Abstract
To contribute to the important task of characterizing the complex multidimensional structure of natural images, a fractal characterization is proposed for the colorimetric organization of natural color images. This is realized from their three-dimensional RGB color histogram, by applying a box-counting procedure to assess the dimensionality of its support. Regular scaling emerges, almost linear over the whole range of accessible scales, and with non-integer slope in log-log allowing the definition of a capacity dimension for the histogram. This manifests a fractal colorimetric organization with a self-similar structure of the color palette typically composing natural images. Such a fractal characterization complements other previously known fractal properties of natural images, some reported recently in their colorimetric organization, and others reported more classically in their spatial organization. Such fractal multiscale features uncovered in natural images provide helpful clues relevant to image modeling, processing and visual perception.
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References
Batlle J., Casalsb A., Freixeneta J., Martía J. (2000) A review on strategies for recognizing natural objects in colour images of outdoor scenes. Image and Vision Computing 18: 515–530
Bex P. J., Makous W. (2002) Spatial frequency, phase, and the contrast of natural images. Journal of the Optical Society of America A 19: 1096–1106
Chapeau-Blondeau F., Chauveau J., Rousseau D., Richard P. (2009) Fractal structure in the color distribution of natural images. Chaos, Solitons & Fractals, 42: 472–482
Chauveau, J., Rousseau, D., & Chapeau-Blondeau, F. (2008). Pair correlation integral for fractal characterization of three-dimensional histograms from color images. In lecture notes in computer science, (vol. LNCS 5099, pp. 200–208). Berlin: Springer.
Chen Y. C., Ji Z., Hua C. (2008) Spatial adaptive Bayesian wavelet threshold exploiting scale and space consistency. Multidimensional Systems and Signal Processing 19: 157–170
Distasi R., Nappi M., Tucci M. (2003) FIRE: Fractal indexing with robust extensions for image databases. IEEE Transactions on Image Processing 12: 373–384
Dong D. W., Atick J. J. (1995) Statistics of natural time-varying images. Network: Computation in Neural Systems 6: 345–358
Faraoun K. M., Boukelif A. (2005) Speeding up fractal image compression by genetic algorithms. Multidimensional Systems and Signal Processing 16: 217–236
Field D. J. (1987) Relations between the statistics of natural images and the response properties of cortical cells. Journal of the Optical Society of America A 4: 2379–2394
Field D. J. (1994) What is the goal of sensory coding?. Neural Computation 6: 559–601
Fisher Y. (1995) Fractal image compression: Theory and applications. Springer, Berlin
Galka A. (2000) Topics in nonlinear time series analysis. World Scientific, Singapore
Gevers T., Smeulders A. W. M. (1999) Color-based object recognition. Pattern Recognition 32: 453–464
Gousseau Y., Roueff F. (2007) Modeling occlusion and scaling in natural images. SIAM Journal of Multiscale Modeling and Simulation 6: 105–134
Grassberger P., Procaccia I. (1983) Characterization of strange attractors. Physical Review Letters 50: 346–349
Hentschel H. G. E., Procaccia I. (1983) The infinite number of generalized dimensions of fractals and strange attractors. Physica D 8: 435–444
Hsiao W. H., Millane R. P. (2005) Effects of occlusion, edges, and scaling on the power spectra of natural images. Journal of the Optical Society of America A 22: 1789–1797
Jacquin A. E. (1992) Image coding based on a fractal theory of iterated contractive image transformation. IEEE Transactions on Image Processing 1: 18–30
Kaplan L. M. (1999) Extended fractal analysis for texture classification and segmentation. IEEE Transactions on Image Processing 8: 1572–1585
Keller J. M., Crownover R. M., Chen R. Y. (1987) Characteristics of natural scenes related to the fractal dimension. IEEE Transactions on Pattern Analysis and Machine Intelligence 9: 621–627
Knill D. C., Field D., Kersten D. (1990) Human discrimination of fractal images. Journal of the Optical Society of America A 7: 1113–1123
Landgrebe D. (2002) Hyperspectral image data analysis. IEEE Signal Processing Magazine 19(1): 17–28
Lian S. (2008) Image authentication based on fractal features. Fractals 16: 287–297
Maggi F., Winterwerp J.C. (2004) Method for computing the three-dimensional capacity dimension from two-dimensional projections of fractal aggregates. Physical Review E, 69(011405): 1–8
Mandelbrot B. B. (1983) The fractal geometry of nature. Freeman, San Francisco
Olshausen B. A., Field D. J. (2000) Vision and the coding of natural images. American Scientist 88: 238–245
Pentland A. P. (1984) Fractal-based description of natural scenes. IEEE Transactions on Pattern Analysis and Machine Intelligence 6: 661–674
Pesquet-Popescu B., Lévy Véhel J. (2002) Stochastic fractal models for image processing. IEEE Signal Processing Magazine 19(5): 48–62
Potlapalli H., Luo R. C. (1998) Fractal-based classification of natural textures. IEEE Transactions on Industrial Electronics 45: 142–150
Ruderman D. L. (1997) Origins of scaling in natural images. Vision Research 37: 3385–3398
Ruderman D. L., Bialek W. (1994) Statistics of natural images: Scaling in the woods. Physical Review Letters 73: 814–817
Schroeder M. (1999) Fractals, chaos, power laws. Freeman, New York
Sharma, G. (eds) (2003) Digital color imaging handbook. CRC Press, Boca Raton
Srivastava A., Lee A. B., Simoncelli E. P., Zhu S. C. (2003) On advances in statistical modeling of natural images. Journal of Mathematical Imaging and Vision 18: 17–33
Theiler J. (1990) Estimating fractal dimension. Journal of the Optical Society of America A 7: 1055–1073
Tolhurst D. J., Tadmor Y., Chao T. (1992) Amplitude spectra of natural images. Ophthalmic and Physiological Optics 12: 229–232
Truong T. K., Kung C. M., Jeng J. H., Hsieh M. L. (2004) Fast fractal image compression using spatial correlation. Chaos, Solitons & Fractals 22: 1071–1076
Wohlberg B., De Jager G. (1999) A review of the fractal image coding literature. IEEE Transactions on Image Processing 8: 1716–1729
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Chauveau, J., Rousseau, D. & Chapeau-Blondeau, F. Fractal capacity dimension of three-dimensional histogram from color images. Multidim Syst Sign Process 21, 197–211 (2010). https://doi.org/10.1007/s11045-009-0097-0
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DOI: https://doi.org/10.1007/s11045-009-0097-0