Recursive Subdivisions of the Plane Yielding Nearly Hexagonal Regions Astrid Lundmark1, Niclas Wadstr¨omer1 and Haibo Li1;2 Coding Group, Dept. of Electrical Engineering, Link¨opings universitet E-mail: fastrid,nic,
[email protected] of Applied Physics and Electronics, Ume˚a universitet. E-mail:
[email protected]
1 Image 2 Dept.
Abstract Hexagonal regions are optimal for subdividing the plane in the sense that the regions are as close as possible to the disk shape while still providing a tessellation of the plane. It is however not possible to do recursive subdivisions, i.e. to divide a hexagon into smaller hexagons. In this paper, hexagon-like fractal regions will be presented. It is possible to decompose a hexagon-like fractal region into a number of smaller but equally shaped hexagon-like regions. Possible applications of hexagon-like fractal regions are cell organization in cell-phone systems and image coding algorithms, including algorithms with position variant resolution.
1 Hexagon-like Fractal Regions As a starter, let us introduce hexagon-like fractal regions which are composed of 7 identical but smaller regions, also called Gosper regions[1]. A bottom-up approach for generating Gosper regions is to recursively merge seven adjacent regions, starting from the grouping of hexagons shown in Figure 1. After each merging operation the regions still lie on a hexagonal grid, which is tilted by the same angle in each iteration. It is however possible to choose if the tilt is going to be in the positive or negative direction, so by choosing alternate positive and negative tilts, the coordinates of the regions will be much easier to handle. After a few iterations, the regions obtain the shape shown in Figure 2. In the general case of hexagon-like fractal regions, which has to our knowledge not been presented earlier in literature, the base tiling consists of S hexagons, see Figure 3. We can relate S to k, where k is the number of edge hexagons along one of the six sides, by S = 3(k2 + k) + 1:
(1)
The ratio of areas of two nearly hexagonal regions at adjacent p regions scale with S, while the lengths scale with R = S. Hexagon-like fractal regions tile the plane, which can be proved e.g. by induction over k. Because of the recur1
Figure 1: A base tiling for subsampling a factor of seven, given an image of hexagonal representation. sive nature of the region generation, it is enough to prove that the base tiling (like the one shown in Figure 1) tile the plane. A Hexagon-like region having k = 2 (and thus S = 19) is shown in figure 4. From generating an Iterated Functions System[2], we can compute the fractal dimension of the border of the regions. The dimension D is given by D=2
log(2k + 1) : log(3(k2 + k) + 1)
(2)
It can be noted that D approaches one as k increases, which for hexagon-like fractal regions means that the regions approach hexagons with increasing k. In Figure 6 an image that has been divided into nearhexagonal regions is shown. In the figure it is also shown how a region can be subdivided into smaller regions.
Figure 4: Hexagon-like fractal region with k = 2, i.e. S = 19.
Figure 2: Gosper region, a hexagon-like fractal region having k = 1, i.e. S = 7.
k 1 2 3 4
S 7 19 37 61
R 2.65 4.36 6.08 7.81
D 1.13 1.09 1.08 1.07
Table 1: Listing for the first values of k (number of edge hexagons along one of the six sides) of S (subsampling factor, area scaling between regions in adjacent hierarchy levels), R (length scaling between regions in adjacent hierarchy levels), and D (fractal dimension). ceptors in the human retina varies with the distance from the center of the retina (fovea) in such a way that a circle centered around the fovea always crosses approximately the same number of light receptors, irrespective of the radius of the circle. The same sampling density variation can be achieved using nearly-hexagonal fractal sub-sampling with space-variant resolution. In Figure 5 space-variant resolution is demonstrated. The upsampling is done with smooth filters to avoid abrupt intensity transitions.
Figure 3: Base tiling pattern for hexagon-like fractal regions, in this case k=2. The shading has been done to illustrate Equation 2.
2 Applications 2.1 Image Coding The compact shape of the regions is an advantage in e.g. vector quantization. In wavelet transform image coding, it is interesting to be able use other subsampling factors than the commonly used subsampling factor of 2. A wider class of hierarchical subsampling schemes, which are suitable for e.g. wavelet decomposition, is described in [3]. Some wavelet filter banks for image coding have been designed [4, 5, 6] for Nearly Hexagonal Regions with a subsampling factor of 7, but compared to the conventional one-dimensional dyadic wavelet filter banks, the area is still very much unexplored. Having variable resolution is desirable in image coding in situations where the viewer's eye movements can be monitored, due to the large variations in visual acuity in different parts of the retina. The density of the light re-
2.2
Cell Management in Mobile Phone Systems
For management of cells in cell phone systems [7] variable resolution is useful for taking care of differences in mobile phone density. Since the total allocated bandwidth in a cell phone systems is usually densely filled with traffic, the main disturbance is not ambient noise but interference from other mobile connections in the system. A way of increasing the capacity is therefore to lower the power in each connection and establish more cells. Doing this in a uniform way for a whole country or continent is not feasible or desirable, since there are large areas with low mobile terminal density which are well served by large cells.
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3 Conclusion A recursive subdivision scheme yielding nearly hexagonal regions has been presented. How to preserve alignment with the original grid has been described. Two application areas have been pointed out, image coding and cell management in mobile telephone systems.
References [1] Martin Gardner. Mathematical games. Scientific American, pages 124–128, 133, December 1976. [2] Michael Barnsley. Fractals Everywhere. Academic Press, 1988. [3] Astrid Lundmark, Niclas Wadstr¨omer, and Haibo Li. Hierarchical subsampling giving fractal regions. Technical Report ISSN 1400-3902, LiTH-ISY-R-2035, Link¨oping University, Sweden, 1998.
Figure 5: Spatial resolution is dependent on distance to the focus of attention. If the eye movements are tracked, the spatial resolution can be adapted to that of the eye. Here three resolution levels are used, the image data that needs to be encoded is considerably reduced. Hexagonlike fractal regions with a subsampling factor of 19 were used.
[4] Edward H. Adelson, Eero Simoncelli, and Rajesh Hingorani. Orthogonal pyramid transforms for image coding. In Russel Hsing, editor, SPIE Visual Communications and Image Processing, volume 845 of Proceedings of the SPIE, pages 50–58. The International Society for Optical Engineering, 1987.
The size of the largest cells is determined by the ambient noise, which here comes into consideration, and the allowed power transmission from a mobile. The variable size can be achieved e.g. by letting the cells have arbitrary shapes, which allows much flexibility, but also leads to much complexity. An attractive alternative is to adopt a hierarchical philosophy where the cells are related in a systematic way. This can make all types of management, e.g. routing and frequency allocation, in the system substantially less complex. The shape of the cells is desired to be as close to circles as possible, but still tessellate the plane. For a nonhierarchical tessellation, hexagons are optimal, but they are not possible to decompose into smaller hexagons. The nearly hexagonal regions described here are essentially as round as hexagons (the border between two cells in all cases being somewhat fuzzy; a connection is not necessarily passed from one cell to the neighbouring while passing a pre-defined spatial border, but rather when the Signal-to-Interference ratio becomes too low in one cell and is significantly better in a neighbouring cell). Nearly hexagonal regions possess the recursive subdivision property, where subsampling factors can be chosen from the ones in Table 1. The properties of tessellating the plane, being recursively subdividable, and having a shape that is close to the disc shape make the nearly hexagonal regions suitable for cells in a mobile telephone network.
[5] Anderew B. Watson and Albert J. Ahumada, Jr. A hexagonal orthogonal-oriented pyramid as a model of image representation in visual cortex. IEEE Transactions on Biomedical Engineering, 36(1):97–106, January 1989. [6] Eero P. Simoncelli and Edward H. Adelson. Non– separable extensions of quadrature mirror filters to multiple dimensions. Proceedings of the IEEE, 78(4):652–664, April 1990. [7] Lars Ahlin and Jens Zander. Principles of Wireless Communication. Studentlitteratur, 1998.
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Figure 6: Illustration of the two applications varying resolution image representation and cell management. Hexagon-like subsampling has been used to subdivide an image of a map. The region of interest has been assigned finer resolution. The map contains 3 different resolutions. In this case one level of finer resolution yields a factor of 61 smaller region area.
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