## Abstract and Applied Analysis

### Poisson Noise Removal Scheme Based on Fourth-Order PDE by Alternating Minimization Algorithm

#### Abstract

To overcome the staircasing effects introduced by the TV regularization in image restoration, this paper investigates a fourth-order partial differential equation (PDE) filter for removing Poisson noise. In consideration of the slow convergence property of the classical gradient descent method, we adopt the alternating minimization algorithm for realizing this scheme. Compared with the corresponding total variation based one, numerical simulations distinctly indicate the superiority of our proposed strategy in handling smooth regions of Poissonian images and improving the computational speed.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 965281, 14 pages.

Dates
First available in Project Euclid: 1 April 2013

https://projecteuclid.org/euclid.aaa/1364845175

Digital Object Identifier
doi:10.1155/2012/965281

Mathematical Reviews number (MathSciNet)
MR2889091

Zentralblatt MATH identifier
1303.65052

#### Citation

Zhou, Weifeng; Li, Qingguo. Poisson Noise Removal Scheme Based on Fourth-Order PDE by Alternating Minimization Algorithm. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 965281, 14 pages. doi:10.1155/2012/965281. https://projecteuclid.org/euclid.aaa/1364845175

#### References

• L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D, vol. 60, no. 1–4, pp. 259–268, 1992.
• Y. Vardi, L. A. Shepp, and L. Kaufman, “A statistical model for positron emission tomography,” Journal of the American Statistical Association, vol. 80, no. 389, pp. 8–37, 1985.
• S. Kido, H. Nakamura, W. Ito, K. Shimura, and H. Kato, “Computerized detection of pulmonary nodules by single-exposure dual-energy computed radiography of the chest–-part 1,” European Journal of Radiology, vol. 44, no. 3, pp. 198–204, 2002.
• E. Bratsolis and M. Sigelle, “A spatial regularization method preserving local photometry for Richardson-Lucy restoration,” Astronomy and Astrophysics, vol. 375, no. 3, pp. 1120–1128, 2001.
• T. Le, R. Chartrand, and T. J. Asaki, “A variational approach to reconstructing images corrupted by Poisson noise,” Journal of Mathematical Imaging and Vision, vol. 27, no. 3, pp. 257–263, 2007.
• Y.-L. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Transactions on Image Processing, vol. 9, no. 10, pp. 1723–1730, 2000.
• S. Didas, J. Weickert, and B. Burgeth, “Properties of higher order nonlinear diffusion filtering,” Journal of Mathematical Imaging and Vision, vol. 35, no. 3, pp. 208–226, 2009.
• F. Li, C. Shen, J. Fan, and C. Shen, “Image restoration combining a total variational filter and a fourth-order filter,” Journal of Visual Communication and Image Representation, vol. 18, no. 4, pp. 322–330, 2007.
• P. Guidotti and K. Longo, “Two enhanced fourth order diffusion models for image denoising,” Journal of Mathematical Imaging and Vision, vol. 40, no. 2, pp. 188–198, 2011.
• M. Lysaker, A. Lundervold, and X.-C. Tai, “Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time,” IEEE Transac-tions on Image Processing, vol. 12, no. 12, pp. 1579–1589, 2003.
• M. R. Hajiaboli, “A self-governing fourth-order nonlinear diffusion filter for image noise removal,” IPSJ Transactions on Computer Vision and Applications, vol. 2, pp. 94–103, 2010.
• M. R. Hajiaboli, “An anisotropic fourth-order partial differential equation for noise removal,” Lecture Notes in Computer Science, vol. 5567, pp. 356–367, 2009.
• X. Liu, L. Huang, and Z. Guo, “Adaptive fourth-order partial differential equation filter for image denoising,” Applied Mathematics Letters, vol. 24, no. 8, pp. 1282–1288, 2011.
• S. Kim and H. Lim, “Fourth-order partial differential equations for effective image denosing,” Electronic Journal of Differential Equations: Conference 17, vol. 17, pp. 107–121, 2009.
• T. Chan, A. Marquina, and P. Mulet, “High-order total variation-based image restoration,” SIAM Journal on Scientific Computing, vol. 22, no. 2, pp. 503–516, 2000.
• D. P. Bertsekas, A. Nedic, and A. E. Ozdaglar, Convex Analysis and Optimization, Tsinghua University Press, 2006.
• C. R. Vogel and M. E. Oman, “Iterative methods for total variation denoising,” SIAM Journal on Scientific Computing, vol. 17, no. 1, pp. 227–238, 1996.
• G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, vol. 147, Springer, New York, NY, USA, 2002.
• D. Wang, Y. Hou, and J. Peng, Partial Differential Equation Based Approach to Image Processing, Science Press, Beijing, China, 2008.
• H.-Z. Chen, J.-P. Song, and X.-C. Tai, “A dual algorithm for minimization of the LLT model,” Advances in Computational Mathematics, vol. 31, no. 1–3, pp. 115–130, 2009.
• A. Chambolle, “An algorithm for total variation minimization and applications,” Journal of Mathematical Imaging and Vision, vol. 20, no. 1-2, pp. 89–97, 2004.
• M. Bergounioux and L. Piffet, “A second-order model for image denoising,” Set-Valued and Variational Analysis, vol. 18, no. 3-4, pp. 277–306, 2010.
• T. Goldstein and S. Osher, “The split Bregman method for ${L}_{1}$-regularized problems,” SIAM Journal on Imaging Sciences, vol. 2, no. 2, pp. 323–343, 2009.
• Y. Huang, M. K. Ng, and Y.-W. Wen, “A fast total variation minimization method for image restoration,” Multiscale Modeling & Simulation, vol. 7, no. 2, pp. 774–795, 2008.
• Y.-M. Huang, M. K. Ng, and Y.-W. Wen, “A new total variation method for multiplicative noise removal,” SIAM Journal on Imaging Sciences, vol. 2, no. 1, pp. 20–40, 2009.
• Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM Journal on Imaging Sciences, vol. 1, no. 3, pp. 248–272, 2008.
• Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.
• P. L. Combettes and V. R. Wajs, “Signal recovery by proximal forward-backward splitting,” Multiscale Modeling & Simulation, vol. 4, no. 4, pp. 1168–1200, 2005.