Abstract

1. What is halftoning?

Thu, 30 Apr 2009 18:08:44 -0400

Halftoning is the technique of converting continuous-tone images, such as those produced by silver halide film cameras, to strictly black and white images for reproduction in binary display devices, such as desktop inkjet printers. The earliest forms of halftoning were used in printers employing photo-lithography [1] and, as illustrated in Fig. 1.1, involved projecting light, from the negative of a continuous-tone photograph, through a mesh screen such as finely woven silk onto a photo-sensitive plate. Bright light, as it passed through a pin-hole opening in the silk screen, would form a large, round spot on the plate. Dim light would form a small spot. Light sensitive chemicals coating the plate would then form insoluble dots that varied in size according to the tones of the original photograph. After processing, the plate would finally have dots, where ink was to be printed, raised slightly above the rest of the plate producing an image such as that in Fig. 1.2.

Figure 1.2 Gray-scale image reproduced as an analog halftone.

In 1880 when the first analog halftoning process was perfected [1], the predominant form of printing was letter press or relief printing, which reproduced monochrome photographs as line drawings created by highly skilled craftsmen. With halftoning, publishers and printing houses had an easy and inexpensive way of reproducing continuous-tone photographs, making photo-lithographic presses the preferred printing technology. Halftoning also made photography a lucrative industry -- leading to a surge in technological innovation for photographic equipment [2].

As halftoning evolved, later versions of the process employed screens made of glass that were coated, on one side, by an opaque substance [2]. A mesh of parallel and equidistant lines were scratched in the opaque surface. A second mesh of parallel and equidistant lines were then scratched in the opaque surface running perpendicular to the original set. Screens would then differ in the number of lines per inch that had been scratched. While finer screens created better spatial resolutions (detail), the quality of the printing press would limit how fine of a mesh could be used. Later still, the glass plate mesh was replaced altogether with a flexible piece of processed film, placed directly in contact with the unexposed lithographic film [1]. This contact screen had direct control of the dot structure (Fig. 1.3) being able to control the screen frequency (the number of lines per inch), the dot shape (the shape of dots as they increase in size from light to dark), and the screen angle (the orientation of lines relative to the positive horizontal axis).

Figure 1.3 The screen frequency, dot shape, and screen angle for an analog halftone pattern.

2. What is AM halftoning?

Wed, 29 Apr 2009 11:38:54 -0400

Today, printing is a far more advanced technology having introduced non-impact printing and with the introduction of desktop publishing. Brought on by advancements in the digital computer [1], the photo-mechanical screening process has, in many instances, been replaced by digital imagesetters. In some cases, printing is no longer binary as continuous-tone dye-sublimation printers are now readily available but due to their speed and material requirements (special papers and inks), have not reached the wide-spread acceptance of four color ink jet or electro-photographic (laser) printers.

In digital printers, the halftoning process of projecting a continuous-tone original through a halftone screen is replaced with a raster image processor (RIP) that converts each pixel of the original image from an intermediate tone directly into a binary dot based on a pixel-by-pixel comparison of the original image with an array of thresholds (Fig. 2.1). Pixels of the original with intensities greater than their corresponding threshold were turned “on” (not printed or printed white) in the final halftoned image while pixels less than their corresponding thresholds were turned “off” (printed or printed black). For large images, the threshold array is tiled end-to-end until all pixels of the original have a corresponding threshold.

Most of the RIPs imitate the halftone patterns of contact screens by employing clustered-dot ordered dithering where the threshold array is small (8x8, 12x12, or 16x16) and is composed of consecutive thresholds arranged along a spiral path radiating outward from the array's center. These arrangements of thresholds result in a single cluster of “off” pixels centered within each tile or cell, forming a regular grid of round dots that vary in size according to tone. These techniques are commonly referred to as amplitude modulated or AM digital halftoning due to their modulating of the size of printed dots. Like contact screens, resulting patterns vary in their screen frequency, dot shape, and screen angle.

2.1 What is the screen frequency?
The screen frequency is the number of lines or rows of clustered-dots per inch of the resulting halftone pattern. Like the original glass plate screens, finer screens create patterns with higher spatial resolutions, but depending on the resolution of the printer (measured in dots per inch), screen frequency is limited by the number of unique gray levels required to reproduce an image without introducing banding artifacts (noticeable transitions between consecutive gray levels). This relationship is defined as:

Shown in Fig. 2.2 is an illustration of the effects of varying the screen frequency on a gray-scale ramp. For reference, studies have shown that AM dots become indistinguishable to the eye at screen frequencies above 200 lpi [3,4].

(a)

(b)

(c)

(d)

Figure 2.2 The effects of varying screen frequency on a gray-scale ramp.

2.2 What is the dot shape?
Dot shape refers to the specific arrangement of thresholds within the dither array, which dictates how clusters vary in both size and shape according to tone. The shape of dots is most clearly recognizable at gray level 1/2 (number of black dots equals the number of white dots), and the most common dot shapes are round, square, and elliptical [5]. Special effect shapes have also been introduced [6,7]. Shown in Fig. 2.3 are examples of four proposed dot shapes.

(a)

(b)

(c)

(d)

Figure 2.3 Various dot shapes proposed for AM halftoning.

2.3 What is the screen angle?
The last parameter of which to classify AM screens is the screen angle or the orientation of screen lines relative to the horizontal axis. This parameter is a function of the human visual system with directional artifacts least noticeable when oriented along the 45˚ diagonal [8]. It follows that for monochrome printing, this screen angle should also be 45˚. Shown in Fig. 2.4 are examples of several screen angles.

(a)

(b)

(c)

(d)

Figure 2.4 The effects of varying screen angle on a gray-scale ramp.

For computational ease, screen angles are typically restricted to rational angles (rise over run equals integer over integer) where each tile is of the same size and shape. Shown in Fig. 2.5 (left) is the division of pixels into a tiling of halftone cells of size 6x6 pixels at a rational angle of 9.5˚. Note how each tile is identically shaped. Irrational angles create a tiling that requires multiple threshold arrays due to varying tile shapes. Fig. 2.5 (right) shows the tiling of cells of size 6x6 pixels at an irrational angle of 16.6˚ where tiles are not identically shaped as indicated by cells labeled A and B. Using irrational angles requires a RIP to generate threshold arrays on the fly to match the size and shape of a particular tile [6]. Rational angles allow for the use of a single array.

Figure 2.5 The tiling of 6x6 halftone cells at (left) a rational screen angle of 9.5˚ and (right) an irrational screen angle of 16.6˚.

Unlike screen frequency and dot shape, the screen angle plays a fundamental role in the elimination of moire, the interference patterns produced by superimposing two or more regular patterns. In color printing, the halftone patterns of cyan, magenta, yellow and black inks are superimposed with AM patterns, composed of regular grids of printed dots, exhibiting moire. Fig. 2.6 shows the moire patterns associated with superimposing just two regular grids. While this interference cannot be altogether eliminated, it is through the screen angles 15˚, 75˚, 0˚, and 45˚ for cyan, magenta, yellow, and black, respectively, that moire is minimized, creating the pleasant rosette pattern of Fig. 2.7.

Figure 2.6 The moire patterns created by offsetting two AM halftone patterns by (top-left) 5˚, (top-right) 10˚, (bottom-left) 15˚, and (bottom-right) 30˚.

Figure 2.7 The rosette pattern created by setting the CMYK channels to screen angles 15˚, 75˚, 0˚, and 45˚ respectively.

3. What is FM halftoning?

Mon, 20 Apr 2009 12:17:01 -0400

Due to freedoms afforded by digital printers, the idea of printing isolated pixels in an effort to minimize halftone visibility (the visibility of the individual dots to a human viewer) emerged as an alternative to clustered-dot dithering. By maintaining the size of printed dots for all gray levels as individual pixels, new dispersed-dot halftoning techniques varied, according to tone, the spacing between printed dots, earning the name frequency modulated or FM halftoning. Early FM halftoning techniques were proposed by Bayer [9] and Bryngdahl [10] and produced an ordered arrangement of isolated dots. These techniques, like AM halftoning schemes, quantized each pixel independently of its neighbors (point process) according to a dither array but with consecutive thresholds dispersed as much as possible. The problem associated with these early FM techniques is that, as in the case of Bayer's dither array (Fig. 3.1 (left)), resulting halftoned images (Fig. 3.1 (right)) suffered from a periodic structure that added an unnatural appearance [11].

3.1 What is error diffusion?
For a better approach to FM halftoning, Floyd and Steinberg [12] proposed error diffusion (Fig. 3.2) as an adaptive technique that quantized each pixel according to a statistical analysis of an input pixel and its neighbors, leading to a stochastic arrangement of printed dots. While this neighborhood process had higher computational complexity, the resulting patterns had apparent spatial resolutions much higher than those achieved by clustered dots (Fig. 3.3); furthermore, as a stochastic patterning of dots, the patterns eliminated the occurrence of the moire that was produced by the superimposing of two or more regular patterns.

Figure 3.2 The error diffusion algorithm.

In error diffusion, each input pixel, , is processed one at a time usually in a left-to-right, row-by-row scanning path. The output pixel is determined by adjusting and thresholding the input pixel such that:

where is the diffused quantization error accumulated during previous iterations as:

with and . Using vector notation, can be expressed as:

where and . For the error filter , Floyd and Steinberg proposed the kernel of Fig. 3.4.

Figure 3.3 Grayscale image halftoned using Floyd's and Steinberg's error diffusion algorithm.

Figure 3.4 Floyd and Steinberg’s original four-weight error filter kernel.

Anastassiou [13] noted that error diffusion was a 2D instance of delta-sigma modulation, and as such, Weissbach et al [14] showed that the power spectrum, , of the output image of error diffusion was related to the power spectrums, and , of the input and error images as:

where:

is a high-pass filter (assuming was low-pass) derived from the 2D discrete Fourier transform of the error filter. Knox [15] later showed that the error image was not white-noise but was strongly correlated with the input image and even contained a linear component of the input image. Being that was a high-pass filter meant that contained an additional high-pass version of the input image . It is due to the relationship between and that error diffusion is considered to be an inherently high-pass filter operation, but even so, edge sharpening is a common operation applied to an image prior to halftoning with error diffusion [16].

3.2 What is blue noise?
What is so appealing about the dither patterns created by error diffusion is that they are composed exclusively of high frequency spectral components. Based on the sensitivity of the human visual system to low-frequency graininess, error diffusion creates halftone patterns far superior to AM halftoning or Bayer's dither, exhibiting much higher spatial detail without adding perceptually disturbing or artificial textures. Ulichney [11] describes these dither patterns, produced by error diffusion, as blue-noise where for a given gray level , the ideal pattern has minority pixels separated in an isotropic manner by an average distance of , the blue-noise principle wavelength, such that

where is the minimum distance between addressable points on the display. The resulting power spectrum then has spectral components with frequencies greater than or equal to , the blue-noise principal frequency, such that

Ulichney uses the term “blue” as a reference to the high frequency spectral components of white light. The term “noise” refers to the randomness of the patterns.

Figure 3.3 The minority pixels of blue-noise separated by an average distance of .

To statistically characterize the spatial arrangement of dots in a blue-noise pattern, Lau et al [17] suggest relying on the spatial and spectral statistics developed for point processes. In this framework, a point process, , is a statistical model governing the location of points in (2-D real space). A sample set, , of is then noted as the set of points , but for some area is a scalar quantity equal to the number of 's in . To relate the framework to digital halftoning, Lau et al define error diffusion as a point process with a binary dither pattern,, representing gray level defining a sample set such that indicates that the pixel is a minority pixel (a white pixel on a black background or a black pixel on a white background).

Figure 3.4 Diagram illustrating the point process , a sample of the process, and the scalar quantity as a function of the subset of .

The quantity is a scalar random variable that can be characterized in terms of its moments. The first-order moment or the expected value of is referred to as the intensity , and its relationship to the gray level is defined as:

For a point process to be stationary, the statistical characteristics of must be invariant to translation. If a process is stationary, then the intensity is constant for all where is the unconditional probability that the sample at location is a minority pixel.

Additional insight into is gained by conditioning the probability distribution of given that a point lies at a given location. The result is a conditional distribution referred to as the Palm distribution [18]. A particular measure under the Palm distribution of is the quantity :

the ratio of conditional probability that a minority pixel exists at given that a minority pixel exists at to the unconditional probability that a minority pixel exists at . , referred to as the reduced second moment measure, may be thought of as the influence of a minority pixel at on the pixel . That is, is a minority pixel at more or less likely to occur because a minority pixel exists at ? If , then the likelihood of a minority pixel at is increased relative to while indicates the likelihood is decreased.

Figure 3.5 A (left) blue-noise dither pattern with its corresponding (center) reduced second moment measure and (right) power spectrum estimate.

For stationary processes, may be written as where is the distance between and and is the direction to from . For a point process to be isotropic, the statistical characteristics of must be invariant to rotation; therefore, if is also isotropic, then . The quantity is commonly referred to as the pair correlation, , defined explicitly as:

the ratio of the expected number of minority pixels located in the ring under the condition that is a minority pixel to the unconditional expected number of minority pixels located in .

In view of Fig. 3.5, blue-noise halftones are characterized in terms of the pair correlation by noting that: (1) few or no neighboring pixels lie within a radius of ; (2) for , the expected number of minority pixels per unit area approaches with increasing ; and (3) the average number of minority pixels within the radius increases sharply near . The resulting pair correlation for blue-noise is therefore of the form in Fig. 3.6 (top) where shows: (a) a strong inhibition of minority pixels near , (b) a decreasing correlation of minority pixels with increasing , and (c) a frequent occurrence of the inter-point distance , the principle wavelength, indicated by a series of peaks at integer multiples of . The principle wavelength is indicated in Fig. 3.6 (top) by a diamond located along the horizontal axis.

Figure 3.6 The (top) pair correlation and (bottom) RAPSD of the ideal blue-noise pattern with principal wavelength and corresponding principal frequency .

In the spectral domain, blue-noise halftones are characterized by the radially averaged power spectral density (RAPSD), , defined as:

the average power within a series of annular rings, , that partition the spectral domain where is the number of frequency samples in the annular ring with center radius [11]. is the power spectrum estimate of the point process, , calculated using Bartlett's method of averaging periodograms [19] such that:

where represents the two dimensional, discrete Fourier transform of the sample , is the total number of pixels in the sample , and is the total number of periodograms being averaged to form the estimate.

Shown in Fig. 3.6 (bottom) is the RAPSD of the ideal blue-noise pattern -- showing three distinct features: (a) little or no low frequency spectral components, (b) a flat, high frequency (blue-noise) spectral region and (c) a spectral peak at cutoff frequency . Ulichney notes that blue-noise dither patterns are better than white-noise (where for all ) due to the absence of any spectral components in the range . Comparing any two halftoning algorithms, the visually preferred is the one that minimizes the amount of spectral content that exists at frequencies below .

Figure 3.7 Jarvis's, Judice's, and Ninke's proposed error filter.

Figure 3.8 Stucki's proposed error filter.

Noting the gray-scale ramps of Fig. 3.9, Floyd's and Steinberg's error diffusion is clearly not the ideal blue-noise halftoning algorithm due to strong diagonal correlations are extreme gray levels ( near 0 and 1) and due to periodic textures at near multiples of 1/4 and 1/3. In light of these artifacts, many modifications to Floyd's and Steinberg's original algorithm have since been introduced. The earliest modifications involved new error filters. Shown in Fig. 3.7 and 3.8 are the Jarvis et al [20] and the Stucki [21] filters with their corresponding gray-scale ramps shown in Fig. 3.9 (b) and (c).

(a) Floyd's and Steinberg's error filter

(b) Jarvis's, Judice's, and Ninke's error filter

(c) Stucki's error filter

(d) Floyd's and Steinberg's error filter with a serpentine raster scan

(e) Ulichney's perturbed filter weight scheme

Figure 3.9 The halftoned gray-scale ramps produced using several variations of error diffusion.

A second approach to modifying error diffusion involved the raster scanning order to which pixels of the input image were processed. Using a serpentine left-to-right and then right-to-left scanning order (Fig. 3.9 (d)) breaks up many of the artifacts found using Floyd's and Steinberg's filter weights near gray level 1/3 and also alleviates some the hysteresis artifacts found near extreme gray levels [11]. Other more radical approaches to raster scans include the use of space-filling curves such as the Hilbert or Peano curves [22].

A third approach to modifying error diffusion, that was first proposed by Ulichney [16], involved randomizing the error-filter weights at each pixel. Specifically, Ulichney prescribed grouping the filter weights into pairs such that weights of similar value where paired together. Ulichney would then add a random value to one member of the pair, subtracting the same value from the other member. In this way, the weights would always sum to 1. To avoid negative filter weights, the maximum magnitude to the random value would be a percentage (like 50%) of the smaller of the two weights within a pair. Shown in Fig. 3.9 (e) is the gray-scale ramp created using Floyd's and Steinberg's filter (paired as 1/16 with 3/16 and 5/16 with 7/16) with a 50% maximum perturbation value.

For a completely different approach to halftoning that generates very pleasing dither patterns, Analoui and Allebach [23] introduced direct binary search (DBS) as to obtain the absolute best arrangement of binary dots for representing a continuous tone image. DBS is an optimization routine that iteratively processes each pixel of the binary image, one at at time, by either swapping the current pixel with one of its eight nearest neighbors or toggling the bit from 1 to 0 or 0 to 1 according to the modeled visual cost between the binary image and the continuous-tone original. If neither a swap nor a toggle reduces the overall visual cost, then the pixel is left unchanged. The algorithm quits when after processing the entire halftoned image, no swaps or toggles occur. Being a steepest descent type optimization, DBS is susceptible to local minimum extrema, and the quality of the final halftone image is affected by the initial or seed dither pattern.

3.3 What is threshold modulation?
Threshold modulation with low level white-noise was also proposed by Ulichney and carried a lower computational cost than modulating multiple error filter weights. This technique was later shown by Eschbach and Knox [24] to be equivalent to adding high-pass filtered white-noise to the input image prior to halftoning. While the resulting halftone patterns did not show any great improvement in their blue-noise characteristics, threshold modulation became a very powerful tool for edge sharpening. Eschbach and Knox showed that modulating the threshold by a scalar multiple of the current input pixel created a halftoned image with sharper edges. They showed that varying the threshold of error diffusion by the function was equivalent to applying ordinary error diffusion to the input image where:

If (a scalar multiple of the current input pixel), then is equivalent to adding a high-pass version of to itself prior to halftoning -- leading to sharper edges in the apparent image . Figure 3.10 shows the resulting halftone images produced by Floyd's and Steinberg's error diffusion on a serpentine raster scan with Knox's and Eschbach's threshold modulation where = 0 and 3.

Figure 3.10 The gray-scale images produced using Floyd's and Steinberg's error filter with a serpentine raster scan illustrating threshold modulation for edge enhancement with (left) = 0 and (right) = 3.

3.4 What are blue noise dither arrays?
A real drawback for error diffusion is its computational complexity. Being a neighborhood operation, error diffusion requires additional storage for intermediate error terms [25]. So a novel advancement in halftoning that evolved from Ulichney's blue-noise model is Void-and-Cluster and blue-noise dither arrays -- dither arrays constructed to produce dither patterns that closely imitate those produced by error diffusion. While typically much larger than those of AM halftoning (128 by 128 or 256 by 256 versus 8 by 8 or 16 by 16), these masks are employed in the exact same manner as clustered-dot ordered dither arrays. Shown in Fig. 3.11 is a sample mask, produced by Void-and-Cluster, and its corresponding halftoned image.

Figure 3.11 A (left) 128 by 128 blue-noise mask and (right) its corresponding gray-scale image.

4. What is printer distortion?

Thu, 16 Apr 2009 15:41:07 -0400

By using FM halftoning schemes, printers maximize their apparent spatial resolution and are relieved of the strict tolerances on screen angles and screen registration. They can also use more and more colors to produce larger color gamuts (the set of achievable colors that can be produced by the printer) [28]. Notably, though, with its associated advantages, FM halftoning has, with few exceptions, only been employed in ink jet printers. The problem is the increased scrutiny placed on the printer's ability to print small, isolated dots.

Noting Fig. 4.1 (a), the ideal display produces dots that completely cover the sample area associated with a given pixel without overlapping neighboring pixels' sample areas. By printing all pixels, perfect black can be obtained. In a real printing device, individual printed dots are round, and in order to produce perfect black, must be large enough as to cover the entire sample area as illustrated in Fig. 4.2 (left) and (left-center) [29]. By overlapping neighboring sample areas, though, the resulting tone is darker than the fraction of all pixels that are printed. Assuming that dots are printed consistently (small variation in size and shape from printed dot to printed dot), this distortion in tone can be corrected by adjusting the intensity level of the input image before halftoning. The amount of compensation depends on the arrangement of printed dots with dispersed-dot (FM) patterns requiring greater degrees of correction than clustered (AM) [30,31]. Ink jet printers are such a device that prints (approximately) round dots that overlap neighboring pixels (Fig. 4.1 (b)). Being able to compensate for distortions introduced by the printing process for any arrangement of dots, ink jet printers can enjoy the benefits associated with FM halftoning.

Figure 4.2 Clusters of (left,right-center) one and (left-center,right) four printed round dots where the dots of (left) and (left-center) cover the entire sample area while the dots of (right-center) and (right) do not cover the corners.

In the electrophotographic printing process, the size and shape of dots varies greatly from printed dot to printed dot (Fig. 4.1 (c)), and this variation can only be minimized when dots are grouped together to form clusters. Images printed using isolated dots tend to show severe tonal distortion and exhibit a great deal of variation in tone across the printed page. Figure 4.3 illustrates the resulting variation in tone across a page for an error diffused halftone representing gray level 7/10 produced by a laser printer set at 1200 dpi. In this figure, the average variation in tone along the vertical axis is plotted along side an image of the page, and the average variation in tone along horizontal is plotted below the image. Due to distortions such as this, unreliable printing devices such as lithographic presses and laser printers continue to use AM halftoning schemes. So a real challenge has come to face researchers as they try to improve the image quality in these types of printers. What they are finding is that halftoning algorithms that cluster same color pixels together, in a random fashion, hold the key by creating patterns that are easier to produce consistently from page to page and by creating color halftone patterns without moire.

Figure 4.3 The resulting printed page produced by a laser printer at 1200 dpi where the gray level 7/10 is produced using error diffusion. The average variations in tone along the horizontal axis and the vertical axis are also shown, plotted alongside and below the picture of the printed page.

5. What is AM-FM halftoning?

Sat, 11 Apr 2009 19:50:22 -0400

AM-FM hybrids (Fig.~\ref{img63_fig}) refer to halftoning algorithms that vary, according to tone, both the size and spacing of dots \cite{lau_arce_gallagher}. While various AM-FM halftoning techniques have been introduced over the years, the best techniques are those that generate green-noise halftone patterns -- binary dither patterns created exclusively of mid-frequency spectral components. The goal of green-noise is to distribute minority pixel clusters in a homogeneous and isotropic fashion. The average size of these clusters can vary with smaller clusters leading to halftoned images with higher spatial resolution and better edge detail and larger clusters leading increased halftone robustness (the ability of the pattern to resist the distortions introduced during the printing process). As a tunable model, green-noise has, as a limiting case, blue-noise, and by using green-noise, binary display devices that were previously restricted to AM halftoning techniques can now combine the maximum dispersion attributes of blue-noise with the clustering attributes of AM halftones \cite{lau_arce_gallagher}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Error Diffusion with Output-Dependent Feedback}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t!]
\centerline{\includegraphics{./IMG_files/img66}}
\caption{The error diffusion with output-dependent feedback algorithm.}
\label{chap5_fig03}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In an effort to address the ``print-ability'' of stochastic dither patterns produced by error diffusion, Levien \cite{levien} added an output-dependent feedback term (Fig.~\ref{chap5_fig03}) such that the coarseness of resulting dither patterns could be tuned through a scalar constant. In this algorithm, the output pixel, $y[n]$, is determined as:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
y[n] & = & \left\{ \begin{array}{lcl}
1 & , & \mbox{if } (x[n]+x_e[n]+x_h[n]) \ge 0 \\
0 & , & \mbox{else}
\end{array} \right.
\label{err_diff_hyst_thr_eqn}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $x_h[n]$ is the hysteresis or feedback term defined as
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
x_h[n] & = & h \sum_{i=1}^{N}a_i \cdot y[n-i]
\label{hyst_trm_eqn}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
with $\sum_{i=0}^{N}a_i=1$ and $h$ an arbitrary constant. Referred
to as the {\em hysteresis constant}, $h$ acts as a tuning parameter with
larger $h$ leading to coarser output textures \cite{levien} as $h$
increases ($h>0$) or decreases ($h<0$) the likelihood of a minority
pixel if the previous outputs were also minority pixels. Eqn.~(\ref{hyst_trm_eqn}) can also be written in vector notation as:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
x_h[n] & = & h {\bf a}^{\rm T} {\bf y}[n]
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where ${\bf a}=[a_1,a_2,\ldots,a_N]^{\rm T}$ and ${\bf y}[n]=[y[n-1],y[n-2],\ldots,y[n-N]]^{\rm T}$. The calculation of the parameters ${\bf y}_e[n]$ and $x_e[n]$ remain unchanged from ordinary error diffusion. Unlike ordinary error diffusion, Levien's algorithm produces strong diagonal textures when coupled with a normal left-to-right raster scan. For this reason, an alternate scanning order such as the serpentine raster scan is mandatory \cite{lau_arce_gallagher}. Shown in Fig.~\ref{chap5_flt00} is the arrangement of two hysteresis and two error filter coefficients first prescribed by Levien.

Green-noise is a statistical model describing the spatial and spectral characteristics of visually pleasing dither patterns composed of a random arrangement of clustered dots. Point process statisticians have long described cluster processes such as those seen in green-noise by examining the cluster process in terms of two separate processes: (i) the parent process that describes the location (centroid) of clusters, and (ii) the daughter process that describes the shape of clusters. In AM halftoning, clusters are placed along a regular lattice, and therefore, variations in AM patterns occur in the cluster shape. In FM halftoning, cluster shape is deterministic, a single pixel. It is the location of clusters that is of interest in characterizing FM patterns. Green-noise patterns, having variation in both cluster shape and cluster location, require an analysis that looks at both the parent and daughter processes.
Looking first at the parent process p, p represents a single sample of the parent process such that p = {xi : i = 1, . . . ,Nc} where Nc is the total number of clusters. For the daughter process d, d represents a single sample cluster of d such that d = {yj : j = 1, . . . ,M} where M is the number of minority pixels in cluster d. By first defining the translation or shift in space Tx(B) of a set B = {yi : i = 1, 2, . . .} by x, relative to the origin, as:
Tx(B) = {yi − x : i = 1, 2, . . .} (1.18)
and then defining di as the ith sample cluster for i = 1, . . . ,Nc, a sample G of the green-noise halftone process G is defined as:
G =Xxi2pTxi(di) =Xxi2p {yji − xi : j = 1, . . . ,Mi}, (1.19)
the sum of Nc translated clusters. The overall operation is to replace each point of the parent sample p, of process p, with its own cluster di , of process d.
In order to derive a relationship between the total number of clusters, the size of clusters and the gray level of a binary dither pattern, Ig is defined as the binary dither pattern resulting from halftoning a continuous-tone discrete-space monochrome image of constant gray level g, and Ig[n] is defined as the binary pixel of Ig with pixel index n. From the definition of (B) as the total number of points of in B, G(Ig) is the scalar quantity representing the total number of minority pixels in Ig, and p(Ig) is the total number of clusters in Ig with p(Ig) = Nc. The intensity, I, being the expected number of minority pixels per unit area can, now, be written as:
I = G(Ig) N(Ig) , (1.20)
the ratio of the total number of minority pixels in Ig to N(Ig), the total number of pixels composing Ig. Given (1.20), ¯M , the average number of minority pixels per cluster in Ig, is:
¯M = G(Ig) p(Ig) = I · N(Ig) p(Ig) , (1.21)
the total number of minority pixels in Ig divided by the total number of clusters in Ig.
Although obvious, (1.21) shows the very important relationship between the total number of clusters, the average size of clusters, and the intensity for Ig. AM halftoning is the limiting case where p(Ig) is held constant for varying I, while FM halftoning is the limiting case where ¯M is held constant for varying I. In addition, (1.21) says that the total number of clusters per unit area is proportional to I/ ¯M . For isolated minority pixels (blue-noise), the square of the average separation between minority pixels (b) is inversely proportional to I, the average number of minority pixels per unit area [11]. By determining the proportionality constant using b = p2 for I = 12 , the relationship between b and I is determined as b = D/p I.
In green-noise, it is the minority pixel clusters that are distributed as homogeneously as possible, leading to an average separation (center-to-center) between clusters (g) whose square is inversely proportional to the average number of minority pixel clusters per unit area, I/ ¯M . Using the fact that limM!1 g = b, the proportionality constant can be determined such that g is defined as:
g = 8><>:D/q(g)/ ¯M , for 0 < g 1/2D/q(1 − g)/ ¯M , for 1/2 < g 1, (1.22)
the green-noise principle wavelength. This implies that the parent process, p, is itself a blue-noise point process with intensity I/ ¯M .
If we assume a stationary and isotropic green-noise pattern, the pair correlation will have the form of Fig. 1.33 (left) given that:
1. Daughter pixels, on average, will fall within a circle of radius rc centered around a parent point such that r2 c = ¯M (the area of the circle with radius rc is equal to the average number of pixels forming a cluster).
2. Neighboring clusters are located at an average distance of g apart.
3. As r increases, the influence that clusters have on neighboring clusters decreases.
The result is a pair correlation that has: (a) a non-zero component for 0 r < rc due to clustering, (b) a decreasing influence as r increases, and (c) peaks at integer multiples of g indicating the average separation of pixel clusters. Note that the parameter rc is also indicated by a diamond placed along the horizontal axis in Fig. 1.33 (left).
In the spectral domain, the placement of clusters g apart leads to a strong spectral peak in P(f) at f = fg, the green-noise principle frequency:
fg =8><>:q(g)/ ¯M /D , for 0 < g 1/2q(1 − g)/ ¯M /D , for 1/2 < g 1. (1.23)
From (1.23) we make several intuitive observations: (i) as the average size of clusters increases, fg approaches DC, and (ii) as the size of clusters decreases, fg approaches fb. Fig. 1.33 (right) illustrates the desired characteristics of P(f) for G showing three distinct features: (a) little or no low frequency spectral components, (b) high-frequency spectral components that diminish with increased clustering and (c) a spectral peak at f = fg.
Noting the gray-scale ramps of Fig. 1.29 which were produced using Levien’s filer kernel with hysteresis constants 0.5, 1.0, and 1.5, Levien’s kernel produces patterns with vertical artifacts at low h (near h = 0) and strong horizontal artifacts at high h (near h = 2). Lau and Arce [33] note that these artifacts can be eliminated by changing the proportions of feedback through the horizontally and vertically aligned feedback weights. Using balanced weights, Lau and Arce are able to produce the gray-scale ramps of Fig. 1.34.

1–3–3 Adaptive Hysteresis
Optimizing the hysteresis constant for a given printing process is achieved by specifying the parameter h according to the desired robustness, but as a constant, error diffusion with output-dependent feedback may, like AM halftoning, sacrifice spatial resolution at certain gray levels for pattern robustness at other levels. In light of this consequence of a hysteresis constant, Lau and Arce [33] propose using an adaptive hysteresis value that varies according to the gray level of the current input pixel. The approach that they first prescribe is to select the minimum h such that the output tone is within a pre-specified tolerance of the input (Fig. 1.35 (left)). A second approach they prescribe is to vary h according to the frequency content of the input image (Fig. 1.35 (right)). In this scheme, the resulting halftoned image will be composed of large clusters in DC regions, where distortions are most noticeable to the human eye, and small clusters near edges where distortions are least noticeable and spatial details require small clusters in order to be preserved.

1–3–4 Green-Noise Masks
In an effort to address the computational complexity of error diffusion with output-dependent feedback, Lau et al [34] introduced an algorithm for the construction of green-noise masks. Just like blue noise masks, green-noise masks offer the absolute minimum in computational complexity – leading to halftone patterns that are faster and less expensive to produce. Figure 1.36 shows a green-noise dither array and its corresponding halftoned image.

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