How much resolution is enough?: Influence of downscaling the pixel resolution of digital images on the generalised dimensions

Fractals are very often illustrated by digital images, this is a consequence of the widespread availability of electronic and computational equipment. These images are always limited by their pixel resolution. Subsequently the determination of dimensions for such images is restricted practically because for their calculation the range of scales is not infinite. Downscaling the pixel resolution of images to get images with reduced resolution worsens the achievable absolute computational results but surprisingly the changes in these values and especially the changes of the courses of the calculated generalised dimensions give very promising results. This is due to the qualitative nature of these changes with respect to pixel downscaling. This opens the possibility of determining the quality of the estimation of the fractal dimensions of any image with any particular pixel resolution. Furthermore the pattern of changes itself caused by downscaling is different between the individual fractals. Several geometrical fractals (Sierpinski gasket, a fern, a Menger gasket and a modified Menger gasket with circles) as well as an example of a measured fractal, the digital images of cancer invasion studies in vitro were investigated to study the influence of downscaling the pixel resolution of digital images on the estimation of fractal dimensions.

[1]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[2]  H. G. E. Hentschel,et al.  The infinite number of generalized dimensions of fractals and strange attractors , 1983 .

[3]  S. Blacher,et al.  Fractal Quantification of the Microvasculature Heterogeneity in Cutaneous Melanoma , 1999, Dermatology.

[4]  Andreas Barth,et al.  Measuring Fractal Dimensions of Cell Contours: Practical Approaches and their Limitations , 1994 .

[5]  R. Pastor-Satorras,et al.  LETTER TO THE EDITOR: Numerical estimates of the generalized dimensions of the Hénon attractor for negative q , 1996 .

[6]  T F Nonnenmacher,et al.  Steroid hormones modify nuclear heterochromatin structure and plasma membrane enzyme of MCF-7 cells. A combined fractal, electron microscopical and enzymatic analysis. , 1998, European journal of histochemistry : EJH.

[7]  Marcin Molski,et al.  ON THE MODIFICATION OF FRACTAL SELF-SPACE DURING CELL DIFFERENTIATION OR TUMOR PROGRESSION , 2000 .

[8]  Losa Ga,et al.  Self-similarity and fractal irregularity in pathologic tissues. , 1996 .

[9]  Cross-talk reduction in confocal images of dual fluorescence labelled cell spheroids. , 1999, Micron.

[10]  M. Bos,et al.  Anomalous multifractal spectrum of aggregating Lennard-Jones particles with Brownian dynamics , 1996 .

[11]  G. D. Lange,et al.  Biological Cellular Morphometry-Fractal Dimensions, Lacunarity and Multifractals , 1998 .

[12]  Jensen,et al.  Erratum: Fractal measures and their singularities: The characterization of strange sets , 1986, Physical review. A, General physics.

[13]  Helmut Ahammer,et al.  Fractal dimension of K1735 mouse melanoma clones and spheroid invasion in vitro , 2001, European Biophysics Journal.

[14]  T. Mattfeldt.  Nonlinear deterministic analysis of tissue texture: a stereological study on mastopathic and mammary cancer tissue using chaos theory , 1997, Journal of microscopy.

[15]  E Claridge,et al.  Shape analysis for classification of malignant melanoma. , 1992, Journal of biomedical engineering.

[16]  A. Arneodo,et al.  Fractal dimensions and (a) spectrum of the Hnon attractor , 1987 .

[17]  Helmut Ahammer,et al.  FRACTAL DIMENSION FOR A CANCER INVASION MODEL , 2001 .

[18]  R. M. Crownover,et al.  Introduction to fractals and chaos , 1995 .

[19]  Marcin Molski,et al.  SELF-SIMILARITY, COLLECTIVITY, AND EVOLUTION OF FRACTAL DYNAMICS DURING RETINOID-INDUCED DIFFERENTIATION OF CANCER CELL POPULATION , 1999 .

[20]  K. Sandau.  A note on fractal sets and the measurement of fractal dimension , 1996 .