Periodic image artifacts from continuous-tonelaser scanners

Paul C. Schubert

Continuous-tone images produced by mechanically scanned analog modulated laser beams are susceptible toimage artifacts in the form of spatially periodic density variations due to machine errors in film transportvelocity, raster scan-line placement, and scan-line intensity. The human eye is particularly sensitive toperiodic patterns and, in ideal conditions, can detect peak-to-peak density variations as small as 0.005 forspatial frequencies around 2-5 cycles/deg. The stringent requirements that this implies for the scannerhardware are derived. Particular attention is paid to the rotating polygon-type scanner, since this devicecurrently provides the best combination of speed, image quality, and cost.

I. Introduction

This paper deals with artifacts in the form of spatial-ly periodic density variations in raster-scanned contin-uous-tone images. In particular, it is assumed that theimages are made of a raster of scan lines which havebeen exposed by a flying spot whose intensity profile isGaussian. The spot intensity is modulated in an ana-log fashion. It is also assumed that the recordingmedium can be characterized solely by the local slopeof the curve of density vs the logarithm of the exposurefor exposure variations about some constant value.

The causes of these artifacts are discussed as well astheir detection bythe observer. It will be seen that themechanical and optical tolerances in the currentlyprevalent mechanical scanners need to be held to strin-gent levels to avoid visible periodic bands in the image.

Section II summarizes data from the literature deal-ing with the contrast sensitivity of the eye. At highspatial frequencies (>5 cycles/deg), the eye experi-ences a falloff in contrast sensitivity due to its imper-fect optics. At low spatial frequencies (<2 cycles/deg)the sensitivity also falls off. This low-frequency be-havior seems not to be attributable to the optics andmust be ascribed to the image processing going on inthe eye and brain.

The author is with 3M Company, Graphics Sector Research Lab-oratory, Graphics Resources Technology Department, St. Paul,Minnesota 55144.

Received 12 February 1986.0003-6935/86/213880-05$02.00/0.© 1986 Optical Society of America.

Section III is a discussion of the three types of ma-chine error that can produce periodic bands in theimage. (1) Velocity variations in the medium trans-port system usually cause bands whose spatial periodis considerably longer than the characteristic perioddefined by the scan-line spacing. (2) Periodic scan-line positioning errors are caused by imperfect beam-scanning hardware, the best example of which is therotating polygon mirror scanner. The beam wobblesup and down from facet to facet due to small (secondsof arc) angular errors in the facet normals. Thus, atwelve-facet mirror, say, would produce a periodicbanding pattern with a period corresponding to twelvescan-line spacings. (3) Periodic variations in scan-lineintensity can be produced, for example, by a polygonscanner with facets of differing reflectivities. Thedensity variations induced by each of these machineerrors is estimated for the case of long-wavelengthbanding, that is, banding with a spatial period muchgreater than the scan-line pitch.

Section IV contains a more detailed analysis of den-sity variations induced by scan-line position errors.Bestenreiner et al.1 have presented numerical resultsfor sinusoidal line-spacing variations. Here a methodis shown for calculating the Fourier components in thedensity for arbitrary repeating patterns of scan-lineposition errors. The same sort of analysis could bemade for arbitrary repeating patterns of scan-line in-tensity variations.

Finally, Sec. V summarizes some conclusions.

II. Visual Contrast Sensitivity

Here are summarized data2 3 generated from obser-vations of test patterns with luminance of the form Lo+ AL sin(27rfx), where f is the spatial frequency of thestripes which run perpendicular to the x direction, and

3880 APPLIED OPTICS / Vol. 25, No. 21 / 1 November 1986

AL/Lo is defined to be the contrast. The measure-ments in Ref. 2 were confined to spatial frequencies forwhich the optics of the eye do not play an importantrole. The interesting falloff in contrast sensitivityobserved at low spatial frequencies (<2 cycles/deg),coupled with the falloff in sensitivity above -5 cycles/deg, indicate that the visual process acts like a band-pass filter with a response peak in the 2-5-cycle/degrange.

To relate the luminance variations in the test pat-terns to density variations that an observer of a piece offilm would see, the following relation is assumed:

AD, = [2/ln(10)]AL/L 0, (1)

where ADpp is the peak-to-peak density variation. Inwhat follows, the threshold contrast figures extractedfrom Refs. 2 and 3 will be changed into threshold peak-to-peak density variations according to this relation.

Figure 1 shows data (open circles) from Fig. 2 of Ref.2. The data have been replotted as density variationsvs spatial frequency in units of cycles/mm. The ob-server is assumed to be at a distance of 300 mm, forwhich 1 cycle/deg = 0.19 cycle/mm. For other viewerdistances, the datum points on the plot must be slidleft or right accordingly. The peak in threshold con-trast sensitivity is clearly seen.

The short curve segment to the right in Fig. 1 is a fitto the data of Fig. 9 of Ref. 3 for spatial frequencies of20-40 cycles/deg or -4-8 cycles/mm for 300-mm view-ing distance. The fitting function was

ADp p = 1.94 X 10-3 exp(O.587f),

C)

_:

0

cam

II-

_j

LO0w

10'

0.1 1

SPATIAL FREQUENCY (cy/mm)

Fig. 1. Contrast sensitivity of the eye. Threshold peak-to-peakdensity vs spatial frequency for viewing distance of 300 mm: open

circles, data from Fig. 2 of Ref. 2; short curve, fit to data from Fig. 9 of

Ref. 3; long curve, overall fit to data.

x

(2)

where f is in units of cycles/mm. The fit reproducesthe data to a few percent.

The smile-shaped curve in Fig. 1 is a least-squares fitto the open circle data plus five points generated fromEq. (2) at 4, 5, 6, 7 and 8 cycles/mm. The fittingfunction was

ADp-p = expla, + a2f + a3 ln(f) + a4 [1n(f)]2 }. (3)

where a, = -5.602, a2 = 0.2614, a = 0.0710, a4 =

0.3886, and f is again in units of cycles/mm. It is Eq.(3) that will be used to summarize the pertinent data inRefs. 2 and 3. It is seen that the minimum detectablepeak-to-peak density variation is between 0.004 and0.005. We will continue to adopt a viewing distance of300 mm in the following so that the more convenientunits of cycles/mm rather than cycles/deg can be used.The curve can be imagined slid to the right or left forviewing distances smaller or greater.

I1l. Description of Banding Artifacts

While some of the results of this paper will be appli-cable to other scanner configurations (e.g., drumtypes), it is in fact the flatbed swept-beam types thatshould be kept in mind in the following. The generalform of such imagers is shown in Fig. 2. The beam isrepetitively swept in the y direction, while the sensitivematerial (the film) moves in the x direction. Ideally,the film is exposed in a series of parallel scan lines withconstant spacing or pitch.

Fig. 2. General form of flatbed raster-type laser scanner.

Spatially periodic variations in this pitch on the filmcan be caused by velocity variations in the film trans-port, facetwise beam position errors transverse to thescan, or machine vibrations which can affect the rela-tive positions of beam and film plane. Figure 3(a)shows schematically the effect of velocity variations onthe intensity pattern presented to the film. A ratherunphysical example is shown in which the transportvelocity varies in a square-wave fashion, causing thescan lines to bunch up where the velocity is lower andto spread apart where it is higher. The top plot showsthe intensity profiles of the scan lines in the x direc-tion. Alternatively, the top plot can be interpreted tobe the individual exposures delivered by each scan ofthe beam. The net exposure is, of course, the sum ofthose from each scan.

The exposure variations caused by long-spatial-wavelength transport velocity variations are easily cal-culated. Denote the average total exposure over theraster by E0. [For a scan pitch less than the full widthat half-maximum (FWHM) of the Gaussian beamspot, the ripple in the exposure in the x direction(perpendicular to the scan) due to spot overlap is lessthan -10%.] The exposure variation AE around E0due to a velocity variation Av is given by

1 November 1986 / Vol. 25, No. 21 / APPLIED OPTICS 3881

l

l

x

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(a)

X

n

(b)

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sures. I will assume here that the y in Eq. (5) has beendetermined using the appropriate raster exposure.

Combining Eqs. (4) and (5), we getAD = [-/ln(10)]Av/v 0 . (6)

As an example of Eq. (6), assume that the transportsystem causes a sinusoidal variation in film velocityresulting in a density variation at a spatial frequencycorresponding to the minimum of the curve in Fig. 1.Take the film contrast y = 2.5. Then, Eq. (6) says thatAD will be less than the threshold value, 0.004-0.005,for Av/vo < 0.004-0.005. In other words, the peak-to-peak variations in film velocity have to be kept below-0.5%o.

Next, we consider the effect of beam wobble on theexposure [Fig.3(b)]. Again, a rather unphysical situa-tion is shown in which the line-position errors occur ina square-wave pattern. The positions of the scanlines, rather than falling equidistant from one another,bunch together when the discrete derivative of the lineposition error is positive and spread out when thederivative is negative.

To make numerical estimates of this effect, let usagain consider the case of a periodic pattern of line-position errors such as might occur with a rotatingpolygon with facet-to-facet angular deviations (pyra-midal error). Assume that this pattern repeats everyM scan lines (M-sided polygon) with M >> 1. Alsoassume that we can approximate this error pattern by asinusoidal function given by

(C)

Fig. 3. (a) Scan-line intensity profiles (top plot) and film-transportvelocity variations. (b) Scan-line intensity profiles and line-posi-tion errors. (c) Scan-line intensity profiles and beam intensity

errors.

Axn = (Axp-p/2) sin(2xrn/M), (7)

where n is an index marking the nth scan line, and Axp-pis the peak-to-peak amplitude of the line-position er-ror. For M >> 1, we can approximate the discretederivative of this function by

dAxnldn = (27r/M)(Axp p/2) cos(2rn/M).AE/E 0 = Av/vo, (4)

where v is the nominal transport velocity.To relate this to the resulting density on the film, we

make the assumption that, for exposure E in the vicini-ty of E0 ,

D = Do + y log(E/E 0 ), (5)

where y is the local photographic contrast of the film,and Do is the density resulting from exposure E. Itshould be pointed out here that the use of Eq. (5) isonly completely justified when the film characteristicsconform with the reciprocity law, namely, that the filmdensity depends only on the exposure (i.e., energy perunit area) and not explicitly on exposure time. Thescanner's line-to-line time is usually orders of magni-tude greater than the dwell time for the laser spot at apoint on the film. Therefore, because the Gaussianintensity profiles of the lines overlap somewhat, apoint on the film receives a group of short exposures,one for each scan line in its vicinity. For materialswhich violate the reciprocity law, this mixing of timescales can lead to rather different sensitometric filmbehavior for raster exposures than for areal flash expo-

(8)

The left-hand side of this equation is just the scanpitch variation as a function of the index n. Thus thepeak-to-peak variation in the scan pitch is just (2r/M)Axp p.

Following our line of reasoning for velocity varia-tions, it follows that the peak-to-peak amplitude of thedensity variations induced by this error is given by

ADp-p = [/ln(10)](27r/M)Axp.p/p, (9)

where p is the nominal scan pitch. As an example ofthe use of Eq. (9), consider a case where M = 12 and o =2.5. Assume that the viewing distance is such that theminimum in the threshold curve of Fig. 1 lies at spatialfrequency 1/(pM). Then, if ADp-p is to be below0.004-0.005, we must have xp-p/p < 0.007-0.009. Fora scan-line pitch of 0.0847 mm (0.0033 in.), for exam-ple, this means that xp-p < 0.6-0.8 m.

One final type of scanner error is shown schematical-ly in Fig. 3(c). A periodic intensity variation such asthis might be caused by facet-to-facet reflectivity vari-ations in a polygon-type scanner. Density variationsinduced by such an error can be estimated using Eq. (6)with velocity v replaced with beam power P on the

3882 APPLIED OPTICS / Vol. 25, No. 21 / 1 November 1986

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right-hand side. For the example cited after Eq. (6),this means that peak-to-peak variations in beam pow-er that result in long-spatial-wavelength banding mustbe kept below -0.5%.

IV. Detailed Analysis of Scan-Line Position Errors

This section shows the way in which the image-density Fourier components are related to the functiondescribing the line-position errors [Fig. 3(b)]. An ex-ample of such a function was shown in Eq. (7) in whicha sinusoidal position error repeats every M scan lines.We can rewrite this equation as follows:

X = Xn + AX,,

= np + (Ax p-/2) sin(2rn/M), (10)

where the xO are the ideal equidistant positions of thescan lines spaced a distance p apart. For this examplethere will be a peak in the Fourier spectrum of thedensity variation at a spatial frequency of 1/Mp.

To make the connection between the Fourier com-ponents of the image density variations and the line-placement errors, we begin by writing the raster expo-sure for constant beam intensity as follows:

E(x) = P0 (2/r)l'2 (wV8 Y)'1n exp[-2(x -Xn) 2/w2], (11)

where P0 is the total beam power striking the film, vs isthe beam spot velocity (in they direction in Fig. 2), andw is the l/e2 radius of the Gaussian spot in the filmplane. For simplicity later on we will assume that theexposure E(x) satisfies the relation E(x) = E(x + Np),where N >> 1. This allows us to use a discrete Fourierrepresentation:

E(x) = 3 F(k.) exp(ikmx), km = 2-rm/(Np),km

m = .. -2,-1,0,1,2,.rNp

F(km) = (Np)_ J E(x) exp(-ikmx)dx. (12)

Now we put the expression for E(x) from Eq. (11) into(12). The result is

N-1

(km) = (Np)-l(Po/vs) exp(-k.,w 2 /8) E exp(-ikmxn). (13)n=O

Equation (13) is actually an approximation, since wehave neglected the contributions to the integral in Eq.(12) due to the Gaussians whose centers lie outside therange (0,Np). For large N this approximation is justi-fied.

The dc component of E(x) is just F(km = 0) or, fromEq. (13), Po/(pv,). In practical continuous-tone sys-tems, the scan-line overlap is such that the km = 0component, which we earlier denoted by Eo, is thedominant one. For the case of no scan-line positionerrors, Eq. (13) yields Fourier components for km val-ues which are integral multiples of 27r/p, that is, spatialfrequencies of fo,2fo,3fo,.. ., where fo = p 1. We cancalculate the density variations in this case using Eq.(5) to make contact with film properties. Note firstthat E(x) can be written in the form

E(x) = 2 E IF(km)I cos(km + 0m),

km

(14)

where the Om values are uninteresting phase factors. Itis the absolute values of the F(km) that will be used tocalculate density variations. For the case of perfectline placement, the peak-to-peak density variationsAD p at spatial frequencies nfo, where n is a positiveinteger, are given by

ADP p(nfo) = 4[y/ln(10)] exp[-(7r2 /2)n2 (c/p) 2 ]. (15)

Consider, for example, the case wherey = 2.5 and (co/p) = 0.849 corresponding to a spot intensity profileFWHM equal to the scan pitch. Equation (15) yieldspeak-to-peak density variations of 0.124 at a spatialfrequency fo and 2.9 X 10-6 at a frequency 2fo. Thevariation at fo is much larger than the minimum de-tectable variation of 0.004-0.005 discussed in Sec. II.However, the scan pitch is usually chosen small enoughso that these variations cannot be resolved by the eyeat normal viewing distances.

For practical cases where the scan lines are notplaced exactly equidistant from one another, addition-al density variation components arise. We shall as-sume that the scanning mechanism produces a patternof scan-line position errors which repeats every Mscanlines. This would occur, for example, in the case of anM-sided polygon scanning mirror. Additional Fouriercomponents in the density variation occur at integermultiples of spatial frequency f = (Mp)-l. Thepeak-to-peak amplitudes ADp pM of these variationsare easily shown to be given by

ADp pM(nfM) = 4M'1[y/ln(10)] exp[- 2 /2)(n/M) 2 (/p) 2 ]

(16)M-1

X E exp(27rinfMxm) .m=O

Equation (9) follows from Eq. (16) by assuming thatAxn << p and that (7r2/2)(M)- 2 (,W/p)2 << 1 and doing anexpansion of the exponential under the summation.

As an example, we consider the case where M = 12, By- 2.5, p = 102 gm (250 scan lines/in.), and /p = 0.849.To demonstrate the surprising sensitivity of the eye toscan-line position variations, we will assume that thescan lines are perfectly placed except that everytwelfth line is out of position by 5 Aim. Then, in Eq.(16), xm = mp for m = 1 to 11, while xm=o = 5 ,im.Figure 4 shows the spectrum of density variations cal-culated using Eq. (16). The shaded region of the fig-ure corresponds to visible image banding at a viewingdistance of 300 mm. Even with a seemingly small line-position error in which only one line in twelve is out by5% of the nominal line spacing, there are several densi-ty components which exceed the threshold for visibili-ty.V. Conclusions

We have seen that the production of continuous-tone images using laser scanners requires stringenttolerances on film-transport velocity, line-placementaccuracy, and beam intensity stability. Representa-tive examples indicate that the transport velocity as

1 November 1986 / Vol. 25, No. 21 / APPLIED OPTICS 3883

-o

:

C)cc

1

..2

C1

n

I

0.1 1.0

SPATIAL FREQUENCY (cv/mm

Fig. 4. Spectrum of density variations for case in which everytwelfth scan line is out of position by 5% of the nominal line spacing.

Viewing distance is assumed to be 300 mm.

well as the beam intensity must be constant to 0.5%peak to peak. Line placement errors must be held to<1% of the nominal spacing.

References

1. F. Bestenreiner, U. Greis, J. Helmberger, and K. Stadler, "Visibil-ity and Correction of Periodic Interference Structures in Line-by-Line Recorded Images," J. Appl. Photogr. Eng. 2, No. 2,86 (1976).

2. D. H. Kelly, "Visual Contrast Sensitivity," Opt. Acta 24, No. 2,10 107 (1977).

3. F. W. Campbell and D. G. Green, "Optical and Retinal FactorsAffecting Visual Resolution," J. Physiol. 181, 576 (1965).

S. R. Forrest of the University of Southern California, photographed by W. J. Tomlinson of BellCommunications Research, at OFC/IGWO-86, Atlanta, in February 1986.

3884 APPLIED OPTICS / Vol. 25, No. 21 / 1 November 1986

l