Generation of self-reproducing fractal patterns using a

JOURNAL OF MODERN OPTICS,

2000,

VOL .

47,

NO.

8, 1469±1474

Generation of self-reproducing fractal patterns using a multiple imaging system with feedback J. COURTIAL and M. J. PADGETT Department of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, Scotland, UK (Received 17 November 1999; revision received 12 January 2000) Abstract. We present a simple optical system for producing self-similar fractal patterns. The key component comprises three adjacent lenses, which form multiple images of a pattern displayed on a monitor. The images are recorded by a camera and displayed as the new pattern on the monitor. Iteration of this process results in an approximation to a self-similar fractal pattern which is independent of the starting image.

1.

Introduction Fractal geometry is one of the fundamental design principles of the natural world, encountered in objects as diåerent as ferns, trees, mountains and clouds. Fractal patterns are usefully characterized in terms of their fractal dimension. A fractal dimension of zero corresponds to a pattern of isolated points, a fractal dimension of one to a pattern of lines and of two to ®lled areas [1]. A sub-category of fractals are `self-similar’ fractals which upon appropriate magni®cation reveal themselves to be multiple copies of themselves. In principle, a photocopier adapted such that it produces, on the same output page, multiple, smaller copies of the original image can produce fractal patterns [2]. Such a device is often called a `multiple reduction copy machine’ (MRCM). By iteratively photocopying successive outputs, it is possible to create many diåerent fractal patterns, like for example a Sierpinski gasket (®gure 1). A complex optical realization of a MRCM with feedback has been proposed before [3]. In this paper we concentrate on a diåerent, extremely simple way of optically realizing a MRCM with feedback: in ®gure 4.27 in [4] a MRCM is sketched as comprising a number of side-by-side lenses. We report here on what we believe to be the ®rst experimental realization of such a device. In our system the lenses form identical, but displaced images of a monitor display. The resulting image is recorded by a detector array, captured by a frame grabber and displayed on the monitor as the new object. The overall system is an optical MRCM with the camera and monitor providing feedback. Running the system results in a pattern with the structure of a self-similar fractal, which is de®ned by the position and magni®cation of the images but is independent of the starting image. One example of a multiple lens component is three identical lenses packed so that their centres lie on the corners of an equilateral triangle [5]. The resulting image comprises three identical versions of the object with their centres lying on Journal of Modern Optics ISSN 0950±0340 print/ISSN 1362±3044 online # 2000 Taylor & Francis Ltd http://www.tandf.co.uk/journals

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J. Courtial and M. J. Padgett

Figure 1.

Figure 2.

Sierpinski gasket.

View through the three adjacent lenses that form the basis of the optical system.

the corners of an equilateral triangle. If the magni®cation is set to 1 =2 then, after the process is completed many times, the resulting image is an approximation to the well-known Sierpinksky gasket. The triple lens optical component is shown in ®gure 2. The overall magni®cation upon each iteration of the pattern is critical in determining the shape and fractal dimension of the pattern. The round trip magni®cation depends both on the optical system and the relative size of the detector array compared to the monitor display. Figure 3 shows the overall optical

Generation of self-reproducing fractal patterns

1471

monitor

side-by-side lenses

de-magnifying lens camera Figure 3.

Complete optical system showing monitor, triple imaging lens, additional demagni®cation lens and CCD camera array.

system. The lenses that comprise the triplet are all of 40 mm focal length. These lenses are situated approximately 1 m from the monitor and the multiple images they produce are re-imaged onto the detector array with a magni®cation of approximately 1 =5, such that on each iteration of the process the overall magni®cation is close to 1 =2. If the output from the camera system is simply connected directly to the monitor then a number of problems arise. The ®rst of these is that as the acquisition rate of the camera is synchronized to the display update rate; a camera directed at its own monitor can result in unwanted banding in the image. The second is that the positive feedback nature of the multiple imaging can result in the ®nal intensity of the image saturating the monitor/camera or falling away to zero. Connected with this last issue is that the monitor display is typically nonlinear in that the brightness of any particular pixel is in¯uenced by those surrounding it. We ease these problems by connecting the camera to a high quality monochrome frame grabber. The resulting image is transferred into computer memory. The intensity values for each pixel are processed such that the dynamic range of the image is reduced and normalized to utilize the full dynamic range of the monitor. This processing ensures that the system iterates to a stable pattern. Figure 4 shows the ®rst nine iterations of the loop after which there is little perceivable diåerence in the images. In this case the initial image was a cloud weather symbol. The ®nal self-sustaining image is independent of the starting pattern and indeed does establish itself even from random noise. Fractals are characterized in terms of their fractal dimension which can be de®ned in a variety of ways. Most common is to consider how many boxes of a

J. Courtial and M. J. Padgett

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Figure 4.

n =1

n =2

n =3

n =4

n =5

n =6

n =7

n =8

n =9

First nine iterations of the image display loop showing the gradual build-up of the self-similar fractal pattern.

given size are required to totally cover the fractal pattern. The gradient of a log±log plot of box number against reciprocal box size gives the fractal dimension. As mentioned earlier, a fractal dimension of one corresponds to a line and of two to a ®lled area. For a grey-scale image this process additionally requires a threshold to be applied to the image. In our case we take this threshold to be close to the saturation value. The `perfect’ self-similar fractal has the same fractal dimension at all scales; in the log±log plot mentioned above this corresponds to all the points lying on a single straight line. In practise, however, the upper scale limit is set by the total size of the image (in our case the size of the monitor) and the lower scale limit by the ®nest detail within the image (in our case the resolution of the optical system). The fractal dimension, or self-similarity dimension, D, of an ideal Sierpinski gasket is calculated according to the formula [1] Dˆ

log …3† ; log …1 =m†

…1†

where m is the magni®cation on each iteration through the optical system. Figure 5 shows graphs of box size against box number after two diåerent iteration numbers for the system from which the fractal dimension of the

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(a) 3.5 3

log10N

2.5 2

D =2

1.5 1 D = 1.10

0.5 0 -2.5

-2

-1.5

-1

-0.5

0

log10 (1/s) (b) 4 D = 1.46

log10N

3

2

1

0 -2.5

-2

-1.5

-1

-0.5

0

log10 (1/s) Figure 5.

Log±log plots of box number, N, against reciprocal box size, 1=s (s in pixels), after (a) 3 and (b) 15 iterations. Insets show the thresholded patterns.

corresponding patterns can be deduced. After three iterations the fractal character of the pattern is not fully established; consequently, for small box sizes the fractal dimension tends to 2, indicating a ®lled area. After many iterations the pattern is established and all the points in the corresponding log±log plot lie on a straight line. Measurement of features within the pattern shows that the magni®cation is 0.45 which should give a fractal dimension of 1.37. From ®gure 5 (b), the measured fractal dimension of 1.46 is in reasonable agreement with this ®gure suggesting

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Generation of self-reproducing fractal patterns

clearly that the resulting pattern has the scaling characteristics of the Sierpinski gasket. The pattern possesses fractal scaling over two orders of magnitude, which is more than many other `real-life’ fractal phenomena [6]. This paper has reported the practical realization of a simple optical system incorporating a multiple lens component to produce a self-similar fractal pattern. Our ultimate goal is to place this component within a laser resonator to form a cavity with a lowest-loss eigenmode with a fractal intensity cross-section. Although the incoherent nature of the gain element diåerentiates this work from the laser, this present work serves to demonstrate the nature of the key optical component.

References

[1] PEITGEN, H.-O., JUÈ RGENS, H., and SAUPE, D., 1992, Chaos and Fractals: New Frontiers of Science (New York: Springer), p. 205. [2] PEITGEN, H.-O., JUÈ RGENS, H., and SAUPE, D., 1992, Chaos and Fractals: New Frontiers of Science (New York: Springer), pp. 23å. [3] SHIH, I.-F., CHANG , D. B., and MOISE, N. L., 1992, Analog Optical Processing for the Construction of Fractal Objects, US Patent 5132831. [4] BARNSLEY, M. F., and HURD, L. P., 1993, Fractal Image Compression (Wellesley, MA: AK Peters Ltd.). [5] COURTIAL, J., 2000, Optics Commun., 174, 235. [6] AVNIR, D., BIHAM, O., LIDAR, D., and MALCAI, O., 1998, Science, 279, 39.