Measurement of the Modulation Transfer Function of Paper

o ren halfto ffect pap To a ently aper ana ar-ta sion c litud eries- wo idere paper, and in the second, the bar target is placed directly on the paper, in close contact. © 1998 Optical Society of America OCIS codes: 100.0100, 100.2810, 110.4100. f o 1. Introduction In recent years there has been renewed interest in modeling the halftone microstructure to better con- trol the colors produced in a halftone image.1–4 Dif- usion of light within the paper has a significant effect n halftone color4; this effect is known as optical dot gain or the Yule–Neilsen effect. Because of diffu- sion, a photon may exit the paper from a different region of the halftone microstructure than that into which it entered the paper. To account rigorously for this effect requires knowledge of the paper’s point- spread function ~PSF! or, equivalently, the paper’s modulation transfer function ~MTF!.5,6 The PSF is a conditional probability density that characterizes photon diffusion, and the MTF is the modulus of the Fourier transform of the PSF.3 In this paper a novel method for measuring the MTF of paper is proposed. The MTF measures the frequency response of an optical system, and over the years a variety of tech- niques have been developed.7 The most common techniques can be divided into three categories: the sine-wave method, the bar-target method, and the edge-gradient method. In all three, an object target with one-dimensional variation in intensity is imaged by the optical system, and the image is analyzed to obtain the MTF. In the sine-wave method, an object target with a sine-wave variation in intensity is imaged, and the MTF is the ratio of image-to-object modulation depth.7,8 This is the most direct method for measur- ing the MTF, but it requires a different target and a different set of measurements for each of the spatial frequencies sampled. Inoue et al.8 have recently measured the paper MTF using the sine-wave method by projecting the image of the sine-wave tar- get onto the paper and measuring the image intensity as a function of position. In the bar-target method, the object has square- wave variation in intensity. There are typically two methods by which the bar-target image is analyzed. In the first,9,10 the MTF is given by the reduction in amplitude of the fundamental frequency component of the bar-target image. In the second,9,11,12 the con- trast transfer function ~CTF! is measured and nu- The author is with Matrix Color, 26 East 33rd Street, New York, New York 10016. Received 20 April 1998; revised manuscript received 22 June 1998. 0003-6935y98y317235-06$15.00y0 © 1998 Optical Society of America 1 November 1998 y Vol. 37, No. 31 y APPLIED OPTICS 7235 Measurement of the modulation transfer function Geoffrey L. Rogers In recent years there has been a control the colors produced in a effect on the halftone color; this e of diffusion, a photon may exit the into which it entered the paper. point-spread function or, equival nique for measuring the MTF of p method uses a bar target, but the In the series-expansion method, b MTF is given by the series-expan the typical method plots the amp varying frequency, whereas the s harmonics for a single target. T series-expansion method are cons f paper ewed interest in modeling the halftone microstructure to better ne image. Diffusion of light within the paper has a significant is known as optical dot gain or the Yule–Neilsen effect. Because er from a different region of the halftone microstructure than that ccount rigorously for this effect requires knowledge of the paper’s , the paper’s modulation transfer function ~MTF!. A new tech- —the series-expansion bar-target technique—is introduced. The lysis more closely resembles that of the edge-gradient technique. rget image data are expanded into a Fourier series, and the paper’s oefficients. It differs from the typical bar-target analysis in that e of the fundamental frequency component for several targets of expansion method plots the amplitude of the fundamental and its possible techniques for measuring the MTF with the bar-target d. In the first, the image of the bar target is projected onto the merically transformed into the MTF. Both methods, like the sine-wave technique, require a different tar- get and a different set of measurements for each spa- s a t f 7 tial frequency sampled. Also, the transformation CTF 3 MTF is strictly correct only for a target of infinite extent.9 Arney et al.5 have measured a mod- ified CTF of paper by placing bar targets in direct contact with the paper and measuring the average reflectance between the bar lines. In the edge-gradient method, the object is a knife edge, and the image data are the edge-spread func- tion.7,13 The MTF is obtained as the Fourier trans- form of the derivative of the edge-spread function. The edge-gradient technique has been the most com- monly used method for measuring the MTF of pa- per.6,14 Engeldrum and Pridham6 projected a knife- edge image on paper and measured the reflectance as a function of position. A major disadvantage to the edge-gradient method is that one must take the de- rivative of noisy data. The noise introduces a bias error that increases with increasing frequency.7 Also, because this method measures the MTF at a single point, paper inhomogeneities tend to make the measurements difficult to reproduce.6 A new technique for measuring the MTF of paper—a series-expansion bar-target technique—is introduced in this paper. In this technique, a bar target is used but the method of analysis more closely resembles that of the edge-gradient technique. In the series-expansion method, bar-target image data are expanded into a Fourier series and the paper’s MTF is given by the series-expansion coefficients; or, equivalently, the MTF is given by the Fourier trans- form of the bar-target image data evaluated at mul- tiples of the target frequency. It differs from the typical bar-target method in that the typical method plots the amplitude of the fundamental frequency component for several targets of varying frequency, whereas the series-expansion method plots the am- plitude of the fundamental and its harmonics for a single target. The typical method uses high- frequency targets so that at most only a few of the harmonics are significant; the series-expansion method requires a low-frequency target so that a large number of harmonics are significant. The series-expansion method essentially measures the edge-spread function, but it has an advantage in that it does not involve taking the derivative. Also, the method measures several edges and therefore aver- ages over paper inhomogeneities. In obtaining the MTF from the harmonics of a single bar target, the series-expansion method is similar to the technique developed by Primot and Chambon15 for measuring the MTF of sampled imaging systems. The experi- mental arrangement required for the series- expansion method is similar to that used by Arney5 for measurements of a modified CTF of paper. Two possible techniques for measuring the MTF with the bar-target series-expansion method are con- sidered in this paper. In the first, developed in Sec- tion 2, the image of the bar target is projected onto the paper and the relative reflectance measured as a 236 APPLIED OPTICS y Vol. 37, No. 31 y 1 November 1998 function of position. In the second technique, devel- oped in Section 3, the bar target is placed directly on the paper, in close contact. The bar target and paper are diffusely illuminated, and the relative reflectance is measured as a function of position. In Section 4, the experimental process is outlined and simulated bar-target image data are analyzed. In the analysis developed below, it is assumed that the MTF of the recording medium and the MTF of the imaging system are together an order of magnitude wider than the MTF of the paper, so the effects of the imaging system and recording medium are ignored. 2. Projected Bar Target The illuminated bar target is imaged onto the paper, and the reflectance is measured as a function of po- sition. The intensity of the light entering the paper, I~x!, is given by I~x! 5 I0t~x! 1 Is, where I0 is the intensity of the light illuminating the target; Is is an evenly distributed background illumi- nation that is due to scatter, etc. ~Is ,, I0!; and t is the quare-wave target function: t~x! 5 ( n tn~x!, (1) with tn~x! 5 51, na # x , Sn 1 1 2Da 0, x $ Sn 1 12Da or x , na , (2) nd a is the bar-target period. The target function ~x! is shown in Fig. 1~a!. The relative reflectance rom the paper at position x is R~x! 5 Rp I0 1 Is *2` ` I~x9!L~x9 2 x!dx9, (3) Fig. 1. ~a! Target function t~x!; ~b! reflectance R~x! for projection method; and ~c! reflectance R~x! for contact method, where, for both, an exponential line-spread function is used with ay^x& 5 50, and a 5 1, b 5 0. where L~x! is the line-spread function and Rp is the paper reflectance. It is assumed that the line- spread function is real, even, and normalized. If the tional to the first moment of the PSF. From this it follows that T t s i line-spread function is a delta function, the relative reflectance between the bars ~t 5 1! is Rp, and from the bars ~t 5 0! is RpIsy~I0 1 Is!. One expresses the target function as a Fourier se- ries: t~x! 5 1 2 1 2 p (n odd ` 1 n sin~vn x!, (4) with the sum for all positive, odd n, and vn 5 2pn a , with a the bar-target period. For simplicity, the phase is chosen such that for n Þ 0 the cosine terms are all zero @Eq. ~2!#. The development does not, however, depend on this particular choice of phase: The phase is arbitrary. Integrating Eq. ~3! with Eq. ~4! to obtain the reflectance R~x!, one must evaluate the integral: * 2` ` L~x9 2 x!sin~vn x9!dx9. Noting that L~x! is even, one can write this as sin~vn x! * 2` ` L~x9!cos~vn x9!dx9, which is equal to sin~vnx!L˜~vn! with L˜~v! the paper’s MTF. One then finds that the reflectance is R~x! 5 a 1 b 2 1 2 p ~a 2 b! ( n odd ` L˜~vn! n sin~vn x!, (5) where one defines a 5 Rp, b 5 Rp Is I0 1 Is . The reflectance is expressed as a Fourier series in Eq. ~5!, and one immediately obtains L˜~vn! 5 Uivn2 * 0 a R~x!exp~2ivn x!dxU 3 ~a 2 b!21, n odd. (6) The MTF is proportional to the Fourier expansion coefficients of the reflectance R~x!. So that there will be a significant number of har- monics in the Fourier expansion of the reflectance, one chooses a bar target such that the half-period is much larger than the lateral scattering length: ^x& ,, ay2, where the scattering length ^x& is propor- L~ay2! 5 L~2ay2! 5 0. (7) The connection between the series-expansion and edge-gradient methods can be shown explicitly by expressing the reflectance R~x! in terms of the edge- spread function and then calculating the MTF. One can integrate Eq. ~3! using Eq. ~2!: R~x! 5 ~a 2 b! ( n *na ~n11y2!a L~x9 2 x!dx9 1 b. (8) The edge-spread function e~x! is related to the line- spread function by e~x! 5 * 2` x L~x9!dx9, (9) from which it follows that * a b L~x9 2 x!dx9 5 e~x 2 a! 2 e~x 2 b!. (10) One finds then for R~x!: R~x! 5 ~a 2 b! ( n $e~x 2 na! 2 e@x 2 ~n 1 1y2!a#% 1 b. (11) o calculate the Fourier series coefficients, one needs o consider R~x! for the range 0 # x # a. That the cattering length is small compared with the period mplies, by Eq. ~7!, e~x! 5 0, if x # 2ay2; e~x! 5 1, if x $ ay2, (12) so within the range 0 # x # a, only three terms contribute, and the reflectance is R~x! 5 ~a 2 b!@e~x! 2 e~x 2 ay2! 1 e~x 2 a!# 1 b. (13) Figure 1~a! shows the target function t~x! and Fig. 1~b! shows the reflectance R~x! given in Eq. ~11! for an exponential line-spread function with scattering length ^x& 5 ay50 and a 5 1, b 5 0. To obtain the MTF, one inserts this expression for R~x!, Eq. ~13!, in Eq. ~6!. There are three terms to integrate: using the fact that n is odd and using Eq. ~12!, one can combine the terms to obtain ~with a 5 1 and b 5 0! L˜~vn! 5 U1 1 ivn * 2a a e~x!exp~2ivn x!dxU. Integrating by parts, one obtains L˜~vn! 5 U* 2` ` de~x! dx exp~2ivn x!dxU, 1 November 1998 y Vol. 37, No. 31 y APPLIED OPTICS 7237 where Eq. ~7! is used to extend the limits of integra- tion to 6`. This is the MTF evaluated at vn in the form as analyzed in the edge-gradient method. One c t t the projected image will be interpreted as a large scattering length. The major disadvantage is that The light passes through the target twice, and the relative reflectance is given by w Rn9~x! 5 51 2 R2n11Sx 2 12 aD, 12 a # x , a , (17) limited noise has been added to data generated using an exponen- tial line-spread function with ay^x& 5 50. Sn 1 7 R~x! 5 Rpt~x! * 2` ` L~x9 2 x!t~x9!dx9, (14) where a uniform illuminance is assumed. There may be some reflectance from the target itself; how- ever, this does not affect the results, as shown below, so it can be ignored. The target function t~x! is the transmittance of the bar target, given by Eq. ~1!. Integrating Eq. ~14! using Eq. ~10!, one can write R~x! as R~x! 5 Rp ( n Rn~x 2 na!, (15) ith Rn~x! defined as Figure 1~c! shows R~x! for the same parameters as are used for the projection R~x! @Fig. 1~b!#. The Fourier expansion coefficients of R~x! are not proportional to the MTF as they are in the projected image method @Eq. ~6!#. However, it is possible to transform R~x! into such a form. As shown in Eq. ~16! and Fig. 1~c!, the areas of the paper covered by the bars of the bar target @t~x! 5 0# provide no infor- mation about the paper @R~x! 5 0#. These points can be discarded with no loss of information. The trans- formation discards the points covered by the bars and Rn~x 2 na! 5 5e~x 2 na! 2 eFx 20, 238 APPLIED OPTICS y Vol. 37, No. 31 y 1 November 1998 an see that the series-expansion method is essen- ially the same as the edge-gradient method, except hat one need not evaluate the derivative. 3. Contact Bar Target There are several advantages to placing the bar tar- get in direct, close contact with the paper rather than projecting its image onto the paper. For one, it is necessary to consider only the MTF of the imaging optics rather than the MTF’s for both the imaging and the projection optics. Also, a slight defocus of 0, x , 0 or x $ a the transformed reflectance R9~x! can be given by R9~x! 5 Rp ( n Rn9~x 2 na!. Figure 2 shows the reflectance R~x! and the trans- formed reflectance R9~x!, using the exponential line- spread function, with noise added. Using Eqs. ~16! and ~17! one finds that, within the range 0 # x # a, R9~x! is R9~x! 5 a@e~x! 2 e~x 2 ay2! 1 e~x 2 a!#. This expression is identical to that for the projected reflectance, Eq. ~13!, with b 5 0. The MTF therefore 1 2DaG, na # x , Sn 1 12Da x , na or x $ Sn 1 12Da . (16) Fig. 2. Simulated reflectance data R~x! for the contact method and the transformed reflectance R9~x! with a 5 1, b 5 0. Band- can be obtained using Eq. ~6! with the R9 data. 4. Experimental Process In the following, simulated data from a contact series-expansion MTF measurement are analyzed. One considers an eight-cycle bar target with a total of 800 data points. The points are generated using Eqs. ~15! with an exponential line-spread function with scattering length ^x& 5 ay50. Zero-mean, band- limited noise is added to the signal. Four of the eight cycles of the data R~x! are shown in Fig. 2. multiple reflections between the target and the paper will tend to increase the scattering length. Calcula- tion shows, however, that this effect is relatively mi- nor, contributing at most a few percent to the measured MTF. The target is in direct, close contact with the paper, and light diffusely illuminates the target and paper. inverts and adds 1 to alternate peaks. With Rn9~x! defined as R2n~x!, 0 # x , 1 2 a There are Np data points per period at equal inter- vals D over M periods ~with M even!. The data points are at positions xk given by t n N (k50 S Np D F o transform of the R9~x! data ~filled diamonds! and the t l a The plots in Ref. 8 show a MTF half-width ~i.e., the frequency for which the MTF is equal to 0.5! of ap- proximately 5 cyclesymm for uncoated paper and a .. 2^x&. t G, 2 MTF obtained by a Fourier transform of the line- spread function ~curve!. Because the simulated data are generated using an exponential line-spread function, the MTF in Fig. 3 is a Lorentzian: MTF ; @1 1 ~2pfd!2#21, with f the spa- ial frequency and d proportional to the scattering ength. In their measurement of paper MTF, Inoue et l.8 fit their data to several model MTF’s ~but not a Lorentzian! and found a best fit for the function @1 1 ~2pfd!2#23y2. To within the degree of accuracy in their measurement, this best-fit function is little dif- ferent from a Lorentzian, although it appears that a Lorentzian would actually give a better fit to their data. xk 5 kD, with k 5 0, 1, 2, . . . , NpM 2 1, and D 5 a Np . The reflectance data are R~k! ; R~xk!. With Eq. ~17! used, the R~k! data are transformed into R9~k!: with 0 # n , M9, with M9 5 My2 the number of periods in the transformed data. The total number of points is N 5 NpM9. Figure 2 shows the simulated reflec- ance data R~x! and the transformed data R9~x!. The MTF is proportional to the discrete Fourier transform of the bar-target data, evaluated at the harmonic frequencies. The discrete Fourier trans- form of R9~x! at vn is rn 5 D M9a (k50 N21 R9~k!exp~ivn xk!. Substituting for D, vn, and xk, one can obtain r 5 1 N21 R9~k!exp i2pnk . (19) R9~k! 5 5R~nNp 1 k!,1 2 RFSn 1 12DNp 1 k Fig. 3. ~a! MTF as determined from the simulated data in Fig. 2 ~filled diamonds! and ~b! MTF obtained as the Fourier transform of he exponential line-spread function with ay^x& 5 50 ~curve!. 1 November 1998 y Vol. 37, No. 31 y APPLIED OPTICS 7239 The bandwidth of R~x!, as can be seen in Eq. ~5!, is the bandwidth of L~x!, which is approximately 1y^x&, so that Np 2a . 1 ^x& , and the number of measured points per cycle must be Np . 2a ^x& , with, as indicated above, 7 cyclesymm for coated paper. The simulated MTF in Fig. 3 is given as a function of fa, where f is the spatial frequency in cycles per unit length and a is the bar-target period. The half-width of the simu- lated MTF is approximately fa ; 8. This corre- sponds to a half-width of 5 cyclesymm using a bar target with period a ; 1.6 mm. Because of the discrete sampling of data, the Nyquist critical frequency must be greater than the data bandwidth to avoid aliasing. The critical fre- quency is fc 5 NP 2a . nNp # k , Sn 1 12DNp Sn 1 1DNp # k , ~n 1 1!Np, (18) rom Eq. ~6!, the MTF is proportional to the modulus f rn: L˜~vn! 5 pnurnu~a 2 b!21. (20) Figure 3 shows the MTF as obtained from the Fourier 5. Conclusion A new procedure for measuring the MTF of paper has been presented. The series-expansion bar-target method has advantages over the typical bar-target, edge-gradient, and sine-wave techniques. The ex- perimental procedure for the series-expansion method is simpler than for the sine-wave or typical bar-target technique in that it requires a single tar- get and a single measurement. 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