a quality measure for halftones obtained by linear and nonlinear

A quality measure for halftones obtained

by linear and nonlinear resamplingD. Van De Ville1∗, K. Denecker1, W. Philips2 and I. Lemahieu1

Ghent University1 ELIS/MEDISIP/IBITECH 2 TELIN

Sint-Pietersnieuwstraat 41, B9000 Ghent, BelgiumE-mail: [email protected]

Abstract—Almost all printing techniques use half-toning to create the illusion of contones on bi-levelprinting devices. Still the most popular halfto-ning technique is classical halftoning, where dotsof varying sizes are placed on a regular screen lat-tice. Since the screen lattice has a different reso-lution and geometry than the original lattice, i.e.,an orthogonal 300 × 300 dpi grid, resampling tech-niques must be used. The quality of the halftonedepends on the amount of aliasing and the amountof blurring due to this resampling. Aliasing canlead to dreaded moir e patterns, while blurringrenders the image unsharp. Traditional linear re-sampling filters must trade-off between these twoproperties. Previous work has presented a newnon-linear resampling technique based on space-frequency analysis to combine the anti-aliasing andedge-preserving properties. Until now, these re-sampling filters were evaluated by visual inspectiononly. This paper proposes a new objective qualitymeasure to evaluate and compare these resamplingfilters in a two-dimensional evaluation space, i.e., ameasure for the amount of aliasing and the amountof sharpness. First, it is shown that existing tech-niques such as inverse halftoning or quality mea-sures for evaluating compressed images are not ap-propriate for these purposes. Next, we present anew quality measure inspired on the human visualsystem which is able to compare an image with itshalftoned result. Experimental results show thatthe non-linear resampling scheme is able to com-bine anti-aliasing and edge-preservation.

Keywords— Image quality measure, human visualsystem, aliasing, halftones, resampling

I. Introduction

Almost all printing techniques today are binary pro-cesses, meaning that they produce bi-level images.Halftoning techniques are required to create the illu-sion of continuous-tone images for a human observer[1]. To achieve the illusion of the original contone,image halftoning techniques rely on the human visualsystem (HVS) to integrate small bi-level features so

∗Dimitri Van De Ville is a Research Assistant of the Fundfor Scientific Research Flanders (FWO Belgium).

that they are perceived as gray values. These tech-niques can be categorized into two groups. The oldercategory, classical halftoning or amplitude modulation(AM) places dots of varying sizes on a regular (screen)lattice. The most recent category, frequency modu-lation (FM) is based on a stochastic distribution ofsmall similar dots.

Classical halftoning is used more often because it isvery robust to ink-spreading distortions. Conceptu-ally, classical halftoning can be considered as a two-stage process. First the image is resampled to anotherlattice (i.e., the screen lattice). Next these values areused to vary the size of the halftone dots. Resamplingcan cause subject-moire due to aliasing when the con-tent of the original image interacts with the screenlattice [2–6]. Such moire patterns are undesirable be-cause they are very annoying and noticeable to the hu-man observer: new frequency components arise witha much lower frequency and (mostly) with a differentorientation than the original component. Resamplingtechniques using linear interpolation functions mustbalance between anti-aliasing (suppressing moire pat-terns) and sharpness (preventing the image from beingblurred) [7, 8]. Previously, we have presented a non-linear resampling scheme which combines the outputof two different linear resampling filters according to alocal risk of aliasing. This method estimates the riskof aliasing using the windowed discrete Fourier trans-form [9] or the Gabor spectrogram [10,11]. This re-sampling scheme makes is possible to combine edgep-reserving and moire-suppressing properties.

Until now, evaluation of these resampling techniqueswas done by visual inspection. Current visual qualitymeasures, such as those presented for coding purposes[12], are not able to compare an original grayscale im-age and a simulation of its halftone (a bi-level imageat high resolution). These methods require both im-ages to be grayscale and of the same dimensions. Onthe other hand, “inverse halftoning” techniques tryto reconstruct the original grayscale image from itshalftone [13, 14]. These techniques also “enhance”the grayscale result and do not incorporate the hu-

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D. Van De Ville, K. Denecker, W. Philips and I. Lemahieu

man visual system (HVS). Both approaches lack theability to compare an original and its halftone regard-ing aliasing and sharpness. This paper presents anew objective quality measure corresponding to theHVS using no prior knowledge about the resamplingor halftoning.

The first section introduces essential theory aboutlattices, sampling, and halftoning. In particular, gra-vure printing is considered as a practical example ofa printing technique using classical halftoning. Next,the new objective quality measure itself is presented.Finally, some experimental results are shown and con-clusions are drawn.

II. Lattices and Halftoning

A. Lattices

A two-dimensional lattice can be characterized bytwo linearly independent vectors r1 and r2. Each lat-tice site is represented by the vector

rm,n = mr1 + nr2, where m,n ∈ Z

= (r1|r2)(mn

)= rm, where m ∈ Z2.

(1)

Thus the lattice is completely specified by the matrixR = (r1|r2). It can also be described by the so-calledVoronoi cell of the lattice R, which is defined as theset of all points that are closer to the origin (00)T

than to any other site of the lattice. The Voronoi cellis often represented by its indicator function:

χR(x) =

1, for x ∈ Voronoi cell,1/k, for x on the edge of the

Voronoi cell,0, for x /∈ Voronoi cell,

(2)

where k equals the number of lattice sites to whichx is equidistant. Note that the Voronoi cell, whenperiodically copied onto all the lattice sites, coversthe entire plane:∑

m∈ZχR(x−Rm) = 1,∀x ∈ R2 (3)

and that their copies do not overlap. In other words,the Voronoi cell can be considered as a “natural pixelshape” which tiles the plane.

The reciprocal lattice is defined as the lattice withmatrix F = (R−1)T The reciprocal lattice plays animportant role in sampling theory. The Voronoi cellof the reciprocal lattice is the two-dimensional variantof the Nyquist frequency, a.k.a. the Nyquist area.If an image contains frequency components outsidethis Nyquist area, resampling to this lattice can causealiasing and subsequently moire patterns.

Images will be denoted as i(x) = i(x1;x2). Thespectrum I(f) of i(x) is its two-dimensional Fouriertransform

I(f) = F{i(x)}(f)

=∫ ∫

R2

i(x) exp(−j2πf · x)dx1dx2.(4)

B. Halftoning

This paper considers a gravure printing setting as anon-trivial practical example of a printing techniqueusing classical halftoning with pure amplitude modu-lation: for every halftone dot, a little notch is engravedwith a computer-driven diamond in the printing plate.The depth of the notch determines the amount of inkit will contain and thus the size of the correspondinghalftone dot. A typical gravure printing screen latticehas a lower resolution and a different hexagonal geom-etry than the usual orthogonal lattices on which theoriginal images are specified. Thus, resampling fromthe original to the screen lattice is a necessary step inthe halftoning process.

The lattice of the original digital image is orthogo-nal: its matrix R and its reciprocal matrix F are

R =(r 00 r

), F =

(1/r 00 1/r

), (5)

where 1/r is the horizontal and vertical resolution ofthe image. On the other hand, the screen lattice is notorthogonal but semiregular hexagonal, i.e., its latticematrix V and reciprocal matrix W are

V =(

0 ba a/2

), W =

(−1/(2b) 1/b

1/a 0

). (6)

For the type of gravure printing considered in thispaper, the values of a and b are 0.2mm and 0.12mm.Consequently, the numerical values of the reciprocalmatrices of the source lattice and the target latticeare

F =(

300 00 300

)dpi, (7)

W =(−105.8 211.6

127 0

)dpi. (8)

Figure 1 shows the Voronoi cells of these lattices andthe corre-sponding reciprocal lattices; χR(x), χF(f),χV(x) and χW(f), are the corresponding indicatorfunctions of the spatial and reciprocal Voronoi cell ofthe source and target lattice, respectively.

The aim of this paper is to evaluate different resam-pling techniques for these lattice settings. Figure 2shows a diagram of a general halftone imaging model[15]. In the case of gravure printing, the halftoner

550 Proceedings of the ProRISC/IEEE workshop

A quality measure for halftones obtained by linear and nonlinear resampling

Fig. 1. The Voronoi cells in the spatial and the spectraldomain in the case of gravure printing halftoning.

ContoneImageImage Halftoner Device

Model VisualHuman

System

Perceived

Fig. 2. Diagram of a halftone imaging model.

resamples the original contone image and computespixel values to be used by the engraving process. Thedevice model takes these values, and simulate the bi-level printing dots by an appropriate threshold matrixon a micro-grid [1] shown in Fig. 3. The generated

Fig. 3. The threshold matrix (10× 10) is used to simulatethe clustered-dot halftoning process on a micro-grid.

high-resolution bi-level image can then be printed ona desktop printer and evaluated by human observers.Figure 4 shows a part of the lena test image repro-duced in this way. This paper proposes to evaluatethe halftoned images by simulating the perception bythe HVS and compare it to the original contone.

III. An Objective Quality Measure

One can think of several approaches using the spe-cific nature of the resampled image. For example, inour case the frequency contents of the resampled im-age can be analyzed using a hexagonal Fourier trans-form [16,17] or use hexagonal high-pass filters andedge detectors [18–20]. Such an approach would be-

Fig. 4. Example of simulated gravure printing halftonewith enlargement.

come very specialized and dependent on the targetlattice and does not incorporate a device model. Fur-thermore, it is still not obvious how to compare tothe original image. Another approach could be to use“inverse halftoning” to reconstruct a grayscale from ahalftone. Most techniques use a low-pass filter withadditional “enhancements” to improve the quality, inparticular the sharpness, of the result. As such, thesemethods do not try to incorporate properties of theHVS.

Therefore, we prefer a more general approach, inde-pendent of the lattices and halftoning technique thatis used, and inspired on the HVS (because we want themeasure to be related to what a human observer sees)and able to compare two images of a different type(grayscale/bi-level) and different dimensions. Figure5 shows a general overview of our new objective qual-ity measure. In the following subsections we explainthe operation stage by stage.

A. The pre-processing stage

Two images i1 and i2 are supplied to the input.Henceforth we suppose i1 is the original image (ref-

T

Σ

Σ1

ξ

x

x

2ξ

x

CSFUP

CSF Sobel

pre-processing output

SobelLPF

|−|

i

i

2

1

~~

main stage

Sobel

Fig. 5. An overview of the objective quality measure.

November 30 – December 1, 2000 551


erence image), while i2 is its halftone. Note that i1is a grayscale image while i2 is a binary image withmuch larger dimensions than i1. The preprocessingstage upsamples the original image i1 to i′1 with thesame dimensions as i2 using a simple interpolation fil-ter (e.g., nearest neighbour interpolation1). In fact,the proposed scheme is defined in the continuous do-main. Since this is unfeasible in practice we use the“micro-grid” of the halftone as an approximation forthe continuous domain.

B. The main stage

The main stage of the quality measure computesseveral processed images which are used by the out-put stage to measure the amount of aliasing and theamount of sharpness.

B.1 Filtering by the contrast sensitivity function

The contrast sensitivity function (CSF) indicates aweighting for different spatial frequencies based on theHVS. A good approximation for the CSF is given by

S(ω) = 1.5 exp(−σ2ω2/2)− exp(−2σ2ω2) (9)

with

σ = 2, ω =2πf60

, f =√f21 + f2

2 , (10)

where f1 and f2 are the horizontal and vertical spatialfrequencies, respectively, in cycles per degree [12,21,22].Additionally, at higher spatial frequencies, the fre-quency response seems to be anisotropic, so a bettermodel is given by [12]

SCSF (f1, f2) = S(ω)O(ω, θ), (11)

with

O(ω, θ) =1 + exp(β(ω − ω0)) cos4(2θ)

1 + exp(β(ω − ω0)), (12)

where θ = arctan(f1, f2) represents the orientationand

β = 8, f0 = 11.13 cycles/degree. (13)

Figure 6 shows the CSF SCSF . Note how frequencycomponents in the diagonal direction with f > f0 arestronger attenuated.

The frequency coordinates of Eq. (11) are in cy-cles/degree. Once the viewing distance and the res-olution are known, it is easy to compute the corre-spondence of cycle/degree to cycle/inch. If the view-ing distance is 7.6 inch and the resolution is 300 dpi,

1This can be seen as a simple model for enlarging the image asit is displayed on a computer display with a square pixel shape(while the viewing distance is increased accordingly).

–2

0

2

4

6

8

10

12

f1

–2

0

24

6

810

12

f2

0

0.2

0.4

0.6

0.8CSF

Fig. 6. The contrast sensitivity function (CSF) indicatesa weighting for spatial frequencies (cycle/degree).

then the Nyquist frequency (i.e., 150 cycles/inch) cor-responds to 20 cycles/degree.

The original images i′1 and i2 are filtered by theCSF. The filter is implemented in the frequency do-main: [23]

I1,CSF (f) = I ′1(f)SCSF (f),I2,CSF (f) = I2(f)SCSF (f).

(14)

Filtering by the CSF models the property of theHVS that small bi-level features are integrated. Thispart can be compared to inverse halftoning [13, 14],except that this approach uses the CSF instead of aGaussian low-pass filter and the images are not sub-sampled to the original dimensions.

B.2 Sobel-filter on the low-pass filtered image

The next part of the main stage produces a binaryimage, which indicates the “real” edges. Edges in animage can originate from borders of objects, or fromhigh-frequency patterns such as textures and grills.The latter kind can introduce moire patterns whenresampled. “Real” edges only refer to the first kind.

For that purpose a low-pass filter (LPF) first sup-presses those (high) frequency components that causemoire patterns. Specifically, the stopband of this filtermust at least contain those frequencies which can notbe represented on the screen lattice, i.e., frequenciesoutside the Nyquist area of the screen lattice. Thefiltered image is written as:

I1,LPF (f) = I1(f)SLPF (f). (15)



In most cases, the CSF (when the viewing distance issufficiently large) can be used as a LPF.

Next, the low-pass filtered image i1,LPF (x) is pro-cessed by a Sobel edge-detector. The Sobel edge-detector first takes the maximum of two high-passfilters:

i1,LPF,SOB(x) = max(iLPF ? sH , i2,LPF ? sV ), (16)

where the ?-operator represents a convolution withthe Sobel kernels

sH =

1 0 −12 0 −21 0 −1

sv = sTH

(17)

.Next, the result of this Sobel-filter is compared to

a threshold T :

i1,T (x) ={

0, i1,LPF,SOB(x) ≤ T1, i1,LPF,SOB(x) > T.

(18)

The binary image i1,T is called the “edge-map”.

B.3 Sobel-filter on the CSF filtered images

The Sobel-response of Eq. (16) is also computed forthe images after being filtered with the CSF. Theseimages, respectively i1,SOB and i2,SOB will be com-pared to each other to assess the sharpness after half-toning.

B.4 Construction of the error image

A general error image represents the absolute dif-ference between both perceived images:

e(x) = |i1,CSF (x)− i2,CSF (x)|. (19)

C. The output stage

The output stage uses the edge-map i1,T to selectthose parts of the processed images which do or do notbelong to “real” edges. For example, the two images

eT (x) = (1− i1,T (x))e(x) (20)i1,CSF,T (x) = (1− i1,T (x))i1,CSF (x), (21)

contains those parts that do not belong to edges, re-spectively of the error image and the perceived orig-inal image. This image pair is used to measure theamount of aliasing by computing the ratio of the en-ergy in both images:

ξ1 =

∑e2T∑

i21,CSF,T

(22)

In other words, the energy in the error image, nor-malized by the original image’s energy, is a measure

TABLE IResults for some test images at viewing distance

7.6” and threshold T = 15.

image resampling method ξ1 ξ2hard cubic conv. 0% 87%hard low-pass 0% 67%lines cubic conv. 100% 0%shirt cubic conv. 13% 88%shirt bilinear 12% 80%shirt low-pass 7% 66%shirt non-linear 7% 83%

for the image difference at non-edge regions, mainlysupposed to be caused by aliasing patterns.

On the other hand, the Sobel-filtered versions of theimages are considered as a measure for the sharpnessof the “real” edges:

i1,SOB,T (x) = i1,T (x)i1,SOB(x), (23)i2,SOB,T (x) = i1,T (x)i2,SOB(x). (24)

Again, the ratio of the energy

ξ2 =

∑i22,SOB,T∑i21,SOB,T

(25)

is a measure for the remaining sharpness in the image.The pairs (ξ1, ξ2) are points in a two-dimensional

evaluation space. They are interpreted as percentageand limited to the range 0 – 100%.

IV. Results

Experiments are undertaken to investigate the feasi-bility of the proposed approach. Halftoned test imagesusing different resampling techniques are compared tothe original test image.

(a) (b)

Fig. 7. (a) Test image “hard”. (b) Test image“shirt”.



(a) Cubic convolution (b) Low-pass (c) Non-linear

(d) i2,CSF (e) e (f) i2,SOB

Fig. 8. Results for the test image “shirt”. (a)-(b) Halftones using linear resampling techniques. (c) Halftone usingnon-linear resampling scheme. (d) The halftoned image (resampled using cubic convolution) after filtering by theCSF. (e) The error image e, moire patterns are most noticable. (f) The halftoned image after the Sobel-filter.

At first, the quality measures of two extreme testimages are compared. The test image “hard”, de-picted in Fig. 7 (a), contains sharp edges but no highfrequency patterns.

Table I indeed shows that the aliasing measure ξ1is 0%. The sharpness measure ξ2 is 87% using cu-bic convolution as resampling filter and drops to 67%using an interpolation function with low-pass charac-teristics. On the other hand, the test image “lines”contains a single sinusoid with a frequency too highfor screen lattice such that the halftoned image con-sists of moire patterns. The measures ξ1 and ξ2 areappropriately 100% and 0%.

Next, the test image “shirt” of Fig. 7 (b) is evalu-ated.

Figure 8 shows the halftoned images using cubicconvolution and low-pass interpolation in (a)-(b). Alinear resampling filter must compromise between edge-preservation and anti-aliasing. Also the quality mea-sures in Table I show less aliasing brings along less

sharpness for linear resampling filters. The non-linearresampled version in Fig. 8 (c) is able to combinethese conflicting properties. The resampling filter islocally adaptive to the frequency content. Both visual

60

65

70

75

80

85

90

95

100

0 2 4 6 8 10 12 14

1

2

Cubic convolutionBilinear

Low-passNon-linear

Fig. 9. Results for the test image “shirt” halftoned usingdifferent resampling techniques.



0

20

40

60

80

100

0 5 10 15 20 25 30 35

Cubic convolution 1

Cubic convolution 2

Low-Pass 1

Low-Pass 2

ThresholdT

Per

cent

age

(%)

(a)

(b)

(d) (e)

(c)

(d) (e)

Fig. 10. (a) Influence of the threshold value T on the qual-ity measures ξ1 and ξ2 for resampling the test image“shirt” using the cubic convolution and low-pass in-terpolation function. Edge-maps for (b) T = 10, (c)T = 15, (d) T = 25, (e) T = 35.

inspection and the objective quality measures confirmthe better result of the non-linear resampling scheme.Figure 8 (d) shows the halftoned image of Fig. 8 (a)after filtering it with the CSF. The error image e isshown in Fig. 8 (e). The Sobel-filtered version ofi2,CSF in Fig. 8 (f) is used to assess the sharpness.

Figure 9 summarizes the results for the test im-age “shirt” by plotting the quality measure for eachhalftoned version as a point in the two-dimensionalevaluation space (aliasing,sharpness). The linear me-thods are connected by a curve. It can be noted thatthe non-linear method is off this curve and closer tothe optimal point (0, 100), where no aliasing occursand all sharpness in conserved.

It is also interesting to investigate the influence ofthe threshold value T . This value is used to computethe edge-map iT . Figure 10 shows several edge-mapsfor the test image “shirt” in (b)-(e). A lower thresh-old value classifies more of the Sobelfiltered image asedges. The plot in Fig. 10 (a) shows how the qualitymeasure changes as function of the threshold value.The sharpness measure experiences only a minor in-fluence of T compared to the aliasing measure. Sincea higher threshold value makes the non-zero regions inthe masked error image eT larger, the aliasing mea-sure contains more than just errors due to aliasingpatterns. However, it is important to note that therelative difference of the values of the quality measuresbetween for example cubic convolution and low-passresampling remains the same. Therefore, a good op-erational range for T is 10–25. The other results pre-sented are obtained by using a threshold value T = 1.

Figure 11 shows the influence of the viewing dis-tance on the quality measure. As the viewing dis-tance increases, the CSF of Fig. 6 gets a smallerpassband and a larger stopband. The aliasing mea-sure decreases for a larger viewing distance, while thesharpness measure increases. This is because the im-ages become more similar. Note that the edge-map isthe same for all viewing distances (the LPF remainsthe same). The other results are obtained using aviewing distance of 7.6” (= 20cm).

0

20

40

60

80

100

4 5 6 7 8 9 10 11

Cubic convolution 1Cubic convolution 2

Low-Pass 1Low-Pass 2

Viewing distance (inch)

Per

cent

age

(%)

Fig. 11. Influence of the viewing distance on the qual-ity measures ξ1 and ξ2 for resampling the test image“shirt” using the cubic convolution and low-pass in-terpolation function.



V. Conclusion

Moire patterns are most undesirable when they ap-pear in halftoned images. Due to aliasing in the re-sampling stage, new low-frequency patterns can arise.Evaluation of halftones resampled using different tech-niques was done previously by visual inspection. Thispaper presents a new objective measure to assess bothsharpness and aliasing of halftones. The approachproduces results which are according to the visualinspection. Furthermore, we evaluate linear resam-pling techniques versus a joint edge-preserving andmoire-suppressing non-linear technique. The objec-tive quality measure makes it possible to compare sev-eral methods on a common basis. The construction isrelatively easy to implement.

Acknowledgments

This work was financially supported by the Fundfor Scientific Research - Flanders (Belgium) througha mandate of Research Assistant (Dimitri Van DeVille).

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a quality measure for halftones obtained by linear and nonlinear

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