1. MESA Lab Synthesis of bidimensional α -stable models with long-range dependence xiaodong sun MESA (Mechatronics, Embedded Systems and Automation) Lab School of Engineering, University of California, Merced E: xsun7@ucmerced.edu Phone:209 201 1947 Lab: CAS Eng 820 (T: 228-4398) sep 22, 2014. Applied Fractional Calculus Workshop Series @ MESA Lab @ UCMerced

2. MESA Lab The paper we talk about Synthesis of bidimensional α -stable models with long-range dependence Beatrice Pesquet-Popescu a, ∗, Jean-Christophe Pesquet b

3. MESA Lab Why need 2D fractal model The motivation for modeling and synthesizing textures with impulsive and long-range dependence (LRD) behaviors are on the following: Segmentation of synthetic or satellite images( high-speckle SAR imagery ) ultrasound medical imaging and astronomical imaging. In computer graphic applications, the generation of 2-D picture realizations( create natural-looking night landscapes) Underwater image modeling (Scattering effect caused by water molecule) Camera internal noise modeling

4. MESA Lab The way to bidimensional α-stable models Generate multivariate stable distribution noise Generate long-range dependence (LRD) behaviors bidimensional α-stable models with long-range dependence

5. MESA Lab Generate multivariate α-stable driving noise According to the proposition 1.7.1 in paper [1]. The α -stable driving noise can be generated [1]G. Samorodnitsky, M.S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman and Hall, New York, 1994.

6. MESA Lab Generate long-range dependence (LRD) behaviors 'fractionally differenced' processes are capable of modelling long-term persistence. 2D discrete-space process with LRD properties can be achieved by a 2D fractional stable process passed a bidimensional filter system. the frequency response of the bidimensional filter can be expressed by

7. MESA Lab Generate long-range dependence α-stable processes Generate 2D α-stable processes X Apply FFT to X,W=fft(X) α-stable noise pass 2D filter H d (). Generate S=W.H d () α-stable process with LRD by using inverse. Ss=ifft2(S)

8. MESA Lab Simulation pcolor α=1.4 d=0.25

9. MESA Lab Simulation contour3 α=1.4 d=0.25

10. MESA Lab Simulation pcolor α=1.6 d=0.3

11. MESA Lab Simulation contour3 α=1.6 d=0.3

12. MESA Lab Simulation pcolor α=1.8 d=0.35

13. MESA Lab Simulation contour3 α=1.8 d=0.35

14. MESA Lab Thank you !