Self-similar random field models in discrete space

Self-similar random fields are of interest in various areas of image processing since they fit certain types of natural patterns and textures. Current treatments of self-similarity in continuous two-dimensional (2-D) space use a definition that is a direct extension of the one-dimensional definition, which requires invariance of the statistics of a random process to time scaling. Current discrete-space 2-D approaches do not consider scaling, but, instead, are based on ad hoc formulations, such as digitizing continuous random fields. In this paper, we show that the current statistical self-similarity definition in continuous space is restrictive and provide an alternative, more general definition. We also provide a formalism for discrete-space statistical self-similarity that relies on a new scaling operator for discrete images. Within the new framework, it is possible to synthesize a wider class of discrete-space self-similar random fields and texture images.

[1]  A. Papoulis.  Linear systems, Fourier transforms, and optics , 1981, Proceedings of the IEEE.

[2]  Heinz-Otto Peitgen,et al.  The science of fractal images , 2011 .

[3]  Mohamed Deriche,et al.  Modelling mammographic images using fractional Brownian motion , 1997, TENCON '97 Brisbane - Australia. Proceedings of IEEE TENCON '97. IEEE Region 10 Annual Conference. Speech and Image Technologies for Computing and Telecommunications (Cat. No.97CH36162).

[4]  J. Lévy Véhel,et al.  Horizons of fractional Brownian surfaces , 2000, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[5]  Lance M. Kaplan.  Extended fractal analysis for texture classification and segmentation , 1999, IEEE Trans. Image Process..

[6]  J. L. Véhel,et al.  Stochastic fractal models for image processing , 2002, IEEE Signal Process. Mag..

[7]  Jan Beran,et al.  Statistics for long-memory processes , 1994 .

[8]  John G. Proakis,et al.  Probability, random variables and stochastic processes , 1985, IEEE Trans. Acoust. Speech Signal Process..

[9]  Conor Heneghan,et al.  Two-dimensional fractional Brownian motion: wavelet analysis and synthesis , 1996, Proceeding of Southwest Symposium on Image Analysis and Interpretation.

[10]  M. Ekstrom,et al.  Two-dimensional spectral factorization with applications in recursive digital filtering , 1976 .

[11]  Wei Zhao,et al.  Discrete-Time continuous-dilation construction of linear scale-invariant systems and multi-dimensional self-similar signals , 1998 .

[12]  P. Martinez,et al.  Texture modelisation by multifractal processes for SAR image segmentation , 1997 .

[13]  C.-C. Jay Kuo,et al.  An improved method for 2-D self-similar image synthesis , 1996, IEEE Trans. Image Process..

[14]  Sharmila Majumdar,et al.  Fractal analysis of bone X-ray tomographic microscopy projections , 2001, IEEE Transactions on Medical Imaging.

[15]  Alex Pentland,et al.  Fractal-Based Description of Natural Scenes , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[16]  Trieu-Kien Truong,et al.  Spectral representation of fractional Brownian motion in n dimensions and its properties , 1995, IEEE Trans. Inf. Theory.

[17]  Leonard T. Bruton,et al.  Simulation of fractal multidimensional images using multidimensional recursive filters , 1994 .

[18]  James M. Keller,et al.  Texture description and segmentation through fractal geometry , 1989, Comput. Vis. Graph. Image Process..

[19]  Kenneth J. Hintz,et al.  Fractional Brownian motion models for synthetic aperture radar imagery scene segmentation , 1993, Proc. IEEE.

[20]  Beatrice Pesquet-Popescu,et al.  Synthesis of nonstationary fields with stationary increments , 1997 .

[21]  M. Taqqu,et al.  Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance , 1995 .

[22]  Zhiyi Chi.  Stationary self-similar random fields on the integer lattice , 2001 .

[23]  Wei Zhao,et al.  Discrete-time models for statistically self-similar signals , 2003, IEEE Trans. Signal Process..

[24]  R. Dekker.  Texture analysis of urban areas in ERS SAR imagery for map updating , 2001, IEEE/ISPRS Joint Workshop on Remote Sensing and Data Fusion over Urban Areas (Cat. No.01EX482).

[25]  B. Pesquet-Popescu,et al.  2D Self-Similar Processes with Stationary Fractional Increments , 1997 .

[26]  I. Miller.  Probability, Random Variables, and Stochastic Processes , 1966 .

[27]  R. Kumaresan,et al.  Isotropic two-dimensional Fractional Brownian Motion and its application in Ultrasonic analysis , 1992, 1992 14th Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[28]  Raghuveer M. Rao,et al.  Scale-based formulations of statistical self-similarity in images , 2004, 2004 International Conference on Image Processing, 2004. ICIP '04..

[29]  B. Pesquet-PopescuandP.  2D Self-Similar Processes with Stationary Fractional Increments , 1997 .

[30]  Raghuveer M. Rao,et al.  Image modeling with linear scale-invariant systems , 1999, Defense, Security, and Sensing.