Digital transmission of halftone pictures

COMPUTER GRAPHICS AND IMAGE PROCESSING

(1974), 3, (195-'202)

Digital Transmission of Halftone Pictures THOMAS S. HUANG Purdue University West Lq~t~lette, Indiana 47907 Communicated b~l A. Rosenfeld Received Januand 28, 1974

We make two contributions in this paper. First, a frequency-domain method is presented for the prediction of flae characteristics of moir6 patterns in sampled halftone pictures. Second, it is demonstrated that conventional efficient coding methods such as run length coding are not particularly suitable for halftone pictures. 1. SAMPLING DENSITY There has been considerable interest lately in the digital transmission of newspaper pages [1]. It has b e e n ~btmd that, for text material, a sampling density of 400 ppi (points per inch) is sufficient to ensure satisfactol3, reproduction quality. For halftone pictures, the required sampling density depends on two factors: (1) how many equivalent brightness levels (i.e., h o w many different halftone dot sizes) would b e desirable; and (2) how dense w e have to sample in order to avoid the appearance of molt6 patterns. No systematic subjective experiments have b e e n conducted with respect to the first question. However, to extrapolate from experiments with continuous-tone pictures [2], it is estimated that 64 levels are sufficient and 32 levels may be satisfactolT in many cases. Leftfh and f., b e the halftone screen density and the sampling density, respectively, in ppi. If f , is k times fi,, t h e n we have k'-' samples per dot, which allows us to create k" dot sizes (by making some of the samples black). For example, for 64 levels, we n e e d f~ = 8fi,; for 32 levels, f~ = 6fi,. With respect to t h e second question, less is known. It is generally assumed that f~ must be at least 10ft, in order to avoid molt6 patterns. H o w e v e r , recent experiments [1] s h o w e d that the factor 10 might be too conservative since no moir6 patterns w e r e observed in any experiment with f~ ~> 8fi,. A discussion of moir6 patterns will be presented in the next section. From the above discussion, then, it seems that fs = 8fi, is a good choice for the sampling density in the case of halftone pictures. A typical value for fi~ is 85 ppi. The corresponding sampling density is fi =fi, = 680 ppi, which is 70% larger than what is n e e d e d for the text material. Since, operationally, it is much simpler to use the same sampling density for both the text material and the halftone pictures, we will be wasting a lot of samples in the text material. Copyright ~ 1974 by Ae,lden)ic Press, Inc. All rights of reproduction in any fin'm reserved.

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PATTERNS W e can gain a great deal of insight into the formation of moir6 patterns by looking at the frequency domain. W e model a halftone screen with dot diameter D and density fi~dots per unit length (Fig. la), by: 2. MOIRI~

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E~(u,v) = (D"/g) • [J,(D/2 2~rp)/D/2 2~'p] w h e r e u and v are spatial frequencies in cycles per unit length, p = ~Z~u'~+ v" and ]1(.) is the Bessel function of the first order.

(2)

D I G I T A L TRANSMISSION O F H A L F T O N E P I C T U R E S

197

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FIG. 2. Fourier transfmza of sampled halhone screen. N o w we sample die halftone screen with an ideal impulse array h ( x , y ) which has a period 1

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Let ~9 be the angle b e t w e e n the sampling raster h ( x , y ) and the halftone screen. Then the Fourier transform of the sampled halftone screen is as shown in Fig. 2, where GD(m'n) are shifted versions of GD. Any strong components of Go('~'n) that fall close enough to the origin in the u - v plane will cause moir6 patterns. The spatial frequency of the moir6 pattern due to a particular component of GDI'~''0 is determined by the vector leading from the origin in the u - v plane to that impulse component, T h e discussion above is based on uniform halftone pictures. The result can b e applied, however, to nonuniform halftone pictures, since in the latter molt6 patterns are most visible in the relatively uniform regions. W e now show some experimental results. A halftone picture 0% = 133 dots per picture width) was sampled at various sampling rates, with 0 = 45°. The resulting sampled pictures are shown in Fig. 3. The halftone picture before scanning is shown in Fig. 4. The sampling rates, as well as the molt6 frequencies, predicted by our theory are given in Table I. For most parts of the halftone picture, we have Dill "~ .66. The relative

198

THOMAS S. HUANG

(a)

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(e) ~IG. 3, Halt:tone picture with varied sampling rates (see Table 1).

FIG. 4. Halftone picture before scanning.

DIGITAL TRANSMISSION OF ttALFTONE PICTURES

199

TABLE [ PREDICTED MOIREFREQUENCIES

Picture no.

Sampling rate (samples per picture width)

Predicted moir6 frequencies (cycles per picture width) and responsible components of Gt~

Fig. 3a Fig. 3b Fig. 3c Fig. 3d Fig. 3e

128 182 246 274 364

60 [1, 1.]; 48 [1, 0] 6 [I, 2]; 9 [2, 0] 58 [1, 1]; 51 [3, 0] 86 [1, 1]; 12 [3, 0] 12 [2, 2]; 18 [4, 0]

strength of the components (re,n) of Go (el. Fig. lb) for this pal~ieular case can be s e e n from the sketch in Fig. 5. It will be seen that the appearance of the moir6 patterns in Fig. 3 confmTns to the predictions in Table 1 and Fig. 5. 3. EFFICIENT CODING Most efficient coding schemes for graphical data are based on the idea of m n l e n g t h coding. In a previous paper [3], we have studied, both theoretically and experimentally, runlength coding and one of its extensions to two dimensions where changes (A) in the positions of the black-white and w h i t e - b l a c k transitions from scan line to scan line are Ixansmitted. It was found that these schemes are very efficient for high resolution (large sampling density) graphical data. This is because the total numbers of bits one needs to transmit for a given picture are, for these two schems, propm~ional to f~ log2 f~. and f~, respectively, rather than to f~ as in the case of direct PCM txansmission. According to the results in [3], we can estimate the efficiency of" run length coding by referring to Fig. 6 where the entropy per sample or cell is plotted against the average m n length a. For the line-to-line A scheme, the entropy p e r sample is approximately 3/a. For a typical piece of text material (from the Wall Street Journal), at a sampling density of 400 ppi, the bandwidth compression ratios for run length coding and the h scheme were found to be 5.5 and 8.3, respectively. Since t h e s e schemes are less efficient for pictures with more runs (and hence

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smaller average run lengths), it is expected that they will give smaller b a n d w i d t h compression ratios for halftone pictures. L e t us assume that w e are scanning a uniform halftone screen at 45 °, as usually is the case in n e w s p a p e r transmission. Let fi, and f` be the screen density and the sampling density, respectively, in ppi and let L be the length of a scan line in inches. Furthermore, let the diameter of the halftone dot b e D, and let x = ~/~ A D

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It can be readily shown that the average run length in samples or cells of the s a m p l e d halftone screen is

a(x) = z / E v ~ ( A / f , ) x + ( 1 - x ) l L ~ ]

for 0 ~ x < 1

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(4) - 1 ], for V ~ < x < 9

F o r a halftone picture in which x varies from dot to dot, w e can use Eq. (4) to estimate the b a n d w i d t h compression ratio or its inverse h, the entropy per cell, in the following way. L e t r=fJfh, and a ( x ) = a(x)/r, w h e r e a(x) is g i v e n in Eq. (4). T h e function a(x) is plotted against x in Fig. 7. L e t p(x) b e the probability distribution of x over the picture, and assume that the variation of x is slow with respect to the screen density. Then, approximately, the average run length over the picture is

a=

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0

F o r any given picture, we can calculate a, then get h ~ from Fig. 6. For the line-to-line A scheme, we have hA = 3/a.

DIGITAL TRANSMISSION OF HALFTONE PICTURES

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We observe from Fig. 7 that for a typical halftone picture, the average run length probably will be around a ~ r. For r = fdfh = 8, then, a ~ 8. W h e n c e , hRL = 0.55 biffcell and hz~ = 0.375 bit/cell In terms of b a n d w i d t h compression ratios: B~L = 1.8 B~ = 2.7 4. A PROPOSAL

A scheme w h i c h would both yield large b a n d w i d t h compression ratio a n d avoid moir6 patterns is to convert the halftone picture to a continuous-tone

202

THOMAS S. ttUANG

p i c t u r e for transmission and reconvert it back to halftone at the receiver. This poses theoretical and practical problems. On the theoretical side, t h e q u e s t i o n arises: U n d e r what conditions can one perform the above conversions and g e t a good quality reconstructed halftone? On the practical side, o n e has to devise simple ways of doing the conversions. Furthermore, w e have to do different things in the halftone picture parts and the text parts of a n e w s p a p e r page. H o w e v e r , let us consider the bandwidth compression aspect of this scheme. S u p p o s e w e require n different dot sizes. Then, to transmit the halftone picture directly, we n e e d n samples per halftone dot, so we have to u s e n bits to transmit each halftone dot. On the other hand, if we convert the halftone picture to a continuous-tone picture in such a w a y that each halftone dot is c o n v e r t e d to a sample with n possible brightness levels, then to transmit each sample by standard P C M requires log2 n bits. Therefore, w i t h o u t any further efficient encoding, w e have already achieved a bandw i d t h compression ratio of n/log2 n. For n = 64, dais ratio is 10.7. REFERENCES 1. ANPA Research Institute Bulletin 949, All-digital newspaper page facsimile tests at the ANFA/RI Research Center, March 18, 1968. 2, T. S, HUANC, PCM picture transmission, IEEE Spectrum, December 1965. 3. T. S. HUANG, Coding of graphical data, in Picture Bandwidth Compression, T. S. HUANG AND O. J. TRETLAK,Eds., Gordon and Breach, 1972.