Noise Error Analysis in Fractal Dimension Estimation of Digital Images

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T. Pant 1,*

1. Indian Institute of Information Technology, Allahabad, UP, India

* Corresponding author.


Received: 22 Feb. 2013 / Revised: 4 Apr. 2013 / Accepted: 15 May 2013 / Published: 28 Jun. 2013

Index Terms

Error analysis, fractal dimension, local fractal dimension, moving window, noisy images, noise models


In present paper the effect of noise and error occurring due to noise in fractal dimension of digital images has been analyzed. For this purpose, three digital images have been used which are added by Gaussian noise, salt and pepper noise and speckle noise. The fractal dimension of both noisy and non-noisy images has been estimated and corresponding error is reported in terms of RMSE. The study shows that noise affects the fractal dimension and there is an increase in fractal dimension due to noise. The average percentage error in fractal dimension has been estimated and reported as an offset for finding actual fractal dimension from noisy images.

Cite This Paper

T. Pant,"Noise Error Analysis in Fractal Dimension Estimation of Digital Images", IJIGSP, vol.5, no.8, pp.55-62, 2013. DOI: 10.5815/ijigsp.2013.08.07


[1]A. P. Pentland, ―Fractal-based description of natural scenes,‖ IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. PAMI-6, pp. 661–674, 1984.

[2]B. B. Mandelbrot, The Fractal Geometry of Nature, WH Freeman and Co., New York, 1982.

[3]W. Sun, G. Xu, P. Gong, and S. Liang, ―Fractal analysis of remotely sensed images: A review of methods and applications,‖ International Journal of Remote Sensing, Vol. 27, pp. 4963–4990, 2006.

[4]T. Pant, D. Singh, and T. Srivastava, ―Advanced fractal approach for unsupervised classification of SAR images,‖ Advances in Space Research, Vol. 45, pp. 1338–1349, 2010.

[5]M. Petrou, and P. G. Sevilla, Image Processing Dealing with Texture, John Wiley and Sons, Ltd., England, 2006.

[6]A. R. Backes, D. Casanova and O. M. Bruno, ―Color texture analysis based on fractal descriptors,‖ Pattern Recognition, Vol. 45, pp. 1984–1992, 2012.

[7]J. J. de Mesquita Sa´ Junior and A. R. Backes, ―A simplified gravitational model to analyze texture roughness,‖ Pattern Recognition, Vol. 45, pp. 732–741, 2012.

[8]M. J. Turner, J. M. Blackledge, and P. R. Andrews, Fractal geometry in digital imaging, Academic Press, Cambridge, Great Britain, 1998.

[9]W. Ju and N. S. -N. Lam, ―An improved algorithm for computing local fractal dimension using the triangular prism method,‖ Computers and Geosciences, Vol. 35, pp. 1224–1233, 2009.

[10]K. Ray, ―Unsupervised edge detection and noise detection from a single image,‖ Pattern Recognition, Vol.46, pp. 2067–2077, 2013.

[11]C. Liu, R. Szeliski, S. B. Kang, C. L. Zitnick, and W. T. Freeman, ―Automatic Estimation and Removal of Noise from a Single Image,‖ IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 30, No. 2, pp. 299–314, 2008.

[12]P. -L. Shui and W. -C. Zhang, ―Noise-robust edge detector combining isotropic and anisotropic Gaussian kernels,‖ Pattern Recognition, Vol. 45, pp. 806–820, 2012.

[13]C. Boncelet, ―Image Noise Models,‖ in A. C. Bovik, (Ed.), Handbook of Image and Video Processing, Academic Press, pp. 325–335, 2000.

[14]R. C. Gonzalez, and R. E. Woods, Digital Image Processing, 2nd ed., Pearson Education, New Delhi, 2002.

[15]T. V. Hoang and S. Tabbone, ―Invariant pattern recognition using the RFM descriptor,‖ Pattern Recognition, Vol. 45, pp. 271–284, 2012.

[16]I. Naseem, R. Togneri, and M. Bennamoun, ―Robust regression for face recognition,‖ Pattern Recognition, Vol. 45, pp. 104–118, 2012.

[17]G. Aubert and J. Aujol, ―A variational approach to remove multiplicative noise,‖ SIAM Journal of Applied Mathematics, Vol. 68, No. 4, pp. 925–946, 2008.

[18]Y. Huang, M. Ng, and Y. Wen, ―A new total variation method for multiplicative noise removal,‖ SIAM Journal on Imaging Sciences, Vol. 2, No. 1, pp. 20–40, 2009.

[19]J. M. Bioucas-Dias and Mário A. T. Figueiredo, ―Multiplicative noise removal using variable splitting and constrained optimization,‖ IEEE Transactions on Image Processing, Vol. 19, No. 7, pp. 1720–1730, 2010.

[20]Y. Han, X. Feng, G. Baciu, W. Wang, ―Nonconvex sparse regularizer based speckle noise removal,‖ Pattern Recognition, Vol. 46, pp. 989–1001, 2013.

[21]T. Pant, ―Implementation of fractal dimension for finding 3D objects: A texture segmentation and evaluation approach,‖ Intelligent Interactive Technologies and Multimedia. Springer Berlin Heidelberg, pp. 284–296, 2013.

[22]R. Lopes, P. Dubois, I. Bhouri, M. H. Bedoui, S. Maouche and N. Betrouni, ―Local fractal and multifractal features for volumic texture characterization,‖ Pattern Recognition, Vol. 44, pp. 1690–1697, 2011.

[23]K. C. Clarke, ―Computation of the fractal dimension of topographic surfaces using the triangular prism surface area method,‖ Computers and Geosciences, Vol. 12 No. 5, pp. 713–722, 1986.

[24]W. Sun, ―Three new implementations of the triangular prism method for computing the fractal dimension of remote sensing images,‖ Photogrammetric Engineering and Remote Sensing, Vol. 72, pp. 373–382, 2005.