m.5) + ( n.5) R if m> and n. 5 Rk if m= k case : ( m +.5) + ( n +.5) Rk case 3: n.5 < nl < n +.5 and.5 < m < m +. 5 m l case 4: n.5 nh < n +. 5 and.5 m < m +. 5 m h case 5: n.5 < nl < n +. 5 and.5 n < n +. 5 n h case 6: m.5 < ml < m +. 5 and.5 m < m +. 5 case 7: m. 5 and m= l case 8: m. 5 and m= h m h where n h = Rk ( m +.5), n l = Rk ( m.5), m h = Rk ( n +.5) and m l = Rk ( n.5). Accrdingly, A m, n, R ) can be determined as ( k f A( m.5, ml, n) ( mh ( m.5) ) + f A( mh, m +.5, n) A( m, n, Rk ) = f A( m.5, m +.5, n) ( mh ( m.5) ) + f A( mh, ml, n) f A(, ml, n) ( mh + f A( mh,.5, n) ) if if if if if if if if case case case 3 case 4 case 5 case 6 case 7 case 8 fr m, n > (A) where q f A( p, q, n) = ( Rk x ( n.5) )dx p (A) A m, n, R ) fr (m,n) s in ther quadrants can be determined by making use f the fllwing symmetric prperty ( k A ( m, n, Rk ) = A( m, n, Rk ) (A3) 6

17 REFERENCES [] R. A. Ulichney, Digital Halftning. Cambridge, MA:MIT Press, 987. [] J. C. Stffel and J. F. Mreland, A survey f electrnic techniques fr pictrial reprductin, IEEE Trans. Cmmunicatin. 9, , 98. [3] R. W. Flyd and L. Steinberg, An adaptive algrithm fr spatial greyscale, Prc. S.I.D. 7(), 75 77, 976. [4] R. A. Ulichney, Dithering with blue nise, Prc. IEEE, vl. 76, pp , Jan [5] B. Klpatzik and C. A. Buman, Optimized errr diffusin fr image display, Jurnal f Electrnic Imaging, (3), 77 9, 99. [6] T. N. Pappas and D. L. Neuhff, Printer mdels and errr diffusin, IEEE Trans. Image Prcessing, vl. 4, pp , Jan [7] R. Levien, Output dependent feedback in errr diffusin halftning," IS&T Imaging Science and Technlgy, pp. 5-8, May 993. [8] D. L. Lau, G. R. Arce, and N. C. Gallagher, Green-nise digital halftning, Prceedings f the IEEE 86, pp , Dec. 998 [9] N. Damera-Venkata and B. L. Evans, Adaptive threshld mdulatin fr errr diffusin halftning, IEEE Trans. Image Prcess., vl., n., pp. 4 6, Jan.. [] N. Damera-Venkata, J. Yen, V. Mnga and B. L. Evans, Hardcpy Image Barcdes Via Blck Errr Diffusin, IEEE Trans. n Image Prcessing, vl. 4, n., pp , 5. [] I. Katsavunidis and C. C. J. Ku, A multiscale errr diffusin technique fr digital halftning, IEEE Trans. n Image Prcessing. Vl.6, N.3, pp ,997. [] Y.H. Chan, A mdified multiscale errr diffusin technique fr digital halftning, IEEE Signal Prcessing Letters, vl. 5, n., pp. 77-8, 998. [3] Y.H. Chan and S. M. Cheung, Feature-preserving multiscale errr diffusin fr digital halftning, Jurnal f Electrnic Imaging, 3(3), pp , 4. [4] Y.H. Fung, K.C. Lui and Y.H. Chan, lw-cmplexity high-perfrmance multiscale errr diffusin technique fr digital halftning, Jurnal f Electrnic Imaging, 6(), (7). [5] Y.H. Fung and Y.H. Chan, Embedding halftnes f different reslutins in a full-scale halftne, IEEE Signal Prcessing Letters, vl. 3, n.3, pp , 6. [6] D. L. Lau and G. R. Arce, Mdern Digital Halftning, Marcel Dekker, New Yrk NY, USA. [7] Zhu Wang and Alan C. Bvik, A Universal Image Quality Index, IEEE Signal Prcessing Letters, vl. 9, n.3, pp.8-84,. 7

18 Figure Captin Figure. Figure. Hw FMED[3] perates in an iteratin cycle The cnceptual mdel f the cluster frmatin with an ideal ring filter in cntinuus spatial dmain. Figure 3. The frequency respnses f (a) square filter H, (b) apprximated ring filter ring filter F. (.8,.8 ) F.8,.8 ) ( and (c) ideal Figure 4. Variance f the frequency cmpnents cvered in an annular ring f any radius r. (a) R =.8 and (b) R =.8. Figure 5. Relatinship between inner radius R, input gray level and cluster size when R / R =. (a) Average f the cluster sizes btained in the simulatin, Mˆ g ; (b) standard deviatin f the cluster sizes btained in the simulatin; (c) cluster size derived by the mdel, M g ; and (d) difference between Mˆ g and M g. Figure 6. Average measured cluster size btained with different cmbinatins f input gray level and inner radius R when (a) R =. 3R, (b) R =. 884R, (c) R = R and (d) R =. R. Figure 7. Parts f the halftning results f a input f cnstant gray-level (a) g=33/55, (b) g=6/55, (c) g=8/55 and (d) g=6/55 with (i) Levien [7], (ii) Damera-Venkata [9], (iii) Damera-Venkata [] and (iv) FMEDg. Figure 8. Hw intra- and inter-cluster dt distributins are measured with D, Δ ( ) and D Δ, λ g + Δ ( ) fr a particular input gray level g. Figure 9. Directinal distributin D, Δ ( ) plts fr the halftning results f a input f cnstant graylevel (a) g=33/55, (b) g=6/55, (c) g=8/55 and (d) g=6/55 with (i) Levien [7], (ii) Damera- Venkata [9], (iii) Damera-Venkata [] and (iv) FMEDg. Figure. Intra-cluster distributin perfrmance Figure. Directinal distributin D Δ, λ g + Δ ( ) plts fr the halftning results f a input f cnstant gray-level (a) g=33/55, (b) g=6/55, (c) g=8/55 and (d) g=6/55 with (i) Levien [7], (ii) Damera-Venkata [9], (iii) Damera-Venkata [] and (iv) FMEDg. Figure. Inter-cluster distributin perfrmance Figure 3. Perfrmance in terms f RAPSD: (a) Levien [7], (b) Damera-Venkata [9], (c) Damera-Venkata [] and (d) FMEDg. Figure 4. Perfrmance in terms f anistrpy: (a) Levien [7], (b) Damera-Venkata [9], (c) Damera-Venkata [] and (d) FMEDg. Figure 5. Testing Images Figure 6. The average MSE v f the testing images f size (a) and (b) 5 5 at different viewing distances (printer reslutin = 6dpi). Figure 7. A crpped regin f riginal Barbara Figure 8. Halftnes prduced with varius algrithms Figure 9. FMEDg utputs btained with (a) R =.8, (b) R =.3, (c) R =.8 and (d) R = 3.. Figure. Number f nn-zer cefficients f ring filter F R, R ) (. Figure A. Area cvered by circle x + y R in a grid unit in the first quadrant. k 8

19 Table captin Table. Table. UQI perfrmance f varius algrithms The executin time f varius algrithms 9

20 Figure. Hw FMED[3] perates in an iteratin cycle Figure. The cnceptual mdel f the cluster frmatin with an ideal ring filter in cntinuus spatial dmain.

21 (a) (b) (c) Figure 3. The frequency respnses f (a) square filter H, (b) apprximated ring filter F and (c) ideal ring filter (.8,.8 ) F. (.8,.8 ) (a) (b) Figure 4. Variance f the frequency cmpnents cvered in an annular ring f any radius r. (a) R =.8 and (b) R =.8.

22 (a) (b) (c) (d) Figure 5. Relatinship between inner radius R, input gray level and cluster size when R / R =. (a) Average f the cluster sizes btained in the simulatin, Mˆ ; (b) standard deviatin f the cluster sizes btained in the simulatin; (c) cluster size derived by the mdel, g M g ; and (d) difference between Mˆ g and M g. (a) (b) (c) (d) Figure 6. Average measured cluster size btained with different cmbinatins f input gray level and inner radius R when (a) R =. 3R, (b) R =. 884R, (c) R = R and (d) R =. R.

23 (i) (ii) (iii) (iv) (a) (b) (c) (d) Figure 7. Parts f the halftning results f a input f cnstant gray-level (a) g=33/55, (b) g=6/55, (c) g=8/55 and (d) g=6/55 with (i) Levien [7], (ii) Damera-Venkata [9], (iii) Damera-Venkata [] and (iv) FMEDg. 3

24 Figure 8. Hw intra- and inter-cluster dt distributins are measured with D ( ) and D ( ) fr a particular input gray level g., Δ Δ, λ g + Δ. 4

25 (i) (ii) (iii) (iv) (a) (b) (c) (d) Figure 9. Directinal distributin D ( ) plts fr the halftning results f a input f cnstant gray-level (a), Δ g=33/55, (b) g=6/55, (c) g=8/55 and (d) g=6/55 with (i) Levien [7], (ii) Damera-Venkata [9], (iii) Damera-Venkata [] and (iv) FMEDg. Figure. Intra-cluster distributin perfrmance 5

26 (i) (ii) (iii) (iv) (a) (b) (c) (d) Figure. Directinal distributin D ( ) plts fr the halftning results f a input f cnstant gray-level (a) Δ, λ g + Δ g=33/55, (b) g=6/55, (c) g=8/55 and (d) g=6/55 with (i) Levien [7], (ii) Damera-Venkata [9], (iii) Damera-Venkata [] and (iv) FMEDg. Figure. Inter-cluster distributin perfrmance 6

27 (a) (b) Figure 3. (c) (d) Perfrmance in terms f RAPSD: (a) Levien [7], (b) Damera-Venkata [9], (c) Damera-Venkata [] and (d) FMEDg. (a) (b) Figure 4. (c) (d) Perfrmance in terms f anistrpy: (a) Levien [7], (b) Damera-Venkata [9], (c) Damera-Venkata [] and (d) FMEDg. 7

28 Huse Lena Mandrill Barbara Bat Peppers Man Figure 5. Testing Images Figure 6. (a) The average MSE v f the testing images f size (a) and (b) 5 5 at different viewing distances (printer reslutin = 6dpi). (b) 8

29 Figure 7. A crpped regin f riginal Barbara (a) Levien [7] (b) Damera-Venkata [9] (c) Damera-Venkata [] (d) Prpsed FMEDg Figure 8. Halftnes prduced with varius algrithms 9

30 (a) R =.8 (b) R =.3 (c) R =.8 (d) R = 3. Figure 9. FMEDg utputs btained with (a) R =.8, (b) R =.3, (c) R =.8 and (d) R = 3.. Figure. Number f nn-zer cefficients f ring filter F R, R ) (. 3

31 Figure A. Area cvered by circle x + y R in a grid unit in the first quadrant. k 3

32 Image size Testing UQI Image [7] [9] [] FMEDg Mandrill Barbara Bat Huse Lena Man Peppers Average Mandrill Barbara Bat Huse Lena Man Peppers Average Table. UQI perfrmance f varius algrithms Testing Time (secnd) Image [7] [9] [] FMEDg Mandrill Barbara Bat Huse Lena Man Peppers Table. The executin time f varius algrithms. 3