The Tug function: A Method of context sensitive dot structuring for digital halftones

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5-1-1990

The Tug function: A Method of context sensitive

dot structuring for digital halftones

Steven Hoffenberg

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Recommended Citation

A Method Of

Context Sensitive Dot Structuring

ForDigital Halftones

by

Steven _Hoffenberg

A project submitted in partial fulfillment of the requirements for the Master of Science degree in the School of _Printing Management and Sciences

of the Rochester Institute of _Technology

May,1990

School of Printing Management and Sciences Rochester Institute of Technology

Rochester, New York

CERTIFICATE OF APPROVAL

MASTER'S PROJECT

This is to certify that the Masters Project of

STEVEN HOFFENBERG

in the Graphic Arts Publishing program, with a concentration in Electronic Publishing, has been approved as satisfactory for the project requirement for the Master of Science degree at the convocation of May, 1990

~~~k)~~~t

_

Project Advisor

M~~J.G~din _

Graduate Program Coordinator

CONTEXT SENSITIVE DoT STRUCTURING FOR DIGITAL HALFTONES

I, Steven Hoffenberg, hereby grant permission to the Wallace

Memorial Library of RIT to reproduce my thesis project in whole or in part. Any reproduction will not be for commercial use or profit.

The author would like to thank the _following individuals and companies for their assistance in this project:

Professor Frank _Cost, project _advisor, without whose interest and encouragement this project would not have been possible;

Tom _Bernard, of Bersearch Information _Services, for PostScript

programming pointers;

Professors Marie Freckleton and Archie Provan, for _enabling extended access to equipment required for _{this project;}

Agfa _Corporation, for _providing to the School of _Printing Management and Sciences the scanner and printer used to produce the final prints of this project;

Adobe _{Systems, Incorporated, for providing} pre-release software used for image data formatting.

Acknowledgments ii

List ofFigures v

ListofTables vii

Abstract viii

Introduction 1

Digital Halftones 3

History 3

Sharpness 4

The _TugApproach 9

PostScript _Halftoning 14

The Spot Function 19

Procedures 24

The _Tug Function 24

The PostScript _Tug Function 36

Imaging Context Sensitive Dots 41

Methodology 53

Image Capture and Conversion 53

PostScript Synthesis and Cell Analysis 55

Image Synthesis 56

Image Evaluation 58

Results 60

Discussion 62

Summary 64

Footnotes 67

Bibliography 70

Software References 72

Appendix A: _Anatomy of a Digital Halftone 74

Appendix B: Computer Program Listings 75

18Spot Functions 76

Pascal TugTest 77

The PostScript _Tug Function 81

PostScript Toggle 89

Appendix C: Agfa P3400PS Printer Specifications 90

Appendix D: _Judging Forms and Results 91

Appendix E: Print Samples 94

Appendix F: Trademark Disclaimer 101

Figure 1: Original Image 5

Figure 2: Image Reconstructed as Photographic Halftone 5 Figure 3: Image Reconstructed as Digital Halftone 6

Figure 4: _{Sharpening by Gray Value Adjustment (Convolution)} 7

Figure 5: Oversampled Input 8

Figure 6: Normal vs. _{Partial-Dot Halftone Structure} 10

Figure 7: _{Sharpening by} The_Tug Approach 10

Figure 8: _Halftoning Sharpness Comparison 11 Figure 9: Clustered and Dispersed Halftone Dot Cells 15

Figure 10: _Screening Grid Problem 18

Figure 11: Spot Function Coordinate System 20

Figure 12: "ShowTheCell" Output 22

Figure 13: Cell Arrangement for The _Tug Function 27

Figure 14: _Tug Function Variables 28

Figure 15: _Neighboring Cell and Spot Function Equations 29

Figure 16: _Tug and Spot Function Consolidation Equations 32

Figure 17: Scrambled Dots 34

Figure 18: "ShowTheTug" Middletone Sample 38

Figure 19: "ShowTheTug" Highlight Sample 39

Figure Bl: "TugTest"

DisplayWindow 77

Figure El: _Portrait, Without_Tug 95

Figure E2: _Portrait, With Tug*

96

Figure E3: Back _Yard, Without _Tug 97

Figure E4: BackYard, With Tug*

98

Figure E5: _Gun, Without _Tug 99

Figure E6: Gun, With Tug*

100

Table 1: Print Execution Times 58

Table 2: _JudgingData _Summary 60

Structuring

by

Steven _Hoffenberg

ABSTRACT

The process of _digitizing images to create halftones _inherently reduces

sharpness through the _averaging of grayness values within cell areas.

Within the context of a resolution independent page description language,

overcoming this reduced sharpness is _{conventionally} addressed _{by adjusting}

the grayness values of cells to create larger or smaller halftone dots where edges are present. Such an approach does not take full advantage of the

capabilities of the output device.

The objective of this project was to design and implement a method of

sharpening digital images by altering the shape and position, rather than the

size, of halftone dots. Such a method can more _accurately represent the

original image and more _closely emulate the characteristics of

photo-mechanically produced halftones.

Within the PostScript page description language, the generation of halftone dots is controlled _by the spot function. A particular typeof spot

function, the _{Tug function,} was developed to control the shape and position

ofhalftone dots based on the grayness value of each cell and its _surrounding

neighbors.

Because the standard PostScript imagingoperators are not designed to

allow halftone dot shapes to be redefined on a cell-by-cell basis, an alternate

method of _generating images was created. A computer program in the

PostScript _programming language was written to perform the requisite image

viewing the effects of the _Tug function.

Three representative photographic images were processed in this

manner. A panel of judges compared the _resulting prints with control prints

processed without the _Tug function but _by the same _imaging method. The

judges'

subjective preferences are _presented, and the relative merits of the

Tug function are discussed.

INTRODUCTION

The purpose of this project is to explore the realm of computer

produced halftones via a resolution independent page description _language,

and to devisea method that can improve its reproduction ofdigitalimages. With the recent advent of _desktop computerdevices for _scanning, image _processing, and _printingof photographic _images, a method that

improves the _quality ofthese printedimages could have sizable impact. An image in computerized form is_by nature digital,_existing atits most

elemental level as a series of electrical signals _{representing only} two values:

one and zero. This is the level at which a computer operates. The processes of

scanningand _digitizing an_{image, however,}tend toreduce its sharpness. This a

commonly recognized _phenomenon, and some of the reasons for its occurrence are explained in this report.

PostScript*

is a computer _programminglanguage optimized for the

purpose of _{communicating} page description signals from computer to output device. It is _currently considered the de facto standard for such a purpose in

the _{desktoppublishing} market.

PostScript contains several operators which the author believes can be

structured in such a _way as to overcome some of the unsharpness introduced

in the image capture process.

While much of the _industry research inthis area is _proprietaryto the

companies _involved, a review of the literature and discussionswith several

industrypersonnel _{specializing in} PostScriptapplications seem to indicate that

the particular approach utilized herein _may not have been _previously

the PostScript _language, the approach takes several orders of magnitude

longer to execute on an output device than a conventional PostScript image

file. This is _contrary to the commercial objective of_reducing printer execution

time.

But the intent of this project is not to create a _commercially saleable

software _product, it is to introduce researchinto an area that appears to have

been _inadequately explored within the realm of _publishingsystems. Most

likely, the industrialapplication of such principles would come at the level of

rasterimage processorhardware and firmware withina _printingdevice.

Regardless of the specific outcome of the present _project, ifthis

investigation leads others to expound on concepts contained _within, and

History

Halftoning techniques have been the dominant methods for printed

reproduction ofphotographic images for more thana century.

On March 4, 1880, The_Daily Graphic newspaper ofNew York _City

published whatis _generallycredited tobe the firstmassprinted halftone,a picture entitled _"Shantytown." That halftone was engineered _byStephen Horgan _usingan etched plate of glass to _{photographically} create an _engraving

with the illusionof continuous tone in animage produced with _onlytwo tones: black inkand newsprintpaper.

According to _Horgan, "Theterm halftone includesall pictures inwhich the lights and shades are defined_by lines and dots of differentsurface areas

madethrough _mechanicallylined screen."1

The termhalftone no longerimplies that picturesmust be made through

mechanically lined screens. _{Photographically} produced vignetted dot contact

screens were _commercially introduced in the 1940's and have sincebecome predominant.

Inthe _1960's, another,less common method of_halftoning was

introduced: that of computer generated pictures. Atthe _time, computer

printers were _only capable of_{producing text characters,} and the earliest

attempts at computer generated pictures were _actually comprised of various

became widely available with _sufficiently abundant pixels (picture elements),

and _sufficiently small _spots, which made computer generated images a practical reality. _This, combined with phenomenal advances inpersonal computer microprocessors and _{desktop scanners,} has led digital _halftoning to thebrink ofpopularity.

(For a clarification of some of the _{halftoning terminology}used within this report, consultAppendix _A.)

Sharpness

The _halftoningprocess, _by its _very nature,reduces the sharpness of an image. Whether produced _by photographic or digital_{means, the} size ofa halftone dot within a cell willbe based on the average value oftransmittance

or reflectance of a given area in the copy.

This area average value is expressed as a halftone dotwhichwill

typically grow fromthe center of the cell outward as the value increases. Even

ifthe portion ofthe original responsible forthe value istowards the edge of the

cell, the halftonedot will growfrom themiddle ofthe cell. This tends todiffuse

the edges of an image.3

Aseries offigureswill _helptoillustrate the point. Figure 1 showsa

portion of an originalimageprior to _halftoningwith a grid thatindicates where the cellswill lay. Theimagecontains both straight and curved edges. Notice thatone ofthe cells, in the centerlower _right, contains areas of_density in

photomechanical _halftone, whether

produced _by contact screen or etched glass _screen, the halftone dots _typically

growfrom the center of _{the cell,}but_they

canbe skewed towards one or more

sides _depending on the position of the valuesintheoriginalimage. Figure2 depicts a generalized view ofhow this

can appear.

This positional _biasing can create

irregularlyshaped _dots, and it ismore

pronounced for middletone dots and

veryslight or non-existentfor highlight

and shadow halftone dots. The effectis

due to thefact that some light

penetrates to the film behind contact or

etched glass screens in between the halftone dots.4 _(Insome

verylight areas, the dots _{may drop} out _completely,but

various exposure techniques can place a just-printabledot in_everycell.)

Also, silver particles tend to react

not _only inresponse to light exposure

and chemical development on

"fl "8 r" i

Jl....l

Image Reconstructed as

This is knownas the _adjacency effect.6

The effect is most apparent when exposed areas are in close _proximity or just _{barely connected,} as would occur

in the middletone dot areas.

The irregularities in the shapes of photomechanicaldots are desirable,

because _they allowthe halftone to _{convey shaping} information.

L^.n.w^.'i^.M.M.M.M.M.j-Figure 3 illustrates howarea

averaging can affect the edges of a scanneddigitalimage. This figure represents a halftone printed at a

0 degree screen angle and a cell output frequency equal to the _scanning input

screen frequency. The result is greater unsharpness, unless corrected _by subsequent image processing. Figure 3

isa_greatlysimplified view,andin actual

practice, the results of a scan depend heavilyon the design ofthe scanner's

optics, mechanics, and electronics (and,

not_{incidentally,}the printer's halftoningsoftware).

Milch (1989) discusses _many ofthe characteristics of a _scanning system that can affect image sharpness. Inparticular, there is a trade-offbetween

obtaining maximumresolution, and _reducingnoise and _granularity inthe original. The size ofthe _scanning apertureis _usually optimized at a sizethat is

Figure 3

Image Reconstructed as Digital

over an even larger area thanin photographic _halftoning, and as a result edges

are even more diffused.

Two approaches are _commonly used to _help sharpen the image.

Strictly speaking, such _sharpeningis not considered image enhancement, as its

purpose isnot to make the printed reproduction better _{than the original,} but

merely toregain detail lost inthe image capture process.8

One approach to _overcoming unsharpness is to perform subsequent

image _processing and readjust the _gray values to compensate for the area

averagingofthescan. An example ofthis is showninFigure4. Compared to

Figures 2and _3, eachhalftone dot thatborders between light and dark areas is

either enlarged or reduced in size to compensate forthe edge diffusion.

Imaging scientists have

developed _many calculations to achieve _{j , j , j}

the desired adjustments. In _{general, the \ j} :

gray value of each halftone cell is _{j |} j.

I

i

ti.

-compared to those ofits _neighboring

cells through mathematical equations

such as the Fouriertransform. _Many

texts, such asJain(1989), detailthose

processing equations.

The term convolution is

sometimes used as a general

descriptionof this effect of

neighborhood comparison and

'"j^^^^^^Khir~

mm 31 t

BSj

Figure 4

computers contain a _{sharpening function which operates in} thismanner. Such

processing, however, can _merelychange the size ofhalftone dots, not the

shape or position ofthe dots within the cells.

The second approach to sharpness is oversampling, or _sampling the

original image at a _frequency greater than the desired screen_frequency of the

printed output. _Oversampling is not_{truly sharpening}as _such, butratherthe

prevention of unsharpening. _Commonly, the scanis at the Nyquistfrequency,

whichis double the maximum desired output_frequency (in boththe vertical

and horizontaldirections).9

Figure 5 illustrates how for such _{oversampling, thegray}value of each

output cellisbased onfour smaller cellsin theoriginalimage. In the _top

example, the four values are combined to create a single halftone dotbased on

the average ofthe values. In _{this case, oversampling} at the input stage can

increase the _accuracy ofthe _gray value_information, somewhat _reducing the diffusion of edges due to area _averagingof a large scanningaperture.

Figure 5

Oversampled Input:

Normal _(top) and Partial-Dot (bottom) Outputs

In the bottom example, each

quadrant of the cell contains a portion of a dot _representing the _gray value

fromthat quadrant. This is knownas partial-dot structure. (Further discussion

Unfortunately, oversamplingcreates larger image data files which

require additional computer _memory for both _processing and storage.

Based on examination oflaserprinter _output, it appears that PostScript

device implementations are capable of partial-dot _structuring when the image

datais oversampled. This occurs _entirely withinthe _{proprietary hardware} and

software of the Adobe PostScript _interpreter, and cannot_{presently be driven}

byexternal software.

It is a _primary purpose ofthis project _{to create, through} downloaded

software,sharper halftones_utilizingapartial-dot typeof_structuring,without

requiringoversampled data on input.

The _Tug Approach

The _oversampling approach applies to the input stage ofthe digital

imagingprocess. The _gray value adjustment approach applies to the

intermediate _processing stages. The _Tugapproach applies to the output stage.

The comparative merits of _oversampling and _gray value adjustment

techniques are not _specifically addressed _here, but it is important to note that

these techniques are not_{mutually exclusive,} and in all _Ukelihood,some

combination of all ofthem would yield the optimumimage quality.

The objective ofthis project is to _develop a digital_halftoning model that

creates sharper images _by varying, on a cell-by-cell_basis, the shape and

position ofthe halftone dots based on the _gray values ofeach cell and its

neighbors.

Inthe realm of high-end electronic dot _{generatingscanners,} some digital

9

Figure 6

Normalvs. Partial-Dot Halftone

Structure handfulofdifferentways basedon _surroundingvalues. This is a typeof

partial-dotstructure.10

Figure 6 shows a comparison of

normal_(left) and partial-dot _(right)

halftone structure. Both cells contain the

same number of blackened _spots, but the

partial-dot is not centered inthe cell and

is shaped as thoughit were comprised of

portions chopped out of quadrants from different sized normal dots.

It is the intent ofthis project to go far beyond _anysuch measures

presentlyemployed, to create custom dot _shaping of near limitless variety Figure 7illustratesthe _Tugapproach for dotstructuring.

The name "Tug" comes from how

the author visualized the process might

work: as though halftone dots have a

magnetic attraction for each _other,

creating a Tug-of-War between each dot and the_{forces pulling}onitfrom its

neighboring cells.

In this scenario, thelarger the

neighboring halftone dot the stronger its

pull. And the largerthe centraldot the

strongerits resistanceto the pull. And if

a weakhalftone dotwere pulled

1,1) ! " __ 1... i

ml

En

MMi

lit

J]

P

i

_i

' ' ! ! ' _! ' Figure 7 Sharpening by

The _Tug Approach

Ideally, this would put halftone dots backwhere _they came frominthe

original.

(The magnetic nature ofthis concept is _purely abstract, andis inno_way

related to actual magnetic forces that _may existin a _{printingdevice.)} The _weighting ofthe resistance to the_tug based on the central spot

strength was _logically originated: _moderatelylarge halftone dots _jumping

around onthe page would more _likelybe _distracting to theviewer. The

validity ofthis assumption remained to be seen.

Figure 8contains reduced size versions ofFigures _{1, 2, 3, 4,}&_7,without

the grids, so that thereader _mayget a sense ofhow some of the _previously

described methods of _halftoning compare.

Notein the _Tug versionhow the dots appear as though_behavinginthis

magnetic way. And noteinparticular, thecellinwhichthe dot has been splitin two.

Original image Digital

Figure 8

Halftoning Sharpness

Comparison

iiii

up

Before proceeding further, it is _necessary to pose several reasonable

questions, and answerthem with abrief review ofthe literature.

Howvisibleis the actualdot structure?

The answer depends on several factors. The _visibility of the dot

structure _obviously varies with the linescreen or cell_frequency, as well as the

viewing distance. Atnormalprint _{viewingdistances,} estimates ofthe

frequencybeyond which the dot structure is no longervisible _varyfrom 100

cellsper_inch11,to 125cellsper_inch12, to 150cells perinch.13

In _many PostScript laser _{printing devices,} the default linescreen is

nominally 60 cells perinch (for reasonsto be explained _below), and the dot

structureis_easilyvisible.

Interestingly, Neugebaueret al. (1962) found that_visibilityofhalftone dot

structure also varieswith the _{gray level uniformity}of animage area and the

pictorial content of the image. In particular, thedot structure was more visible

in areas ofuniform tone thanin areas with fine detail.

Why not use 150 linescreen?

Simply, the higherthe _linescreen, the fewer the _graylevels thatcanbe

reproduced. For _example, in a printer with addressable resolution of300 spots

perinch at 60 cells perinch (and 0 degree screen angle), each cell contains 25

pixels ina 5 x5 matrix. In theory,this canreproduce 26 levels of_{gray (0} to 25

spots canbe blackened). At 150 cells per_{inch, however,} each will contain_only

4pixels ina 2x2_matrix,and can _onlyreproduce 5levels ofgray. A good

descriptionand illustration ofthis trade offof_graylevels forlinescreen is

Tests _byHamilton (1988) indicate that the actual number ofdistinct _gray

levels that a laser printer can reproduceis _{significantly} less than the

mathematically possible number.14

For most 300-spots-per-inch _printers, a 60 linescreen is considered to be

the least detrimentaltrade _off, and most such printers use that number as the

default.

Is dot_{shaping really} desirable?

John Seybold and Dressier _{(1987) state,} "To appear continuous, a

halftone must contain dots ofboth varying shapes and sizes."15

Jonathan Seybold and Tribute (1988) state, "Ideally,where there is an

abrupt black/white _transition, a halftone dot should _convey shape or contour

information as wellas a tonal value. TheAdobe PostScript RIP [raster image

processor] doesn't appear to be _doing this."16

Roth (1988)sums itup:

. . . thereisone thingthatPostScript interpreters can'tdo with cell shapes which canbe done with photographic screening,

andit_maybe thebiggest flaw in thewhole scenario. . . . where

light areasborder dark areas . . . thephotographic halftone cells

are not _regularly shaped; they'repear-shaped effectively. This

results in _{very sharp} edges and _crisphalftones. ThePostScript

halftones don'tshare thateffect.17

These last two references contain samples from high resolution

imagesetters at high _linescreen, which _may have been produced from files with

little or no oversampling. Itshould be noted that the relative degree of

oversampling varies with theoutput linescreen even ifthe image data remains

constant. The same files might haveproduced partial-dots if_they had been

PostScript _Halftoning

The PostScript language was designed _by Adobe Systems Incorporated

with the expressed purpose of _creating a standard page description language

for electronic printing.18 _{The language}

was introduced in_1985,and madeits

commercial debut with the Apple LaserWriter printer for the Macintosh

computer.

Three books _by Adobe are considered the standard references for the

language. _Among PostScript _{programmers, these} books are _generally

identified_by thecolor oftheircovers. Thisauthor will referto them as such.

The PostScript Language Reference Manual _{(Adobe, 1985a),} a.k.a. The Red

Book, contains the public definitionof the languageand its operators. This is

the single most essential book for anyone interested in PostScript

programming, and most other literature aboutthe language will refer the

reader back to it. The PostScript Language Tutorial and Cookbook (Adobe,

1985b), a.k.a. The BlueBook, is the companion volume _offeringexamples and

explanations of how the language is used. PostScript Language Program

Design _{(Adobe, 1988),}a.k.a. The Green_Book, is intended _primarilyforsoftware

developers who willcreate programs and printerdrivers whose outputwillbe

page descriptions in PostScript.

A _key tenet of PostScript is device independence: a program written in

PostScript should be executable on _any PostScript-compatible output device.

Since different output devices can have different addressable _{resolutions, by}

corollary another _key tenet of PostScript is resolution independence: a

It is these two aspects ofthe language thathave _largely enabled it to become widely accepted. It is also these two aspects that_largely dictate the

process of _{printing halftone images throughPostScript.}

Figure 9

Clustered and Dispersed

Halftone Dot Cells For digital_imagingin general, there

are two basic methods of _constructing halftones: _{clustered-dot,} and

dispersed-dot.19 _{Figure 9 illustrates}

the two types. In both cases, 18 out of 64spots are

blackened, but the distributionof the blackened spotsis_markedly different.

The traditional photomechanical halftone process is the most obvious

example of clustered-dot image formation. Infact, in most common _{usage, the}

termhalftone implies the clustered-dot formation. Partial-dot _structuringis considered to _belong in this category.

Some electronic _printing processes can use the dispersed-dot

formation. The process of_specifying the order inwhich spots are activated is

called dithering. In most common usage, the term_dithering implies the dispersed-dot formation.

Dispersed-dot formations, when _{appropriately} configured, can _convey

halftoning without the appearance of a regular_screening pattern.

Ulichney (1987) demonstrates the profound effect that variationsin

dithering can have on image appearance. Based on a comprehensive

dispersed-dot halftones are _{clearly preferable,} and clustered-dot techniques should _only

be used when the _printing process cannot accommodate isolated pixels.20

Unfortunately, in most situations where PostScript isemployed the

process cannot _easily accommodate isolated pixels.

The resolution independence of the language presents one problem:

Howcanindividualpixels be _uniquelyaddressed whenitis noteven known

how many pixels eachhalftone cell willcontain? Evenifresolution

independence is ignored and the addressable resolution of the output device

is incorporated into a PostScript program, there remain other difficulties.

Hamilton (1988) compared clustered-dot and dispersed-dot techniques

on a PostScript printer ofknown addressable resolution, the Apple

LaserWriter at 300spots perinch. He found thatdispersion ofhalftone dots

lead to unusual_{non-linearity} of tone reproduction. In cases with

dispersed-dot _formation, an increase in blackened spots lead to a decrease in _density for

middletone values. In one case with cells _containing25 addressable _pixels,

cells with 18 blackened spots contained _{approximately} the same _density as

those with _only 6 blackened spots. These effects are _largely attributed to

electrostatic interactions between charged toner particles.

Anothermajor problem with dispersed-dot _halftoning crops _up when

such pages are used as _copy for subsequent printed reproduction. Image

capture, platemaking, and most conventional_printing processes_virtually

require the use of a clustered dot. The high contrast emulsionsfor films and

plates are _likely to _drop out isolated spots, _{dramatically altering} reproduction

used as proofs for subsequent high resolution _{imagesetting,} or are themselves thefinalprintedwork.

The Adobe _{PostScript interpreter is}

clearly designed to support

clustered-dot _{formations. The thrust}of this project is to determine ifit canbe externallydriven tosupport a type of partial-dot _structuring,which can realize some of the advantages of _both clustered-dot and _{dispersed-dot formations.}

Severalsoftware mechanisms for_halftoning are _{built into PostScript. At}

themostbasic _{level, halftoning} is controlled _by the set screen operator. The

operator takes three operands: the linescreen _frequency; the screen _angle; and a procedure called the spot_function, which controlsthe orderin which _device

pixels willbe turned onto builda halftone dot.

In all PostScript _{printers, defaults for these parameters are pre-set.}

Precise implementation of the set screen operator is _{device-dependent,} because the actual screen grid is defined in device pixels.21

In most_cases, a

PostScript programdoes not haveto implement newsettings. In _fact, it is

generallydiscouraged.22'23

Atypicaluse ofthe setscreen operator might appearas follows:

60 45 _{dup mul exch _dup mul add 1 exch _sub} setscreen

PostScript uses post-fix _notation, where the operands precede the

operator. In the above _line, 60 isthe requested linescreen_frequency, 45 degrees is the requested screen _angle, and the sequence within the { }, is the

spotfunction.

In PostScript the screen angle is applied as a clockwise rotation from the

The _frequency and angle are _requested, but will not _necessarily be

delivered _bythe outputdevice. This is due to thevagaries of_matching

user-specified halftonecells to those that a printer is _physically capable ofimaging. Anyhalftoning desiredon a devicewith afixed output grid is limitedinthe

selection of angles and frequencies that can be produced _using thatgrid.

Figure 10 illustrates how the problem appears in a 300 spots perinch

printerwith a requested _frequency of60. Inthe _group offourcells at 0 degree

angle _(left),each cell contains 25 pixels, butwhenthe cells are rotated to45

degrees (center) two ofthe cells contain 25 pixels and the other two contain24

pixels. The PostScriptinterpreter mechanisms are structured to provide a

repeatable pattern of cells all _containingthe samenumber of pixels. This

enables a seamless_tiling ofcells.24

Figure 10

Screening Grid Problem

To achieve this seamless halftoning, either the _frequency or the screen

typicallychanged_by the printer'sPostScript _interpreter, unbeknownst to the

user, to53cells perinchwith each cell_containing32 pixels(Figure _10,right). This is somewhatironicin that the defaultsettings for the LaserWriter

are 60 cells per inch _frequencywith a 45 degree screen_angle, and the device

cannot _actuallyrender at its own defaults.

Similar problems can force the adjustment of the screen _angle, orboth

the _frequency and _angle, when requested values cannot be matched to the

devicegrid.

Roth ₍₁₉₈₈₎ details some PostScript programs that canassist a userin

determining the frequencies and screen angle combinations that _any particular device can _actually achieve.

The Spot Function

Of _primary interest in this project is the spot function operand for the

setscreen operator. It is the spotfunction whichenables PostScript to

dictate dot shapes ina resolution independent manner.

Forthe spot_function, each halftone cell is considered to contain its own

miniature coordinate system ofxand _yaxes from-1 to +1,with(0, 0) at the

center ofthe cell. This is depictedin Figure 11.

The spotfunction is an executable procedure, that takes as its input the

xand_y coordinates ofeach pixelin a halftone cell, and outputs a single value thatdetermines the_priority in whichthat pixel willbe blackened to form a spot

as the _grayvalue ofthe cell variesfromwhite toblack.25 The outputvalues

1.1) y (1 1

l

+

-,-

-*- -a m-

-a a i

a a

a a _T a a

-1.-1) (1,-1)

Figure 11

Spot Function Coordinate System value _{for any given pixel compared to the values ofthe other}

pixels, the higher its _priority to be blackened.

Whenthe setscreen operator is called, the

spotfunction is executed and all the pixels inthe

cells are prioritized. This is simplified _bythe

repeatable nature of the seamless cells; the

calculations _only must be performed once. Most

spot functions (except _{those for}special _effects) yield

a _priority that creates some form of _centrally

clustered halftone dot. _Usually the spot closest to

thecenter willbe blackened _first,and thosein the

corners willbe blackened last.

Example:

60 45 _{dup mul exch _dup mul add 1 exch sub} setscreen. Thexand_y values are fed in with_y on_topofthe stack. This procedure

doesthe following: duplicatesy,multiplies it_byitself, exchanges thex and_y,

duplicates the x, multipliesit_byitself, adds the twoproducts _together, and

subtracts the sumfrom 1. (For specifics on the PostScript operators and the

stack-oriented nature ofthe _language, consult The Red Book.)

The effect ofthisparticular spot functionis to rankpixels _bytheir

distancefrom the center of _{the cell,} and the _resultingcluster will grow as a

circular patch centered in the cell.

In some PostScript devices the default spot function is one which

creates dot growth in a diamond-shaped pattern that resembles the more

APostScript program _byTom Bernard called

"ShowTheCell,"

listed in

the book Real World PostScript _{(Roth, 1988),} is _extremely useful in _evaluating

the effects of _anygiven spot function. Without "ShowTheCell" to ascertain

actuallydot structuring, this entire project would have been much more

difficult.

"ShowTheCell"

can be downloaded to a PostScript printer forexecution

via anapplication such as SendPS _(Adobe, 1986). The output from the printer

is a numerical and graphical _displayof the coordinates and spot function

valuesforeachpixelinthe cell.

Figure 12 shows a page produced _by a modified version of

"ShowTheCell,"

executed on a300 spotsper inch Apple LaserWriterPlus, _using

the setscreen parameters _previously described. Theimportant modification

wasto_flip the verticalaxis ofthecell.

In both theLaserWriter Plus and the Agfa P3400 PSprinter used in this

project, the_y axis ofthe device space is inverted relative to user space. This

places the negative_y values at the _topofthe cell. _Purelyfor thesake of_clarity

in_evaluatingcell structures for this project, the_y axis wasflipped backto

showing positive values at the _top ofthe cell. (Consult The Red Book for

21 -0.8.0.8 -0.28 22 -0.399,0.8 0.199 23 0.0.0.8 0.359 24 0.399,0.8 0.199 25 0.8,0.8 : -0.28 * 16 S,,.,,..,,,,,,-0.8,0.399 ...

17 ....-0.399,0.399 18 0.0.0.399 19 0.399,0.399 20 0.8,0.399 0.199 0.68 0.84 0.68 0.199

i 11 -0.8,0.0 0.359 12 -0.399,0.0 0.84 13 0.0,0.0 1.0 ah 14 0.399,0.0 0.84 am 15 0.8.0.0 : 0.359 W 6 -0.8,-0.399 |; 0.199 7 -0.399,-0.399 0.68 8 0.0,-0.399 0.84 9 0.399,-0.399 0.68 10 0.8,-0.399 0.199

1 2 3 4 5

-0.8.-0.8 -0.399. -0.8 0.0,-0.8 0.399,-0.8 0.8,-0. -0.28 0.199 0.359 0.199 -0.28

(-1.-1) . ,

, (1.-1)

60 0 Default pixelcount x.y value 3*> Figure 12 'ShowTheCell" Output --.%4*

%, \ s;, . W

4%^o

^

*^

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t\i.,

AC'-^'o

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S\"""- -ot of

""-,

In "ShowTheCell"

output, foreach_{pixel, three} lines ofnumbers appear:

the pixelcount; thex and_{y coordinates;} and the spot function value forthat

pixel. The pixelcount numbers the pixelsin the orderinwhich _theyare

presented to the spot function.

The program also presents the true cell _frequency and true screen angle

that the deviceisusing. Thisis all_extremelyuseful informationforanyone

experimentingwith setscreen.

The program generates variable sized black circles to indicate the

relative preference based onthe spotfunction value. The larger the circle the

higher_rankingthe pixelis inthepriority. Thisgives avisual_displayofhowdot

growth willoccurwithin_{the cell,} butitcan alsobesomewhat misleading.

Some viewers of output produced for this project believed that this indicated

the device could producevariable sized spots at the pixellevel. This is not the

case.

For _{this study, the}main interest isthe spotfunction value.

It is the spot function which will be used to implement the concept of

the _Tug approach, to alter the shape and position ofhalftone dots based on

PROCEDURES

Priorto _creating a PostScript-produced image _{containing halftone} dots

of_varying shapes and _{positions, two} majortasks (and a host of minor _ones)

needed to be addressed.

The first major task was to _develop a method of_determining what each

halftonedot should be shaped _like, based on the known _grayvalues.

The second major task was to _develop a method of_imaging such dots

ina real picture on a PostScript printer. This taskwas_by no means_trivial, and

was _actually the more precarious ofthe two.

The _Tug Function

Initially, theauthor hadin mind theconcept of_employing a fixed

number of dot _structures, and _designing an analysis method to choose which

should be used. Of _{those structures,} 18 differentones were determined to be

useful for_conveying sharpnessand contour information. These involved

favoring and _{disfavoringthe corners, edges,} diagonals, and axes ofthe cell.

Relatively simple functionsto produce each were written and tested with

"ShowTheCell."

The code forthese 18spot functions is listedin Appendix B.

Itbecame apparent, however, that a more _{theoretically} elegant

technique would be to write a custom spot function for each cellbased on its

gray value and those ofits neighboring cells. Atfirst this seemed beyond

feasibility. But ideas that are onthe verge ofthe possible havea _way of

naggingintheperiphery ofconsciousness: _"Ifit were_possible,how mightit

Eventually a series of concepts were drawn _up forwhat spotfunction

values would be desirable invarious parts ofthe cell forseveral sample

scenarios.

For _instance, if allthe cells on one side ofthe nominal cell were _relatively

dark and all the cells on the opposite side were _{relativelylight,} the spot

functionvalues onthe darkside shouldbe _higher, and thoseon the light side

should be _{lower, pushing}thehalftone dot towards the darker side.

But, ifcells onboth sidesweredarkand the nominalcell werelight, the

spot function valuesshouldbe higher onboth_sides, and lowerin the middle,

splittingthehalftone dotin two.

And, ifall thecells were of_{approximately} thesame _{value, the} default

spotfunction value should be employed to retain a _centrally clustered dot.

These decisions were _entirely subjective. But when in doubt as to how a

particular dotshould _behave, the author referred to a diagram, such as forthe

original imagein Figure 1, to guidethe process.

Itwas also decided that no attempt would be made to account for

varying screen angles. The equations would _{only be} intended forangles at

which cellboundaries all _squarely coincide. Due to devicepixel grid matching,

this occurs _only at0 degrees and multiples of 45 degrees,but the assignment of

variables based on image datain a linear arrayis muchmore complicated at

45 degree multiples.

Once the desired spot function values for the sample scenarios were

established, some sample equations were writtenwhich would generate the

In effect this was _{reverse-engineering} the normal process. The spot

functionvalues were _{empiricallyderived,} and the equations werethen_logically

derived to fit the values. The author considers the _{understanding} ofthis tobe

very important. It can neverbe demonstrated that these _Tug equations are

"correct"

in _any absolute _sense; it can_only be demonstrated that _they

accomplish what theauthor intended them to do.

The task of _testingequations _by hand for various sample values turned

outto be _{extremely tedious,}even for hypotheticalcells with _only 13 pixels.

Using "ShowTheCell" for the calculations was ruled _out, because the program

takes several minutes _{to execute,} and more _{significantly, the} author was

concerned that in _{using PostScript's} atypical post-fix notation, the act of

designingthe equations could not _easily be integrated into the act oftheir

programming.

A _testingprogram was writtenin the _{Pascal programminglanguage,}

using Turbo Pascal Macintosh (Borland, v. 1.1). A_listing of this Pascal program,

labelled "TugTest,"

alongwith a screen capture ofthe_runningprogram

window, Figure Bl, appearinAppendix B. Sucha program was crucialto this

project.

The _testing program displays spot function values fora hypotheticalcell

containing 81 pixels (9 x 9). The program provedto be quitefast and efficient.

With "TugTest," itwas _possible,over the course of several _{weeks, to} evaluate

an estimated 200 variations on the equations.

In general, theequations were firsttested withvarious extreme values

for the variables, then ifthe performance seemed desirable, intermediate

The program had demonstrated that the initial set of equations was not

always _producing desired results. Withsome of the _grayvalue combinations

for whichthe equations had been_{written, the} results were not as anticipated.

Aftermuch trial and _error, new equations were rewritten into a program

function _using if/else type statementsto _qualifythe portion of the cellupon

which each equation acts. The _{resulting function behaves mostly} as desired

under a wide assortment of cell_gray values. Itis _interesting to note thatthe final equations are much simpler than the author had anticipated at the outset

(althoughtheir _simplicity belies the thought process required to determine that

these were the ones to accomplish the task).

The final form ofthe functionis discussedbelow.

Figure 13 shows the basicarrangement of cells forthe _Tug function.

A

B

C

D

--?-E

F

G

H

-1,-1)

z

Figure 13 Cell Arrangement for The _Tug Function (1,-D

The _gray value ofthe central cellis a assignedto variable_Z, and the _gray

values ofthe _neighboringcells are assigned to thevariables A through H as

the inverse ofthe PostScript default of_assigning0 =black _{and 1} = _white, but

the method employed here better suits the concept of halftone dots _tugging

on each other: a higher number creates a stronger tug.

The coordinates in cell Z are depicted in the standard spotfunction

manner on_x,y axes from-1 to +1.

Thevariables for the _Tugfunctionare summarizedin Figure 14.

Variables.._....Range

x,y ....[-1..11 pixel coordinates in Z cell A,B,C,D,

E,F,G,H...[0..1] gray values ofneighbor cells, white=0, black=l Z ...J0..11 grayvalueofZcell

Ti,Tr,Tt,Tb ...[0..11 Tugvaluestowards edges (left,right, top,bottom) Ttl,Ttr,

TblTbr...

....[0..1] Tug values towards corners

s ...J-1..11 PostScriptspotfunction valuefor pixelx,y

T ...M..11 normalized total Tug

N ...[0..11 weighting and thresholdingfactor forTug

st ...[-1..1] finalTugfunctionvalueforZ,at pixelx,y

Figure 14

TugFunction Variables

The values ofthe eight _neighboring cells and the _existing spotfunction

are first combined with_{the x, y}coordinatesin a series ofnine equations [1 - 9].

These equations are shown in Figure 15with graphic depictions of their actions

onthe variables, wherelargermarks indicatepixels withlargerequation _values,

smaller marks indicate smaller equation values, and blank areasindicate

[1] forx<0andy>0: _Tti=A* Ixl *

ly

(forother_x,y : _Tti= ₀₎

[2] for_y> ₀and I yI > IxI: Tt=B*lyl*(lyl-lxl) (forother_x,y: _Tt =0)

[3] forx>₀and _y>_{0: Ttt}=C * IxI * I y I (forother_x,y : _Ttr= 0)

[4] forx<0 and IxI > I y I: _Ti=D* I_xI *(IxI - I

yI ) (forother_x,y : _Ti= 0)

[5] forx>0and Ixl > _{lyl: Tr}= E* Ixl *(lxl -lyl)

(forother_x,y: _Tr=₀₎

[6] forx<0and_y<0 : Tbi= F* I_xI * I y I (forother_x,y: _Tbi= 0)

[7] fory<0and lyl > _{Ixl: Tb}=G* lyl *(lyl -Ixl)

(forother_x,y: Tb=₀₎

[8] forx>0and_y<0 : _Tbr=H* IxI * I_yI (forother_x,y : Tbr= ₀₎

[9] S= 1 -(x2

+ _y2₎ [typicalspot function]

Figure 15

NeighboringCell and Spot Function Equations

-.

.

Equations [1-8] are qualified to act_only on pixelsinspecific parts ofthe

nominal _cell, and eachwill yield a valuebetween 0and 1. These equations will

not decrease the _Tug function values. This is the pivotal distinction between

the original set ofequations and the final set. Attempts to increase the values

in some parts of the cell and decrease the values in other parts tended to

create effects that cancelled each other out more often than desired.

It canbe seen that theseeight equations are oftwo basic types. One

type increases the result based on the product of the absolute values ofx and

y. This increases the_Tug function value the closera pixelis to the corner and fosters cell growth from the corner.

Earlier versions of those equations used the average of the absolute

values [(IxI +I_yI )/2]. Thistended to create dotgrowth_alongthe edges in

addition to through the middle of _{the quadrant,} whichwas deemed to be

undesirable for an equation based on the value of a _cornering cell.

The other type of equation increases the result based on the product of

the absolute value of x or_y and the difference betweenthe absolute values ofx

and y. This increases the_Tugfunction value the closer a pixel isto the intersection of an axis and anedge, and fosters cellgrowth from thatpoint.

Earlier versions of those equations used the absolute value of just one

of the coordinates, creatingcell growth _{evenly along}the edge. When

combined with the corner equations, this overemphasized corners.

The set of equations [1-8] as shown here creates _evenly increased corner

and edge values when the _grayvalues of all the surrounding cells areequally

Equation [9] calculates a typical_centrally clustered spotfunction value

and must yield valueswithinthe range -1 to +1. Anyvalid spot function canbe

used inplace of equation [9]. In _PostScript, this equation does not have to be

specified, the _existingspot functioncanbe retrieved with a call to

currentscreen command thatreturns the setscreen parameters thatare

alreadyin effect,which wouldbe the defaults unless _specifically redefined

previously inthe program.

The inclusion of the spot function component achieves the desired

effect of_{leaving every}halftone dot as specified _by the _existing spotfunction

unless it is _actively tugged into some other position.

When combined in proper _{proportions, these} nine equations can create

virtually anyimaginable halftone dot shape and position.

Three subsequent _equations, shown in Figure _16, combine the effects of

the _neighboringcells [10], calculatethe _{weighting factor for}the

neighbors'

effects [11],and thenweight the values of the spotfunction and the

neighbors'

effects toyield the final_Tugfunction valueforeachpixel [12]. Equation [10] adds the valuesfrom [1

-8] and normalizes the suminto

the -1 to +1 range requiredfor spot functions. This represents the total_tug

from the neighbors.

Eventhough there are eight equations for _neighboringcells, most pixels

in the cell are acted upon_{by only} two ofthem. Theexceptions are the pixels

that border the areas _affected, and in these cases no more than one equation

willhave a value greaterthan 0. In allcases, for_anyvalues of cells

AthroughH, the sum of values from equations [1

[10] T= _(Tt

+Tb +_Ti +_Tr +_Tti+ _Ttr+ _Tbi+Tbr):(-2-l

[11] N=

Va

|

|

|

_|

_|

)

[12] St= (T *_{N+ S}* _Z) / 2

Figure 16

Tug and _{Spot Function Consolidation Equations}

Equation [11] produces the N value from the square root ofthe mean of

the absolute value of the difference betweenthe cell and the neighbors. This

value isused as the_weighting factor forthe _tug componentin equation [12].

The value_St is the result returned_by the_Tugfunction.

Therelative weightings ofthe _tug(T) from the neighbors, and the spot

function _(S) in equation [12] were the mosttroublesome partsin the derivation

of all the equations. Italso involved the greatest degree ofsubjective

assessment. A simple _averagingofS and Twas ruled out, as this would not

allow dots to be placed at the edges or corners ofthe cell under _any

conditions.

Initially, the _tug componentwas weightedby the weakness ofthe _gray

value incellZ, _usingthe term (1

-Z) to _multiplybyT. (Inthat case, thesum

was not divided by2, sinceas Z_increased, [1

-Z] decreased, and the sum was

always withinthe-1 to +1 range.) Whenactualprintedimages were first

generated forthis project,it was apparentthatwith the (1

-Z) weighting,

difference atall in _grayvaluebetweenZand A through Hwould pull the dotto

the extreme edges ofthe cell. This occurred _only in areas of almost uniform

highlights, and appeared as somewhat _randomlyscrambled halftone dots.

Clearlythis artifact was undesirable when seen within animagewhich also

contained more _regularly positioned dots.

Therefore, rather than_weighting the T component _bythe weakness of Z,

it was determined that the _weighting should be based onthe strength ofthe

neighbors (N). Copious attempts at equations foranN value included: the

average neighbor_gray value; the largest neighbor _gray value; the absolute

value ofthe average of the differences, the average ofthe absolute values of

the differences, and the largest difference between Z and its neighbors, suchas

(A

-Z), (B

-Z), etc.; the differences between opposing neighbor cells, such as

(A

-H), (B

-G), etc., and _{the sums, the products, the} squares and the square

roots of _many ofthese quantities. Theroot mean difference equation [11] used

in the latest version of the _Tug function seemedto produce the best

compromise between desirable sharpness and undesirable artifacts. This was

purelyajudgmentcall.

Asan example, inearly versions of the equations, which weighted the T

value_bythe weakness ofZ [i.e. (1

-Z)], any not-quite-smooth highlight area

produced scrambled dots as shown in Figure 17.

Changingthe _{weighting factor} to anN value equal to the root mean

difference equation altered the function so that all halftone dots stay centered

With Z-Weighted _Tug

With N-Weighted_Tug

Figure 17

The _weightingof the _existing spot function_(S) based onthe magnitude

ofZ enacts the concept of a halftone dot _resistingthe _tug of its neighbors

based on itsown strength. Highlighthalftone dots can be displaced anywhere

inthe cell; middletone dots will remain centralbutcanbe skewed in _any

direction; and shadowdots will _mostly remainas specified _bythe spot

function. _Again, this choice was _conceptually based onthe model ofthe _Tug

function as established at the outset.

The entire foundation ofthisfunction is based on the need to _only

specify relative rather thanspecific spot function values. For some values of

the cells, the entire range ofpossiblespotfunction values from -1 to +1 is used,

but for _manycell_{values, only} narrow portions ofthe range are used. In

general, the range is determined _by a complex interactionbetween the

magnitude of the _gray value ofthe central cell and the sum of the differences of

the_grayvalues oftheneighborcells. Thisvariationinrange is _primarilya

result ofthe _{positive-only} effectofthe equations for_neighboring cell values.

The author encourages readers to use the Pascal "TugTest"

program

(listed in AppendixB) toexperiment withthe various _gray levelvaluesfor the

cells, and to experiment with the equations themselves to assess the effect of

The PostScript _Tug Function

The task of_writing the equations into a PostScript spot function

program was _relativelystraightforward. The_Tugfunctionas written in

PostScriptappears in AppendixB, within the program _{labelled "The} PostScript

TugFunction."

Itwasnotknown inadvanceifthefunction wouldexecute, even ifthe

code were _{syntactically} correct. All the spot functions the author _previously

had seenor writtenhimselfwere _fairly simpleprocedures: one ortwo linesof

mostly arithmetic operators. The _Tugfunction _clearly was farmore complex.

Itwas not even knownifvariables other than the xand _y coordinates ofa pixel

couldbe used within a spot function procedure. Itwas not knownifvariables

could be defined withinthe procedure. And it was notknown ifvariables defined outside the function could be used within the procedure.

Although there was no particular reason to believe that the code

sequence would not execute within the context of a spot function parameter for setscreen, there wasalso no precedent_indicatingthatitwould. Without

knowing the specifics ofthe _underlyingmechanisms used in PostScript

interpreter implementations (which are _proprietaryto Adobe _{Systems, Inc.),}

these concerns could notbe taken for granted.

Some simple _feasibility tests were performed. The Macintosh

application Lasertalk (Emerald CitySoftware, v. 1.0) was used to debug and

download "ShowTheCell" PostScript files toApple LaserWriter Plus andAgfa

P3400PSprinters, using various spot functionmodifications. Suchtests

included _defining variables withinthe spot _{function, using}variables _defined

function. In all cases the results were as anticipated forthe code, and no

unusual limitations appeared to exist on the use of operators in the spot

function.

Using the_Tugfunctionfor_defining thespot within

"ShowTheTug"

(amodified version of _{"ShowTheCell"),} various combinations of valuesfor a

nominal cell and its neighbors were downloaded to the LaserWriter. Figure 18

and Figure 19 are samples ofthe pages produced as output. These

representations were compared to those obtained on-screen with the Pascal

"TugTest"

to _verifythat the _Tugfunctionwas_{functioning properly}and no

obvious errors were made in the post-fixnotation of the PostScript

programming.

The representations inFigure 18and Figure19 indicatethe _prioritywith

whichthe pixels in the cell willbe blackened as _previouslydescribed for

"ShowTheCell"

output.

These samples_only show how a single cell would be imaged. With

numerous cells layed inplace, however, the effect _may be somewhat different

21 -0.8,0.8 -0.021 16 -0.8,0.399 0.093 11 -0.8,0.0 0.113 -0.8,-0.399 0.036 -0.8,-0.E -0.135 22 -0.399,O.E 0.093 17 -0.399,0.399 0.051 12 -0.399,0.0 0.0B5 23 0.0,o.s 0.113 0.0,0.399 0.085 13 0.0,0.0 0.076 -0.399,-0.399 0.0,-0.399 0.023 0.057 2 -0.399,-0.8 -0.02 3 0.0,-O.E -0.001 24 0.399,O.E 0.036 19 0.399.0.399 0.023 14 0.399,0.0 0.057 0.399.-0.399 -0.005 4 0.399,-0.8 -0.077 25 0.8,0.8 -0.135 20 0.8.0.399 -0.02 15 0.8,0.0 -0,001 10 0.8,-0.399 -0.077 5 0.8,-0,( -0.25

requested_frequency=60,requested angle=₀

truefrequency=_60.0,trueangle=0

spotfunction=TUG

pixelcount

x.y

spotfunctionvalue

A 1.00 B 0.80 c 0.60 D 0.80 Z 0.60 E 0.40 F 0.60 G 0.40 H 0.20 Cell Neighbors Figure 18 "ShowTheTug" Middletone Sample

-0.8,0.8 0.06 22 -0.399 0.027 0.8 23 0.0,0.8 -0.037 24 0.399, -0.134 0.8 25 0.8.0.8 -0.263 16 -0.8,0.399 0.027 17 -0.399 -0.147 0.399 is 0.0,0.399 -0.171 * 19 0.399, -0.227 0.399 20 0.8,0.399 -0.134 11 -0.8,0.0 -0.037 12 -0.399, -0.171 0.0 13 0.0,0.0 -0.216 14 0.399, -0.171 0.0 15 0.8,0.0 -0.037 "-- "">" "' 6 -0.8,-0.399 -0.134 7 -0.399, -0.227 -0.399 ;i 8 0.0,-0.399 -0.171 9 0.399. -0.147 -0.399 10 0.8,-0.399 0.027 . f 1 -0.8,-0.8 -0.263 2 -0.399 -0.134 -0.8 3 0.0.-0.8 -0.037 4 0.399, 0.027 -0.8 5 0.8.-0.8 0.06

requested_frequency= _60,requested angle.0 pixelcount true_frequency=60.0 trueangle=₀

*.y

spotfunction=TUG spot unction value

A 1.00 B 0.60 c 0.20 D 0.60 Z 0.20 E 0.60 F 0.20 G 0.60 H 1.00 Cell Neighbors Figure 19 "ShowTheTug" Highlight Sample

Usingthe _{Tug function,} a tremendous number ofdot shapings is

possible _by

merely changing the values ofthe nine_{variables, Z,}and Ato H.

In normal PostScript_{imaging applications,} each ofthe nine _graylevels

can_beone of256values,_yielding atheoretical potentialof

2569

(4.7x _IO21)

possible spot functions. In _any particular _{implementation,however,} the actual

number achievable is miniscule _by comparison.

Fora devicewith 300 addressable spots per_{inch, using} a 60 linescreen

(25 pixelsper_{cell), there} are_only

225

(33.5million) possible combinationsof

pixel spots thatcouldbe "on" simultaneously. (Thereare25! [factorial]or

25

1.5 x 10 possible orders in which _they could be activated, but once the pixels

to be activated are _determined, the order in which this is accomplished is

irrelevant.)

Inthe case ofthe_{Tug function,} since one ofthe variables is itself the

gray value ofthe cell _{being imaged, many} of the possible combinations cannot

be used. Forexample, ifthe _gray valueis_relativelylarge,all combinations that

leave thecenter of the cell _empty can never occur. The author will leave the

calculation of such possibilities to an ambitious reader.

More importantly, many ofthe possible combinations will appear

identical on the printed page. For a laser printer, theactual number of

discernable halftone dot shapes would depend on the _many interactions of

laser, photoconductive surface, toner particles, paper substrate, and the

viewing distanceand vision of theviewer. Hamilton(1988) gives someinsight

into mathematical possibilities vs. actual capabilities of a laser printer.

Clearly the _Tugfunction will notbethe limitingfactor. Itmight evenbe

It also seems apparent that the _quantity and _variety of possible dot

shapes _may far exceed those possible through what has been called

partial-dotstructuring. That _is,the_Tug functioncan produce structures thatare not

merelyportions of regularhalftone dotschopped out _along a rectangular grid.

(Referbackto Figures 5 and _6.) For_instance, thehighlight sample shownin

Figure 19 _probably could notbe produced _bypartial-dot methods _currently in

use, except perhaps in the case ofextreme _oversampling on the order of25

sample values to 1 cell. The_Tugfunction accomplishes this with no

oversampling whatsoever.

The term partial-dot seems inadequate to describe the possible dot

structures that can becreated _by the _Tugfunction. _Therefore, project advisor

Professor Frank _Cost, has proposed the term context sensitive dot _structuring

as a generic term to describe the phenomenon of _shaping a dot based on its

neighboringcells.

Imaging Context Sensitive Dots

From the_many

"ShowTheTug"

samples the authorhad generated, it

appeared that on a _purely hypothetical basis, the Tug function could createthe

individual halftone dot shapes as desired bythe author.

In order to _bring the _Tug function beyond the realm of the _merely

abstract, it was _necessary to reproduce photographic images _utilizing the

function in situ. This was much more problematic than itmight seem.

The standard method of_{printing gray}scale images through PostScript is

with the image operator. Theoperator takes five operands: the number of