# Misregistration sensitivity in clustered-dot color halftones

**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

#### Open Access

## Abstract

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.

## Introduction

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.^{4- 5} 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.^{6- 7} 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 process^{9} 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.

## Related Work

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,^{13- 14} the validity of the Demichel equations for the common rotated screen configurations has not received much attention until recently.

Rogers^{15} 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 Hersch^{16} 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 Rogers^{15} and Amidror and Hersch.^{16- 17} 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.

## Color Halftone Misregistration Analysis Framework

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.

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 $\Lambda k$ and a binary halftone spot function $sk(x)$,^{20- 21} where $x=[x,y]T$ represents the spatial coordinates. $\Lambda k$ represents the 2D periodicity of the $k$’th halftone separation and is mathematically defined as^{22}

*linearly independent*vectors $v1k=[vx1k,vy1k]T$ and $v2k=[vx2k,vy2k]T$ as its columns. Thus, $\Lambda k$ 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 $\Lambda k$ 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 $\Lambda k$, 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 as

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 is

^{24- 25}although, empirically, values of $\gamma >2$ 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 $c=c1\u2026cK$, 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 $2K\u22121$, we can rewrite Eq. 4 as

Now if we denote by $\beta k$ 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 have

We extend this notation and use $\beta k1\u2026km$ to represent the fractional area covered by the colorants $k1\u2026km$ (parts of which are also possibly covered by additional colorants), where $k1\u2026km$ is a set of distinct colorant indices drawn from ${1,\u2026,K}$. For uniqueness, we consider only subscripts $k1\u2026km$ indices arranged in ascending order, that is $k1<k2<\cdots <km$. Then, just as with the individual colorant areas, these fractional areas can be represented in terms of the Neugebauer primary areas as

The above system of equations can be inverted to obtain an expression for the fractional areas of the primaries $ac1\u2026cK$ in terms of the fractional areas $\beta k1\u2026km$ for the colorant combinations. One can see that $a1\u20261=\beta 1\u2026K$ and

Due to the equivalence of the Neugebauer primary fractional areas $ac1\u2026cK$ to the colorant overlap areas $\beta k(c)$, invariance properties with respect to displacement misregistration established for one are applicable to the other. We therefore proceed by obtaining expressions for the areas $\beta k(c)$. Note that we adopt this approach because it is easier to directly obtain expressions for the terms $\beta k(c)$, as opposed to the Neugebauer primary areas $ac$.

To compute $\beta k(c)$, consider the overlay of the halftones that constitute $k(c)$. The function

*spatial average*of $hk(c)(x;dk(c))$.

If each of the matrices $Vki\u22121Vkj$ has only rational numbers as their elements for all pairs $ki\u2260kj$ in $k(c)$, the intersection of the lattices $\Lambda k(c)=\u2229k\u220ak(c)\Lambda k$ is a 2D lattice,^{27- 28} 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 $\Lambda k(c)$ 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 as

*digital halftone screens*, that is, screens defined on a common discrete periodic grid for any $k,l\u220a{1,\u2026,K}$, $Vk\u22121Vl$ 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 $\u2223Uk(c)\u2223\u2192\u221e$. 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 that

*DC term*of the frequency spectrum of the overlay of its constituent colorants. The Fourier transform of Eq. 9 yields

*reciprocal lattice*of $\Lambda k$ (see Ref.

^{22}, pp. 23–24),

^{28}which is represented by

Using these results we see that (see Appendix ^{a})

Then, Eq. 11 can be computed as

*singular*.

^{16}Note that only terms indexed by variables of the summation symbol in Eq. 17 depend on the interseparation misregistration amounts.

## Conditions for Color Misregistration Insensitivity

Denote by $Ravg(0)(\lambda )$ and $Ravg(d)(\lambda )$, the average spectra of “prints” with interseparation displacements $0$ (perfect registration) and $d=[d1,\u2026,dK]$ (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 $Ravg(\lambda )$ is affected by interseparation misregistration as we show in the following subsections. We also observe here that the terms ${\beta k}k=1K$, corresponding to the total fractional area covered by the individual colorants, are independent of misregistration.