# Image Pyramids as a New Approach for the Determination of Fractal Dimensions

### Michael Mayrhofer-Reinhartshuber, Philipp Kainz, Helmut Ahammer

#### Abstract

The consideration of different scales and the application of fractal methods on digital images is of high importance if real world objects are investigated. In this context the fractal dimension is an important parameter to characterize structures and patterns. An accurate understanding of them is obligatory if significant and comparable results should be obtained. Recently a new method using an image pyramid approach was compared to the very popular Box Counting Method. The intriguing results showed that a trustable value for the fractal dimension could be obtained in much faster computational times compared to traditional Box Counting algorithms. In addition to these results of this new approach, which is only applicable to binary (black/white) images, we present developments toward the application to grey value/color images. Especially the determination of the grey value surface and the interpolation used to downscale the images seem to have major influence on the results achieved.

#### References

- Vaney, T. T. J., and Tritthart, H. A. (2003). How much resolution is enough? Influence of downscaling the pixel resolution of digital images on the generalised dimensions. Physica D, 181:147-156.
- Ahammer, H. and Mayrhofer-R., M. (2012). Image pyramids for calculation of the box counting dimension. Fractals, 20:281.
- Appleby, S. (1996). Multifractal characterization of the distribution pattern of the human population. Geographical Analysis, 28(2):147-160.
- Asvestas, P., Matsopoulos, G. K., and Nikita, K. S. (1998). A power differentiation method of fractal dimension estimation for 2-d signals. Journal of Visual Communication and Image Representation, 9:392-400.
- Bisoi, A. K. and Mishra, J. (2001). On calculation of fractal dimension of images. Pattern Recognition Letters, 22:631-637.
- Biswas, M. K., Ghose, T., Guha, S., and Biswas, P. K. (1998). Fractal dimension estimation for texture images: A parallel approach. Pattern Recognition Letters, 19:309-313.
- Chaudhuri, B. B. and Sarkar, N. (1995). Texture segmentation using fractal dimension. IEEE Transactions on Pattern Analysis and Machine Intelligence, 17:72-77.
- Chinga, G., Johnsen, P. O., Dougherty, R., Berli, E. L., and Walter, J. (2007). Quantification of the 3d microstructure of sc surfaces. J Microsc, 227(Pt 3):254-265.
- Clarke, K. C. (1986). Computation of the fractal dimension of topographic surfaces using the triangular prism surface area method. Computers & Geosciences, 12:713- 722.
- Hausdorff, F. (1918). Dimension und äußeres Maß. Mathematische Annalen, 79(1-2):157-179.
- Higgs, R. (2011). Imaging: Fractal analysis for chd mortality. Nature Reviews Cardiology, 8(2):62.
- Jelinek, H. F., Ristanovic, D., and Milosevic, N. T. (2011). The morphology and classification of alpha ganglion cells in the rat retinae: a fractal analysis study. J Neurosci Methods, 201(1):281-287.
- Jin, X. C., Ong, S. H., and Jayasooriah (1995). A practical method for estimating fractal dimension. Pattern Recognition Letters, 16:457-464.
- Keller, J. M., Chen, S., and Crownover, R. M. (1989). Texture description and segmentation through fractal geometry. Computer Vision, Graphics, and Image Processing, 45:150-166.
- Lee, W.-L. and Hsieh, K.-S. (2010). A robust algorithm for the fractal dimension of images and its applications to the classification of natural images and ultrasonic liver images. Signal Processing, 90:1894-1904.
- Lopes, R. and Betrouni, N. (2009). Fractal and multifractal analysis: A review. Medical Image Analysis, 13:634- 649.
- Losa, G. A., Ieva, A. D., Grizzi, F., and Vico, G. D. (2011). On the fractal nature of nervous cell system. Frontiers in Neuroanatomy, 5.
- Mandelbrot, B. B. (1977). Fractals: Form, Chance and Dimension. W. H. Freeman & Company, 1st edition.
- Mandelbrot, B. B. (1983). The Fractal Geometry of Nature : Updated and Augmented. W. H. Freeman & Company, New York.
- Normant, F. m. c. and Tricot, C. (1991). Method for evaluating the fractal dimension of curves using convex hulls. Phys. Rev. A, 43:6518-6525.
- Paumgartner, D., Losa, G., and Weibel, E. R. (1981). Resolution effect on the stereological estimation of surface and volume and its interpretation in terms of fractal dimensions. J Microsc, 121(Pt 1):51-63.
- Peleg, S., Naor, J., Hartley, R., and Avnir, D. (1984). Multiple resolution texture analysis and classification. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-6(4):518-523.
- Pickover, C. A. and Khorasani, A. (1986). Fractal characterization of speech waveform graphs. Computers & Graphics, 10(1):51-61.
- Pruess, S. A. (1995). Fractals in the Earth Sciences, chapter Some remarks on the numerical estimation of fractal dimension, pages 65-75. Plenum Press.
- Romand, S., Wang, Y., Toledo-Rodriguez, M., and Markram, H. (2011). Morphological development of thick-tufted layer v pyramidal cells in the rat somatosensory cortex. Frontiers in Neuroanatomy, 5.
- Russell, D. A., Hanson, J. D., and Ott, E. (1980). Dimension of strange attractors. Physical Review Letters, 45:1175-1178.
- Sarkar, N. and Chaudhuri, B. B. (1992). An efficient approach to estimate fractal dimension of textural images. Pattern Recognition, 25:1035-1041.
- Sarkar, N. and Chaudhuri, B. B. (1995). Multifractal and generalized dimensions of gray-tone digital images. Signal Processing, 42:181-190.
- Shelberg, M., Lam, N., and Moellering, H. (1983). Measuring the fractal dimension of surfaces. Proceedings of the Sixth International Symposium on ComputerAssisted Cartography, 6:319-328.
- Sullivan, A. C., Hunt, J. P., and Oldenburg, A. L. (2011). Fractal analysis for classification of breast carcinoma in optical coherence tomography. J Biomed Opt, 16(6):066010.
- Sun, W. (2006). Three new implementations of the triangular prism method for computing the fractal dimension of remote sensing images. Photogrammetric Engineering & Remote Sensing, 72(4):373-382.
- Tang, Y. Y., Tao, Y., and Lam, E. C. M. (2002). New method for feature extraction based on fractal behavior. Pattern Recognition, 35:1071-1081.
- West, G. B. (2012). The importance of quantitative systemic thinking in medicine. Lancet, 379(9825):1551-1559.

#### Paper Citation

#### in Harvard Style

Mayrhofer-Reinhartshuber M., Kainz P. and Ahammer H. (2013). **Image Pyramids as a New Approach for the Determination of Fractal Dimensions** . In *Proceedings of the 2nd International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,* ISBN 978-989-8565-41-9, pages 239-243. DOI: 10.5220/0004325902390243

#### in Bibtex Style

@conference{icpram13,

author={Michael Mayrhofer-Reinhartshuber and Philipp Kainz and Helmut Ahammer},

title={Image Pyramids as a New Approach for the Determination of Fractal Dimensions},

booktitle={Proceedings of the 2nd International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,},

year={2013},

pages={239-243},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0004325902390243},

isbn={978-989-8565-41-9},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 2nd International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,

TI - Image Pyramids as a New Approach for the Determination of Fractal Dimensions

SN - 978-989-8565-41-9

AU - Mayrhofer-Reinhartshuber M.

AU - Kainz P.

AU - Ahammer H.

PY - 2013

SP - 239

EP - 243

DO - 10.5220/0004325902390243