Pulse-density modulation on rastered media: combining pulse-density modulation and error diffusion

708 J. Opt. Soc. Am. A/Vol. 7, No. 4/April 1990

Pulse-density modulation on rastered media: combiningpulse-density modulation and error diffusion

Reiner Eschbach

Xerox Corporation, Webster Research Center, 800 Phillips Road 128-27E, Webster, New York 14580

Received April 15, 1989; accepted December 7, 1989

An approach to generating a pulse-density modulation (PDM) on rastered media is presented. The procedurecombines techniques of generating a PDM on spatial nonquantized media with the standard error-diffusionalgorithm. The algorithm produces homogeneous pulse distributions along with a good reproduction of fine detailof the object.

1. INTRODUCTION

A wide range of applications require the representation ofcontinuous-valued input functions on binary media. Thisrequirement can arise from the hardware used, as in therepresentation of data on matrix printers, binary electronicdisplays, and metal films, or from system considerationssuch as noise insensitivity and the ability to reproduce andcopy the data without loss of information.

An additional constraint often imposed by hardware orsoftware is the quantization in the spatial domain, or raster-ing. This quantization is inherent in all digital systemsalthough it is not always obvious, as in cases when theaddressability of the system is much finer than the smallestfeature or pulse that it is able to produce. The address-ability in most cases, however, is approximately equal to thesmallest producible feature or pulse. It is this relationshipthat is generally assumed when the term pixel is used andthat is assumed throughout this paper.

A wide variety of coding methods exist that can be used todisplay data on a binary and rastered medium. Examplesare pulse-width modulation' and modifications such as or-dered dither2 -4 and numerical methods such as minimumaverage error,5 error diffusions 9 (ED), and statistical cod-ings.10 The numerical methods can be regarded as a sort ofpulse-density modulation (PDM) because the input func-tion is represented by pulses or pixels of the identical sizewith relative spacing dependent on the input. Each outputpixel is determined without the requirement for the pixels toform certain patterns or to cluster into areas as is done inpulse-width modulation or in ordered-dither techniques. Ifthe numerical methods are viewed as an implicit approach toPDM, the modification of PDM,l 2 so that it can be used onrastered media, is suggested.

The approach to extending the definition of a PDM torastered media described in this paper uses an integrationconcept to generate a PDM1" for some part of the image andto use error diffusion for other parts of the image that wouldresult in a poor image representation if coded directly as aPDM. This guarantees a homogeneous pulse distribution inareas where the PDM approximations hold and a fast andconvenient way to process the other areas. The combina-tion of PDM and ED is especially attractive because the ED

can, in the one-dimensional (1-D) case, be described as aspatially quantized PDM.13 The two-dimensional (2-D) EDhas not yet been analytically described, but the close relationbetween 1-D and 2-D ED suggests that the 2-D ED algo-rithm has similarities to a PDM. The combination of thetwo methods, which originated from different applications,permits the preservation of the advantages of both methodswhile reducing their shortcomings.

2. PULSE-DENSITY MODULATION ANDERROR DIFFUSION

PDM is a way to code a continuous-input function by usingpulses of fixed size at arbitrary positions in space. No spa-tial quantization is present in the binary output. In displayor printing applications the decoding of the continuous-input information is performed by the visual system and canroughly be regarded as a low-pass operation. Figure 1 showsthe result of applying PDM to a constant-input function.This application results in a homogeneous pulse distribu-tion.

ED also produces pulses of fixed size, but in contrast toPDM the pulse positions are quantized to give an address-ability equal to the pulse size. This quantization is due tothe fact that ED was defined for devices that are divided intopixels.6

The combination of the two algorithms, which have differ-ent underlying assumptions, is shown in the following para-graphs.

A. Generation of a Two-Dimensional Pulse-DensityModulationDifferent methods that can generate a 2-D PDM are avail-able.1'1 2 A modification of the method described in Ref. 11has been used in all examples. The method and its modifi-cations will be briefly described for clarity.

In the 2-D PDM a pulse of fixed size is set whenever theintegral of the input over an area Am reaches a certain valuedetermined by the output pulse:

J J I(x', y') dx'dy' = SIO,Am

(1)

where I(x', y') is the input function, Am is the integration

0740-3232/90/040708-09$02.00 © 1990 Optical Society of America

Reiner Eschbach

Vol. 7, No. 4/April 1990/J. Opt. Soc. Am. A 709

in which neighboring pixels are integrated or, in this case,summed. Note that the integration areas are assumed tocontain a large number of pixels (50-100), so that the resultcan be regarded as continuous in space. The transition to asmall number of pixels inside each integration area and theresulting artifacts will be treated in Subsection 2.1.2. Thevalues of the pixels determined by the road map are addeduntil the total value SIo is reached or exceeded. The algo-

...............................

Fig. 1. Result of the PDM applied to a constant-input function.

i j

o 0 0

i

i

0

-1

-1

0

0

0

2

0

(a) (b)Fig. 3. Area coverage for the integration areas that resemble halfcircles (a) and triangles (b). The half-circle integration areas leadto a fish-scale pattern, and the triangular areas to a diamond-shapedpattern.

o a1 0

0 1

-1 0

2 0

1 1

0 2

-1 1

Fig. 2. Integration areas in the form of a half circle (a) and atriangle (b). The first eight elements of the road map are listed tothe right of the integration areas.

area, S is the size of the pulse, and I0 is a normalizationconstant [Eq. (1) appears as Eq. (6) in Ref. 11]. Additionalconstraints have to be fulfilled by the integration areas Am.The representation of the input function by a pulse densityrequires that individual integration areas Am do not overlap.Constant-input functions should generate areas that coverthe whole input region with an isotropic center-to-centerspacing and an isotropic changing of the area size for differ-ent input values. In general these constraints cannot com-pletely be met for all inputs. The incorporation of square-like integration areas was shown in Ref. 11.

1. Modification of the Integration AreasThe method of generating the integration areas has beenmodified to a map-oriented approach. The input array isprocessed starting from the top left corner, which is thentaken as the center of origin for a second, moving coordinatesystem. The integration area is described as a sort of roadmap in this second coordinate system; it specifies the order

Fig 4 Result of applying the PDM to a gray wedge with spatialquantization. The pulse distribution is homogeneous for low inten-sities (I d 1/4), which is equivalent to a spatial quantization muchfiner than the average center-to-center spacing but gets more andmore inhomogeneous for higher intensities.

a

b

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areas generate. Disturbances of the form of the area by achanging input function do not generate strongly differentarea shapes and have no obvious effect on following integra-tion areas.

2. Pulse-Density Modulation on a Rastered MediumDifferent possibilities exist to approximate a PDM on arastered medium. In the first approach a PDM could becalculated without any underlying raster (or with a raster ofhigher resolution than the pulse size so that the spatialquantization can be neglected), and the resultant pulse dis-tribution is subsequently quantized. In the second ap-

4, { 9 , A 2 t k gproach the PDM would be performed directly on the rasterthat corresponds to the output. This second approach will

~ .. gJ |!:Y~ g t l M tbe used in the following paragraphs.X~, . f X .~ ..2 ..@. gThe introduction of the spatial quantization violates the

Bi55 *j.'<I k Sbil.' .~. .B. .' i II. GI~via.[p4 assumptions underlying the PDM in different ways. First,1X55>m/{5|5.9¢1 I ')05t.\& §>0>e ;;.g ~ the pulse position is no longer arbitrary in space but coarsely

quantized, which leads to the possible introduction of un-@J'| ..... *'gs55.5 ................. ~wanted textures. Second, the integration areas reduce to a

few pixels, which reduces the possibility of isotropic areasand area scaling. The third effect of the coarse quantization

.- <-,5-*.... .. .--5...... is that the integral in Eq. (1) cannot be exactly solved be-cause an increase of the integration (in this case summation)area from n to n + 1 pixels generally exceeds SIo. This effect

:: ::. .: : :: ..... ::: : . . : .is compensated for by calculating the error that is made insolving Eq. (1) and distributing it over the border of the area,

.... ~~~~. ... ... iscmestdfrb:acltn h ro hti aei

Fig. 5. Pulse distribution generated by applying the ED algorithmon the same input as in Fig. 4. The low- and high-intensity regionsexhibit an inhomogeneous pulse distribution, whereas the medium-intensity regions yield a good pulse distribution.

rithm then moves to the next unprocessed pixel of the image,which is again used as the center of origin for the road map.Note that the next unprocessed pixel is determined by itsposition in the input image (row, column) and will, in mostcases, not be the next pixel on the road map, which is sortedcorresponding to its distance from the current center of theroad map. In this way, the whole input array is sequentiallyprocessed. Figure 2 shows a half circle and a triangle asexamples for two different integration areas along with therelative i and j coordinates of the first eight pixels specifiedby the road map. The road maps are generated by a numeri-cal sorting of the list (i, j) of relative target pixels. Thesorting is done by corresponding to z values produced by theformula

z = all (2)

for the half circle and by the formula

z = il + IjI (3)

for the triangular integration area. Pixels with identical zvalues are sorted by decreasing i value. Figure 3 shows theresulting integration areas for a constant-input function.The integration areas described by a half circle result in thefish-scale pattern of Fig. 3(a), and the triangular-integrationareas result in the diamond pattern of Fig. 3(b). The advan- Fig. 6. Combining PDM and ED by applying them to differenttage of these two integration areas over the squarelike inte- intensity regions of the image leads to a visible seam, which can begration area lies in the stability of the patterns that the two seen as a horizontal line for the gray wedge.

Reiner Eschbach


(a)

(b) - b \ be " O A v A d

Fig. 7. (a) Intensity-dependent switching between PDM and EDleads to the adaptation of the integration areas to the borderline(marked by the arrow) between PDM and ED. (b) Resetting allborder areas eliminates this effect and leads to a more irregulartransition between PDM and ED and thus to a less visible seam.

i.e., over all unprocessed pixels touching the area. Thesepixels can easily be found by taking the next target pixels ofthe road map. In the examples shown, this procedure wassimplified by distributing the error over the next four pixelsof the road map, instead of finding all border pixels. Thisapproach distributes an error that is made inside a wholearea onto its boundary and differs from ED, in which theerror of individual pixels is distributed. Figure 4 shows theresult of applying PDM to a gray wedge of 256 pixels and 512scan lines with 0 - I 1, in which the algorithm generateswhite pulses on a black background. From Fig. 4 it is obvi-ous that the PDM algorithm produces a homogeneous pulsedistribution in the low-input regions in which the resultingareas extend over several pixel sizes. This effect is to beexpected because the assumption underlying PDM is that nospatial quantization of the pulse position is present. Thisassumption is violated in our example for the higher-inputvalues, leading to area sizes of less than two pixels (for inputvalues > 1/2) and thus to a strong spatial quantization. Forthe limit of area sizes equal to 1 pixel, I 1, the PDMdegenerates into the ED algorithm.

The problem of small areas with few pixels can be reducedby splitting the input into one bright (I > 1/2) and one dark(I 1/2) part and by processing the bright part as PDM,using black pulses on a white background, and processingthe dark part, using white pulses on a black background.This is possible because raster resolution and pulse size areidentical in the situation described here, i.e., dark areas arerepresented by white pixels on a background of black pixels,

and bright areas are represented by black pixels on a back-ground of white pixels. In this case, all occurring areas areat least the size of two pixels. From Fig. 4, however, it isevident that the approximations made in the algorithm giveonly a reasonable PDM for I _ 1/4 or, because of symmetry,for I _ 3/4. The input region 1/2 < I < 3/4 has to be treatedwith a different approach for a PDM or, as is done here, witha different algorithm.

B. Error DiffusionA possible choice for an algorithm that can be used in theinput region 1/4 < I < 3/4 is the ED algorithm described inRef. 4. In the ED algorithm the value of the pixel at (i, j) iscompared to a threshold. When the value exceeds thethreshold, a 1 is set in the output, otherwise a 0 is set. Thedifference between the value before thresholding and afterthresholding is defined as error. This error is divided intofour parts and added to future input values to produce modi-fied values. These modified input values are themselvescompared to the threshold, and the algorithm sequentiallyprocesses the input in a row-by-row (or column-by-column)fashion. The weighting factors for the error originally pro-posed in Ref. 4 are 7/16 for pixel (i + 1, j), 1/16 for (i + 1, j +1), 5/16 for (i, j + 1), and 3/16 for pixel (i - 1, j + 1), using thesame coordinate system as Fig. 2, so that the sum of theweighting factors equals 1. Other weighting factors havebeen suggested, 8"14 but the original factors have been used in

Fig. 8. Modified transition scheme between PDM and ED showsno visible seam at 1/4 and 3/4 for the gray wedge and a betterrendition of low- and high-input regions than the ED in Fig. 5.

Reiner Eschbach

Reiner Eschbach712 J. Opt. Soc. Am. A/Vol. 7, No. 4/April 1990

(a)

(b)

, -, :-- .:*. .. .Ye.;* !; *. :: :* .: *

Fig. 9. (a) Result of applying the PDM on a gray-tone image. The pulse distribution is homogeneous in the low- and high-intensity regions,and fine details are well reproduced. (b) The result of applying the ED algorithm to the same input is shown. Here fine details are wellreproduced, but the low- and high-intensity areas show strong wormlike artifacts.

....... . . ..... ..


all examples in this paper. The ED algorithm is especiallypromising for the combination with the PDM because the1-D version can be described as a sampled PDM13 and be-cause there is a close relationship between the 1-D and 2-DED algorithms. Figure 5 shows the result of the ED algo-rithm applied to the same input as in Fig. 4. In this case,input values I > 1/4 are well represented by the algorithm,whereas values I 1/4 result in a nonhomogeneous pulsedistribution. A comparison of Figs. 4 and 5 shows that thePDM produces good results in regions where the ED algo-rithm shows artifacts and that the ED produces good resultsin regions where the assumptions of the PDM algorithm areviolated.

C. Blending Pulse-Density Modulation and ErrorDiffusionThe comparison of Figs. 4 and 5 suggests processing inputpixels with values I < 1/4 and I > 3/4 with the PDM algo-rithm and values 1/4 I 3/4 with the ED algorithm.Other values might be used as separation points between thetwo methods, but 1/4 and 3/4 were used in the experimentsof Section 3.

Several problems have to be solved to blend the PDM andED algorithms without a visible seam at locations where theswitching between the algorithms occurred. Figure 6 showsthe example of such an unwanted seam. Here, a gray wedge(0 _ I _ 1/4) was processed by the PDM for I _ 1/8 and theED for I > 1/8 with a clearly visible seam between the twoareas. The separation here was performed at 1/8 becausethe effect is much more clearly visible at 1/8 than at 1/4

where the normal switching would occur. The effect iscaused by the pulse location inside the integration areas,where every pulse is located at the center of the area. Thisleads to a separation of the pulses from the border and thusto a visible seam. Another problem arises when the integra-tion area is smaller than the area size required to fulfill Eq.(1). In this case the pixels cannot be processed with PDMalthough the local input values would require this process-ing..

These problems can be solved by an addition to the area-growing scheme. The algorithm that determines whetherthe next pixel given by the road map is an unprocessed pixelin the correct value region also determines whether the areatouches pixels with values outside the specified value region.All areas touching parts of the image with values outside thespecified value region are marked as border areas. All bor-der areas are then reset and afterward processed with the EDalgorithm. Figure 7(a) shows the integration areas Am thatare generated by using the triangular-integration areas ofFig. 2(b) on a wedge as input function. Here, the effect ofthe switching is clearly visible as a horizontal line at theswitching point, indicated by an arrow. The formation ofthe horizontal line is caused by the area-growing procedure.The integration areas grow in the diamond pattern untilthey touch a zone with values outside the specified valueregion. The area then grows along the border of the zoneand creates a reproduction of the borderline. All areas thattouch the line are reset to give Fig. 7(b), in which the hori-zontal line is replaced by an irregular jagged edge. Theeffect of this modification is shown in Fig. 8(a), where the

The large white areas have been processed by the ED algorithm.

Reiner Eschbach

qw:lrl

CAP

Fig. 10. Integration areas for the PDM of Fig. 9(a).


Fig. 11. Pulse distribution generated by the PDM for the input of Fig. 9(a). Dark areas are processed as white pixels on black background andbright areas as black pixels on white background.

Fig. 12. Pulse distribution generated by the ED for the medium-input levels of Fig. 9(a).

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(a) (b)Fig. 13. (a) Fourier spectrum of Fig. 9(a). The separation between signal (low-pass portion) and noise clouds is independent on the direction.(b) The transform of Fig. 9(b) shows a nonuniform separation of signal and noise clouds and a bad separation along the diagonal from top left tobottom right, which is caused by the so-called worms. The data in (a) and (b) have been nonlinearly normalized (identical for both cases) toshow the structure of the noise clouds.

seam has been dramatically reduced. The seam is furtherreduced for standard, realistic-input objects that have notbeen created inside a computer because the boundary be-tween the dark, medium, and bright regions will alreadyhave an irregular shape. The elimination of the borderareas automatically solves the problem of an insufficientintegration area, i.e., when only a few pixels are in the re-quired input value region, because those areas are alwaysborder areas. The opposite approach could have been takenby permitting the integration areas to protrude into themedium-input region. This modification can be imple-mented by checking only the starting pixel for the integra-tion and neglecting the regional separation for the road map.Figure 8 shows the result of applying the PDM algorithm onthe same input as in Fig. 5. The artifacts of the ED algo-rithm were clearly visible in Fig. 5, while the result obtainedwith the PDM algorithm in Fig. 8 shows a more homoge-neous pulse distribution without disturbing artifacts.

3. EXPERIMENTAL RESULTS

The algorithm was applied on a scanned test image to verifythe prediction that the adaptation of PDM to a rasteredmedium produces homogeneous bright and dark areas in theimage while retaining the good characteristics of ED for themedium-input levels. The image of Fig 9(a) was processedwith the modified PDM algorithm, which led to a homoge-neous pulse distribution in the dark portions of the hair andbackground. No visible seam or false contour is generatedby the blending of the PDM with the ED. Figure 9(b) showsthe same image processed with the standard ED algorithm.Strong wormlike artifacts can be seen in the hair and the

dark background. Figure 10 shows the integration areasgenerated by the PDM algorithm, and Fig. 11 shows thecorresponding pulse distribution. Note that the dark areasare processed as white pulses on a black background and thebright areas as black pulses on a white background. Figure12 shows the result of applying the ED algorithm on theremainder of the image. The summation of Figs. 11 and 12resulted in the final image that is shown in Fig. 9(a). Figure13 shows the Fourier transforms of a 512 X 512 section of theimages in Fig. 9. The noise-free area around the dc peak inFig. 13(b) has the characteristic form for an image binarizedwith the ED algorithm. The separation between the noiseclouds and the dc peak is large along one diagonal (goingfrom lower left to upper right) and small or disappearingalong the other diagonal. This behavior is typical for non-homogeneous pulse distributions. The spectrum of Fig.9(a) shown in Fig. 13(a) exhibits a more homogeneous distri-bution of the noise cloud. The separation between noiseand signal is large for all directions, and the cloud does notextend into the signal. Note that Figs. 13(a) and 13(b) havebeen nonlinearly normalized to show the structure of theclouds.

4. CONCLUSIONS

PDM and ED concepts have been combined to produce aPDM algorithm on a rastered medium. The procedure com-bines the positive features of both PDM and ED by using thetwo algorithms on those parts of the image where they arebest adapted. This leads to a separation of the input imageinto three parts. The low- and high-input regions are codedwith PDM, and the medium-input levels are processed by

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the ED algorithm. This separation leads to a homogeneouspulse distribution in the low- and high-input regions and tothe suppression of artifacts in that region. Medium-inputlevels in the above-described procedure are processed by theED algorithm leading to a faithful reproduction of the graylevel and a good reproduction of fine detail. The combina-tion of the two methods is done in a way that the transitionfrom one algorithm to the other is generally invisible. Thistransition was achieved by adapting the transition point tothe area contours. In this way, no distinctive borders be-tween the regions are established.

The described algorithm offers an alternative to existingbinarization procedures, and it can be used advantageouslyin situations in which the input has to be coded with alimited number of pixels, as in the case of computer displaysand printers.

ACKNOWLEDGMENTS

I thank Keith T. Knox, Paul G. Roetling, Judith E. Stine-hour, and Myra VanInwegen for many useful discussions.Special thanks go to Judith Stinehour for help in preparingthe figures.

REFERENCES

1. J. C. Stoffel and J. F. Moreland, "A survey of electronic tech-niques for pictorial image reproduction," IEEE Trans. Com-mun. C-29, 1898-1925 (1981).

2. C. N. Judice, J. F. Jarvis, and W. H. Ninke, "Using ordereddither to display continuous tone pictures on an ac plasmapanel," Proc. Soc. Inf. Disp. 15, 161-169 (1974).

3. B. E. Bayer, "An optimum method for two-level rendition ofcontinuous-tone pictures," in IEEE 1973 International Confer-ence on Communications (Institute of Electrical and Electron-ics Engineers, New York, 1973), Vol. 1, pp. 26-11-26-15.

4. 0. Bryngdahl, "Halftone images: spatial resolution and tonereproduction," J. Opt. Soc. Am. 68, 416-422 (1978).

5. M. R. Schroeder, "Images from computers," IEEE Spectrum 6,66-78 (1969).

6. R. W. Floyd and L. Steinberg, "An adaptive algorithm for spa-tial grey-scale," Proc. Soc. Inf. Disp. 17, 75-77 (1976).

7. J. A. G. Hale, "Dot spacing modulation for the production ofpseudo grey pictures," Proc. Soc. Inf. Disp. 17, 63-74 (1976).

8. C. Billotet-Hoffmann and 0. Bryngdahl, "On the error diffu-sion technique for electronic halftoning," Proc. Soc. Inf. Disp.24, 253-258 (1983).

9. G. S. Fawcett and G. F. Schrack, "Halftoning techniques usingerror correction," Proc. Soc. Inf. Disp. 27, 305-308 (1986).

10. J. Frank and R. Groff, "On statistical coding of two-tone imageensembles," Proc. Soc. Inf. Disp. 17, 102-110 (1976).

11. R. Eschbach and R. Hauck, "Binarization using a two-dimen-sional pulse-density modulation," J. Opt. Soc. Am. A 4, 1873-1878 (1987).

12. R. Eschbach and R. Hauck, "A 2-D pulse density modulation byiteration for halftoning," Opt. Commun. 62, 300-304 (1987).

13. R. Eschbach and R. Hauck, "Analytic description of the 1-derror diffusion technique for halftoning," Opt. Commun. 52,165-168 (1984).

14. P. Stucki, "MECCA-A multiple-error correction computationalgorithm for bilevel image hardcopy reproduction," IBM Re-search Rep. RZ 1060 (IBM Zurich Laboratory, Zurich, 1981).

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Pulse-density modulation on rastered media: combining pulse-density modulation and error diffusion