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Misregistration sensitivity in clustered-dot color halftones

[+] Author Affiliations
Basak Oztan

University of Rochester, Department of Electrical and Computer Engineering, Rochester, New York 14627-0216

Gaurav Sharma

University of Rochester, Department of Electrical and Computer Engineering, and Department of Biostatistics and Computational Biology, Rochester, New York 14627-0126

Robert P. Loce

Xerox Research Center Webster, Xerox Corporation, Webster, New York 14580

J. Electron. Imaging. 17(2), 023004 (June 05, 2008). doi:10.1117/1.2917517
History: Received July 19, 2007; Revised October 25, 2007; Accepted November 14, 2007; Published June 05, 2008; June 09, 2008
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Halftoned separations of individual colorants, typically cyan, magenta, yellow, and black, are overlaid on a print substrate in typical color printing systems. Displacements between these separations, commonly referred to as “interseparation misregistration”, can cause objectionable color shifts in the prints. We study this misregistration-induced color shift for periodic clustered-dot halftones using a spatiospectral model for the printed output that combines the Neugebauer model with a periodic lattice representation for the individual halftones. Using Fourier analysis in the framework of this model, we obtain an analytic characterization for the conditions for misregistration invariance in terms of colorant spectra, periodicity of the individual separation halftones, dot shapes, and misregistration displacements. We further exploit the framework in a hybrid analytical-numerical simulation that allows us to obtain quantitative estimates of the color shifts due to misregistration, thereby providing a characterization for these shifts as a function of the optical dot gain, halftone periodicities, spot shapes, and interseparation misregistration amounts. We present simulation results that demonstrate the impact of each of these parameters on the color shift and demonstrate qualitative agreement between our approximation and experimental data.

Halftoning is a method of encoding a continuous tone (contone) image using a reduced number of quantization levels, typically two, such that at normal viewing distances, the perception of the printed or displayed halftone image closely approximates that of the contone image. Color halftoning is used in most printing processes for the production of hardcopy prints. The halftoning operation is applied to the colorant image separations or channels, which usually correspond to the cyan (C), magenta (M), yellow (Y), and black (K) colorants. The halftoning operation produces a bilevel output for each separation and the printing process prints these colorant image separations in overlay on a white substrate such as paper.45 Halftoned images exercise only two extremes of a printing device’s response corresponding to on/off states that represent a saturated amount of colorant or no colorant, respectively. Through process controls, it is easier to maintain these extremes in a stable condition over time, whereas, maintaining the stability of the printer response at intermediate levels is much more demanding. For a number of printing technologies, halftoning thus enables the production of prints that are relatively stable from print to print in color and tones and low in noise.

In an idealized, noninteracting, printing process, the halftoning algorithm for each colorant separation could be designed independently.67 In actual practice, however, there are interactions among the colorant separations that must be considered jointly in order to provide acceptable image quality. The interactions are best understood by considering that the light reflected from the paper to the observer is transmitted through the layer of toner or ink of each colorant separation. At a given point on the printed image, the resulting transmittance for each individual wavelength of light is, to first-order, the multiplication of transmittances from each colorant. For any wavelength that is absorbed by more than one colorant, this multiplicative phenomenon produces an interaction that varies spatially according to the halftone pattern produced by the halftoning algorithm. The spatial implications can be understood by recognizing that sum and difference frequency components are produced in the Fourier spectrum [i.e., a spatial Fourier spectrum per wavelength in the (visible) electromagnetic spectrum] via the convolution property of the Fourier transform [The convolution theorem (see Ref. 8, p. 108) states that the Fourier transform of the convolution/multiplication of two signals is the multiplication/convolution of their Fourier transforms]. These additional spatial frequency components could appear as undesirable texture or moiré in the printed image.

Due to the halftone interactions just described, most color halftoning methods in practice use halftone image structures that are designed to minimize the negative visual impact of low spatial frequency components that are the sum or difference of frequency components of the individual halftone separation images. A component at zero spatial frequency (commonly referred to as the “DC component”) that arises from the sum or difference of other nonzero frequency components has, however, often not been considered undesirable. One reason for this apparent anomaly is that the color control of the printer eliminates the impact of a fixed DC component through the process used to map desired color values to printer CMYK combinations. However, the color response of a printer at DC (and in fact at any frequency) depends not only on the frequency components in the individual separation halftones but also on their relative phase. Spatial misregistration between the halftone colorant separations corresponds to a change in relative phase and can, therefore, alter the value of the DC term, which manifests as a color shift. Some amount and orientation of misregistration between halftone image separations is unavoidable due to various operations within a printing process such as mechanical paper transport, paper shrinkage, and misalignment of imagers. In the present work, we investigate this interseparation misregistration-induced color shift by examining the zero spatial frequency component produced in the overlay. Our goal is to investigate the impact of misregistration both analytically and by means of a simulation framework.

The preceding discussion is applicable to almost all halftoning methods that process colorant separations independently. Several techniques have been proposed for the halftoning of individual separations (or equivalently monochrome images) that fall under one of the following three categories:5 (1) point processes, (2) neighborhood algorithms, and (3) iterative methods. The characteristics of the printing technology, specific application requirements, and computational complexity are some of the common factors that impact the choice of a halftoning technique. Due to their stability and predictability, clustered-dot halftones are commonly used in the two primary methods of high volume printing: lithography and xerography. Clustered-dot halftoning evolved from Talbot’s original photographic screening process9 and, for digital imagery, is typically accomplished by pixel-by-pixel thresholding against a periodic halftone threshold array. Conventional clustered-dot halftones are considered amplitude modulated (AM) signals in the sense that different gray levels are reproduced by varying the size of halftone spots while keeping their periodicity constant. In the present paper, we restrict our attention to methods used for xerographic and offset printers and, thus, consider only clustered-dot halftones in our analysis of color variation with interseparation misregistration.

The remainder of this paper is organized as follows. Section 2 reviews the literature on this problem and connects it to our contributions. In Sec. 3, we develop a characterization of halftone color sensitivity to interseparation misregistration using Fourier analysis in a lattice framework. In Sec. 4, the conditions under which the average color is invariant to displacement misregistration are described. Next, in Sec. 5, we incorporate an analytic model for the halftone screens in our lattice framework to develop a numerical model for quantitative estimation of color shifts due to misregistration in clustered-dot color halftones. We present results from the model along with comparisons to experimental data in Sec. 7. A discussion of the additional practical implications and extensions is included in Sec. 8. Based on that development, we also propose a metric for evaluating misregistration sensitivity of two-colorant halftone configurations. Finally, Sec. 9 summarizes the main conclusions.

The average spectrum of a halftone image may be modeled (to first-order) using the Neugebauer equations.10a,10b,11 Using these equations, we can readily see that if the areas of overlap between the colorant separations can be modeled as statistically random, that is, satisfy the Demichel equations,12a,12b the average spectrum is independent of the interseparation alignment. Therefore, randomization of the interseparation overlaps eliminates the problem of misregistration-induced color shift in color halftones. With this motivation, rotated halftone screens, in which the halftones for different separations are rotated relative to each other, are commonly used in practice. Though alternatives to the Demichel equations have been proposed for nonrotated halftone configurations,1314 the validity of the Demichel equations for the common rotated screen configurations has not received much attention until recently.

Rogers15 examined the validity of Demichel equations for the superposition of two halftone screens with circular halftone spots, where each screen has two orthogonal frequency vectors and the four frequency vectors for the screens have equal magnitudes. In this scenario, he demonstrated that the equations hold only for certain angular separations between the screens. Rogers achieved this result by deriving an expression for the neighbor distribution of halftone spots and determining the conditions under which this distribution is statistically random.

Amidror and Hersch16 extend the work to the superposition of an arbitrary number of screens. They present a general proposition that characterizes the failure of the Demichel equations in the Fourier domain by a singular configuration of frequency vectors [We provide a precise mathematical definition of the term singular in the next section]. Using computer simulations on a high resolution pixel grid, they numerically demonstrate the validity of the proposition. In particular, they verify that the conventional 30-deg angular separation equifrequency CMK halftone configuration (see Ref. 6, pp. 339–341) used in lithographic printing is not invariant to color misregistration. In related work,17 they also consider the stability of the rosette structure, also known as microstructure, resulting from the overlap of multiple screens and demonstrate that the microstructure is sensitive to layer misregistration for singular configurations and insensitive for nonsingular configurations.

In the first part of our work, we build on the foundations of Rogers15 and Amidror and Hersch.1617 We cast the general K-screen superposition problem in a lattice framework that has previously been used for the frequency analysis of halftone superpositions. Each halftone separation is modeled as a periodic function with periodicity determined by a two-dimensional (2D) lattice. Using Fourier analysis in our lattice framework, we obtain mathematical expressions for the average halftone color (spectrum) as a function of screen periodicities, halftone spots, interseparation screen displacements, and Neugebauer primary spectra. The analysis provides a comprehensive framework for understanding the conditions for color insensitivity to displacement misregistration. Conventionally, it is believed that misregistration insensitivity is achieved if either the colorants have nonoverlapping absorption bands,6 or if the halftone periodicities meet a “nonsingularity” condition.16 In addition to validating these known criteria, our analysis reveals additional situations under which insensitivity to misregistration is achievable despite these conditions being violated. In particular, we demonstrate that in some scenarios, the halftone spot functions may provide invariance to misregistration. The analysis also reveals the existence of nontrivial (different from an integral number of lattice periods) displacements for which the invariance holds.

In the second part of our work, we address the quantitative estimation of the color shift induced by misregistration. For this purpose, we use the Neugebauer model in a semianalytic simulation that exploits the analysis from the first part to obtain spatial domain expressions for the average spectrum of the print expressed in terms of colorant overlap areas. The latter are computed analytically for common dot shapes. When combined with measured Neugebauer primary spectra, these allow for a quantification of the color shift for different misregistrations. Prior work in this area has also been based on the Neugebauer model with estimation of the area coverages for the Neugebauer primaries performed either experimentally by using a planimeter for the dot-on-dot geometry,18 or through computer simulations, where halftone images are generated on a high resolution digital grid and areas computed by pixel-counting methods.1,19,16 The experimental approach has only been applied in a limited context due to measurement challenges. The pixel-counting simulation approaches become quite memory intensive for certain configurations and can also suffer from a limitation in accuracy due to the use of a finite grid. Our semianalytic methodology allows the model to operate without the computationally expensive high resolution simulation, thereby enabling evaluation of numerous screen configurations. Comparison of the framework against experimental data with independent measurement of misregistration demonstrates good qualitative agreement.

For our analysis, we assume that color printing is accomplished by halftoning K individual colorant separations, where K=4 for the typical CMYK scenario, and printing these in overlay. In Fig. 1, we schematically illustrate the overall process for a typical clustered-dot color halftoning with CMY colorants. We model individual colorant clustered-dot halftones in terms of a lattice that represents their periodicity and a spot function that represents the shape of the halftone dots. The average color for the printed overlay of these halftones is obtained by using the spectral Neugebauer model.10a,10b The model computes the spatially averaged reflectance for the print as a weighted average of reflectances of all possible overlays of the colorants on the substrate, where the overlays are referred to as the Neugebauer primaries and the weights correspond to the fractional areas of the Neugebauer primaries. A change in these fractional areas due to interseparation misregistration is the primary source of color shifts when the colorants have overlapping absorption bands. We therefore derive our model in this framework and obtain expressions for the average reflectance spectrum and the fractional areas, which we use in turn to characterize the conditions under which the average spectrum is invariant to displacement misregistration.

Individual Colorant Halftones

A halftone image hk(x) generated for the k’th colorant plane of a constant gray-level contone image can be modeled as the convolution of a planar lattice Λk and a binary halftone spot function sk(x),2021 where x=[x,y]T represents the spatial coordinates. Λk represents the 2D periodicity of the k’th halftone separation and is mathematically defined as22Display Formula

1Λk={VknknkZ2},
where Z denotes the set of integers and Vk=[v1kv2k] is a 2×2 real-valued matrix, with two 2×1linearly independent vectors v1k=[vx1k,vy1k]T and v2k=[vx2k,vy2k]T as its columns. Thus, Λk is the (discrete) set of all integer linear combinations of the vectors v1k and v2k in R2. The vectors v1k and v2k represent a basis for the lattice Λk and for any point Vknk in the lattice, the vector nk=[nkx,nky]T is the representation of the point in the lattice with respect to the basis Vk.

The halftone spot function sk(x) is confined within a unit cell of Λk, denoted by Uk, and takes values 1 or 0 corresponding to the situation that ink k is, or is not, deposited at the position x. hk(x) can accordingly be written asDisplay Formula

2hk(x)=sk(x)*nkδ(xVknk),
where δ(x) is the Dirac delta function. Displacement misregistration of the k’th separation by the vector dk=[Δxk,Δyk]T is readily incorporated in this representation by replacing sk(x) with sk(xdk), where Δxk and Δyk are the registration errors along the x and y axes, respectively. The halftone separation hk(x) can accordingly be written asDisplay Formula
3hk(dk)(x)=sk(xdk)*nkδ(xVknk).

Spectral Neugebauer Model

On a color print, these multiple halftone image separations are overlaid, typically producing all possible 2K overlays of K colorants. The colors associated with each of the 2K overlays are referred to as the Neugebauer primaries. Using the Yule-Nielsen (YN) modified Neugebauer model,10a,10b,11,23 the average spectrum of the printed halftone isDisplay Formula

4Ravg(λ)=(i=02K1aiRi1γ(λ))γ,
where ai and Ri(λ) are the fractional area coverage and the spectral reflectance of the i’th Neugebauer primary, respectively, and γ is the empirical YN correction factor. This factor accounts for optical dot gain, that is, the scattering of the absorbed light within the paper due to which halftone spots appear larger than their physical size on the paper. Paper quality, spectral characteristics of the colorants, halftone periodicities, and spot shapes are some of the elements that affect optical dot gain. Generally 1γ2 is accepted as physically meaningful,2425 although, empirically, values of γ>2 often provide better agreement with data, particularly for high frequency printers.26

To obtain expressions for the Neugebauer primary areas, we first represent these areas in terms of an alternative but equivalent (in the sense that either is obtainable from the other) set of areas that are more readily amenable to analysis. Figure 2 illustrates this alternate representation, which we describe next. For notational convenience, in this process, we index each of the 2K possible combinations of the K colorants by a K-bit binary index string c=c1cK, where ck=1 indicates the presence of the k’th colorant and ck=0 its absence in the combination. Interpreting the string as the binary representation of a Neugebauer primary index between 0 and 2K1, we can rewrite Eq. 4 asDisplay Formula

5Ravg(λ)=(c1=01cK=01ac1cKRc1cK1γ(λ))γ.

Now if we denote by βk the fraction of the total area covered by the k’th colorant (which is possibly covered by additional colorants and, therefore, distinct from the area for the corresponding Neugebauer primary), we haveDisplay Formula

6βk={c1cKck=1}ac1cK.

We extend this notation and use βk1km to represent the fractional area covered by the colorants k1km (parts of which are also possibly covered by additional colorants), where k1km is a set of distinct colorant indices drawn from {1,,K}. For uniqueness, we consider only subscripts k1km indices arranged in ascending order, that is k1<k2<<km. Then, just as with the individual colorant areas, these fractional areas can be represented in terms of the Neugebauer primary areas asDisplay Formula

7βk1km={c1cKck1=1,,ckm=1}ac1cK.

The above system of equations can be inverted to obtain an expression for the fractional areas of the primaries ac1cK in terms of the fractional areas βk1km for the colorant combinations. One can see that a11=β1K andDisplay Formula

8ac1cK={1cI+(c)acifc1cK=00,βk(c)cI+(c)acotherwise,
where the notation k(c)=k1km denotes the string of indices {kl}l=1m for which ckl is nonzero, and I+(c) is the set of all indices that include the nonzero indices ckl and at least one additional (distinct) nonzero index [Note that each element of I+(c) denotes a combination of colorants that include the nonzero m colorants indicated by c as a subset and at least one additional nonzero colorant]. The above relation [Eq. 8] can also be seen from Fig. 2, which illustrates the relation between the areas ac and βk(c) for K=3 with CMY colorants.

Due to the equivalence of the Neugebauer primary fractional areas ac1cK to the colorant overlap areas βk(c), invariance properties with respect to displacement misregistration established for one are applicable to the other. We therefore proceed by obtaining expressions for the areas βk(c). Note that we adopt this approach because it is easier to directly obtain expressions for the terms βk(c), as opposed to the Neugebauer primary areas ac.

Fractional Areas of Colorant Combinations

To compute βk(c), consider the overlay of the halftones that constitute k(c). The functionDisplay Formula

9hk(c)(x;dk(c))=kk(c)hk(dk)(x)
indicates the spatial locations covered by the colorants in k(c), taking a value 1 if x is covered by the colorants in k(c) and 0 otherwise, where dk(c)=[dk1,,dkm] are the displacement vectors of the individual separations that constitute k(c). The fractional area βk(c) is the spatial average of hk(c)(x;dk(c)).

If each of the matrices Vki1Vkj has only rational numbers as their elements for all pairs kikj in k(c), the intersection of the lattices Λk(c)=kk(c)Λk is a 2D lattice,2728 whose periodicity may be represented in terms of a basis matrix Vk(c). The basis matrix Vk(c) can be obtained using a least common right multiple computation for integer matrices (see Ref. 27 pp. 35–38).29

As it can be seen from the illustration in Fig. 3, a period of Λk(c) includes at least one period of each of the constituent lattices. The overlay hk(c)(x;dk(c)) of the constituent halftones is then periodic over this lattice and thus the spatial average is obtained asDisplay Formula

10βk(c)=1Uk(c)xUk(c)hk(c)(x)dx,
where Uk(c) denotes a unit cell of Λk(c) and Uk(c) the area of this unit cell. We note that for digital halftone screens, that is, screens defined on a common discrete periodic grid for any k,l{1,,K}, Vk1Vl has only rational entries and thus the above assumption holds for any colorant combination in c. Situations in which the assumption does not hold may be viewed as a limiting case for out analysis, where Uk(c). Our ensuing frequency domain analysis [particularly Eq. 17] still holds for these situations even though the basis matrix Vk(c) does not exist.

From the Fourier transform properties it follows thatDisplay Formula

11βk(c)=Hk(c)(0),
where Hk(c)(u) represents the Fourier transform of hk(c)(x) and u=[u,v]T denotes the coordinates in frequency space. In other words, βk(c) is the DC term of the frequency spectrum of the overlay of its constituent colorants. The Fourier transform of Eq. 9 yieldsDisplay Formula
12Hk(c)(u)=Hk1(dk1)(u)**Hkm(dkm)(u),
where Hk(dk)(u) represents the Fourier transform of the halftone image hk(dk)(x). Let Sk(u) represent the Fourier transform of the halftone spot function sk(x). Applying the shift and convolution property of the Fourier transform on Eq. 3, Hk(dk)(u) can be written asDisplay Formula
13Hk(dk)(u)=1VkSk(u)exp(2πjdkTu)nkδ(uWknk),
where the Fourier transform of the “comb” function nkδ(xVknk) takes nonzero values on the elements of the reciprocal lattice of Λk (see Ref. 22, pp. 23–24),28 which is represented byDisplay Formula
14Λk*={Wknk=(Vk1)TnknkZ2},
where Wk=(Vk1)T represents the basis matrix for the reciprocal lattice Λk*.

Using these results we see that (see Appendix a)Display Formula

15Hk(c)(u)=nk1nkmkk(c)(1VkSk(Wknk)exp(2πjdkTWknk))δ(ulk(c)Wlnl),
which can only take nonzero values if u=kk(c)Wknk. Let Nk(c) represent the set {(nk1,,nkm)i=1mWkinki=0}, which includes the indices of all the frequency vectors that sum up to the zero vector.

Then, Eq. 11 can be computed asDisplay Formula

16βk(c)=Hk(c)(0)=(nk1,,nkm)Nk(c)kk(c)1VkSk(Wknk)exp(2πjdkTWknk)=kk(c)1VkSk(0)+(nk1,,nkm)Nk(c)\{0}kk(c)1VkSk(Wknk)exp(2πjdkTWknk)
Display Formula
17=kk(c)βk+(nk1,,nkm)Nk(c)\{0}kk(c)1VkSk(Wknk)exp(2πjdkTWknk),
where Nk(c)\{0} denotes the elements in Nk(c) with the exclusion of the all-zero vector 0. If Nk(c)\{0} is a nonempty set, then the overlay of the halftones is said to be singular.16 Note that only terms indexed by variables of the summation symbol in Eq. 17 depend on the interseparation misregistration amounts.

Denote by Ravg(0)(λ) and Ravg(d)(λ), the average spectra of “prints” with interseparation displacements 0 (perfect registration) and d=[d1,,dK] (misregistered), respectively. A difference in these terms represents a misregistration-induced color shift [Strictly speaking, this is a shift in the average spectrum that will typically produce a corresponding color shift]. In this section, we consider the conditions under which these terms do not differ, yielding insensitivity to color misregistration in the average color. In Eq. 5, there are two elements that affect the value of these terms: spectral reflectances of the Neugebauer primaries, and fractional area coverages of the Neugebauer primaries. The former is affected by the spectral interactions of the colorants (inks) in their absorption bands of the spectra, the latter is a function of individual halftone separation periodicities, halftone spots, and the inter-separation misregistrations as shown in Eq. 17. These terms define conditions under which Ravg(λ) is affected by interseparation misregistration as we show in the following subsections. We also observe here that the terms {βk}k=1K, corresponding to the total fractional area covered by the individual colorants, are independent of misregistration.