Determination of point-spread function of paper substrate based on light-scattering simulation D. Modrić,1,* K. Petric Maretić,1 and Aleš Hladnik2 1
Faculty of Graphic Arts, Zagreb, Croatia
2
University of Ljubljana, Faculty of Natural Sciences and Engineering, Department of Textiles, Ljubljana, Slovenia *Corresponding author: [email protected] Received 17 June 2014; revised 27 September 2014; accepted 15 October 2014; posted 16 October 2014 (Doc. ID 214148); published 12 November 2014
The objective of this work was to establish the relationship between the calculated subsurface scatteredphoton distribution and the mathematical quantity known as point-spread function (PSF). Photon distribution of subsurface scattered light was calculated using the Monte Carlo method developed for describing reflectance and opacity of paper and of images printed on paper. The obtained normalized photon distribution made it possible to separate optical and mechanical components of dot gain for the paper-ink system. In the presented method of obtaining the reflectance profile of a screen element, the PSF convolves with a modelled reflectance profile of that element. It was found that the PSF can be better approximated by means of the Lorentzian function when compared to the Gaussian profile that was used in the past research on this topic. © 2014 Optical Society of America OCIS codes: (170.0110) Imaging systems; (170.3010) Image reconstruction techniques; (170.3660) Light propagation in tissues. http://dx.doi.org/10.1364/AO.53.007854
1. Introduction
Quality of any printed product strongly depends on the interaction of the incident light and the substrate, i.e., paper. Modeling and simulation of a substrate’s properties and behavior using computer assisted tools requires the knowledge of the transport of light in the medium, which in turn depends on the substrate thickness, bulk, as well as surface structure. These tools offer the possibility for experimenting with different combinations of paper components without physically producing the real paper and printing on it, which is often expensive and time consuming. Radiation transfer theory describes the interaction of radiation with the medium, which scatters and absorbs light. Paper, the substrate on which we are focusing in this work, has a stochastic three1559-128X/14/337854-09$15.00/0 © 2014 Optical Society of America 7854
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dimensional (3D) network structure that is mainly composed of plant fibers, where cellulose and hemicellulose molecules are mutually linked by hydrogen bonds. Other main ingredients include fillers, sizing agents, various functional additives, water, and air. Properties of papermaking fibers depend on the type of plant being used (softwood, hardwood, nonwood fibers, recovered paper) and the characteristics of the technological process, known as pulping (mechanical, chemical, semi-chemical), which transforms the plant fibers into the raw material for paper production. Individual scatterers within a paper sheet, which include fibers and particles of added chemicals, are so close to each other that multiple scattering becomes important. Particles comprising these components scatter light to various degrees making paper a complex medium for optical measurements and simulations. The fibers are partly compressed and/or kinked. Effects due to other irregularities, such as surface roughness and micro defects, are assumed to be averaged out by the
multiple scattering of the photons; however, they do influence scattering efficiency. In the scattering theory one starts with a single scattering where ensembles are investigated that have a sufficiently small number of particles whose mutual distance is large enough so that each particle suffers no interaction with neighboring particles. In a single scattering it is assumed that the same input electric field acts on each particle in the ensemble where the influence of other particles on the scattered field in the ensemble can be neglected. We treat an ensemble of scatterers in such a manner that the scattering property of the ensemble is simply a sum of the contributions of individual scatterers [1]. Paper structure can be described statistically, by means of material and geometrical distributions, or explicitly as a fiber network. Green et al. [2] proposed a stochastic model for the fiber structure of paper and simulated multiple scattering based on geometrical optics. Similar models were presented by Coppel and Edstrom [3] and Neuman et al. [4]. None of these simulation models took into account the alignment of fibers in paper. They were instead based on a pre-defined statistical fiber network before the light-scattering simulations were performed. Linder and Lofqvist [5] generated the structure parallel with each scattering event. Because they focused on the effects of light scattering from fiber orientation, the fine particles were excluded from their study. Since our model is completely stochastic in nature, we do not model effects of a fiber web and other local structural distortions that appear in paper. Our approach [6] to photon propagation utilizes the Monte Carlo method applied on paper substrate and differs from the above mentioned approaches. It is based on the work of Phral et al. [7] and Wang et al. [8]. The method applies a stochastic model in which the expected value of the variable (or combination of several variables) is equivalent to the value of physical quantity to be determined. Expected value is determined as the mean value of multiple mutually independent samples that represent the specified random variable. To construct a series of independent samples we used randomly generated numbers that follow the distribution of these variables. Our simulation of propagation of photons offers a flexible but rigorous approach to the transport of photons in a medium such as paper. It is governed by the local rules of photon propagation that are, in the simplest case, expressed as probability distributions determining two basic quantities—the step size (mean free path) between two points of the photon-substrate interaction (absorption and/or scattering) and the scattering angle under which a photon leaves the point of interaction. Since the method is statistical, it involves the calculation of a large number of photon paths and is therefore very computationally expensive. The main purpose of our study is to show that the paper point-spread function can be better described
using the Lorentzian function than Gaussian or an exponential function. This suggests that the underlying mechanisms of the transport of light in paper can be somewhat different from those that are generally accepted. Motivation for these simulations was to describe and predict the transport of light through a medium such as paper and to see the effect of the subsurface light scattering on the optical dot gain. Monte Carlo simulations of this type are based on molecular optical properties that are considered to be uniformly spread over small parts of the paper volume (e.g., cellulose fibers, fillers, adhesives, etc.). It should be noted that the simulation does not include the details of radiation distribution of energy within the cellulose fibers. 2. Background
It is well known that any real imaging process tends to blur the image. Perfect reproduction isn’t possible because the information loss is immanent to any real imaging system. In addition, the processes of printing and viewing an image are generally nonlinear. Individual dots of a given raster grayscale image cannot be distinguished from one another if the print is viewed from an appropriate distance. This is the consequence of our visual system’s characteristic to integrate the observed image. It means reflectance is related to a statistical weighting factor F, which is a relative ratio between the surface area covered by screen dots and the surface of paper. It is evident that F can span values between 0 for a non-printed paper and 1 for paper completely covered by ink. Murray and Davies have modelled the reflectance, R, of a half-tone image as RF FRi 1 − FRp ;
(1)
where Ri and Rp denote the reflectances of printed paper and of non-printed paper, respectively [9]. Deviation from this simple linear model that occurs in practice is known as the Yule and Nielsen’s effect [10]. Based on the experience, to describe this effect, the authors have suggested the following modification of Eq. (1): 1 1 n RF FRni 1 − FRnp
n ∈ 1; ∞;
(2)
where n is an empirical constant selected so that the calculated reflectance matches the experimentally obtained one. It is also known as the calibration constant for the printing system. Reflectance predicted by Eq. (2) is generally higher than the measured one. 3. Point-Spread Function of Paper and Dot Gain
The image of a perfect point source can never be as sharp as the point source itself. The point-spread function (PSF) is a term denoting the average distance, l, that light travels in a substrate before it returns to the surface as reflected light. PSFx; y is, 20 November 2014 / Vol. 53, No. 33 / APPLIED OPTICS
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thus, probability density function ℘ℜ x − x0 ; y − y0 that describes a probability that a photon, which enters the substrate, paper, leaves it on the same surface at a distance x; y from the entrance point x0 ; y0 . If the diameter of the input light beam is small, in our case it is zero, with regard to the relative distance of the scattered light in the paper, then the deflection of luminous flux in a radial p direction, r x2 y2 , from the entry point is a direct measure of the PSFx; y. Our idea was to develop an optical dot gain model as a function of several easily obtainable parameters such as scattering or absorption coefficients. Numerous papers explain the effects of photons’ spreading in a medium using PSF [11,12]. One of the earliest works using a microscopic approach was carried out by Ruckdeschel and Hauser [13]. The authors analyzed the Yule–Nielsen’s model by means of a lateral light scattering within the paper using the PSF. They presented a physical analysis of optical dot gain and defined the reflectance at a given location x; y on the substrate, Rx; y Eq. (3). Our approach was to use PSF to calculate the magnitude of optical dot gain by combining it, by means of convolution, with a function that describes the distribution of half-tone dot pattern, Tx; y, which is the ink transmittance pattern on paper at point x; y. Our work is based on the idea presented by Arney et al. [14] to include the ink transmittance term into the overall reflectance expression. In this approach, the incident light is assumed to pass through the ink with no scattering, be partially absorbed, and then scatter in the substrate and exit passing through the ink layer being partially absorbed one more time. Light falling on a screen element partially goes through it and may leave the paper due to subsurface scattering at a location where there is no screen element. Similarly, a considerable percentage of an incoming light beam, due to subsurface scattering, hits a screen dot from below and is considered to be absorbed by the ink itself. However, a small portion of light passes through the dot and contributes to the overall transmittance. All these contributions taken together are responsible for a characteristic ink transmittance pattern. The applied model demonstrates that the expected reflectance of raster images can be estimated using the following equations: Rx; y Rp · Tx; y ZZ · ℘R x − x0 ; y − y0 Tx0 ; y0 dx0 dy0 ;
(3)
Rx; y Rp · Tx; y · Tx; y PSFx; y;
(4)
1 R A 7856
ZZ Rx; ydxdy;
(5)
x;y
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where Tx; y T ink λ ∈ h0; 1i, Tx; y 0 if x; y is covered with ink, or Tx; y 1 if the ink is absent. In the expression Tx; y T ink λ variable λ refers to the light wavelength or the ink color and we assume that transmittance T ink λ is independent of position. Where there is a dot, light is partially absorbed in the ink; otherwise it passes through without any change. This can be described by the transmission matrix T 0 whose elements t0;ij have the value τ0 for the screen elements and 1 elsewhere. Rp x; y is the reflectance of the substrate and A is the surface of the paper sheet on which mean reflectance is measured [15]. Equation (4) is a concise way of expressing the physics of optical dot gain, but the practical implementation of this equation can be very complex. Generally, computer solution of the convolution integral involves the application of a two-dimensional Fourier transform (FT); see Eq. (6). According to the convolution theorem, integration, i.e., summation, in spatial domain is a simple multiplication in Fourier domain. Operator * in the Eqs. (4) and (6) is a convolution operator where convolution is derived by multiplying Fourier transform of functions T and PSF, and then performing inverse Fourier transform (iFT) of the obtained product as shown in Eq. (6): Tx; y PSFx; y iFTfFTTx; y · FTPSFx; yg: (6) As mentioned above, in the literature one encounters several approaches describing lateral light scattering in the paper using PSF. Most of these functions are determined empirically [13] or by assuming specific types of functions [16,17]. However, some authors have focused on numerical simulation [18], measuring the microscopic reflectance [19] or radiation diffusion [20]. Two formulations have been proposed: Gaussian line-spread function LSFx [21], and exponential PSF [22,23]. LSFx is the line-spread function, and it provides a direct measure of the lateral distance light scatters from the illuminated edge. The relationship between the PSF and LSF is as follows: Z LSFx
∞ −∞
PSFx; ydy:
(7)
Since the PSF can also be recognized as a probability density, some authors tried to describe the lateral scattering within the paper using stochastic models. In recent years significant effort has been put into modeling the interaction of light with substrate and ink to be able to achieve better tone-value predictions of a paper substrate printed with screen elements. An important effect that occurs in addition to mechanical dot gain is optical dot gain, which has its origin in the scattering and diffusion of photons within a paper substrate [11,24].
The model applied in our considerations yields the PSF in such a way that it is in fact experimentally determined, as discussed below. Theoretically, PSF depends only on the characteristics of the substrate and is independent of the applied ink. This is not completely true because the diffusion of photons in a substrate is a function of several processes: multiple internal scattering on paper components and/or absorption on paper components and ink, internal reflection on the border between the paper and air (coating), etc. Consequently, we have to conclude that the applied ink on the surface of paper also affects the internal reflection and, in turn, the PSF itself. From the above discussion it can be summarized that the PSF is a transfer characteristic of a system for a specific input information and the corresponding output. 4. Monte Carlo Model of Subsurface Light Transport
In this section we briefly present the Monte Carlo path tracing method that simulates the events of multiple scattering. Monte Carlo modeling of light transport in a scattering medium is a standard simulation technique, and voluminous literature exists on the development of various types of Monte Carlo models. We will introduce a model that simulates the transport of light using a volumetric mapping of photons. We would like to describe a typical trajectory of a single photon packet (Fig. 1). Each step between photon positions (dots) is variable and equals lnζ∕μa μs where ζ is a random number and μa and μs are the absorption and scattering coefficients, respectively; see Table 1. The scattering phase function represents the angular distribution of light intensity scattered by a particle (in our case paper component) at a given wavelength and is given with value g, which is called the anisotropy of scattering. This is the averaged cosine scattering angle according to the used Henyey–Greenstein phase function that approximates Mie-scattering on the particles
Fig. 1. Calculated trajectories of individual photon packets in a medium based on our model. Input photon packets at the upper surface can be approximated as being one-dimensional and perpendicular to the paper surface.
having dimensions comparable with the wavelength of light that illuminates. The weight of the photon is decreased from an initial value of 1 as it moves through the medium, and equals an after n steps, where a is the albedo a μs ∕μa μs . Assigning statistical weight to a photon enables us to treat photons as packets [26] of a hundred photons, for example. The idea of implementing the concept of photonic packets is governed by the law of conservation of energy. Rather than tracing one photon at a time as nature does, we can consider a source emitting packets of many photons. When the photon strikes the surface, a fraction of the photon weight escapes as reflectance and the remaining weight is internally reflected and continues to propagate. Eventually, the photon weight drops below a threshold level and the simulation for that photon is terminated. In this example, termination occurred when the last significant fraction of remaining photon weight escaped at the surface at the position indicated by the circle (°) (see Fig. 3). Since the paper medium is composed of several components, it is necessary to include simulation scattering and/or absorption on each component. At the end of each step it is randomly determined at which component interactions occur, taking into account the percentage of each component. This was done so that the percentage of individual components was equated with the weight factor that takes care of how many times the scattering or absorption occurs on the respective component. Many photon trajectories (104 to 106 ) were typically calculated to yield a statistical description of photon distribution in the medium. As a simple example of our simulations, we calculated typical trajectories of individual photons (Fig. 1). The history of individual photons is simulated using random numbers, accurate representation of the probability of photon interaction with the medium and the exact model of problem 3D geometry. Our Monte Carlo simulation deals with the transport of an infinitely narrow beam of photons incident upon the multilayer substrate. We simulate illumination incident normally on a point, and to minimize noise 106 wave packets are simulated in each run. We chose the wavelengths of 400, 500, 600, and 700 nm. Each layer, which is considered as infinitely long, is described with the following parameters: mean thickness, refractive index, absorption coefficient μa , scattering coefficient μs , and anisotropy factor g (Henyey–Greenstein scattering phase function). Our model has either one or three layers of a substrate, i.e., without or with a coating on both sides of the paper. Each layer has accompanying absorption and scattering coefficients. In comparison with the dimensions of the spatial distribution of photons, the investigated substrate is considered infinitely large. Monte Carlo simulations of this type are based on the macroscopic optical properties of paper that are assumed to uniformly extend across the paper (e.g., cellulose fibers, fillers, adhesives, etc.). Our simulation does not take into account local distribution 20 November 2014 / Vol. 53, No. 33 / APPLIED OPTICS
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Table 1.
Numerical Data for our “Paper” Consisting of Five Components: Filler, Mechanical Pulp, Chemical Pulp, Coating and Air [25]
Component Filler Mechanical pulp Chemical pulp Coating Air
Weight Ratio, % Asymmetry Factor (g) 14 30 27 17 12
0.7 0.5 0.75 0.02 0
Scattering Coefficient (μs ), m2 ∕kg
Absorption Coefficient (μa ), m2 ∕kg
Wavelength (λ), nm
Wavelength (λ), nm
400
500
600
700
400
500
600
700
25 25 25 30 0
25 75 108 30 0
25 70 115 30 0
25 70 110 30 0
0.5 29 29 0.02 0
0.5 6 6 0.02 0
0.5 1 1 0.02 0
0.5 0.5 0.5 0.02 0
of radiation energy within, for example, cellulose fibers. History of the photons’ motion is recorded until the photon either leaves the medium or is absorbed during an interaction with the medium. Absorption at the individual component is determined by comparing a randomly generated number with albedo. The following results were calculated based on some typical values of paper component properties that were obtained from the literature and used already in our previous work [27]. We chose a trade-off paper composition that corresponds to the available data while still being simple enough not to consume excessive computer time. Our paper consisted of four components: pulp (chemical and mechanical), filler (CaCO3 ), sizing agent (AKD), and air (although the number of components was not limited by the software) in the amounts represented as weight ratios shown in Table 1. Of course, the composition of the actual papers was measured. We measured the reflectance on three types of printing paper commonly used in digital printing: Arcoprint 120 g∕m2 , Navigator 80 g∕m2 and Splendorgel 115 g∕m2 . We then compared it with the modelled one, which is a commercial secret, but the composition used in our modelled paper gave fairly good results, as demonstrated by the modelled reflectance profiles of paper prints [25]. The assumption was that the kaolin coating layer (n 1.55) is homogeneous and that its thickness is uniform over the entire surface of paper. We used a coating thickness of 5% (i.e., 0.01431 mm) of paper thickness which was d 0.2862 mm. This corresponds to the thickness of a commercial paper Splendorgel 115 g∕m2 on which the line screen elements were printed. Our profile was generated as a result of the contribution of all wavelengths to simulate white light in a way that we adjusted their contributions to the standard source D65. In Fig. 1 one can see that there is a one-dimensional (1D) beam of photon packets entering the paper at its top surface. Under such conditions no initial specular reflection on the paper surface takes place. Figure 1 shows that the photons can undergo multiple scattering events before they experience either transmission, absorption or reflection, or their return to the incoming surface. Multiple scattering, according to the simulation, causes attenuation 7858
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(weight reduction) of photon packets. This leads to an interesting finding that, given the particular paper composition and its values of the scattering and absorption coefficients, light absorption into the paper is far more significant than expected. Absorption values can, according to the simulation, reach as high as 30% of the initial value. Another important fact that follows from our experiments is that there are virtually no multiple reflections [28] between the paper-air boundary surfaces. Their number is negligible, so any description that uses these reflections as a major argument in an attempt to model the Yule–Nielsen’s effect, does not reflect real situation and fails to generate acceptable results. It is important to emphasize that our simulation does not take into account the wave nature of light, and that it ignores its phase and polarization. As mentioned in the Introduction, the motivation for this simulation was to predict and describe the transport of light through a medium such as paper and to determine the influence of the light propagation on the optical dot gain. Photons are reflected many times on their way through the medium and, consequently, phase and polarization very quickly become random and therefore play only a minor role in energy transport. 5. Determination of PSF
The basic ideas of modeling the reflectance of screen elements were presented earlier in the text. We modelled the profile of a printed line using Eq. (3) or Eq. (4), in which the influence of optical properties of paper and ink components is taken into account. Starting from the very definition of PSF we calculated its shape as a function of paper parameters. By modeling line profiles we wanted to examine the individual contribution of each parameter such as scattering and absorption coefficients of paper components, weight percentage of each component, asymmetry parameters, type and thickness of the coating layer (where applicable), the refractive index of the coating layer and its thickness, type and form of the paper surface, type of ink and its physical properties, and many other parameters that affect, to a greater or lesser degree, the collective effect known as optical dot gain. Previous approaches lacked the complexity necessary for the realistic description of the system and their authors used approximations that did not
Table 2. Comparison of Calculated (w C ) and Fitted (w L ) Parameter w for the Uncoated and Coated Paper
Uncoated Coated
wC (mm)
wL (mm)
Δw (mm)
0.19463 0.14579
0.19279 0.13966
0.00156 0.00613
always have a sound physical basis. Then at the end of the 1970s, Yule and Nielsen suggested that the LSF should be described by means of Gaussian distribution [21]. Our model, based on stochastic approach, shows that for a given set of parameters, PSF can be represented by Lorentzian distribution which we put, for simplicity, in the origin of the coordinate system as shown in Fig. 2. The obtained function is independent of its position in the incident plane, so it can be considered as shift invariant. Function PSFx; y represents the flux density as a function of rectangular coordinates on the image plane and describes how a sharp point is spread out due to the physics of the imaging process. The origin is the location of the ideal image spot. If we plot the intensity of the spot, we obtain a 1D PSF profile. The curve, calculated data in Fig. 2, was generated by calculating the profile radial symmetry and averaging over the annulus using a predefined radial step. The argument for the cylindrical symmetry approximation (rotation invariant) of the obtained profile can be deduced from Fig. 3. This image was obtained using more than 20.000 calculated points, i.e., photons (only a fraction of them are shown), which is a relatively small number, but illustrates cylindrical symmetry of the calculated subsurface scattered light beam fairly well. Each circle in the figure represents the exit position of a photon packet. Distribution of the scattered light depicted in Fig. 2, when normalized using the surface area under the curve, represents probability distribution. By definition, to get the PSF we have to normalize
Fig. 2. Calculated radial distribution of the subsurface scattered input light beam of photon packages that penetrate the uncoated paper surface at the point (0, 0) and Lorentzian and Gaussian fitting functions.
Fig. 3. Planar distribution of outgoing photon packets illustrating approximation of cylindrical symmetry for the calculated subsurface scattered light beam.
the calculated area of radial distribution to unity. The analytical form of PSF is found by fitting the obtained distribution by Lorentzian function. Consequently, the resulting PSF can be considered as if obtained experimentally: 1 1 w∕2 PSFr ; R π r − rc 2 w∕22
(8)
where rc is the location of the center of distribution and w is a parameter specifying the width of the profile (see below). Constant R is defined by the definition of PSF as a probability, so that 1 w∕2 R dr: 2 2 −∞ π r − rc w∕2 Z
∞
(9)
Figure 2 shows that there is a very good agreement (Pearson linear correlation coefficient R > 99%) between the calculated values, the PSF, and the Lorentzian function. It can also be seen that the fitting using the Gaussian model is poorer. With the earlier defined convolution, Eq. (3) was the basis for modeling the profile lines in Fig. 4. Shape of Tx; y, ink transmittance of the screen element—in our case line—on paper, was, in our approximation, modelled as being rectangular with maximum and minimum reflectance values corresponding to averaged measurements and by neglecting the effects of edges occurring due to mechanical dot gain. To print a printed form with a linear sample—a line—we used a previously calibrated digital electro-photographic machine, Indigo Turbo Stream 1000. The printed line width was measured by image analyzer PIAS. Figure 4 shows a comparison between the measured and the calculated profiles. We compared the Lorentzian profile obtained by the Monte Carlo method with the usual Gaussian profile of the PSF. For comparison, a model profile with which we 20 November 2014 / Vol. 53, No. 33 / APPLIED OPTICS
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Fig. 4. Comparison of measured and calculated (Lorentz, Gauss) reflectance profiles with the model profile for the line screen element. The nominal line width is d 1 mm, while the measured width is dm 1.06 mm. The dash-bordered regions are shown separately in Figs. 6(a) and 6(b).
approximated product Rp · Tx; y (transmittance sample of ink on paper multiplied by reflectance of paper itself) is also shown. The resolution of printed screen elements can be defined as the width within which the overall PSF drops to one half of the maximum value, which is called full width at half-maximum (FWHM). If the object consists of two ideal points, just a distance FWHM apart, there is a fair chance that they will be separated in the image. Both Gaussian and Lorentzian functions contain only a single parameter, w, specifying the FWHM of the profile, which must be adjusted in order to fit the data. Additional evidence that the Lorentzian function gives a better fit to the measured profile than the Gaussian function is provided when focusing on the two particular regions of Fig. 4: the profiles and the curves between 75% and 87% reflectance Fig. 5(a) and around 5% reflectance Fig. 5(b). Figure 5(b) shows that the Lorentzian PSF better describes the nature of optical dot gain compared to the Gaussian function. In the region characterized by the reflectance close to 5% it is difficult to determine the exact position of the edge due to the fact that, as the analysis indicates, the line is somewhat narrower than the desired one, which is probably the effect of the applied printing technique. Further comparison of the generated PSF between the coated and the uncoated papers reveals that there are differences in the parameter w values for the two paper types due to the presence of coating. This suggests that the parameter w somehow depends on the composition of the paper surface (refractive index). We conducted the same analysis by varying paper thickness and ratio of individual components of paper, but the results of these experiments did not confirm the influence of these two quantities on the parameter w. Since our profile depends only on a single parameter, w, we tried to find a potential explanation for our observation that the modelled half-width profile 7860
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Fig. 5. Detail of the calculated (Lorentzian and Gaussian fit). (a) Measured profiles in the far wing region and (b) in the region where the reflectance is close to 5%. Also shown is the approximation model of Rp Tx; y profile used in Eqs. (3) and (4).
w is different for coated and uncoated paper, as shown in Fig. 6(a). Figure 6(a) shows a comparison of modelled distribution profiles of scattered radiation for a coated and uncoated paper. It is apparent that the intensity distribution for the coated paper is characterized by a noticeably narrower parameter w (see Table 2). This is the result of multiple internal reflections at the paper-coating border. Consequently, transmission is also reduced and the number of photons returning to the incoming surface is lower. Of course, this is not the case with the uncoated paper where the scattering is substantial. In Maretić Petric et al. [29] we discussed in more detail which of the two observed functions, Lorentz or Gaussian, gives a better description of the LSF and why this is the case. Given that the obtained correlation coefficients were extremely high and very similar, a rating quality model was made using Akaike information criterion. The Akaike information criterion is a measure of the relative quality of a statistical model for a given set of data and provides a means for model selection. A more elaborate report
Fig. 6. Comparison of modeled radial distributions of subsurface scattered radiation for coated and uncoated paper. (a) Calculated distributions using Monte Carlo approach and (b) values normalized to the same peak value to enable comparison of parameters w.
on these experiments will be published in the forthcoming article. 6. Conclusions
Our method based on the Monte Carlo approach describes the local rules of photon subsurface propagation, which are expressed, in the simplest case as the probability distributions that describe the size of the photon shift between the position of two events (interactions), and the direction angles of the photon trajectory (in the case of scattering). Involved interactions are the scattering and absorption of the photon package on each paper component. Descriptions using photonic packages, instead of single photons, were adopted due to the law of energy conservation. The idea of this study was to demonstrate that the calculated subsurface distribution of scattered photons corresponds to PSF, since the PSF gives the probability for an entering photon to emerge out at a remote point on the same surface. We recognize our modelled distribution of these photons (photon packets), after the standardization area under the curve on to unit, as PSF. We modelled
the distribution fit with Lorentzian and Gaussian function in order to obtain an analytical form of PSF. Our attempt to describe reflectance of the printed paper surface is based on Eq. (3) with which we modelled PSFx; y using the Monte Carlo method. PSF and, consequently, MTF describe lateral light scattering in paper and is used to describe color and tone reproduction in printed half-tone images. In the presented method of obtaining a reflectance profile of any screen element, the PSF convolves with a modelled reflectance profile of the bitmap element with some simple geometric assumptions. In our case, the modelled profiles gave satisfactory results of the reflection in the far wing of the line profile edge. This is the result of the fact that our model describes only the optical dot gain, and mechanical dot gain in the model line profile was included simply as a broader original line. Further evidence that the Lorentzian function efficiently describes optical dot gain is provided if we let the line width go to zero. In this case the mechanical dot gain disappears (no screen element), and what remains is the optical dot gain, which we describe by Lorentzian function. We demonstrated that our method adequately models optical component of dot gain, indicating that we were able to successfully separate optical and mechanical components of dot gain. In addition, the method of convolving PSF with model profile lines showed better approximation of the measured PSF profile with Lorentzian (i.e., Cauchy distribution) than with the Gaussian function. Our work shows that the modelled distribution is better described with a Lorentzian (Cauchy) function. Comparison with measured data also confirms the fact that the Lorentz function gives a better description of the PSF. We can conclude that the method appropriately describes the optical dot gain phenomenon, and that we can, from the measured reflectance line profile using deconvolution, obtain valuable information about the nature and morphology of mechanical dot gain. Separation of mechanical and optical dot gain will enable us to optimize the printing process, but also aid in producing paper substrate with the specified optical and mechanical properties. On the other hand, the knowledge of the origin and behavior of the total dot gain can be used to eliminate blurred edges using an appropriate image analysis procedure such as image sharpening or denoising. A possible physical reason for obtaining a Lorentz profile of PSF lies in Huygens’ principle, which says that one can determine the intensity by assuming that the light is re-emitted from any line between the source and the target, while the Cauchy–Lorentz distribution describes the normalized intensity of light on a line from a point source. The final distribution of photons leaving the substrate after n steps, obtained as result of the contributions in all points of the path, has a Lorentzian shape because the sum of n independent Cauchy distributions is a Cauchy distribution scaled by a factor of n. We believe that 20 November 2014 / Vol. 53, No. 33 / APPLIED OPTICS
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the interpretation of the physical background reflects the fact that the paper components, absorption and scattering on them and perhaps some other parameters, are not entirely mutually independent. The method provides a satisfactory description of the line profiles and also gives guidelines for future research on the impact of various factors, not only on the design of the paper itself, but also on other important aspects of printing technology such as paper-ink interactions, quality control, accurate recalibration of printers, etc. This work was supported by the Croatian Ministry of Science, Education and Sports under Grant No. 128-1281955-1960. References 1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983). 2. K. Green, L. Lamberg, and K. Lumme, “Stochastic modeling of paper structure and Monte Carlo simulation of light scattering,” Appl. Opt. 39, 4669–4683 (2000). 3. L. G. Coppel, P. Edström, and M. Lindquister, “Open source Monte Carlo simulation platform for particle level simulation of light scattering from generated paper structures,” in Proceedings of Papermaking Research Symposium, E. Madetoja, H. Niskanen, and J. Hämäläinen, eds. (Kuopio University, 2009). 4. M. Neuman, L. G. Coppel, and P. Edström, “Point spreading in turbid media with anisotropic single scattering,” Opt. Express 19, 1915–1920 (2011). 5. T. Linder and T. Löfqvist, “Anisotropic light propagation in paper,” Nord. Pulp Pap. Res. J. 27, 500–506 (2012). 6. D. Modrić, R. Beuc, and S. Bolanča, “Monte Carlo modeling of light scattering in paper,” J. Imaging Sci. Technol. 53, 020201 (2009). 7. S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE, 5, 102–111 (1989). 8. L. Wang, S. L. Jacques, and L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995). 9. A. Murray, “Monochrome reproduction in photoengraving,” J. Franklin Inst. 221, 721–744 (1936). 10. J. A. C. Yule and W. J. Neilsen, “The penetration of light into paper and its effect on half-tone reproduction,” in TAGA Proceedings (Technical Association of the Graphic Arts, 1951), pp. 65–76.
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11. D. R. Wyble and R. S. Berns, “A critical review of spectral models applied to binary color printing,” Color Res. Appl. 25, 4–19 (2000). 12. J. S. Arney, C. D. Arney, and M. Katsube, “An MTF analysis of papers,” J. Imaging Sci. Technol. 40, 19–25 (1996). 13. F. R. Ruckdeschel and O. G. Hauser, “Yule-Nielsen effect in printing: a physical analysis,” Appl. Opt. 17, 21, 3376–3383 (1978). 14. J. S. Arney, J. Chauvin, J. Nauman, and P. G. Anderson, “Kubelka-Munk theory and the MTF of paper,” J. Imaging Sci. Technol. 47, 339–345 (2003). 15. P. Emmel and R. D. Hersch, “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000). 16. G. Fischer, J. Rodriguez-Giles, and K. R. Scheuter, “Ein physikalisches Modell für die Beschreibung von Lichtstreuprozessen,” Die Farbe 30, 199–220 (1982). 17. G. L. Rogers, “Optical dot gain: lateral scattering probabilities,” in Recent Progress in Digital Halftoning II, R. Eschbach, ed. (Society for Imaging Science, 1999), Chap. V, pp. 495–500. 18. L. Yang, S. Gooran, and B. Kruse, “Simulation of optical dot gain in multichromatic tone reproduction,” J. Imaging Sci. Technol. 45, 198–204 (2001). 19. I. Shinichi, T. Norimichi, and M. Yoichi, “Analyzing CTF of print by MTF of paper,” J. Imaging Sci. Technol. 42, 572–576 (1998). 20. G. L. Rogers, “Optical dot gain in a halftone print,” J. Imaging Sci. Technol. 41, 643–656 (1997). 21. J. A. C. Yule, D. J. Howe, and J. H. Altman, “The effect of the spread function of paper on halftone reproduction,” TAPPI J. 50, 337–344 (1967). 22. P. G. Engeldrum and B. Pridham, “Application of turbid medium theory to paper spread function measurements,” TAGA Proc. 47, 339–352 (1995). 23. S. Gustavson, “Color gamut of halftone reproduction,” J. Imaging Sci. Technol. 41, 283–290 (1997). 24. G. L. Rogers, “Optical dot gain: lateral scattering probabilities,” J. Imaging Sci. Technol. 43, 341–345 (1998). 25. K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind (Cambridge, 1997). 26. H. Kahn and T. E. Harris, Estimation of particle transmission by random sampling, Vol. 12 of National Bureau of Standards Applied Mathematics Series (1951), pp 27–30. 27. D. Modrić, K. Petric Maretić, and A. Hladnik, “Modeling spatial reflection from an uncoated printing paper using Monte Carlo simulation,” Nord. Pulp Pap. Res. J. 27, 968–975 (2012). 28. F. R. Clapper and J. A. C. Yule, “The effect of multiple internal reflections on the densities of halftone prints on paper,” J. Opt. Soc. Am. 43, 600–603 (1953). 29. K. Maretić Petric, M. Milković, and D. Modrić, “Akaike information criterion in the edge analysis of the screen element,” Technical Gazette 20, 441–447 (2013).
Faculty of Graphic Arts, Zagreb, Croatia
2
University of Ljubljana, Faculty of Natural Sciences and Engineering, Department of Textiles, Ljubljana, Slovenia *Corresponding author: [email protected] Received 17 June 2014; revised 27 September 2014; accepted 15 October 2014; posted 16 October 2014 (Doc. ID 214148); published 12 November 2014
The objective of this work was to establish the relationship between the calculated subsurface scatteredphoton distribution and the mathematical quantity known as point-spread function (PSF). Photon distribution of subsurface scattered light was calculated using the Monte Carlo method developed for describing reflectance and opacity of paper and of images printed on paper. The obtained normalized photon distribution made it possible to separate optical and mechanical components of dot gain for the paper-ink system. In the presented method of obtaining the reflectance profile of a screen element, the PSF convolves with a modelled reflectance profile of that element. It was found that the PSF can be better approximated by means of the Lorentzian function when compared to the Gaussian profile that was used in the past research on this topic. © 2014 Optical Society of America OCIS codes: (170.0110) Imaging systems; (170.3010) Image reconstruction techniques; (170.3660) Light propagation in tissues. http://dx.doi.org/10.1364/AO.53.007854
1. Introduction
Quality of any printed product strongly depends on the interaction of the incident light and the substrate, i.e., paper. Modeling and simulation of a substrate’s properties and behavior using computer assisted tools requires the knowledge of the transport of light in the medium, which in turn depends on the substrate thickness, bulk, as well as surface structure. These tools offer the possibility for experimenting with different combinations of paper components without physically producing the real paper and printing on it, which is often expensive and time consuming. Radiation transfer theory describes the interaction of radiation with the medium, which scatters and absorbs light. Paper, the substrate on which we are focusing in this work, has a stochastic three1559-128X/14/337854-09$15.00/0 © 2014 Optical Society of America 7854
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dimensional (3D) network structure that is mainly composed of plant fibers, where cellulose and hemicellulose molecules are mutually linked by hydrogen bonds. Other main ingredients include fillers, sizing agents, various functional additives, water, and air. Properties of papermaking fibers depend on the type of plant being used (softwood, hardwood, nonwood fibers, recovered paper) and the characteristics of the technological process, known as pulping (mechanical, chemical, semi-chemical), which transforms the plant fibers into the raw material for paper production. Individual scatterers within a paper sheet, which include fibers and particles of added chemicals, are so close to each other that multiple scattering becomes important. Particles comprising these components scatter light to various degrees making paper a complex medium for optical measurements and simulations. The fibers are partly compressed and/or kinked. Effects due to other irregularities, such as surface roughness and micro defects, are assumed to be averaged out by the
multiple scattering of the photons; however, they do influence scattering efficiency. In the scattering theory one starts with a single scattering where ensembles are investigated that have a sufficiently small number of particles whose mutual distance is large enough so that each particle suffers no interaction with neighboring particles. In a single scattering it is assumed that the same input electric field acts on each particle in the ensemble where the influence of other particles on the scattered field in the ensemble can be neglected. We treat an ensemble of scatterers in such a manner that the scattering property of the ensemble is simply a sum of the contributions of individual scatterers [1]. Paper structure can be described statistically, by means of material and geometrical distributions, or explicitly as a fiber network. Green et al. [2] proposed a stochastic model for the fiber structure of paper and simulated multiple scattering based on geometrical optics. Similar models were presented by Coppel and Edstrom [3] and Neuman et al. [4]. None of these simulation models took into account the alignment of fibers in paper. They were instead based on a pre-defined statistical fiber network before the light-scattering simulations were performed. Linder and Lofqvist [5] generated the structure parallel with each scattering event. Because they focused on the effects of light scattering from fiber orientation, the fine particles were excluded from their study. Since our model is completely stochastic in nature, we do not model effects of a fiber web and other local structural distortions that appear in paper. Our approach [6] to photon propagation utilizes the Monte Carlo method applied on paper substrate and differs from the above mentioned approaches. It is based on the work of Phral et al. [7] and Wang et al. [8]. The method applies a stochastic model in which the expected value of the variable (or combination of several variables) is equivalent to the value of physical quantity to be determined. Expected value is determined as the mean value of multiple mutually independent samples that represent the specified random variable. To construct a series of independent samples we used randomly generated numbers that follow the distribution of these variables. Our simulation of propagation of photons offers a flexible but rigorous approach to the transport of photons in a medium such as paper. It is governed by the local rules of photon propagation that are, in the simplest case, expressed as probability distributions determining two basic quantities—the step size (mean free path) between two points of the photon-substrate interaction (absorption and/or scattering) and the scattering angle under which a photon leaves the point of interaction. Since the method is statistical, it involves the calculation of a large number of photon paths and is therefore very computationally expensive. The main purpose of our study is to show that the paper point-spread function can be better described
using the Lorentzian function than Gaussian or an exponential function. This suggests that the underlying mechanisms of the transport of light in paper can be somewhat different from those that are generally accepted. Motivation for these simulations was to describe and predict the transport of light through a medium such as paper and to see the effect of the subsurface light scattering on the optical dot gain. Monte Carlo simulations of this type are based on molecular optical properties that are considered to be uniformly spread over small parts of the paper volume (e.g., cellulose fibers, fillers, adhesives, etc.). It should be noted that the simulation does not include the details of radiation distribution of energy within the cellulose fibers. 2. Background
It is well known that any real imaging process tends to blur the image. Perfect reproduction isn’t possible because the information loss is immanent to any real imaging system. In addition, the processes of printing and viewing an image are generally nonlinear. Individual dots of a given raster grayscale image cannot be distinguished from one another if the print is viewed from an appropriate distance. This is the consequence of our visual system’s characteristic to integrate the observed image. It means reflectance is related to a statistical weighting factor F, which is a relative ratio between the surface area covered by screen dots and the surface of paper. It is evident that F can span values between 0 for a non-printed paper and 1 for paper completely covered by ink. Murray and Davies have modelled the reflectance, R, of a half-tone image as RF FRi 1 − FRp ;
(1)
where Ri and Rp denote the reflectances of printed paper and of non-printed paper, respectively [9]. Deviation from this simple linear model that occurs in practice is known as the Yule and Nielsen’s effect [10]. Based on the experience, to describe this effect, the authors have suggested the following modification of Eq. (1): 1 1 n RF FRni 1 − FRnp
n ∈ 1; ∞;
(2)
where n is an empirical constant selected so that the calculated reflectance matches the experimentally obtained one. It is also known as the calibration constant for the printing system. Reflectance predicted by Eq. (2) is generally higher than the measured one. 3. Point-Spread Function of Paper and Dot Gain
The image of a perfect point source can never be as sharp as the point source itself. The point-spread function (PSF) is a term denoting the average distance, l, that light travels in a substrate before it returns to the surface as reflected light. PSFx; y is, 20 November 2014 / Vol. 53, No. 33 / APPLIED OPTICS
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thus, probability density function ℘ℜ x − x0 ; y − y0 that describes a probability that a photon, which enters the substrate, paper, leaves it on the same surface at a distance x; y from the entrance point x0 ; y0 . If the diameter of the input light beam is small, in our case it is zero, with regard to the relative distance of the scattered light in the paper, then the deflection of luminous flux in a radial p direction, r x2 y2 , from the entry point is a direct measure of the PSFx; y. Our idea was to develop an optical dot gain model as a function of several easily obtainable parameters such as scattering or absorption coefficients. Numerous papers explain the effects of photons’ spreading in a medium using PSF [11,12]. One of the earliest works using a microscopic approach was carried out by Ruckdeschel and Hauser [13]. The authors analyzed the Yule–Nielsen’s model by means of a lateral light scattering within the paper using the PSF. They presented a physical analysis of optical dot gain and defined the reflectance at a given location x; y on the substrate, Rx; y Eq. (3). Our approach was to use PSF to calculate the magnitude of optical dot gain by combining it, by means of convolution, with a function that describes the distribution of half-tone dot pattern, Tx; y, which is the ink transmittance pattern on paper at point x; y. Our work is based on the idea presented by Arney et al. [14] to include the ink transmittance term into the overall reflectance expression. In this approach, the incident light is assumed to pass through the ink with no scattering, be partially absorbed, and then scatter in the substrate and exit passing through the ink layer being partially absorbed one more time. Light falling on a screen element partially goes through it and may leave the paper due to subsurface scattering at a location where there is no screen element. Similarly, a considerable percentage of an incoming light beam, due to subsurface scattering, hits a screen dot from below and is considered to be absorbed by the ink itself. However, a small portion of light passes through the dot and contributes to the overall transmittance. All these contributions taken together are responsible for a characteristic ink transmittance pattern. The applied model demonstrates that the expected reflectance of raster images can be estimated using the following equations: Rx; y Rp · Tx; y ZZ · ℘R x − x0 ; y − y0 Tx0 ; y0 dx0 dy0 ;
(3)
Rx; y Rp · Tx; y · Tx; y PSFx; y;
(4)
1 R A 7856
ZZ Rx; ydxdy;
(5)
x;y
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where Tx; y T ink λ ∈ h0; 1i, Tx; y 0 if x; y is covered with ink, or Tx; y 1 if the ink is absent. In the expression Tx; y T ink λ variable λ refers to the light wavelength or the ink color and we assume that transmittance T ink λ is independent of position. Where there is a dot, light is partially absorbed in the ink; otherwise it passes through without any change. This can be described by the transmission matrix T 0 whose elements t0;ij have the value τ0 for the screen elements and 1 elsewhere. Rp x; y is the reflectance of the substrate and A is the surface of the paper sheet on which mean reflectance is measured [15]. Equation (4) is a concise way of expressing the physics of optical dot gain, but the practical implementation of this equation can be very complex. Generally, computer solution of the convolution integral involves the application of a two-dimensional Fourier transform (FT); see Eq. (6). According to the convolution theorem, integration, i.e., summation, in spatial domain is a simple multiplication in Fourier domain. Operator * in the Eqs. (4) and (6) is a convolution operator where convolution is derived by multiplying Fourier transform of functions T and PSF, and then performing inverse Fourier transform (iFT) of the obtained product as shown in Eq. (6): Tx; y PSFx; y iFTfFTTx; y · FTPSFx; yg: (6) As mentioned above, in the literature one encounters several approaches describing lateral light scattering in the paper using PSF. Most of these functions are determined empirically [13] or by assuming specific types of functions [16,17]. However, some authors have focused on numerical simulation [18], measuring the microscopic reflectance [19] or radiation diffusion [20]. Two formulations have been proposed: Gaussian line-spread function LSFx [21], and exponential PSF [22,23]. LSFx is the line-spread function, and it provides a direct measure of the lateral distance light scatters from the illuminated edge. The relationship between the PSF and LSF is as follows: Z LSFx
∞ −∞
PSFx; ydy:
(7)
Since the PSF can also be recognized as a probability density, some authors tried to describe the lateral scattering within the paper using stochastic models. In recent years significant effort has been put into modeling the interaction of light with substrate and ink to be able to achieve better tone-value predictions of a paper substrate printed with screen elements. An important effect that occurs in addition to mechanical dot gain is optical dot gain, which has its origin in the scattering and diffusion of photons within a paper substrate [11,24].
The model applied in our considerations yields the PSF in such a way that it is in fact experimentally determined, as discussed below. Theoretically, PSF depends only on the characteristics of the substrate and is independent of the applied ink. This is not completely true because the diffusion of photons in a substrate is a function of several processes: multiple internal scattering on paper components and/or absorption on paper components and ink, internal reflection on the border between the paper and air (coating), etc. Consequently, we have to conclude that the applied ink on the surface of paper also affects the internal reflection and, in turn, the PSF itself. From the above discussion it can be summarized that the PSF is a transfer characteristic of a system for a specific input information and the corresponding output. 4. Monte Carlo Model of Subsurface Light Transport
In this section we briefly present the Monte Carlo path tracing method that simulates the events of multiple scattering. Monte Carlo modeling of light transport in a scattering medium is a standard simulation technique, and voluminous literature exists on the development of various types of Monte Carlo models. We will introduce a model that simulates the transport of light using a volumetric mapping of photons. We would like to describe a typical trajectory of a single photon packet (Fig. 1). Each step between photon positions (dots) is variable and equals lnζ∕μa μs where ζ is a random number and μa and μs are the absorption and scattering coefficients, respectively; see Table 1. The scattering phase function represents the angular distribution of light intensity scattered by a particle (in our case paper component) at a given wavelength and is given with value g, which is called the anisotropy of scattering. This is the averaged cosine scattering angle according to the used Henyey–Greenstein phase function that approximates Mie-scattering on the particles
Fig. 1. Calculated trajectories of individual photon packets in a medium based on our model. Input photon packets at the upper surface can be approximated as being one-dimensional and perpendicular to the paper surface.
having dimensions comparable with the wavelength of light that illuminates. The weight of the photon is decreased from an initial value of 1 as it moves through the medium, and equals an after n steps, where a is the albedo a μs ∕μa μs . Assigning statistical weight to a photon enables us to treat photons as packets [26] of a hundred photons, for example. The idea of implementing the concept of photonic packets is governed by the law of conservation of energy. Rather than tracing one photon at a time as nature does, we can consider a source emitting packets of many photons. When the photon strikes the surface, a fraction of the photon weight escapes as reflectance and the remaining weight is internally reflected and continues to propagate. Eventually, the photon weight drops below a threshold level and the simulation for that photon is terminated. In this example, termination occurred when the last significant fraction of remaining photon weight escaped at the surface at the position indicated by the circle (°) (see Fig. 3). Since the paper medium is composed of several components, it is necessary to include simulation scattering and/or absorption on each component. At the end of each step it is randomly determined at which component interactions occur, taking into account the percentage of each component. This was done so that the percentage of individual components was equated with the weight factor that takes care of how many times the scattering or absorption occurs on the respective component. Many photon trajectories (104 to 106 ) were typically calculated to yield a statistical description of photon distribution in the medium. As a simple example of our simulations, we calculated typical trajectories of individual photons (Fig. 1). The history of individual photons is simulated using random numbers, accurate representation of the probability of photon interaction with the medium and the exact model of problem 3D geometry. Our Monte Carlo simulation deals with the transport of an infinitely narrow beam of photons incident upon the multilayer substrate. We simulate illumination incident normally on a point, and to minimize noise 106 wave packets are simulated in each run. We chose the wavelengths of 400, 500, 600, and 700 nm. Each layer, which is considered as infinitely long, is described with the following parameters: mean thickness, refractive index, absorption coefficient μa , scattering coefficient μs , and anisotropy factor g (Henyey–Greenstein scattering phase function). Our model has either one or three layers of a substrate, i.e., without or with a coating on both sides of the paper. Each layer has accompanying absorption and scattering coefficients. In comparison with the dimensions of the spatial distribution of photons, the investigated substrate is considered infinitely large. Monte Carlo simulations of this type are based on the macroscopic optical properties of paper that are assumed to uniformly extend across the paper (e.g., cellulose fibers, fillers, adhesives, etc.). Our simulation does not take into account local distribution 20 November 2014 / Vol. 53, No. 33 / APPLIED OPTICS
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Table 1.
Numerical Data for our “Paper” Consisting of Five Components: Filler, Mechanical Pulp, Chemical Pulp, Coating and Air [25]
Component Filler Mechanical pulp Chemical pulp Coating Air
Weight Ratio, % Asymmetry Factor (g) 14 30 27 17 12
0.7 0.5 0.75 0.02 0
Scattering Coefficient (μs ), m2 ∕kg
Absorption Coefficient (μa ), m2 ∕kg
Wavelength (λ), nm
Wavelength (λ), nm
400
500
600
700
400
500
600
700
25 25 25 30 0
25 75 108 30 0
25 70 115 30 0
25 70 110 30 0
0.5 29 29 0.02 0
0.5 6 6 0.02 0
0.5 1 1 0.02 0
0.5 0.5 0.5 0.02 0
of radiation energy within, for example, cellulose fibers. History of the photons’ motion is recorded until the photon either leaves the medium or is absorbed during an interaction with the medium. Absorption at the individual component is determined by comparing a randomly generated number with albedo. The following results were calculated based on some typical values of paper component properties that were obtained from the literature and used already in our previous work [27]. We chose a trade-off paper composition that corresponds to the available data while still being simple enough not to consume excessive computer time. Our paper consisted of four components: pulp (chemical and mechanical), filler (CaCO3 ), sizing agent (AKD), and air (although the number of components was not limited by the software) in the amounts represented as weight ratios shown in Table 1. Of course, the composition of the actual papers was measured. We measured the reflectance on three types of printing paper commonly used in digital printing: Arcoprint 120 g∕m2 , Navigator 80 g∕m2 and Splendorgel 115 g∕m2 . We then compared it with the modelled one, which is a commercial secret, but the composition used in our modelled paper gave fairly good results, as demonstrated by the modelled reflectance profiles of paper prints [25]. The assumption was that the kaolin coating layer (n 1.55) is homogeneous and that its thickness is uniform over the entire surface of paper. We used a coating thickness of 5% (i.e., 0.01431 mm) of paper thickness which was d 0.2862 mm. This corresponds to the thickness of a commercial paper Splendorgel 115 g∕m2 on which the line screen elements were printed. Our profile was generated as a result of the contribution of all wavelengths to simulate white light in a way that we adjusted their contributions to the standard source D65. In Fig. 1 one can see that there is a one-dimensional (1D) beam of photon packets entering the paper at its top surface. Under such conditions no initial specular reflection on the paper surface takes place. Figure 1 shows that the photons can undergo multiple scattering events before they experience either transmission, absorption or reflection, or their return to the incoming surface. Multiple scattering, according to the simulation, causes attenuation 7858
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(weight reduction) of photon packets. This leads to an interesting finding that, given the particular paper composition and its values of the scattering and absorption coefficients, light absorption into the paper is far more significant than expected. Absorption values can, according to the simulation, reach as high as 30% of the initial value. Another important fact that follows from our experiments is that there are virtually no multiple reflections [28] between the paper-air boundary surfaces. Their number is negligible, so any description that uses these reflections as a major argument in an attempt to model the Yule–Nielsen’s effect, does not reflect real situation and fails to generate acceptable results. It is important to emphasize that our simulation does not take into account the wave nature of light, and that it ignores its phase and polarization. As mentioned in the Introduction, the motivation for this simulation was to predict and describe the transport of light through a medium such as paper and to determine the influence of the light propagation on the optical dot gain. Photons are reflected many times on their way through the medium and, consequently, phase and polarization very quickly become random and therefore play only a minor role in energy transport. 5. Determination of PSF
The basic ideas of modeling the reflectance of screen elements were presented earlier in the text. We modelled the profile of a printed line using Eq. (3) or Eq. (4), in which the influence of optical properties of paper and ink components is taken into account. Starting from the very definition of PSF we calculated its shape as a function of paper parameters. By modeling line profiles we wanted to examine the individual contribution of each parameter such as scattering and absorption coefficients of paper components, weight percentage of each component, asymmetry parameters, type and thickness of the coating layer (where applicable), the refractive index of the coating layer and its thickness, type and form of the paper surface, type of ink and its physical properties, and many other parameters that affect, to a greater or lesser degree, the collective effect known as optical dot gain. Previous approaches lacked the complexity necessary for the realistic description of the system and their authors used approximations that did not
Table 2. Comparison of Calculated (w C ) and Fitted (w L ) Parameter w for the Uncoated and Coated Paper
Uncoated Coated
wC (mm)
wL (mm)
Δw (mm)
0.19463 0.14579
0.19279 0.13966
0.00156 0.00613
always have a sound physical basis. Then at the end of the 1970s, Yule and Nielsen suggested that the LSF should be described by means of Gaussian distribution [21]. Our model, based on stochastic approach, shows that for a given set of parameters, PSF can be represented by Lorentzian distribution which we put, for simplicity, in the origin of the coordinate system as shown in Fig. 2. The obtained function is independent of its position in the incident plane, so it can be considered as shift invariant. Function PSFx; y represents the flux density as a function of rectangular coordinates on the image plane and describes how a sharp point is spread out due to the physics of the imaging process. The origin is the location of the ideal image spot. If we plot the intensity of the spot, we obtain a 1D PSF profile. The curve, calculated data in Fig. 2, was generated by calculating the profile radial symmetry and averaging over the annulus using a predefined radial step. The argument for the cylindrical symmetry approximation (rotation invariant) of the obtained profile can be deduced from Fig. 3. This image was obtained using more than 20.000 calculated points, i.e., photons (only a fraction of them are shown), which is a relatively small number, but illustrates cylindrical symmetry of the calculated subsurface scattered light beam fairly well. Each circle in the figure represents the exit position of a photon packet. Distribution of the scattered light depicted in Fig. 2, when normalized using the surface area under the curve, represents probability distribution. By definition, to get the PSF we have to normalize
Fig. 2. Calculated radial distribution of the subsurface scattered input light beam of photon packages that penetrate the uncoated paper surface at the point (0, 0) and Lorentzian and Gaussian fitting functions.
Fig. 3. Planar distribution of outgoing photon packets illustrating approximation of cylindrical symmetry for the calculated subsurface scattered light beam.
the calculated area of radial distribution to unity. The analytical form of PSF is found by fitting the obtained distribution by Lorentzian function. Consequently, the resulting PSF can be considered as if obtained experimentally: 1 1 w∕2 PSFr ; R π r − rc 2 w∕22
(8)
where rc is the location of the center of distribution and w is a parameter specifying the width of the profile (see below). Constant R is defined by the definition of PSF as a probability, so that 1 w∕2 R dr: 2 2 −∞ π r − rc w∕2 Z
∞
(9)
Figure 2 shows that there is a very good agreement (Pearson linear correlation coefficient R > 99%) between the calculated values, the PSF, and the Lorentzian function. It can also be seen that the fitting using the Gaussian model is poorer. With the earlier defined convolution, Eq. (3) was the basis for modeling the profile lines in Fig. 4. Shape of Tx; y, ink transmittance of the screen element—in our case line—on paper, was, in our approximation, modelled as being rectangular with maximum and minimum reflectance values corresponding to averaged measurements and by neglecting the effects of edges occurring due to mechanical dot gain. To print a printed form with a linear sample—a line—we used a previously calibrated digital electro-photographic machine, Indigo Turbo Stream 1000. The printed line width was measured by image analyzer PIAS. Figure 4 shows a comparison between the measured and the calculated profiles. We compared the Lorentzian profile obtained by the Monte Carlo method with the usual Gaussian profile of the PSF. For comparison, a model profile with which we 20 November 2014 / Vol. 53, No. 33 / APPLIED OPTICS
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Fig. 4. Comparison of measured and calculated (Lorentz, Gauss) reflectance profiles with the model profile for the line screen element. The nominal line width is d 1 mm, while the measured width is dm 1.06 mm. The dash-bordered regions are shown separately in Figs. 6(a) and 6(b).
approximated product Rp · Tx; y (transmittance sample of ink on paper multiplied by reflectance of paper itself) is also shown. The resolution of printed screen elements can be defined as the width within which the overall PSF drops to one half of the maximum value, which is called full width at half-maximum (FWHM). If the object consists of two ideal points, just a distance FWHM apart, there is a fair chance that they will be separated in the image. Both Gaussian and Lorentzian functions contain only a single parameter, w, specifying the FWHM of the profile, which must be adjusted in order to fit the data. Additional evidence that the Lorentzian function gives a better fit to the measured profile than the Gaussian function is provided when focusing on the two particular regions of Fig. 4: the profiles and the curves between 75% and 87% reflectance Fig. 5(a) and around 5% reflectance Fig. 5(b). Figure 5(b) shows that the Lorentzian PSF better describes the nature of optical dot gain compared to the Gaussian function. In the region characterized by the reflectance close to 5% it is difficult to determine the exact position of the edge due to the fact that, as the analysis indicates, the line is somewhat narrower than the desired one, which is probably the effect of the applied printing technique. Further comparison of the generated PSF between the coated and the uncoated papers reveals that there are differences in the parameter w values for the two paper types due to the presence of coating. This suggests that the parameter w somehow depends on the composition of the paper surface (refractive index). We conducted the same analysis by varying paper thickness and ratio of individual components of paper, but the results of these experiments did not confirm the influence of these two quantities on the parameter w. Since our profile depends only on a single parameter, w, we tried to find a potential explanation for our observation that the modelled half-width profile 7860
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Fig. 5. Detail of the calculated (Lorentzian and Gaussian fit). (a) Measured profiles in the far wing region and (b) in the region where the reflectance is close to 5%. Also shown is the approximation model of Rp Tx; y profile used in Eqs. (3) and (4).
w is different for coated and uncoated paper, as shown in Fig. 6(a). Figure 6(a) shows a comparison of modelled distribution profiles of scattered radiation for a coated and uncoated paper. It is apparent that the intensity distribution for the coated paper is characterized by a noticeably narrower parameter w (see Table 2). This is the result of multiple internal reflections at the paper-coating border. Consequently, transmission is also reduced and the number of photons returning to the incoming surface is lower. Of course, this is not the case with the uncoated paper where the scattering is substantial. In Maretić Petric et al. [29] we discussed in more detail which of the two observed functions, Lorentz or Gaussian, gives a better description of the LSF and why this is the case. Given that the obtained correlation coefficients were extremely high and very similar, a rating quality model was made using Akaike information criterion. The Akaike information criterion is a measure of the relative quality of a statistical model for a given set of data and provides a means for model selection. A more elaborate report
Fig. 6. Comparison of modeled radial distributions of subsurface scattered radiation for coated and uncoated paper. (a) Calculated distributions using Monte Carlo approach and (b) values normalized to the same peak value to enable comparison of parameters w.
on these experiments will be published in the forthcoming article. 6. Conclusions
Our method based on the Monte Carlo approach describes the local rules of photon subsurface propagation, which are expressed, in the simplest case as the probability distributions that describe the size of the photon shift between the position of two events (interactions), and the direction angles of the photon trajectory (in the case of scattering). Involved interactions are the scattering and absorption of the photon package on each paper component. Descriptions using photonic packages, instead of single photons, were adopted due to the law of energy conservation. The idea of this study was to demonstrate that the calculated subsurface distribution of scattered photons corresponds to PSF, since the PSF gives the probability for an entering photon to emerge out at a remote point on the same surface. We recognize our modelled distribution of these photons (photon packets), after the standardization area under the curve on to unit, as PSF. We modelled
the distribution fit with Lorentzian and Gaussian function in order to obtain an analytical form of PSF. Our attempt to describe reflectance of the printed paper surface is based on Eq. (3) with which we modelled PSFx; y using the Monte Carlo method. PSF and, consequently, MTF describe lateral light scattering in paper and is used to describe color and tone reproduction in printed half-tone images. In the presented method of obtaining a reflectance profile of any screen element, the PSF convolves with a modelled reflectance profile of the bitmap element with some simple geometric assumptions. In our case, the modelled profiles gave satisfactory results of the reflection in the far wing of the line profile edge. This is the result of the fact that our model describes only the optical dot gain, and mechanical dot gain in the model line profile was included simply as a broader original line. Further evidence that the Lorentzian function efficiently describes optical dot gain is provided if we let the line width go to zero. In this case the mechanical dot gain disappears (no screen element), and what remains is the optical dot gain, which we describe by Lorentzian function. We demonstrated that our method adequately models optical component of dot gain, indicating that we were able to successfully separate optical and mechanical components of dot gain. In addition, the method of convolving PSF with model profile lines showed better approximation of the measured PSF profile with Lorentzian (i.e., Cauchy distribution) than with the Gaussian function. Our work shows that the modelled distribution is better described with a Lorentzian (Cauchy) function. Comparison with measured data also confirms the fact that the Lorentz function gives a better description of the PSF. We can conclude that the method appropriately describes the optical dot gain phenomenon, and that we can, from the measured reflectance line profile using deconvolution, obtain valuable information about the nature and morphology of mechanical dot gain. Separation of mechanical and optical dot gain will enable us to optimize the printing process, but also aid in producing paper substrate with the specified optical and mechanical properties. On the other hand, the knowledge of the origin and behavior of the total dot gain can be used to eliminate blurred edges using an appropriate image analysis procedure such as image sharpening or denoising. A possible physical reason for obtaining a Lorentz profile of PSF lies in Huygens’ principle, which says that one can determine the intensity by assuming that the light is re-emitted from any line between the source and the target, while the Cauchy–Lorentz distribution describes the normalized intensity of light on a line from a point source. The final distribution of photons leaving the substrate after n steps, obtained as result of the contributions in all points of the path, has a Lorentzian shape because the sum of n independent Cauchy distributions is a Cauchy distribution scaled by a factor of n. We believe that 20 November 2014 / Vol. 53, No. 33 / APPLIED OPTICS
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the interpretation of the physical background reflects the fact that the paper components, absorption and scattering on them and perhaps some other parameters, are not entirely mutually independent. The method provides a satisfactory description of the line profiles and also gives guidelines for future research on the impact of various factors, not only on the design of the paper itself, but also on other important aspects of printing technology such as paper-ink interactions, quality control, accurate recalibration of printers, etc. This work was supported by the Croatian Ministry of Science, Education and Sports under Grant No. 128-1281955-1960. References 1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983). 2. K. Green, L. Lamberg, and K. Lumme, “Stochastic modeling of paper structure and Monte Carlo simulation of light scattering,” Appl. Opt. 39, 4669–4683 (2000). 3. L. G. Coppel, P. Edström, and M. Lindquister, “Open source Monte Carlo simulation platform for particle level simulation of light scattering from generated paper structures,” in Proceedings of Papermaking Research Symposium, E. Madetoja, H. Niskanen, and J. Hämäläinen, eds. (Kuopio University, 2009). 4. M. Neuman, L. G. Coppel, and P. Edström, “Point spreading in turbid media with anisotropic single scattering,” Opt. Express 19, 1915–1920 (2011). 5. T. Linder and T. Löfqvist, “Anisotropic light propagation in paper,” Nord. Pulp Pap. Res. J. 27, 500–506 (2012). 6. D. Modrić, R. Beuc, and S. Bolanča, “Monte Carlo modeling of light scattering in paper,” J. Imaging Sci. Technol. 53, 020201 (2009). 7. S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE, 5, 102–111 (1989). 8. L. Wang, S. L. Jacques, and L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995). 9. A. Murray, “Monochrome reproduction in photoengraving,” J. Franklin Inst. 221, 721–744 (1936). 10. J. A. C. Yule and W. J. Neilsen, “The penetration of light into paper and its effect on half-tone reproduction,” in TAGA Proceedings (Technical Association of the Graphic Arts, 1951), pp. 65–76.
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