The Effectiveness of Discrete Voronoi Division

in Fractal Image Compression

Takayasu FUCHIDA and Sadayuki MURASHIMA

Department of Information and Computer Science, Kagoshima University1-21-40 Kohrimoto, Kagoshima-shi, Kagoshima 890-0065 Japan

Email: {fuchida,mura}@ics.kagoshima-u.ac.jp

Abstract–In this paper, we show the effectivenessof discrete Voronoi division as the shape of ranges infractal image compression. Normally, a square shape isused for a range of fractal image compression. On theother hand, we’ve proposed the way to use a discreteVoronoi region as a range. It is thought that a Voronoirange has more flexible shape than a square range, butthe effectiveness of discrete Voronoi division for fractalimage compression has not been investigated yet. Wecompress some images using these two methods andcompare the values of error between range and domainby using computer simulations.

1. Introduction

Fractal image compression is one of the way of im-age compression using self-similarity contained in theoriginal image [1, 2]. It has attracted attentions as amethod of high compression ratio as much as JPEGmethod or more. In the normal fractal image com-pression method, a square shape is used as a range.But it is hardly thought that the square range wouldbecome the best fitting shape to the natural image. Aflexible shape – like a polygonal shape – has complexfeatures and it was thought that is could not be usedfor the shape of range in fractal image compression.

Recently we have proposed a way to compress an im-age by using discrete Voronoi division as range divisionin fractal image compression [5]. Discrete Voronoi di-agram can be constructed from only the informationof kernel points. So we can realize flexible ranges byusing discrete Voronoi diagram.

The Voronoi range may be more flexible than squarerange but it is not clear how the Voronoi range effec-tive.

So we compare the two methods – square rangeand Voronoi range – and calculate the value of errorsbetween range and domain. By comparing these val-ues, we can know how the obtained range is worthy toextract the self-similarity from the given image.

Figure 1: Relationship between domain and range

2. Fractal image compression using discreteVoronoi division

2.1. Fractal image compression

“Fractal” is a coined word by Mandelbrot and itmeans self-similarity [3]. The fractal image com-pression method compresses an image using the self-similarity contained in the original image. Concretely,a given image is first divided into the many areas calledrange which does not overlap with each other. Next,for each range, we search the domain which has (nor-mally) twice the size of the range. At this time, thedomain whose similarity to the range is the highest issearched. Here, the similarity is calculated by usingRMS metric as shown later. Fig.1 shows the relation-ship between a range and a domain. The domain mayoverlap with each other.

Such a system like this is called partitioned iter-ated function system (PIFS). Former researches of frac-tal image compression, almost all cases were using asquare range. This method is called quad tree method.In this method, if the value of RMS error of a rangewas not enough small, the range was divided into 4square ranges. By repeating this process, a range isdivided small gradually.

2004 International Symposium on NonlinearTheory and its Applications (NOLTA2004)

Fukuoka, Japan, Nov. 29 - Dec. 3, 2004

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The contractive map from domain to range can berepresented by using affine transformation. Generalaffine transformation of ith range wi is shown in for-mula (1).

wi

⎡⎣ xyz

⎤⎦ =⎡⎣ 1/2 0 0

0 1/2 00 0 si

⎤⎦⎡⎣ xyz

⎤⎦+⎡⎣ aibiti

⎤⎦ (1)

Where x, y are the coordinates of domain pixel andz is the brightness of this pixel. Normally, becausethe size of domain is twice the size of range, the affineparameters of resize, rotation and shear are fixed toµ1/2 00 1/2

¶. So a domain pixel means the block of

2× 2 pixels and the position of this pixel is upper-leftcoordinates of this block, and the brightness means thesum of brightnesses of these 4 pixels. The parametersai and bi represent parallel translation, and si and tidecide the conversion of brightness value.When the intensities of pixels in the range are

r1, r2, · · · , rn, and the intensities of pixels in the do-main are di, d2, · · · , dn, the RMS error Ei can be cal-culated as follows.

Ei =nXj=1

(sidj + ti − rj)2 (2)

where n is the number of pixel contained in the range.We can seek si and ti to minimize this Ei. The

minimum of Ei occurs when the partial derivativeswith respect to si and ti become zero as follows.

∂Ei∂si

= 0,∂Ei∂ti

= 0 (3)

This value Ei will become settled if any domain isspecified. The domain where the value of Ei is possiblethe smallest is an optimum domain.

2.2. Fractal image compression using discreteVoronoi division

Voronoi diagram is a diagram which divide spaceinto the influence areas of kernel points. Kernel pointis a generator of Voronoi diagram. A point which ex-ists in one Voronoi region is a point whose nearestkernel point is that region’s kernel point.A discrete Voronoi diagram is a Voronoi diagram

which is made on the digitized space. The discreteVoronoi region is a set of pixels whose nearest kernelpoint is that region’s kernel point. If we got a discreteVoronoi diagram, we could know which kernel pointis the nearest one from any point of that image in aninstant.When an original image was given, we can construct

a discrete Voronoi diagram on that image. And also

Figure 2: Contractive mapping from domain to range

we can use these discrete Voronoi regions as the rangesof fractal image compression.In this case, we can use the affine transformation

from a domain to a range like the case of the squarerange. Fig.2 shows the affine transformation in thecase of using discrete Voronoi range. In this figure,Ri(x, y) is a pixel of ith range, and Di(2x, 2y) isthe corresponding domain to that range. BRi andBDi are the boundary rectangle of the range anddomain respectively, and x, y are the coordinates inthe boundary rectangle. We denote the sum of 4pixels Di(2x, 2y), Di(2x + 1, 2y), Di(2x, 2y + 1) andDi(2x+ 1, 2y + 1) as D

∗i (x, y).

Then the RMS error Ei can be calculated as follows.

Ei =Xx,y∈Vi

(siD∗i (x, y) + ti −Ri(x, y))2 (4)

In this formula, Vi is the discrete Voronoi region of ithrange.

2.3. The way of range division

Generally, as a range becomes smaller, it becomeseasier to find a good domain. So we will want to dividethe image into small ranges but a lots of ranges causesbad compression ratio. Because the quantities of in-formation which should be stored into the compressedfile will also increase as the number of ranges increase.So when we would divide one range, we should di-

vide a range whose RMS error is larger than any otherranges.The strategy of dividing range is as follows.When the square range is used, we divide the origi-

nal image into square ranges roughly, then we calculatethe RMS error for all ranges. After this, we select nwranges and divide them into 4 ranges. We repeat thisprocess until the number of ranges reached over nm.

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When the Voronoi range is used, range division isdecided by the position of kernel points. And if weadded a kernel point to arbitrary point on the im-age, the shape of several ranges which exist aroundthis point would be changed by this addition. So af-ter adding a kernel point, we will have to reconstructthe discrete Voronoi diagram to obtain new Voronoiranges. Therefore we take the following strategy todecide the range division.

1. First, we distribute ni kernel points on the orig-inal image randomly and divide the whole imageinto ni initial ranges.

2. Optimum domains are searched for every ranges.At this time, the value of RMS error for each rangewill be calculated.

3. if the number of range reached to nm then end.

4. We select nw ranges whose RMS errors are largerthan any other ranges and remove the kernelpoints of these ranges and add two kernel pointsin each removed range randomly.

5. Reconstruct the discrete Voronoi diagram by us-ing new kernel points.

6. go to 2.

By taking this strategy, we can decide the numberof range freely. But since this algorithm constructsdiscrete Voronoi diagram many times, it is requireda effective construction method of discrete Voronoidiagram. In this research, we used the incremen-tal method on digitized space to construct a discreteVoronoi diagram quickly [4].

3. Computer simulations

In order to compare the difference of two form ofrange, we performed several computer simulations. Weused 3 images whose size is 512× 512, – Lenna, Pep-pers and Baboon. These are the standard images fordata compression.Simulation environments are as follows. CPU is

Pentium4 3G Hz and Memory is 1.5G Byte, WindowsXP Professional. Programming language is Gnu C++.Fig.3(a) and (b) show examples of range division by

using square ranges and Voronoi ranges respectively.In both divisions, they have almost 10000 ranges. Itturns out that the Voronoi ranges more concentrateon the detailed parts of Lenna than the square ranges.(For example, on the part of hair.)Fig.4 shows the result of comparing the quality of

images to the compression ratio in 3 cases. The hori-zontal axis shows the compression [bpp], and the ver-tical axis shows the PSNR [dB]. The original image is8 [bpp]. PSNR is calculated by following formula,

(a) Range division using square range.

(b) Range division using Voronoi range.

Figure 3: Range division of both way

PSNR = 20 log10256

RMSE. (5)

From these figures, it is found that when the com-pression ratio is greater than 0.4 [bpp], the method us-ing Voronoi range is better than conventional method.But as the compression ratio becomes smaller, thequality of restored image of conventional method be-comes slightly better than the proposed method.Since the proposed method is more complicated

than the conventional method, it takes much compres-sion time than the conventional one. The conventionalmethod takes about 50 seconds when the compressionratio is 0.4 [bpp], but the proposed method takes about500 seconds in same ratio. Voronoi method is slowerthan conventional method about 10 times in this case.Some reasons can be considered about this. One rea-

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son is the complexity of the form of Voronoi range.The flexibility also means the complexity. Thereforewe have to more times of calculation to obtain theRMS error of Voronoi range than square range. In thecase of square range, we can use pre-calculated cumu-lations to calculate the RMS error but it is impossiblefor Voronoi ranges. For these reasons, the proposedmethod takes much time.

4. Conclusions

In this paper, we compared two way of range divi-sion in fractal image compression. When the com-pression ratio is over 0.4 [bpp], it was found thatthe Voronoi range could realize higher quality thansquare range. But when the compression ratio be-comes smaller than it, the quality of restored imagebecomes slightly worse. One of the reason is thatsince the Voronoi range is more flexible than the squarerange then it can be arranged at the complicated partsof original image.On the other hand, the proposed method requires

more calculation time to compress image. Since theVoronoi range has a complex form then we can notuse the pre-calculated cumulations to obtain the RMSerrors.

Acknowledgements

The portion of this study has been funded by theGrants-in-Aid for Scientific Research No.14780233 ofthe Japanese Ministry of Education, Culture, Sports,Science and Technology.

References

[1] Yuval Fisher Editor, Fractal Image Compression– Theory and Application –, Springer, 1994.

[2] M.Nelson, J.L.Gailly, The Data Compression BookSecond Edition, M&T Books, 1995.

[3] B.B. Mandelbrot, The Fractal Geometry of Nature,2nd edition, W. H. Freeman and Co., San Fran-cisco, California, 1982.

[4] T.Fuchida, H.Nakamura, K.Mori andS.Murashima, An Efficient Method to Con-struct a 2-Dimensional Discrete Voronoi Diagramby Adding Kernel Points One by One, IEICETrans A, Vol.J85-A No.5 pp.571—583, May 2002.(in Japanese)

[5] T.Fuchida, H.Nakamura, K.Mori andS.Murashima, Fractal Image Compression us-ing Discrete Voronoi Tessellation, Proceedings ofInternational Symposium on NOLTA 2002, Vol.2pp.711—714, Dec. 2002.

(a) result of ’Baboon’

(b) result of ’Lenna’

(c) result of ’Peppers’

Figure 4: Results of computer simulations

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