554 J. Opt. Soc. Am. A/Vol. 10, No. 4/April 1993

Quantization noise in electronic halftoning

Manfred Broja and Olaf Bryngdahl

Department of Physics, University of Essen, 4300 Essen 1, Germany

Received May 13, 1992; accepted October 29, 1992

Several classes of electronic halftone techniques are described that can be applied to binarize graytone images.The difference between the quantized and the corresponding analog image constitutes the quantization noise.The characteristics of the quantization noise can be influenced by the parameters that determine the halftoneprocess. Iterative methods show high flexibility. In particular, they permit control of the image spectrumduring the quantization process.

1. INTRODUCTION

In digital electronic imaging the pictorial information issampled and quantized12 and is determined by the hard-ware used. However, the image characteristics can beextensively influenced by software.

In this paper we consider different electronic methodsthat convert graytone images into quantized graylevelones. Such processes can be interpreted as coding ofgraytone images. Decoding procedures require con-straints on the quantization noise. It is helpful to charac-terize the quantization noise introduced by nonlinearcoding methods, as it is an important factor at decoding.

Situations are described in which the number of quan-tization levels is reduced to the minimum, i.e., the twolevels 0 (dark) and 1 (bright). The halftone (binarizationof graytones) techniques introduce quantization (binariza-tion) noise, the characterization of which is strongly de-pendent on the particular procedure applied.

The choice of halftone method permits the selection of aparticular image-transfer characteristic. Transfer func-tions of imaging procedures are conventionally and con-veniently described in spectral-frequency terms. Thefeature that we concentrate on here is the possibility ofcontrolling the quantization noise. The resulting noisespectra are treated for some basis methods.

2. DESCRIPTION OF ELECTRONICHALFTONE IMAGES

In order to describe properties and requirements of thehalftone procedure, we consider two alternatives: theimage-plane distribution and the spatial-frequency spec-trum of the image. The constraints that are due to thehardware can be formulated in the x= (x, y) plane ofthe image. The conditions placed on the properties ofthe imaging can be defined in the u = (u,v) plane of thespectrum. The effects caused by the conversion of thegraytone image f(x) into the binary image g(x) appearas binarization effects in the image plane and as aquantization-noise spectrum in the frequency plane. Thefrequency spectra F(u) and G(u) of f(x) and g(x) are ob-tained by Fourier transformation , i.e., F(u) = SPf(x)and G(u) = S;g(x).

If no deviation or only a slight deviation between the

low-pass characteristics of f(x) and g(x) is desired, the re-quirement to be fulfilled is that

F(u) G(u) (1)

in the low-frequency region, D, or that the mean-squareerror o2 be approximately equal to zero, i.e., that

| IF(u) - G(u)I2du = 2 0 (2)

when integrated over the same region. With this require-ment, only high-frequency changes of f(x) are allowedwhen it is being converted to a binary image, g(x).

A comparatively high resolution of the output system isassumed here. If we keep this in mind, the chosen con-straint of conformity between the low-pass portions of f(x)and g(x) appears feasible. However, for media with lowerresolution, the total spectral content is important becausethe corresponding image details will be resolved. In thissituation, modified metrics must be considered 3- 0; e.g., ametric may be chosen by the application of appropriateconstraints that are adapted to the characteristics of thevisual system.

In what follows, we concentrate on the characteristics ofthe quantization noise generated by different proceduresand therefore do not present the various existing metrics.

In a particular halftone situation the existing con-straints can be taken into account by a proper choice ofbinarization procedure. We present a way to characterizeand compare existing methods that is based on the con-figuration of the quantization-noise spectrum introducedby the halftone process. The graytone image of Fig. 1was used in the experiments.

3. QUANTIZATION NOISE RESULTINGFROM HARDCLIP

To convert a digital graytone image, f(m), where m=(m, n) are the sampling points, to a binary one, g(m), f(m)is compared pixel by pixel with a threshold value."

The binarization of graytone images introduces quan-tization noise q(m):

g(m) = f(m) + q(m).

0740-3232/93/040554-07$05.00 C 1993 Optical Society of America

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To predict features of the quantization noise, we ap-proximate the nonlinear function h by a series, e.g.,arctan(x), which is motivated by the application of thetransform method'12 in communication theory. The non-linear function h[[(x)], with h[[(x)] = 0 outside a fre-quency region, D, can be expressed as

h[[(x)] = iaj[*f(x)] . (8)j=O

The Fourier spectrum of the image binarized by the appli-cation of Eq. (6) can, according to Eq. (8), be given in theform of a weighted sum of j repeated convolutions of thegraytone image spectrum F(k) with itself; i.e.,

G(k) = > a F(k)]. (9)j=O

Introduction of this expression into Eq. (5) results in

Q(k) = (a, - 1)F(k) + > aj~jF(k)], (10)j=O

which indicates that F(k) and the coefficients aj have tobe known in order for Q(k) to be determined. For a quali-tative discussion they do not need to be determined ex-plicitly. In the case of an image containing only low

Fig. 1. (a) 5122-pixel graytone image used in the halftone illus-trations, (b) amplitude of its Fourier spectrum with suppresseddc peak.

The quantization noise in the image is

q(in) = g(in) - (i). (4)

We obtain by the discrete Fourier transform the corre-sponding frequency spectrum of the noisea

QWk = G(k) - F(k), (5)

where k = (k, 1) are the discrete sampling points in the spectrum plane.

In order to describe the quantization noise, we considerthe binarization with a fixed threshold, t. The binaryimage can be expressed in the form

g(in) = /2{11 + sgn[[ (i) - ],(6)

with

sgn(~) = i for (7)

An example of a hardclip, according to Eq. (6) is illustrated in Fig. 2 (a). This type of binarization does not fulfill con-ditions (1) and (2); this is discussed below. Equation (6) is Fig. 2. (a) 512 2-pixel halftone image by hardclip at constanta nonlinear relation between [(in) and g(in), and a depen- threshold 0.5, (b) amplitude of its quantization-noise spectrumdence of G(u) on F(u) cannot be explicitly determined, with suppressed dc peak.

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556 J. Opt. Soc. Am. A/Vol. 10, No. 4/April 1993

frequencies, the repeated convolutions spread the noisecontribution and enhance it in the low-frequency region.This effect is illustrated in Fig. 2(b), where the intensitydistribution, Q(k)l 2, is shown. A row of the quantization-noise spectrum IQ(k,0)12 is reproduced in Fig. 8(c) below.

4. CHARACTERISTICS OF ELECTRONICHALFTONE METHODS

One can apply different types of halftone methods to meetrequirements placed on the binarization of graytoneimages, viz., to relate. the values of f(m) to the two-levelquantized g(m) and to make F(k) and G(k) coincide in alow-frequency region, D.

A. Carrier-Concept ProceduresBinarization with a constant threshold as discussed inSection 3 results in a poor graytone reproduction with re-spect to detail resolution, as a consequence of the differ-ence in the low-pass contributions of f(m) and g(m). Toimprove the graytone resolution, i.e., to shift the quantiza-tion noise to higher frequencies, we may use a techniquebased on the carrier' concept.3'16

The image, f(m), may be compared pixelwise with avarying threshold, c(m). We assume that

(11)c(m) = c(m) + t.

The binarized distribution of the halftoned image is

g(m) = /2{1 + sgn[f(m) - c(m) - t]}.

The effect of the carrier on the convolution is clear inthe high-frequency region of the spectrum. Especiallystriking are the noise contributions at frequencies givenby the carrier. The j-times-repeated convolution of[F(k) - C(k)] with itself also causes contributions at mul-tiples of the carrier frequency. From this considerationone can assume that the spectral components of the quan-tization noise achieve appreciable amplitudes in the high-frequency region for binarization by a carrier methodcompared with amplitudes achieved by a straight hardclip.This tendency has been confirmed by Kermisch andRoetling.'7

B. Error-Correction and Error-Diffusion ConceptsAnother class of halftone process comprises the error-correction and error-diffusion principles. 8 9 It is a se-quential binarization method with a noise characteristicthat differs from that of the carrier methods. The basicidea is to take into account those errors that are introducedlocally by the binarization. The sequential processingallows the error made at a certain pixel to be corrected bybeing considered at the next pixels to be binarized.

The process consists of regarding a corrected gray-tone image

f(0) = f(0) (18)

(12)

This is identical to the binary image formed by comparingthe modified image

(im) = (m) - c(m), (13)

i.e., the superposition of a graytone image f(m), with acarrier c(m), where t is a constant threshold. Introductionof Eq. (13) into Eq. (12) results in

g(m) = /2{1 + sgn[f(m) - t]}, (14)

which is obtained from Eq. (6) by replacingf(m) with (m).Thus the qualitative description of the quantization

noise, presented in Section 3, can be applied for carrierhalftone methods as well. The Fourier spectrum of themodified image of Eq. (13) is

F(k) = F(k) - Ck,

and we obtain

Q(k) = G(k) - F(k) = G(k) - F(k) - C(k).

(15)

(16)

Equation (10) takes now the form

Q(k) = (a, - 1)[F(k) - C(k)] + Eaj{j*[F(k) - C(k)]}.

(17)

In Fig. 3 a digitally binarized graytone image made byusing a carrier (clustered dither) method is shown togetherwith its quantization-noise spectrum introduced by the bi-narization. A plot of a scan line through the quantization-noise spectrum Q(k, 0)12 is shown in Fig. 8(b), below.

Fig. 3. (a) 5122-pixel halftone image by the carrier procedure;the pyramid carrier period is 42 pixels. (b) Amplitude of thequantization-noise spectrum with suppressed dc peak.

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Vol. 10, No. 4/April 1993/J. Opt. Soc. Am. A 557

To indicate the tendency of the shape of the quantization-noise spectrum, we may consider the one-dimensionalcontinuous case.20 The process is now described by

f(x) = f(x) + e(x - A),

e(x) =[(x) - g(x),

where A indicates the diffusion of the error.transformed versions of these equations are

F(u) = F(u) + E(u)exp(-i2iruA),

E(u) = F(u) - G(u).

(22)

(23)

The Fourier-

(24)

(25)

From Eqs. (24) and (25) we obtain

Q(u) = G(u) - F(u) = E(u)[exp(-2iruA) - 1]

= -i2E(u)exp(-iuA)sin(7ruA). (26)

Fig. 4. (a) 5122-pixel halftone image by the error-diffusionmethod,'9 (b) amplitude of its quantization-noise spectrum withsuppressed dc peak.

at the binarization step

g(m) = /2 {1 + sgnf(m) - t]}. (19)

O indicates the starting point of the process. Theerror made,

e(m) = (m) - g(m), (20)

is distributed to not-yet-processed neighbor pixels accord-ing to

From Eq. (26) it is evident that the amplitude Q(u) of thequantization-noise spectrum is essentially determined byE(u)sin(iruA). E(u) is dependent on the graytone imagef(x) and on the chosen distribution of the error andsin(ruA); on the other hand, it is independent off(x).

The indicated parameters permit flexibility and themodification of the quantization noise introduced by thebinarization. 2 1

12 3

The carrier concept as well as the error-diffusion con-cept allows for numerous variations of the original ideas." 2

Even combinations of carrier and error-diffusion conceptsare possible. 2 3 2 4

C. Example of an Iterative Halftone ConceptThe procedure-dependent parameters, e.g., carrier periodand error distribution, allow us to influence the quantiza-tion noise formed by the binarization. In principle it ispossible to optimize requirements on the quantizationnoise by an appropriate choice of parameters. 3 25 Thenonlinear relation between the graytone image f(m) andthe binary one g(m) [see Eqs. (14) and (19)] in generalcauses problems in the analytical treatment.

However, it is possible to apply numerical methods forbinarization in which the Fourier spectrum can be usedexplicitly to control the quantization noise.

A number of algorithms in the class of iterative halftoneprocedures have been suggested, e.g., the iterative-Fourier-

f(m) = f(m) + E y(o,p)e(m - o,n - p),Op

graytone image f(m)(21)

where y(o,p) indicates the weight distribution of thediffused errors. When choosing the range op, one mustpermit the transport of errors only to pixels not yet quan-tized. This procedure is repeated sequentially over allimage pixels.

To define the process, we must state the distribution, y,of the error; the sequential order of the binarization; andthe threshold t.

In Fig. 4 an example of halftoning with the use of theerror-diffusion algorithm'9 is illustrated together with itsquantization-noise spectrum. In Fig. 8(c) below, a row ofthe quantization-noise spectrum Q(k, 0)12 is reproduced.

Fj(k)

tj=j+1operation

in spectrum

Gj (k)

Fig. 5. Flow chart of theused in the experiments.

9 -1 Fj(k) �1�-> f(m)

operationW in image

g (m)gj (m)

binary image g(m)

iterative-Fourier-transform algorithm

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transform, 26 direct-binary-search,2 7 simulated-annealand neural-network algorithms.25' 29

As an example, the iterative-Fourier-transformrithm is given here. The algorithm is illustrated in FiA graytone image f(m) is binarized with the use ofhalftone process. The binary image go(m) is Fottransformed into Go(k). The portion resulting fquantization noise Qo(k) = Go(k) - F(k) is in gerdistributed all over the spectrum. To require a noise.low-pass portion, D, we replace Q0 (k) with a modinoise term,

Q0(k) 0Qo(k)

for (k,l) E D

for (k,l) t D

The spectrum becomes

F,(k) = F(k) + Qo(k),

where the contribution in the low-pass region, D, is idcal to the corresponding spectrum of f(m). The in,Fourier transform results in fi(m), which fulfills the strum requirements, but in general it is not binary.serves as the beginning of the next iteration cycle.goal of the iterative-Fourier-transform algorithm issynthesis of the Fourier pair g(m) and G(k) = S;with constraints in the (m, k) plane.

The iterative algorithm tends to stagnate after acycles when a straight hardclip according to Eq. (6) fobinarization is applied. Different ways of performingbinarization step have been suggested to overcomestagnation. Instead of the constant threshold, t, oneapply a local random value, t(m), i.e., a random carIn this way the stagnation can be avoided.0 Thestraint in the image is then given by

gj(m) = /2{1 + sgn[[j(m) - tj(m)]},

ing,8

1lgo-'g. 5.any

trier

With the image-plane-operation constraint

gj(M) = {[f(M)1

for [,(m) cfor c, < f,(m) < 1 - ,for f,(m) 21 - cj

EruLi in the iteration algorithm, where cj is a constant that isieral increased from 0 to 0.5 as the procedure proceeds, a binary-free image is obtained when c = 0.5 is reached. In general,LIfied gj(m) is not binary for cj < 0.5. With monotonically in-

creasing cj's the stepwise operation according to Eq. (31) isconverted to the direct binarization.

(27) Halftone images obtained by using the iterative al-gorithms illustrated above are shown in Figs. 6 and 7. InFigs. 8(d) and 8(e) a row of the quantization-noise spectraQ(k, 0)12 is reproduced. Besides the examples mentionedand shown, there are a magnitude of imaginable possibili-

(28) ties for spectrally weighting the quantization noise (see,for example, Ref. 32).

bnti-rersespec- 5. COMMENTS

Them) The typical characteristics of the quantization noise inThe the procedures discussed in this paper are elucidated fur-the ther in Fig. 8.

Am),

fewr theIg thethismayier.l

!con-

(29)

where 0 s t(m) c 1 is determined by a random-numbergenerator for each pixel (m, n) and each iteration cycle j.Thus the iteration procedure becomes a combination of aniterative-Fourier-transform algorithm and aspects of aMonte Carlo method." Experimental investigations ofthis method have confirmed the avoidance of stagnation,but some problems in fulfilling the spectral constraints re-main.3 0 A modified binarization prescription can beused to circumvent these difficulties:

gj(m) 1= l /211 + sgn[fj(m) -tj(m)]l

1

for f[(m) < cfor c c f(m) 1 -cfor [L(m) >1 - c

(30)

c is a constant with the restriction 0 c c c 0.5. Forc s f(m) ' 1 - c the binarization is performed with arandom threshold, and otherwise a direct conversion isperformed to 0 and 1. This iterative synthesis of a half-tone image depends on the performance of an image bi-na4ization in each iteration cycle.

A step-by-step introduction of the binarization can alsobe applied for the synthesis of the desired Fourier pair.32

Fig. 6. (a) 5122-pixel halftone image by the iterative procedurewith a random carrier, (b) amplitude of its quantization-noisespectrum with suppressed dc peak. The 2562-pixel area of thespectrum was controlled.

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nique. An essential feature of the iterative procedure isthat adjustment to different requirements is possiblewhen constraints in the image as well as in the spectrumare considered during the binarization. In particular,control of the Fourier spectrum of the image during thebinarization permits direct influence on the quantizationnoise formed [see Figs. 8(d)and (e)]. Typical of the itera-tive halftoning concept is its high flexibility. The itera-tive binarization method forms an extension of existinghalftoning techniques whereby spectral control is a power-ful asset.

Combinations of the above-demonstrated basis methodsfrequently are applied. The quantization-noise charac-teristic is then modified accordingly, but typical featuresof its configurations remain.

IQ(k,0)|

a

Fig. 7. (a) 5122-pixel halftone image by the iterative procedurewith a stepwise introduction of binarization, (b) amplitude of itsquantization-noise spectrum with suppressed dc peak. The2562-pixel area of the spectrum was controlled.

The central characteristic considered here is the quan-tization noise introduced by the binarization method usedto achieve a halftone image of a graytone one. From theanalysis and examples it is evident that the different pro-cedures can be characterized as follows.

The use of periodic carriers for halftoning causes dis-tinct noise contributions around those frequencies thatare given by the period of the carrier [see Fig. 8(b)]. Theerror-diffusion algorithm results in frequency-dependent-weighting quantization noise [see Fig. 8(c)]. The errordistribution, the specific path of the sequential process-ing, and the choice of threshold determine the noise char-acteristics. In principle one can achieve an optimizationby appropriate parameter values to fulfill the requirementsplaced on the quantization noise. The example of con-straint mentioned above demonstrates how to obtain thelowest possible contribution of quantization noise in a low-pass region. From an optimization we expect to obtainthe parameters that will give the quantization noise thedesired properties. Because of the nonlinear relation be-tween f(m) and g(m), approximations are used in general,and from them it is possible to reach qualitative optimiza-tions, which in turn determine the parameters of ahalftoning procedure." 20,'25

Another powerful approach to meeting the requiredhalftone constraints is the application of an iterative tech-

- 256

b

l ~ ~~~ Il

0 k 255Fig. 8. Scanned row across quantization-noise spectrumIQ(k, 0)12 for a, hardelip at constant threshold 0.5; b, pyramid car-rier; c, error-diffusion method; d, iterative procedure with ran-dom carrier; e, iterative procedure with stepwise introduction ofbinarization.

'11v

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ACKNOWLEDGMENT

We are thankful to M. Tluk for his help with theillustrations.

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M. Broja and 0. Bryngdahl