GRAPHICAL MODELS AND IMAGE PROCESSING

Vol. 59, No. 1, January, pp. 49–53, 1997ARTICLE NO. IP960414

NOTE

Adaptive Ordered DitherYUEFENG ZHANG1

Software Technology Group, Honeywell Hi-Spec Solutions, 343 Dundas Street, Suite 100, London, Ontario, Canada N6G 1V5

Received March 4, 1996; revised September 16, 1996; accepted October 11, 1996

along a space-filling curve over the image. Unfortunately,this method tends to blur image details if the cluster size isOrdered dither (C. N. Judice, J. F. Jarvis, and W. H. Ninke,

in Proceedings of the S.I.D. 15, 4 (Fourth Quarter), 1974, reasonably large. This problem is solved in the stochasticpp. 161–169.) is an efficient and widely used halftoning tech- screening dithering method [6] by performing an adaptivenique. One disadvantage with this method is that it produces variation of the cluster size. Like its predecessor [5], how-blurred images. Recently, a space-filling curve ordered dith- ever, this method is computationally expensive and not suit-ering approach (Y. Zhang, submitted for publication.) has been able for parallel processing.proposed that improves the ordered dither in image quality by Recently, Zhang [8] proposed a space-filling curve or-generating clustered dot patterns along a space-filling curve

dered dither that can significantly improve the space-fillingover the image. In this method the cluster size is fixed in thecurve clustering method [5] in both speed and image qual-process of halftoning. This paper presents a new ordered dith-ity. In this method, the pixels of the image are divided intoering method, called adaptive ordered dither, that does halfton-classes by subdividing the space-filling curve over the imageing by using a space-filling curve to perform an adaptive varia-into segments as the space diffusion [9] does. Then, liketion of the cluster size. Experimental results demonstrate thatin ordered dither [1], the class numbers associated withthe new method can significantly improve the space-filling

curve ordered dither in revealing image details. 1997 Academic Press the pixels of the image are used as the dithering thresholdsof these pixels. However, this method is still limited inproducing pleasing images due to a difficulty with choosingan appropriate cluster size (see Section 2.2).

1. INTRODUCTION We observe that, similar to the stochastic screening dith-ering method [6], if the space-filling curve ordered dither

Ordered dither [1] is an efficient digital halftoning [8] is combined with a strategy of adaptive clustering, amethod for transforming a gray scale image into a binary new ordered dithering method results. The new methodimage. In this method, only one comparison is needed for can significantly improve the space-filling curve orderedhalftoning one pixel in an image and all the pixels in the dither [8] with respect to image quality.image can be halftoned simultaneously. The major draw-back of ordered dither is that the quality of a halftoned 2. THE NEW METHODimage is usually poor compared with the halftoning resultsproduced by other methods [2, 7, 8]. Recently some ap- In this section, we present a new ordered dither method,

called adaptive ordered dither, that inherits the advantagesproaches [3, 4] were proposed in the literature that can im-of the space-filling curve ordered dither [8] and the stochas-prove the quality of ordered dither image. However, like thetic screening dithering method [6]. In particular, like theoriginal ordered dither [1], these methods fail to producespace-filling curve ordered dither, to avoid regular patternpleasing images on a device (e.g., laser printer) that cannotartifacts, the new method assigns dithering thresholds to thecontrol individual dots (pixels) precisely. Velho and Gomespixels of the image along a space-filling curve. Similar to the[5] proposed a space-filling curve clustering approach thatstochastic screening dithering [6], to improve rendition ofavoids this difficulty by generating clustered patterns of dotsgray shades while preserving image details, the new methodchanges the cluster size according to the frequency that the1 The author is currently under contract with Motorola Inc. at Motorolaintensities of the pixels of the image change along the space-Building IL-27, MS 2d7, 1501 West Shore Drive, Arlington Heights,

IL 60004. filling curve.

491077-3169/97 $25.00

Copyright 1997 by Academic PressAll rights of reproduction in any form reserved.

50 YUEFENG ZHANG

2.1. Constant Clustering

In the space-filling curve ordered dither [8], ditheringthresholds are assigned to the pixels of the image along aspace-filling curve by subdividing the curve into congruentsegments. This is achieved by replicating a one-dimensionaldithering threshold array over the space-filling curve. In thismanner the cluster size is fixed in the process of halftoning.For convenience, clustering with constant cluster size iscalled ‘‘constant clustering’’ in this paper.

As an example of constant clustering along a space-filling curve, Fig. 1 shows how a Hilbert curve [8] is usedto assign dithering thresholds to the pixels of an 8 3 8image by using a cluster size of 16. Figure 1a shows adiscrete Hilbert curve over the image. A traversal orderingof the image by the Hilbert curve is shown in Fig. 1b. InFig. 1c, the pixels of the image are associated with ditheringthresholds by replicating the one-dimensional array of 0,1, . . . , 15 over the curve.

2.2. Adaptive Clustering

There is a difficulty in choosing an appropriate clustersize f if the value of f is fixed in the process of halftoning.On one hand, if f is too small, the tone of the resultingimage can be poor. On the other hand, if f is too large,the resulting image can be grainy and thus blur out imagedetails. We observe that, by performing an adaptive varia-tion of the cluster size f as the stochastic screening ditheringmethod [6] does, a good rendition of gray shades can beachieved while the image details are preserved and thus,the above difficulty is avoided.

In the stochastic screening dithering method [6], thecluster size f changes exponentially with the gradient mag-nitude of the intensities of the pixels of the image. Experi-mental results show that if this strategy of adaptive cluster-ing is combined with the space-filling curve ordered dither[8], the quality of the resulting image is not quite controlla-ble. We observe that the following strategy of adaptiveclustering works well with the space-filling curve ordereddither [8] in revealing image details.

Given an n 3 n image I(n 3 n) and a space-filling curveover this image, I can be represented as a one-dimensionalarray of pixels according to the traversal ordering of thespace-filling curve, that is, I(i) (or pixel i) represents thepixel of the image with traversal ordering number i. Con-sider a clustered dot pattern (a sequence of adjacent dots(pixels) with respect to the traversal ordering) starting atpixel i. Like the stochastic screening dithering method [6],we can use an estimation of the frequency that the pixel in-tensities of the image change along the space-filling curveto determine the cluster size of this dot pattern. In this paperthe frequency is estimated by counting the number of pixelsstarting at pixel i that have the same or ‘‘similar’’ intensity FIG. 1. Assigning dithering thresholds to the pixels of an 8 3 8 imagevalues as pixel i. This can be achieved by counting the num- along a Hilbert curve.

ADAPTIVE ORDERED DITHER 51

ber of pixels j satisfying the condition uI( j ) 2 I(i)u # d, whered is a tolerance for handling small pixel intensity change.

After the cluster size f has been determined, the ditheringthresholds D( j ), j 5 i, i 1 1, . . . , i 1 f 2 1, of the dots (pixels),i, i 1 1, . . . , i 1 f 2 1 can be determined as follows: D( j ) 5j 2 i, j 5 i, i 1 1, . . . , i 1 f 2 1. In this manner the pixels ofthe entire image can be associated with dithering thresholdsalong the space-filling curve starting at pixel 0.

Using the above method to determine the cluster size,large cluster size (large dot patterns) will result in the areasof constant (or slowly changing) pixel intensity in the imageand small cluster size (small dot patterns) will occur in theareas of image details (see Fig. 6). To keep the cluster sizereasonably small to avoid over-blurring the image details, amaximum bound is applied to the cluster size f in this paper.

2.3. Parallel Processing

A disadvantage of the method of adaptive clusteringpresented in Section 2.2 is that the dithering thresholdsare assigned to the pixels of the image in a sequentialorder. To determine the dithering thresholds in parallel,

FIG. 2. The 256 3 256 image of the boat scene halftoned by thewe can subdivide the space-filling curve over the imagespace-filling curve clustering method with a cluster size of 8 pixels.

into segments of length l and then apply the strategy ofadaptive clustering described in Section 2.2 to each of thesegments simultaneously. In this paper we set the segment

ties gradually change. In particular, when the value of llength l to the maximum bound of the cluster size.increases, the resulting halftoned image becomes moregrainy. If the value of d increases, the range of the pixel2.4. Comparison with the Space-Filling Curveintensities that are considered as ‘‘similar’’ extends. In theOrdered Ditherextreme case, d can equal the maximum bound of all the

In both the space-filling curve ordered dither [8] and the possible pixel intensities (1 in this paper). In this case, theadaptive ordered dither presented in this paper, the dith- adaptive ordered dither becomes the space-filling curveering thresholds of the pixels of the image are determined ordered dither [8] with the cluster size of l. From thisby subdividing a space-filling curve over the image into seg- point of view, the space-filling curve ordered dither can bements. The main difference between these two methods is considered as a special case of the adaptive ordered dither.that the size of the segments (i.e., the cluster size) in thespace-filling curve ordered dither is a constant, while the 3. EXPERIMENTAL RESULTScluster size in the new method changes with the runninglength of the pixels that have similar intensities along the In this section, a 256 3 256 real image of a boat scene

is used to demonstrate the performance of the adaptivespace-filling curve. Both the adaptive cluster size and theconstant cluster size have advantages of their own. The main ordered dither presented in this paper. Figure 2 shows the

result of halftoning the image by the space-filling curveadvantage of the adaptive cluster size over the constant clus-ter size is that the resulting ordered dither can produce bet- clustering method [5]. The result of applying the space-

filling curve ordered dither [8] to the image is shown inter images with respect to image details. The advantage oftheconstantclustersize is that theresultingditheringthresh- Fig. 3. In both Figs. 2 and 3, the cluster size is 8. By

comparing Figs. 2 and 3, we can see that the space-fillingolds array is independent of the image to be halftoned.In the space-filling curve ordered dither [8], the quality curve ordered dither is much better than the space-filling

curve clustering method in image quality.of a halftoned image is completely determined by the clus-ter size. In the adaptive ordered dither, however, the qual- We observe that Fig. 3 can also be created by applying

the new adaptive ordered dither presented in this paperity of a halftoned image can be controlled by both thebound l of the cluster size and the tolerance d of the pixel to the source image with the cluster size bound l 5 8 and

the pixel intensity tolerance d equal the maximum valueintensities. The value of l controls the overall graininessof the halftoned image and the value of d determines the of all the possible pixel intensities (1 in this paper). This

is consistent with the claim described in Section 2.4 thatgraininess of the areas in the image where the pixel intensi-

52 YUEFENG ZHANG

FIG. 3. The 256 3 256 image of the boat scene halftoned by the FIG. 4. The 256 3 256 image of the boat scene halftoned by thespace-filling curve ordered dither with a cluster size of 8 pixels. adaptive ordered dither with a cluster size bound of 8 pixels and a pixel

intensity tolerance of 0.

the space-filling curve ordered dither [8] can be considered be an advantage in some applications since it avoids theartifacts of snakes appearing in Fig. 7. The main advantageas a special case of the adaptive ordered dither.of the new method over error diffusion is that it is parallel-Figure 4 shows the halftoned image of the boat sceneizable and computationally cheaper.generated by the adaptive ordered dither with l 5 8 and

d 5 0. It can be seen by comparing Figs. 3 and 4 that theadaptive ordered dither significantly improves the space-filling curve ordered dither [8] in revealing fine image de-tails (see the curved rigging immediately above the cabinof the boat on the right-hand side of the image). Figure 5shows the halftoned image of the boat scene produced bythe adaptive ordered dither with l 5 8 and d 5 2/256. Bycomparing Fig. 5 with Fig. 4, we can observe that increasingthe value of d improves the rendition of the shades of theimage while preserving image details. The image in Fig. 6is halftoned by the adaptive ordered dither with l 5 16and d 5 5/256. We can see by comparing this figure withFigs. 3, 4, or 5 that when the cluster size bound l increases,the resulting image becomes more grainy. We can alsoobserve from Fig. 6 that due to adaptive clustering in theprocess of halftoning, small clustered dot patterns appearin the areas of image details (e.g., the rigging) and largeclustered dot patterns occur in the areas where the intensi-ties of the pixels change slowly (see the clouds in the sky).

The result of halftoning the image by the error diffusion[2] is shown in Fig. 7. By comparing this figure withFig. 5 (or 4), we can see that the adaptive ordered ditheris comparable with the error diffusion in revealing image

FIG. 5. The 256 3 256 image of the boat scene halftoned by thedetails. Figure 5 looks a little bit fuzzy compared to adaptive ordered dither with a cluster size bound of 8 pixels and a pixelFig. 7 in the large uniform areas of the image. This might intensity tolerance of 2/256.

ADAPTIVE ORDERED DITHER 53

A comparison of the space-filling curve ordered ditherwith the conventional ordered dither [1] has been describedin [8]. Experimental results show that the space-fillingcurve ordered dither [8] improves the conventional or-dered dither significantly. The adaptive method presentedin this paper furthers the improvement.

4. CONCLUSION

In this paper, an adaptive ordered dither was presented.This method inherits the advantages of the space-fillingcurve ordered dither [8] and the stochastic screening dith-ering method [6].

Like the space-filling curve ordered dither, in the newmethod, dithering thresholds are assigned to the pixels ofthe image under halftoning by subdividing a space-fillingcurve over the image into segments. This makes it possibleto produce images with clustered dot patterns efficiently.This kind of image proves to be suitable for a binary devicethat cannot control isolated dots precisely.

In the space-filling curve ordered dither [8], the space-filling curve is subdivided into segments of the same length.

FIG. 7. The 256 3 256 image of the boat scene halftoned by error dif-One difficulty with this method is that it is hard to get a fusion.balance between the image tone and details. This difficultyis avoided in this paper by using a strategy of adaptive

the pixel intensities are constant or change slowly. In thisclustering similar to the stochastic screening ditheringmanner the adaptive ordered dither can produce imagesmethod [6]. By using adaptive clustering, small clusteredwith both fine tones and details.dot patterns result in the areas of image details and large

clustered dot patterns occur in the areas of the image whereREFERENCES

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