1.

Introduction

[1] 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 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.

2.

Related Work

The average spectrum of a halftone image may be modeled (to first-order) using the Neugebauer equations.^{10, 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,¹² 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.

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.^{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.

3.

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.¹⁰ 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.

Fig. 1

Modeling framework for analyzing the effect of interseparation misregistration on the color (average reflectance spectrum) of periodic clustered-dot halftone prints. (Color online only.)

3.2.

Spectral Neugebauer Model

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

Eq. 4

$$R_{avg}\left(\lambda \right)={\left(\sum _{i=0}^{2^K-1}{a}_i{R}_i^{\frac{1}{\gamma }}\left(\lambda \right)\right)}^{\gamma },$$

where $a_i$ and $R_i\left(\lambda \right)$ are the fractional area coverage and the spectral reflectance of the $i$ ’th Neugebauer primary, respectively, and $\gamma$ 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 $1\le \gamma \le 2$ is accepted as physically meaningful,^{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 $2^K$ possible combinations of the $K$ colorants by a $K$ -bit binary index string $\mathbf{c}=c_1\dots c_K$ , where $c_k=1$ indicates the presence of the $k$ ’th colorant and $c_k=0$ its absence in the combination. Interpreting the string as the binary representation of a Neugebauer primary index between $0$ and $2^K-1$ , we can rewrite Eq. ⁴ as

Eq. 5

$$R_{avg}\left(\lambda \right)={(\sum _{c_1=0}^1\cdots \sum _{c_K=0}^1{a}_{c_1\dots c_K}{R}_{c_1\dots c_K}^{1∕\gamma }\left(\lambda \right))}^{\gamma }.$$

Fig. 2

Illustration of the relation between the Neugebauer primary areas $a_{\mathbf{c}}$ and colorant overlap areas ${\beta }_{\mathbf{k}\left(\mathbf{c}\right)}$ for $K=3$ with CMY colorants. (Color online only.)

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

Eq. 6

$${\beta }_k=\sum _{\{c_1\dots c_K\mid c_k=1\}}{a}_{c_1\dots c_K}.$$

We extend this notation and use ${\beta }_{k_1\dots k_m}$ to represent the fractional area covered by the colorants $k_1\dots k_m$ (parts of which are also possibly covered by additional colorants), where $k_1\dots k_m$ is a set of distinct colorant indices drawn from $\{1,\dots ,K\}$ . For uniqueness, we consider only subscripts $k_1\dots k_m$ indices arranged in ascending order, that is $k_1<k_2<\cdots <k_m$ . Then, just as with the individual colorant areas, these fractional areas can be represented in terms of the Neugebauer primary areas as

Eq. 7

$${\beta }_{k_1\dots k_m}=\sum _{\{c_1\dots c_K\mid c_{k_1}=1,\dots ,c_{k_m}=1\}}{a}_{c_1\dots c_K}.$$

The above system of equations can be inverted to obtain an expression for the fractional areas of the primaries $a_{c_1\dots c_K}$ in terms of the fractional areas ${\beta }_{k_1\dots k_m}$ for the colorant combinations. One can see that $a_{1\dots 1}={\beta }_{1\dots K}$ and

Eq. 8

$$a_{c_1\dots c_K}=\{\begin{array}{cc}\hfill 1-\sum _{\mathbf{c}∊{\mathcal{I}}^{+}\left(\mathbf{c}\right)}{a}_{\mathbf{c}}\hfill & \hfill \mathrm{if}{c}_1\dots c_K=0\dots 0,\hfill \\ \hfill {\beta }_{\mathbf{k}\left(\mathbf{c}\right)}-\sum _{\mathbf{c}∊{\mathcal{I}}^{+}\left(\mathbf{c}\right)}{a}_{\mathbf{c}}\hfill & \hfill \mathrm{otherwise},\hfill \end{array}$$

where the notation $\mathbf{k}\left(\mathbf{c}\right)=k_1\dots k_m$ denotes the string of indices $\{k_l{\}}_{l=1}^m$ for which $c_{k_l}$ is nonzero, and ${\mathcal{I}}^{+}\left(\mathbf{c}\right)$ is the set of all indices that include the nonzero indices $c_{k_l}$ and at least one additional (distinct) nonzero index [Note that each element of ${\mathcal{I}}^{+}\left(\mathbf{c}\right)$ denotes a combination of colorants that include the nonzero $m$ colorants indicated by $\mathbf{c}$ 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 $a_{\mathbf{c}}$ and ${\beta }_{\mathbf{k}\left(\mathbf{c}\right)}$ for $K=3$ with CMY colorants.

Due to the equivalence of the Neugebauer primary fractional areas $a_{c_1\dots c_K}$ to the colorant overlap areas ${\beta }_{\mathbf{k}\left(\mathbf{c}\right)}$ , invariance properties with respect to displacement misregistration established for one are applicable to the other. We therefore proceed by obtaining expressions for the areas ${\beta }_{\mathbf{k}\left(\mathbf{c}\right)}$ . Note that we adopt this approach because it is easier to directly obtain expressions for the terms ${\beta }_{\mathbf{k}\left(\mathbf{c}\right)}$ , as opposed to the Neugebauer primary areas $a_{\mathbf{c}}$ .

3.3.

Fractional Areas of Colorant Combinations

To compute ${\beta }_{\mathbf{k}\left(\mathbf{c}\right)}$ , consider the overlay of the halftones that constitute $\mathbf{k}\left(\mathbf{c}\right)$ . The function

Eq. 9

$$h_{\mathbf{k}\left(\mathbf{c}\right)}(\mathbf{x};{\mathbf{d}}_{\mathbf{k}\left(\mathbf{c}\right)})=\prod _{k∊\mathbf{k}\left(\mathbf{c}\right)}{h}_k^{\left({\mathbf{d}}_k\right)}\left(\mathbf{x}\right)$$

indicates the spatial locations covered by the colorants in $\mathbf{k}\left(\mathbf{c}\right)$ , taking a value 1 if $\mathbf{x}$ is covered by the colorants in $\mathbf{k}\left(\mathbf{c}\right)$ and 0 otherwise, where ${\mathbf{d}}_{\mathbf{k}\left(\mathbf{c}\right)}=[{\mathbf{d}}_{k_1},\dots ,{\mathbf{d}}_{k_m}]$ are the displacement vectors of the individual separations that constitute $\mathbf{k}\left(\mathbf{c}\right)$ . The fractional area ${\beta }_{\mathbf{k}\left(\mathbf{c}\right)}$ is the spatial average of $h_{\mathbf{k}\left(\mathbf{c}\right)}(\mathbf{x};{\mathbf{d}}_{\mathbf{k}\left(\mathbf{c}\right)})$ .

If each of the matrices ${\mathsf{V}}_{{\mathit{k}}_i}^{-1}{\mathsf{V}}_{k_j}$ has only rational numbers as their elements for all pairs $k_i\ne k_j$ in $\mathbf{k}\left(\mathbf{c}\right)$ , the intersection of the lattices ${\mathbf{\Lambda }}_{\mathbf{k}\left(\mathbf{c}\right)}={\cap }_{k∊\mathbf{k}\left(\mathbf{c}\right)}{\mathbf{\Lambda }}_k$ is a 2D lattice,^{27, 28} whose periodicity may be represented in terms of a basis matrix ${\mathsf{V}}_{\mathbf{k}\left(\mathbf{c}\right)}$ . The basis matrix ${\mathsf{V}}_{\mathbf{k}\left(\mathbf{c}\right)}$ 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 ${\mathbf{\Lambda }}_{\mathbf{k}\left(\mathbf{c}\right)}$ includes at least one period of each of the constituent lattices. The overlay $h_{\mathbf{k}\left(\mathbf{c}\right)}(\mathbf{x};{\mathbf{d}}_{\mathbf{k}\left(\mathbf{c}\right)})$ of the constituent halftones is then periodic over this lattice and thus the spatial average is obtained as

Eq. 10

$${\beta }_{\mathbf{k}\left(\mathbf{c}\right)}=\frac{1}{\mid {\mathcal{U}}_{\mathbf{k}\left(\mathbf{c}\right)}\mid }{\int }_{\mathbf{x}∊{\mathcal{U}}_{\mathbf{k}\left(\mathbf{c}\right)}}{h}_{\mathbf{k}\left(\mathbf{c}\right)}\left(\mathbf{x}\right)d\mathbf{x},$$

where ${\mathcal{U}}_{\mathbf{k}\left(\mathbf{c}\right)}$ denotes a unit cell of ${\mathbf{\Lambda }}_{\mathbf{k}\left(\mathbf{c}\right)}$ and $\mid {\mathcal{U}}_{\mathbf{k}\left(\mathbf{c}\right)}\mid$ 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 $k,l∊\{1,\dots ,K\}$ , ${\mathsf{V}}_{\mathit{k}}^{-1}{\mathsf{V}}_l$ has only rational entries and thus the above assumption holds for any colorant combination in $\mathbf{c}$ . Situations in which the assumption does not hold may be viewed as a limiting case for out analysis, where $\mid {\mathcal{U}}_{\mathbf{k}\left(\mathbf{c}\right)}\mid \to \infty$ . Our ensuing frequency domain analysis [particularly Eq. ¹⁷] still holds for these situations even though the basis matrix ${\mathsf{V}}_{\mathbf{k}\left(\mathbf{c}\right)}$ does not exist.

Fig. 3

Example of lattice intersection. Panels (a) and (b) show points on lattices ${\mathbf{\Lambda }}_1$ and ${\mathbf{\Lambda }}_2$ , respectively. The larger blue circles in (c) show the intersection points of the overlaid lattices and the region inscribed by the dashed lines shows a unit cell of the intersection denoted by ${\mathcal{U}}_{12}$ .

From the Fourier transform properties it follows that

Eq. 11

$${\beta }_{\mathbf{k}\left(\mathbf{c}\right)}=H_{\mathbf{k}\left(\mathbf{c}\right)}\left(\mathbf{0}\right),$$

where $H_{\mathbf{k}\left(\mathbf{c}\right)}\left(\mathbf{u}\right)$ represents the Fourier transform of $h_{\mathbf{k}\left(\mathbf{c}\right)}\left(\mathbf{x}\right)$ and $\mathbf{u}=[u,v{]}^T$ denotes the coordinates in frequency space. In other words, ${\beta }_{\mathbf{k}\left(\mathbf{c}\right)}$ is the DC term of the frequency spectrum of the overlay of its constituent colorants. The Fourier transform of Eq. ⁹ yields

Eq. 12

$$H_{\mathbf{k}\left(\mathbf{c}\right)}\left(\mathbf{u}\right)=H_{k_1}^{\left({\mathbf{d}}_{k_1}\right)}\left(\mathbf{u}\right)*\dots *H_{k_m}^{\left({\mathbf{d}}_{k_m}\right)}\left(\mathbf{u}\right),$$

where $H_k^{\left({\mathbf{d}}_k\right)}\left(\mathbf{u}\right)$ represents the Fourier transform of the halftone image $h_k^{\left({\mathbf{d}}_k\right)}\left(\mathbf{x}\right)$ . Let $S_k\left(\mathbf{u}\right)$ represent the Fourier transform of the halftone spot function $s_k\left(\mathbf{x}\right)$ . Applying the shift and convolution property of the Fourier transform on Eq. ³, $H_k^{\left({\mathbf{d}}_k\right)}\left(\mathbf{u}\right)$ can be written as

Eq. 13

$$H_k^{\left({\mathbf{d}}_k\right)}\left(\mathbf{u}\right)=\frac{1}{\mid {\mathsf{V}}_{\mathit{k}}\mid }{S}_k\left(\mathbf{u}\right)\exp (-2\pi j{\mathbf{d}}_k^T\mathbf{u})\sum _{{\mathbf{n}}_k}\delta (\mathbf{u}-{\mathsf{W}}_k{\mathbf{n}}_k),$$

where the Fourier transform of the “comb” function ${\sum }_{{\mathbf{n}}_k}\delta (\mathbf{x}-{\mathsf{V}}_k{\mathbf{n}}_k)$ takes nonzero values on the elements of the reciprocal lattice of ${\mathbf{\Lambda }}_k$ (see Ref. 22, pp. 23–24),²⁸ which is represented by

Eq. 14

$${\mathbf{\Lambda }}_k^{*}=\{{\mathsf{W}}_k{\mathbf{n}}_k={\left({\mathsf{V}}_k^{-1}\right)}^T{\mathbf{n}}_k\mid {\mathbf{n}}_k∊{\mathbb{Z}}^2\},$$

where ${\mathsf{W}}_k=({\mathsf{V}}_k^{-1}{)}^T$ represents the basis matrix for the reciprocal lattice ${\mathbf{\Lambda }}_k^{*}$ .

Using these results we see that (see Appendix A)

Eq. 15

$$H_{\mathbf{k}\left(\mathbf{c}\right)}\left(\mathbf{u}\right)=\sum _{{\mathbf{n}}_{k_1}}\dots \sum _{{\mathbf{n}}_{k_m}}\prod _{k∊\mathbf{k}\left(\mathbf{c}\right)}\left(\frac{1}{\mid {\mathsf{V}}_k\mid }{S}_k\left({\mathsf{W}}_k{\mathbf{n}}_k\right)\exp (-2\pi j{\mathbf{d}}_k^T{\mathsf{W}}_k{\mathbf{n}}_k)\right)\delta (\mathbf{u}-\sum _{l∊\mathbf{k}\left(\mathbf{c}\right)}{\mathsf{W}}_l{\mathbf{n}}_l),$$

which can only take nonzero values if $\mathbf{u}={\sum }_{k∊\mathbf{k}\left(\mathbf{c}\right)}{\mathsf{W}}_k{\mathbf{n}}_k$ . Let ${\mathcal{N}}_{\mathbf{k}\left(\mathbf{c}\right)}$ represent the set $\left\{\right({\mathbf{n}}_{k_1},\dots ,{\mathbf{n}}_{k_m})\mid {\sum }_{i=1}^m{\mathsf{W}}_{k_i}{\mathbf{n}}_{k_i}=\mathbf{0}\}$ , which includes the indices of all the frequency vectors that sum up to the zero vector.

Then, Eq. ¹¹ can be computed as

Eq. 16

$${\beta }_{\mathbf{k}\left(\mathbf{c}\right)}=H_{\mathbf{k}\left(\mathbf{c}\right)}\left(\mathbf{0}\right)=\sum _{({\mathbf{n}}_{k_1},\dots ,{\mathbf{n}}_{k_m})∊{\mathcal{N}}_{\mathbf{k}\left(\mathbf{c}\right)}}\prod _{k∊\mathbf{k}\left(\mathbf{c}\right)}\frac{1}{\mid {\mathsf{V}}_k\mid }{S}_k\left({\mathsf{W}}_k{\mathbf{n}}_k\right)\exp (-2\pi j{\mathbf{d}}_k^T{\mathsf{W}}_k{\mathbf{n}}_k)=\prod _{k∊\mathbf{k}\left(\mathbf{c}\right)}\frac{1}{\mid {\mathsf{V}}_k\mid }{S}_k\left(\mathbf{0}\right)+\sum _{({\mathbf{n}}_{k_1},\dots ,{\mathbf{n}}_{k_m})∊{\mathcal{N}}_{\mathbf{k}\left(\mathbf{c}\right)}\backslash \left\{\mathbf{0}\right\}}\prod _{k∊\mathbf{k}\left(\mathbf{c}\right)}\frac{1}{\mid {\mathsf{V}}_k\mid }{S}_k\left({\mathsf{W}}_k{\mathbf{n}}_k\right)\exp (-2\pi j{\mathbf{d}}_k^T{\mathsf{W}}_k{\mathbf{n}}_k)$$

Eq. 17

$$=\prod _{k∊\mathbf{k}\left(\mathbf{c}\right)}{\beta }_k+\sum _{({\mathbf{n}}_{k_1},\dots ,{\mathbf{n}}_{k_m})∊{\mathcal{N}}_{\mathbf{k}\left(\mathbf{c}\right)}\backslash \left\{\mathbf{0}\right\}}\prod _{k∊\mathbf{k}\left(\mathbf{c}\right)}\frac{1}{\mid {\mathsf{V}}_k\mid }{S}_k\left({\mathsf{W}}_k{\mathbf{n}}_k\right)\exp (-2\pi j{\mathbf{d}}_k^T{\mathsf{W}}_k{\mathbf{n}}_k),$$

where ${\mathcal{N}}_{\mathbf{k}\left(\mathbf{c}\right)}\backslash \left\{\mathbf{0}\right\}$ denotes the elements in ${\mathcal{N}}_{\mathbf{k}\left(\mathbf{c}\right)}$ with the exclusion of the all-zero vector $\mathbf{0}$ . If ${\mathcal{N}}_{\mathbf{k}\left(\mathbf{c}\right)}\backslash \left\{\mathbf{0}\right\}$ 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.

4.

Conditions for Color Misregistration Insensitivity

Denote by $R_{avg}^{\left(\mathbf{0}\right)}\left(\lambda \right)$ and $R_{avg}^{\left(\mathbf{d}\right)}\left(\lambda \right)$ , the average spectra of “prints” with interseparation displacements $\mathbf{0}$ (perfect registration) and $\mathbf{d}=[{\mathbf{d}}_1,\dots ,{\mathbf{d}}_K]$ (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 $R_{avg}\left(\lambda \right)$ is affected by interseparation misregistration as we show in the following subsections. We also observe here that the terms $\{{\beta }_k{\}}_{k=1}^K$ , corresponding to the total fractional area covered by the individual colorants, are independent of misregistration.

4.2.

Periodicity Sufficiency Condition

Consider the expression in Eq. ¹⁷ in Sec. 3.3 for the fractional area of the colorant combination $\mathbf{k}\left(\mathbf{c}\right)$ . If ${\mathcal{N}}_{\mathbf{k}\left(\mathbf{c}\right)}\backslash \left\{\mathbf{0}\right\}$ is an empty set, or in other words, none of the fundamental frequency vectors or their harmonics sum up to the zero vector, then the corresponding fractional area ${\beta }_{\mathbf{k}\left(\mathbf{c}\right)}$ does not depend on the displacements $\{{\mathbf{d}}_k{\}}_{k∊\mathbf{k}\left(\mathbf{c}\right)}$ . If this property ( ${\mathcal{N}}_{\mathbf{k}\left(\mathbf{c}\right)}\backslash \left\{\mathbf{0}\right\}$ empty) holds for every possible colorant combination, then the equivalence between the Neugebauer primary areas and areas of colorant combinations ensures that the Neugebauer primary areas $a_{c_1\dots c_K}$ do not depend on the displacements $\{{\mathbf{d}}_k{\}}_{k=1}^K$ . Therefore, in this scenario, the average spectrum of the printed halftone $R_{avg}\left(\lambda \right)$ is invariant to interseparation misregistration. A set of halftone screens for which the above property [ ${\mathcal{N}}_{\mathbf{k}\left(\mathbf{c}\right)}\backslash \left\{\mathbf{0}\right\}$ empty for all $\mathbf{k}\left(\mathbf{c}\right)$ ] holds is referred to as a nonsingular halftone configuration.¹⁶

Let us visualize this case by examining the conventional 30-deg angular separation equifrequency CMK halftone screen overlay, which is commonly used in lithographic printing systems. Let ${\mathsf{V}}_1$ , ${\mathsf{V}}_2$ , and ${\mathsf{V}}_3$ , which are formed by the basis vectors shown in Fig. 4 , represent the basis matrices for the lattices ${\mathbf{\Lambda }}_1$ , ${\mathbf{\Lambda }}_2$ , and ${\mathbf{\Lambda }}_3$ for C, M, and K separations, respectively. We first consider the superposition of any two of these separations—for example, C and M. To compute ${\beta }_{12}$ , we first consider the set of indices ${\mathcal{N}}_{12}\backslash \left\{\mathbf{0}\right\}$ in Eq. ¹⁷. From the frequency domain basis matrices shown in Fig. 4, it can be seen that there do not exist any $({\mathbf{n}}_1,{\mathbf{n}}_2)∊{\mathbb{Z}}^2$ that can satisfy ${\mathsf{W}}_1{\mathbf{n}}_1+{\mathsf{W}}_2{\mathbf{n}}_2=\mathbf{0}$ . Thus, ${\mathcal{N}}_{12}\backslash \left\{\mathbf{0}\right\}$ is an empty set and ${\beta }_{12}$ is invariant to interseparation misregistration. Similarly, it can be shown that ${\mathcal{N}}_{13}\backslash \left\{\mathbf{0}\right\}$ and ${\mathcal{N}}_{23}\backslash \left\{\mathbf{0}\right\}$ are also empty sets and consequently ${\beta }_{13}$ and ${\beta }_{23}$ are also invariant to interseparation misregistration. Note that, in these cases, ${\beta }_{\mathbf{k}\left(\mathbf{c}\right)}$ terms are only determined by the first term in Eq. ¹⁷, which is the multiplication of the DC terms in each individual separation that constitutes $\mathbf{k}\left(\mathbf{c}\right)$ and, therefore, the statistical randomness condition assumed by Demichel equations is satisfied.

Fig. 4

Basis vectors for CMK colorants in conventional screens. (Color online only.)

Now consider overlay of all three of the separations and the ${\beta }_{123}$ . In this case, we can see that ${\mathcal{N}}_{123}\backslash \left\{\mathbf{0}\right\}$ is not an empty set and the overlay is singular. For example, ${\mathbf{n}}_1=[-1,0{]}^T$ , ${\mathbf{n}}_2=[1,0{]}^T$ , and ${\mathbf{n}}_3=[0,1{]}^T$ is a member of ${\mathcal{N}}_{123}\backslash \left\{\mathbf{0}\right\}$ . The set ${\mathcal{N}}_{123}\backslash \left\{\mathbf{0}\right\}$ has an infinite number of elements and color sensitivity to misregistration is expected for the overlay of these three separations. Figure 5 illustrates this behavior of the conventional CMK halftone overlay. The configuration of Fig. 5, known as the clear-centered rosette, transforms into the so-called dot-centered rosette configuration of Fig. 5 when each of the separations is displaced by half its lattice period. The change from Fig. 5 and 5 illustrates the substantial change in the microstructure due to interseparation misregistration in a singular overlay.

Fig. 5

Sensitivity of CMK overlap in conventional screens to misregistration. A half period displacement of each of the screen transforms the microstructure from the clear-centered rosette of (a) to the dot-centered rosette in (b). (Color online only.)

4.3.

Spot Function Dependence

From Eq. ¹⁷, we observe that if the set ${\mathcal{N}}_{\mathbf{k}\left(\mathbf{c}\right)}\backslash \left\{\mathbf{0}\right\}$ is nonempty, this alone does not ensure that $R_{avg}^{\left(\mathbf{d}\right)}\left(\lambda \right)$ and $R_{avg}^{\left(\mathbf{0}\right)}\left(\lambda \right)$ differ, because ${\beta }_{\mathbf{k}\left(\mathbf{c}\right)}$ terms are functions of the constituent halftone spots and the interseparation misregistration amounts. Depending on these, interseparation misregistration may still have no effect on the average color of the halftone if the summation in Eq. ¹⁷ is either zero or remains constant as the displacement $\mathbf{d}$ is varied. This condition holds trivially when colorant coverages take on only values of 0 or 100%, but it can also hold for nontrivial cases, as we illustrate next by means of an example.

Consider an overlay of two separations with lattices ${\mathbf{\Lambda }}_1$ and ${\mathbf{\Lambda }}_2$ having the basis matrices

$${\mathsf{V}}_1=\left[\begin{array}{cc}\hfill M\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill M\hfill \end{array}\right]\mathrm{and}{\mathsf{V}}_2=\left[\begin{array}{cc}\hfill M\hfill & \hfill M\hfill \\ \hfill M\hfill & \hfill -M\hfill \end{array}\right],$$

respectively. Let us define the corresponding halftone spot functions $s_1\left(\mathbf{x}\right)$ and $s_2\left(\mathbf{x}\right)$ as shown in Fig. 6 within unit cells of the constituent lattices outlined by the dashed lines.

Fig. 6

Spot functions $s_1\left(\mathbf{x}\right)$ and $s_2\left(\mathbf{x}\right)$ exhibit misregistration invariance despite a potentially sensitive geometry.

These functions can also be represented as

Eq. 19

$$s_1\left(\mathbf{x}\right)=\mathrm{rect}\left(\frac{x-y}{M}\right)\mathrm{rect}\left(\frac{x+y}{M}\right)$$

and

Eq. 20

$$s_2\left(\mathbf{x}\right)=\mathrm{rect}\left(\frac{x}{M}\right)\mathrm{rect}\left(\frac{y}{M}\right).$$

The respective Fourier transforms of these functions can be written as

Eq. 21

$$S_1\left(\mathbf{u}\right)=\frac{M^2}{2}\mathrm{sinc}\left(\frac{u-v}{2}M\right)\mathrm{sinc}\left(\frac{u+v}{2}M\right)$$

and

Eq. 22

$$S_2\left(\mathbf{u}\right)=M^2\mathrm{sinc}\left(uM\right)\mathrm{sinc}\left(vM\right).$$

The summation in Eq. ¹⁷ requires the value of the previous functions at the frequency coordinates ${\mathsf{W}}_1{\mathbf{n}}_1$ and ${\mathsf{W}}_2{\mathbf{n}}_2$

Eq. 23

$$S_1\left({\mathsf{W}}_1{\mathbf{n}}_1\right)=S_1\left(\frac{1}{M}\left[\begin{array}{c}\hfill n_{1x}\hfill \\ \hfill n_{1y}\hfill \end{array}\right]\right)=\frac{M^2}{2}\mathrm{sinc}\left(\frac{n_{1x}-n_{1y}}{2}\right)\mathrm{sinc}\left(\frac{n_{1x}+n_{1y}}{2}\right)$$

and

Eq. 24

$$S_2\left({\mathsf{W}}_2{\mathbf{n}}_2\right)=S_2\left(\frac{1}{2M}\left[\begin{array}{c}\hfill n_{2x}+n_{2y}\hfill \\ \hfill n_{2x}-n_{2y}\hfill \end{array}\right]\right)=M^2\mathrm{sinc}\left(\frac{n_{2x}+n_{2y}}{2}\right)\mathrm{sinc}\left(\frac{n_{2x}-n_{2y}}{2}\right).$$

The indices ${\mathbf{n}}_1$ and ${\mathbf{n}}_2$ for the summation in Eq. ¹⁷ are defined by the condition

Eq. 25

$${\mathsf{W}}_1{\mathbf{n}}_1+{\mathsf{W}}_2{\mathbf{n}}_2=\frac{1}{M}\left[\begin{array}{c}\hfill n_{1x}+\frac{n_{2x}+n_{2y}}{2}\hfill \\ \hfill n_{1y}+\frac{n_{2x}-n_{2y}}{2}\hfill \end{array}\right]=\mathbf{0}.$$

The previous relation implies that only indices such that $n_{2x}$ or $n_{2y}$ are both odd or both even can contribute terms in ${\mathcal{N}}_{12}\backslash \left\{\mathbf{0}\right\}$ . However, in these cases, the value of the sinc functions in Eq. ²⁴ is zero. As Eq. ¹⁷ suggests, this ensures ${\beta }_{12}$ is the multiplication of the fractional area coverages of the individual separations, and therefore, the characteristics of the halftone spots can define a condition to ensure that $R_{avg}\left(\lambda \right)$ is insensitive to interseparation misregistration. In Fig. 7 , we show an example using this configuration in which colorant overlap area does not change even though the microstructure changes due to interseparation misregistration. We note that for this specific configuration, the invariance under misregistration depends only on the spot function $s_2\left(\mathbf{x}\right)$ and holds for any arbitrary choice of the spot function $s_1\left(\mathbf{x}\right)$ .

Fig. 7

Halftone images generated by using the periodicities and spot functions shown in Figs. 6 and 6 for C and M colorants, respectively. Observe that Neugebauer primary areas do not change for this configuration with change in misregistration displacement. (Color online only.)

4.4.

Invariant Misregistrations

Now we consider the invariant misregistration combinations, that is, the separation displacements for which we are assured zero color shift with respect to the perfectly registered print. It is readily seen from Eq. ¹⁷ that if the displacement of the $k$ ’th separation is a point on the corresponding lattice, the term ${\mathbf{d}}_k^T{\mathsf{W}}_k{\mathbf{n}}_k$ is integer-valued and the result of the summation is identical to that for a perfectly registered halftone. This represents the trivial case when the interseparation displacements are matched to the separations’ lattice periodicities. Invariance is, however, also achievable for nontrivial displacements, as we illustrate next. We assume in our analysis that for the lattices under consideration, the basis matrices satisfy the constraint ${\mathsf{V}}_{k_i}^{-1}{\mathsf{V}}_{{\mathit{k}}_j}∊{\mathbb{Q}}^{2\times 2}$ , where $\mathbb{Q}$ is the set of rational numbers so that the intersection and sum lattices are defined for any subset of lattices.^{27, 28}

Consider, again, an overlay of two separations with lattices ${\mathbf{\Lambda }}_1$ and ${\mathbf{\Lambda }}_2$ . In this case, Eq. ¹⁷ simplifies to

Eq. 26

$${\beta }_{12}={\beta }_1{\beta }_2+\sum _{({\mathbf{n}}_1,{\mathbf{n}}_2)∊{\mathcal{N}}_{12}\backslash \left\{\mathbf{0}\right\}}\frac{1}{\mid {\mathsf{V}}_1\mid \mid {\mathsf{V}}_{2\mid }}{S}_1\left({\mathsf{W}}_1{\mathbf{n}}_1\right)S_2\left({\mathsf{W}}_2{\mathbf{n}}_2\right)\exp [-2\pi j({\mathbf{d}}_1^T{\mathsf{W}}_1{\mathbf{n}}_1+{\mathbf{d}}_2^T{\mathsf{W}}_2{\mathbf{n}}_2)].$$

If the terms $\exp [-2\pi j({\mathbf{d}}_1^T{\mathsf{W}}_1{\mathbf{n}}_1+{\mathbf{d}}_2^T{\mathsf{W}}_2{\mathbf{n}}_2\left)\right]$ in the above summation are unity for all $({\mathbf{n}}_1,{\mathbf{n}}_2)∊{\mathcal{N}}_{12}\backslash \left\{\mathbf{0}\right\}$ , then we can readily see that the value of ${\beta }_{12}$ is equal to the value obtained for the perfectly registered case $({\mathbf{d}}_1={\mathbf{d}}_2=\mathbf{0})$ . We readily see that this happens if the terms $({\mathbf{d}}_1^T{\mathsf{W}}_1{\mathbf{n}}_1+{\mathbf{d}}_2^T{\mathsf{W}}_2{\mathbf{n}}_2)$ are integer-valued for all $({\mathbf{n}}_1,{\mathbf{n}}_2)∊{\mathcal{N}}_{12}\backslash \left\{\mathbf{0}\right\}$ . Suppose $({\mathbf{n}}_1,{\mathbf{n}}_2)∊{\mathcal{N}}_{12}\backslash \left\{\mathbf{0}\right\}$ , that is, they are nonzero integer vectors satisfying ${\mathsf{W}}_1{\mathbf{n}}_1+{\mathsf{W}}_2{\mathbf{n}}_2=\mathbf{0}$ . Then, ${\mathsf{W}}_2{\mathbf{n}}_2=-{\mathsf{W}}_1{\mathbf{n}}_1$ and $({\mathbf{d}}_1^T{\mathsf{W}}_1{\mathbf{n}}_1+{\mathbf{d}}_2^T{\mathsf{W}}_2{\mathbf{n}}_2)=({\mathbf{d}}_1-{\mathbf{d}}_2{)}^T{\mathsf{W}}_1{\mathbf{n}}_1=({\mathbf{d}}_2-{\mathbf{d}}_1{)}^T{\mathsf{W}}_2{\mathbf{n}}_2$ . Now ${\mathsf{W}}_2{\mathbf{n}}_2∊{\mathbf{\Lambda }}_2^{*}$ and $-{\mathsf{W}}_1{\mathbf{n}}_1∊{\mathbf{\Lambda }}_1^{*}$ , hence $\mathbf{w}={\mathsf{W}}_2{\mathbf{n}}_2=-{\mathsf{W}}_1{\mathbf{n}}_1∊{\mathbf{\Lambda }}_1^{*}\cap {\mathbf{\Lambda }}_2^{*}$ and vice versa if $\mathbf{w}∊{\mathbf{\Lambda }}_1^{*}\cap {\mathbf{\Lambda }}_2^{*}$ , then there exists a $\mathbf{w}={\mathsf{W}}_1{\mathbf{n}}_1^{\prime }=-{\mathsf{W}}_2{\mathbf{n}}_2^{\prime }$ , whence $({\mathbf{n}}_1^{\prime },{\mathbf{n}}_2^{\prime })∊{\mathbb{Z}}^2$ such that ${\mathsf{W}}_1{\mathbf{n}}_1^{\prime }+{\mathsf{W}}_2{\mathbf{n}}_2^{\prime }=\mathbf{0}$ . From the definition of the reciprocal lattice, it follows that $({\mathbf{d}}_1-{\mathbf{d}}_2{)}^T{\mathsf{W}}_1{\mathbf{n}}_1$ is an integer if and only if ${\mathbf{d}}_2-{\mathbf{d}}_1$ is an element of $({\mathbf{\Lambda }}_1^{*}\cap {\mathbf{\Lambda }}_2^{*}{)}_{\hspace{0.5em}}^{*}$ , that is, the reciprocal lattice of ${\mathbf{\Lambda }}_1^{*}\cap {\mathbf{\Lambda }}_2^{*}$ . Because $({\mathbf{\Lambda }}_1^{*}\cap {\mathbf{\Lambda }}_2^{*}{)}_{\hspace{0.5em}}^{*}={\mathbf{\Lambda }}_1+{\mathbf{\Lambda }}_2$ , ${\mathbf{d}}_1-{\mathbf{d}}_2∊{\mathbf{\Lambda }}_1+{\mathbf{\Lambda }}_2$ defines a sufficient condition that ensures ${\beta }_{12}$ is invariant to misregistration, where

Eq. 27

$${\mathbf{\Lambda }}_1+{\mathbf{\Lambda }}_2\stackrel{\mathrm{def}}{=}\{{\mathbf{x}}_1+{\mathbf{x}}_2\mid {\mathbf{x}}_1∊{\mathbf{\Lambda }}_1,{\mathbf{x}}_2∊{\mathbf{\Lambda }}_2\}$$

is the sum lattice of ${\mathbf{\Lambda }}_1,{\mathbf{\Lambda }}_2$ . One can also infer that the sum lattice ${\mathbf{\Lambda }}_1+{\mathbf{\Lambda }}_2$ defines the periodicity of color shift in the interseparation displacement. The color shift for any interseparation displacement ${\mathbf{d}}^{12}={\mathbf{d}}_1-{\mathbf{d}}_2$ is equal to the color shift obtained with an equivalent displacement ${\mathbf{d}}_e^{12}$ lying in the unit cell of ${\mathbf{\Lambda }}_1+{\mathbf{\Lambda }}_2$ . The equivalent displacement ${\mathbf{d}}_e^{12}$ is “ ${\mathbf{d}}^{12}$ modulo ${\mathbf{\Lambda }}_1+{\mathbf{\Lambda }}_2$ ,” that is, the unique vector in the set $\{{\mathbf{d}}^{12}+{\mathsf{V}}_{{\mathbf{\Lambda }}_1+{\mathbf{\Lambda }}_2}\mathbf{n}\mid \mathbf{n}∊{\mathbb{Z}}^2\}$ that lies in the unit cell of the sum lattice ${\mathbf{\Lambda }}_1+{\mathbf{\Lambda }}_2$ , where ${\mathsf{V}}_{{\mathbf{\Lambda }}_1+{\mathbf{\Lambda }}_2}$ denotes the basis matrix for ${\mathbf{\Lambda }}_1+{\mathbf{\Lambda }}_2$ . A basis matrix for ${\mathbf{\Lambda }}_1+{\mathbf{\Lambda }}_2$ can be computed as the greatest common left divisor (gcld) of the basis matrices for ${\mathbf{\Lambda }}_1$ and ${\mathbf{\Lambda }}_2$ .

In the general $K$ -separation scenario, as shown in Appendix B, the invariant misregistrations are characterized as constraints on the pairwise displacements expressed mathematically as

Eq. 28

$${\mathbf{d}}_i-{\mathbf{d}}_j∊{\mathbf{\Lambda }}_i+{\mathbf{\Lambda }}_j\mathrm{for}\mathrm{all}\mathrm{pairs}i\ne j\mathrm{in}\{1,\dots ,K\}.$$

In addition, one can infer this also defines the periodicity of the misregistration-induced color shift in the space of interseparation displacement vectors ${\mathbf{d}}_i-{\mathbf{d}}_j$ , for all $i,j∊\{1,\dots ,K\}$ . Thus, if $({\mathbf{d}}_1,\dots ,{\mathbf{d}}_K)$ and $({\mathbf{d}}_1^{\prime },\dots ,{\mathbf{d}}_K^{\prime })$ are two misregistration displacement vectors, their misregistration-induced color shift is equal if for all pairs of separations $i$ and $j$ , $({\mathbf{d}}_i-{\mathbf{d}}_j)$ is congruent with $({\mathbf{d}}_i^{\prime }-{\mathbf{d}}_j^{\prime })$ modulo the sum lattice ${\mathbf{\Lambda }}_i+{\mathbf{\Lambda }}_j$ , that is, $({\mathbf{d}}_i-{\mathbf{d}}_j)-({\mathbf{d}}_i^{\prime }-{\mathbf{d}}_j^{\prime })∊{\mathbf{\Lambda }}_i+{\mathbf{\Lambda }}_j$ . In Fig. 8 , we show color misregistration invariant configuration examples generated using the halftone periodicities shown in Fig. 3.

Fig. 8

Invariant misregistrations. Panel (a) shows perfect registration, and (b), (c), and (d) show nontrivial misregistrations under which average color is invariant. The top half of each subfigure shows an integral number of periods of the intersection lattice and the lower half shows the displacement misregistration vector ${\mathbf{d}}_2$ on the lattice ${\mathbf{\Lambda }}_2$ . Note that ${\mathbf{d}}_1=\mathbf{0}$ and ${\mathbf{d}}_2∊{\mathbf{\Lambda }}_1+{\mathbf{\Lambda }}_2$ for all configurations and Neugebauer primary areas are invariant to misregistration in these cases. (Color online only.)

5.

Quantitative Evaluation of Misregistration-Induced Color Shifts

The analysis of the preceding section characterized the conditions under which misregistration sensitivity may be encountered without quantifying the amount of color shift (i.e., sensitivity). In this section, we consider a simulation model that builds upon the analysis already presented and allows quantitative evaluation of color misregistration sensitivity. For this purpose, we compute the average reflectance using our model of Eq. ⁵ for two cases: one corresponding to perfectly registered separations and the other for the shift in consideration. These values may then be transformed to the approximately perceptually uniform Commission Internationale d’ Eclairage L*a*b (CIELAB) color space.³⁰ The color difference resulting from the misregistration can then be obtained in $\mathrm{\Delta }{E}_{ab}^{*}$ units as the Euclidean distance between the CIELAB pairs. We are interested in the dependence of this color difference on the halftone lattices $\{{\mathbf{\Lambda }}_k{\}}_{k=1}^K$ , the colorant area coverages $\{{\beta }_k{\}}_{k=1}^K$ , and the separation displacements $\{{\mathbf{d}}_k{\}}_{k=1}^K$ . The overall system for quantitative estimation of the color shift is illustrated in Fig. 9 . By repeating the process for different values of these elements as well as Neugebauer primary spectra, and YN parameter $\gamma$ , we can obtain quantitative estimates of the misregistration-induced color shift as a function of these parameters. The process requires a calculation of the colorant overlap areas ${\beta }_{\mathbf{k}\left(\mathbf{c}\right)}$ , which we consider next.

Fig. 9

Simulation model for quantitative estimation of color shift induced by interseparation misregistration for a clustered-dot color halftone.

5.2.

Halftone Spot Model

The computation in Eq. ²⁹ requires the halftone spot functions $\left\{s_k\right(\mathbf{x}){\}}_{k∊\mathbf{k}\left(\mathbf{c}\right)}$ for the respective separations. In practice, these functions are produced as a result of thresholding the contone image for the $k$ ’th separation ${\tau }_k\left(\mathbf{x}\right)$ against a threshold function for that separation $T_k\left(\mathbf{x}\right)$ . For most reasonable threshold functions, the spot function $s_k\left(\mathbf{x}\right)$ may be uniquely determined by the corresponding colorant area coverage ${\beta }_k$ . For orthogonal screens, a useful threshold function was defined by Pellar and Green^{31, 32} as

Eq. 30

$$T\left(\mathbf{x}\right)=\cos \left(2\pi f_{x}x\right)+\cos \left(2\pi f_{y}y\right),$$

where $f_x$ and $f_y$ are the screen frequencies, $x$ and $y$ are the respective spatial coordinates (along two orthogonal spatial directions) [In digital halftoning, this threshold function is usually defined as a discrete array on a discretized unit cell of the corresponding lattice, we will however find the analytic representation more convenient for our purposes]. For an arbitrary (possibly nonorthogonal) halftone, whose periodicity is given by the lattice ${\mathbf{\Lambda }}_k$ , we generalize the equation to obtain the threshold function

Eq. 31

$$T_k\left(\mathbf{x}\right)=\cos \left(2\pi \mid {\mathbf{w}}_1^k\mid x^{\prime }\right)+\cos \left(2\pi \mid {\mathbf{w}}_2^k\mid y^{\prime }\right),$$

where ${\mathsf{W}}_k=\left[{\mathbf{w}}_1^k{\mathbf{w}}_2^k\right]=\left[\begin{array}{cc}\hfill w_{x_1}^k\hfill & \hfill w_{x_2}^k\hfill \\ \hfill w_{y_1}^k\hfill & \hfill w_{y_2}^k\hfill \end{array}\right]$ is the basis matrix of ${\mathbf{\Lambda }}_k^{*}$ . The transformation

Eq. 32

$$\left[\begin{array}{c}\hfill x^{\prime }\hfill \\ \hfill y^{\prime }\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill \cos \left({\varphi }_k^x\right)\hfill & \hfill -\sin \left({\varphi }_k^x\right)\hfill \\ \hfill \sin \left({\varphi }_k^y\right)\hfill & \hfill \cos \left({\varphi }_k^y\right)\hfill \end{array}\right]\left[\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right]$$

represents a shearing of the coordinate system that ensures a period of Eq. ³¹ maps to a unit cell of ${\mathbf{\Lambda }}_k$ , and ${\varphi }_k^x=\arctan (w_{y_1}^k∕w_{x_1}^k)$ and ${\varphi }_k^y=(w_{y_2}^k∕w_{x_2}^k)$ .

Individual colorant halftone separations may be obtained by thresholding the contone value for the colorant channel against the corresponding threshold function. Specifically,

Eq. 33

$$h_k\left(\mathbf{x}\right)=\{\begin{array}{cc}\hfill 1\hfill & \hfill \mathrm{if}{\tau }_k\left(\mathbf{x}\right)>T_k\left(\mathbf{x}\right),\hfill \\ \hfill 0\hfill & \hfill \mathrm{otherwise},\hfill \end{array}$$

defines the $k$ ’th halftone separation. For spatially constant contone images $\left[{\tau }_k\right(\mathbf{x})={\tau }_k,\forall \mathbf{x}]$ the resulting halftone corresponds to the model of Eq. ², however, the corresponding spot functions $\left\{s_k\right(\mathbf{x}){\}}_{k=1}^K$ do not allow for a closed form computation of the convolution terms in Eq. ²⁹ and these must therefore be obtained by computer simulation.

For quantitative evaluation of the colorant overlap area ${\beta }_{\mathbf{k}\left(\mathbf{c}\right)}$ , we consider a finely sampled representation of the unit cell ${\mathcal{U}}_{\mathbf{k}\left(\mathbf{c}\right)}$ and corresponding to each pixel of this sampled representation generate binary halftone separation values using Eq. ³³. The area ${\beta }_{\mathbf{k}\left(\mathbf{c}\right)}$ may then be obtained by counting the fraction of pixels at which the colorants overlap.

Accurate computations using this methodology require high resolution grids. In addition, in some configurations, the size of ${\mathcal{U}}_{\mathbf{k}\left(\mathbf{c}\right)}$ may become unwieldy for the generation of images. We therefore consider a further simplification and in practice use either of these two approaches as appropriate.

5.2.1.

Simplified analytic halftone spot model

The computation and storage requirements can be significantly reduced by using simple analytic spot functions that provide realistic approximations to actual halftone spot shapes for which overlap areas are obtained more readily from geometric relations. For this simplified model, we assume ${\mathbf{w}}_1^k$ and ${\mathbf{w}}_2^k$ are orthogonal to each other and $\mid {\mathbf{w}}_1^k\mid =\mid {\mathbf{w}}_2^k\mid$ because orthogonal equifrequency halftone screens are commonly employed in clustered-dot color halftoning. Under this assumption, we examine the dependence of the spot function on the area coverage ${\beta }_k$ as shown in Fig. 10 , where a linear contone ramp (ranging from white to black) has been thresholded using the function of Eq. ³¹. We observe that the gray levels progressing from highlight to midtone can be modeled as growing black spots on a white background up to $\beta =0.5$ . From midtone to shadow gray levels, the halftone spots can be modeled as shrinking white holes on a black background, where the holes are displaced by half the lattice period relative to the black spots. Based on the observed shapes in Fig. 10, we model these spots and holes as circles in highlights and shadows, and squares in midtone gray levels. A closeness metric is employed to determine the fractional area coverage at which the model switches between the circle and square approximations. Based on the symmetry, it suffices to consider area coverages $\beta ∊[0,0.5]$ to select the switch point. For a given $\beta$ , let $s^{\beta }\left(\mathbf{x}\right)$ denote the spot obtained from Eq. ³³ with the corresponding area coverage. The error in approximating $s^{\beta }\left(\mathbf{x}\right)$ by a circular or square spot is then evaluated as the area that lies in $s^{\beta }\left(\mathbf{x}\right)$ or the approximating spot function, but not both (also called the symmetric difference or XOR). The shape (circle or square) providing the closest approximation is then used to approximate $s^{\beta }\left(\mathbf{x}\right)$ as illustrated in Fig. 11 . For the gray level used in the figure, we can see that the closer approximation for this specific case corresponds to the circle. Overlap area difference between $s^{\beta }\left(\mathbf{x}\right)$ and the closest circle and square approximation is shown in Fig. 11 as a function of the area coverage $\beta$ . As expected, in highlights and shadows, circular spots provide a closer approximation than square spots and vice versa in midtones. At approximately $\beta =0.35$ , the error curves for square and circular spots intersect. Therefore,for $\beta ∊[0.35,0.65]$ , we use the square halftone spot approximation for $s^{\beta }\left(\mathbf{x}\right)$ , and for $\beta ∊[0,0.35)\cup (0.65,1]$ , we use the circular approximation.

Fig. 10

Spot growth with the threshold function defined in Eq. ³¹ when $\mid {\mathbf{w}}_1\mid =\mid {\mathbf{w}}_2\mid$ , and ${\mathbf{w}}_1$ and ${\mathbf{w}}_2$ are orthogonal.

Fig. 11

Circle and square spot approximations to halftone spot function $s^{\beta }\left(\mathbf{x}\right)$ . Panel (a) shows an example of best approximations, (b) shows the error for the best circle and square approximations to $s^{\beta }\left(\mathbf{x}\right)$ as a function of fractional area coverage $\beta$ , where the error corresponds to the area lying in $s^{\beta }\left(\mathbf{x}\right)$ or the approximation, but not both.

Assuming the halftone spot of a separation is either a circle or a square, a composite halftone image is composed of the overlay of these shapes at different displacements. The intersection of circles and squares can be found using simple algebraic methods,^{15, 33} and once these points are determined, it is easy to compute the overlap area of these halftone spots geometrically. In particular, circles and squares are convex shapes, thus their intersection with other circles or squares also generates a convex region. Neighboring vertices of an intersection region are always connected to each other by either a straight line or a circular arc depending on the shape of its constituent halftone spots. Thus, an intersection region may be composed of a polygon, circular segments, or both.

For the purpose of illustration, we show the intersection of three halftone spots (two circles and a square) in Fig. 12 . In this figure, the intersection region is formed by the polygon $P_1\to P_2\to P_3\to P_4\to P_1$ and two circular segments attached to this polygon, whose secant lines are $\overline{P_2{P}_3}$ and $\overline{P_3{P}_4}$ . Each of these regions is shown in Fig. 12 using different shades of gray. The areas for these regions are readily computed, from which, the overlap area is directly obtained by summing the three indicated areas together. Within the unit cell ${\mathcal{U}}_{123}$ , there are multiple instances of halftone spots of separations $1,2,$ and $3$ . The aforementioned computation is repeated for each possible selection that incorporates a spot from each of these separations. The sum of the areas from these individual computations is divided by the area of the unit cell ${\mathcal{U}}_{123}$ to obtain ${\beta }_{123}$ .

Fig. 12

An example of three intersecting halftone spots showing computation of areas that contribute to ${\beta }_1,{\beta }_2,{\beta }_3$ , and ${\beta }_{123}$ .

This geometric model does not have a resolution constraint and computes significantly faster than the pixel-counting approach. We quantify this computational advantage in Table 5 in Sec. 8.2.

Table 5

Average computation time required to compute Neugebauer primary fractional areas by using pixel-counting and proposed method. A 2.8-GHz Intel Pentium 4 PC system with 2 GBytes of main memory running on Microsoft Windows XP was used for timing simulations in MATLAB 7.0.1.

		Pixel-Count Method		Algebraic Method
β1	β2	tavg (sec)	ΔEab*	tavg (sec)	ΔEab*
0.4	0.4	5.88	1.74	0.22	1.63
0.3	0.3	5.88	0.35	0.13	0.46
0.4	0.3	5.88	0.69	0.33	0.60

6.

Model Parameters

Parameter values necessary for an exercising model of Fig. 9 can be obtained from experimental data. We briefly outline this process here. The spectra of $2^K$ Neugebauer primaries can be directly measured from prints of patches of each of the primaries. Single-colorant ramps for each of the $K$ separations are used to relate the digital control values that drive the printer to the respective area coverages $\{{\beta }_k{\}}_{k=1}^K$ . For a single-colorant halftone image $h_k\left(\mathbf{x}\right)$ , the fractional area coverage ${\beta }_k$ is derived from the Neugebauer estimate of the spectrum using a least-squares procedure.³⁴ For a single-colorant image, Eq. ⁵ becomes

The previous procedure requires the value of the YN coefficient $\gamma$ . The optimal value of $\gamma$ is selected from a set of candidate values by determining the value that minimizes the average mean-square error between the predicted reflectance value in Eq. ⁵ and the measured reflectance values over the complete set of colorant ramps.

7.

Simulation and Experimental Results

Our experimental setup used a xerographic CMYK printer with an addressability of $4800\times 600\mathrm{dpi}$ [For convenience, we will state our lattice basis matrices in terms of a printer having identical addressability of $600\mathrm{dpi}$ along each of the two orthogonal directions. Equivalent numbers for the $4800\times 600\mathrm{dpi}$ case are readily obtained by multiplying the first-rows of the basis matrices by a factor of 8 corresponding to the ratio of addressabilities in the horizontal and vertical directions]. The 16 Neugebauer primaries for the printer were measured from printed targets. Digital halftone screens with angular orientations close to the conventional analog screens were used as the primary halftones for our study although alternative halftone configurations were also investigated. The lattice periodicities for these halftones are specified by the basis matrices listed in Table 1 . These orthogonal screens are oriented as C at ${\tan }^{-1}(1∕3)\simeq 18.4\mathrm{deg}$ , M at ${\tan }^{-1}\left(3\right)\simeq 71.6\mathrm{deg}$ , Y at ${\tan }^{-1}\left(0\right)=0\mathrm{deg}$ , and K at ${\tan }^{-1}\left(1\right)=45\mathrm{deg}$ . These angles are chosen to approximate the conventional 30-deg difference between the frequency vectors of analog C, M, and K screens shown in Fig. 4. Thus, this configuration is referred to as the digital conventional configuration throughout this section. In addition to this configuration, we also explore two-colorant dot-on-dot/dot-off-dot halftone configurations.⁴ These configurations employ the same lattice periodicity for the two-colorant separations and maximize/minimize the overlap between the halftone spots of the two separations, respectively. The dot-on-dot configuration is obtained by using the same threshold function for the two separations. If the threshold function for one of the separations is displaced by half a period, a corresponding dot-off-dot configuration is obtained. The latter thus represents a misregistered version of the former. This specific halftone configuration and misregistration typically leads to the largest change in color.^{1, 35} It is, therefore, particularly helpful for studying the effects of parameters other than the halftone periodicities. In our experiments, the basis matrix corresponding to the black (K) separation of Table 1 is used when referring to dot-on-dot and dot-off-dot geometries.

Table 1

CMYK halftone periodicities for the digital conventional configuration (for printer addressability 600×600dpi ; lpi=lines per inch).

As outlined in Sec. 6, single-colorant ramps of the C, M, Y, and K colorants that ranged in area coverage from 0 to 100% were printed and utilized to estimate the relationship of digital CMYK values to the colorant fractional areas $\{{\beta }_k{\}}_{k=1}^4$ . The YN parameter $\gamma$ was estimated as 1.4 and used throughout for the digital conventional halftone configuration. Color computations were performed using the CIE D50 standard illuminant³⁰ and with a white point corresponding to the unprinted paper.

We carry out several simulations in which we investigate the change in average halftone color due to interseparation misregistration as a function of various experimental parameters. Specifically, we consider the impact of optical dot gain, colorant combinations, periodicity of the individual halftone separations, colorant area coverages, and misregistration amounts. We present results that explore various subsets of these factors. In doing so, we focus our attention on two-screen combinations because this allows us to explore the remaining parameter space more comprehensively and present results visually. We also consider specific questions of interest for three-screen CMK combinations toward the end of this section.

7.1.

Optical Dot Gain

We consider the impact of optical dot gain on color misregistration sensitivity by evaluating color shifts (using the methodology of Fig. 9) for different values of the YN parameter $\gamma$ . In general, an increase in the optical dot gain parameter $\gamma$ tends to reduce the magnitude of misregistration-induced color shifts. This is in agreement with physical intuition. As the optical dot gain increases, there is greater mixing of light entering the paper through the differently colored regions corresponding to the various Neugebauer primaries. This mixing reduces the dependence of the average spectrum/color on interseparation misregistration.

In Fig. 13 , we illustrate this dependence on $\gamma$ for the sensitive CM dot-on-dot halftone configuration under the situation, where the fractional area coverages ${\beta }_1$ and ${\beta }_2$ corresponding to the C and M separations are equal. The abscissa of the graph in Fig. 13 represents this fractional area coverage and the ordinate represents the change in color in $\mathrm{\Delta }{E}_{ab}^{*}$ units produced when one of the separations is displaced by half the lattice period (producing a dot-off-dot configuration). The different plots on the graph represent the color change due to this misregistration for different values of the YN parameter $\gamma$ . Note that for a given area coverage, as $\gamma$ increases, the amount of color shift reduces. This effect is quite significant when an increase in $\gamma$ from $1.0$ to $2.5$ reduces the worst case misregistration-induced color shift from 40 to approximately 21 in $\mathrm{\Delta }{E}_{ab}^{*}$ units. Also, in the absence of YN correction $(\gamma =1)$ , the corresponding curve in this figure is highly asymmetric with respect to the midtone gray level 0.5. In particular, shadow tones are significantly more sensitive to misregistration than the symmetrically located values in the highlight tones (about 0.5). As the YN parameter $\gamma$ increases, these curves become more symmetric and the amount of color shifts are also reduced. By examining differences in Neugebauer primary areas and spectra, instead of the $\mathrm{\Delta }{E}_{ab}^{*}$ difference in CIELAB coordinates, we see that the CIELAB cube-root nonlinearity is the primary source of the asymmetry, which is in effect compensated by the higher $\gamma$ values.

Fig. 13

Effect of optical dot gain on the amount of change in average color between the dot-on-dot and dot-off-dot configurations of C $(k=1)$ and M $(k=2)$ colorants when both colorants have equal fractional area coverages.