References
- P. Blomgren, T.F. Chan, P. Mulet, and C. Wong, Total variation image restoration: Numerical methods and extensions, Proceedings of IEEE International Conference on Image Processing, III (1997), pp. 384–387.
- T. Chan, A. Marquina, and P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput. 22 (2000), pp. 503–516. doi: 10.1137/S1064827598344169
- T.F. Chan, J. Shen, and L. Vese, Variational PDE models in image processing, Not. AMS 50 (2003), pp. 14–26.
- T.F. Chan and X.C. Tai, Identification of discontinuous coefficients in elliptic problems using total variation regularization, SIAM J. Sci. Comput. 25 (2003), pp. 881–904. doi: 10.1137/S1064827599326020
- Q. Chen, P. Montesinos, Q.S. Sun, P.A. Heng, and D.S. Xia, Adaptive total variation denoising based on difference curvature, Image Vis. Comput. 28 (2010), pp. 298–306. doi: 10.1016/j.imavis.2009.04.012
- A. Chambolle and P.L. Lions, Image recovery via total variation minimization and related problems, Numer. Math. 76 (1997), pp. 167–188. doi: 10.1007/s002110050258
- N. Chumchob, K. Chen, and C. Brito-Loeza, A new variational model for removal of combined additive and multiplicative noise and a fast algorithm for its numerical approximation, Int. J. Comput. Math. 90 (2013), pp. 140–161. doi: 10.1080/00207160.2012.709625
- R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, Interscience Publishers, New York, 1953.
- R. Deriche and O. Faugeras, Les EDP en traitement des images et vision par ordinateur, Tech. Rep. 2697, INRIA, November 1995.
- S. Didas, J. Weickert, and B. Burgeth, Stability and local feature enhancement of higher order nonlinear diffusion filtering, in Pattern Recognition, Lecture Notes in Computer Science, W. Kropatsch, R. Sablatnig, and A. Hanbury (Eds.), Vol. 3663, Springer, Berlin , 2005, pp. 451–458.
- S. Didas, J. Weickert, and B. Burgeth, Properties of higher order nonlinear diffusion filtering, J. Math. Imag. Vis. 35 (2009), pp. 208–226. doi: 10.1007/s10851-009-0166-x
- X. Feng, R. Glowinski, and M. Neilan, Recent developments in numerical methods for fully nonlinear second order partial differential equations, SIAM Rev. 55 (2013), pp. 205–267. doi: 10.1137/110825960
- O. Ghita, D.E. Ilea, and P.F. Whelan, Adaptive noise removal approach for restoration of digital images corrupted by multimodal noise, IET Image Process. 6 (2012), pp. 1148–1160. doi: 10.1049/iet-ipr.2010.0587
- J.B. Greer and A.L. Bertozzi, H1 Solutions of a class of fourth order non-linear equations for image processing, in V. Chepyzhov, M. Efendiev, A. Miranville, R. Temam (Eds.), Discrete and Continuous Dynamical Systems 2004, special issue in honor of Mark Vishik, 10 (1) (2004), pp. 349–366.
- J.B. Greer and A.L. Bertozzi, Traveling wave solutions of fourth order PDEs for image processing, SIAM. J. Math. Anal. 36 (2004), pp. 38–68. doi: 10.1137/S0036141003427373
- G. Gilboa, N. Sochen, and Y.Y. Zeevi, Variational denoising of partly textured images by spatially varying constraints, IEEE Trans. Image Process. 15 (2006), pp. 2281–2289. doi: 10.1109/TIP.2006.875247
- R. Liu, Z. Lin, W. Zhang, K. Tang, and Z. Su, Toward designing intelligent PDEs for computer vision: An optimal control approach, Image Vis. Comput. 31 (2013), pp. 43–56. doi: 10.1016/j.imavis.2012.09.004
- 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 Trans. Image Process. 12 (2003), pp. 1579–1590. doi: 10.1109/TIP.2003.819229
- M. Lysaker and X.-C. Tai, Iterative image restoration combining total variation minimization and a second-order functional, Int. J. Comput. Vis. 66 (2006), pp. 5–18. doi: 10.1007/s11263-005-3219-7
- Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures, American Mathematical Society Boston, MA, USA, 2001.
- M. Nikolova, Weakly constrained minimization: Application to the estimation of images and signals involving constant regions, J. Math. Imaging Vis. 21 (2004), pp. 155–175. doi: 10.1023/B:JMIV.0000035180.40477.bd
- K. Papafitsoros and C.B. Schönlieb, A combined first and second order variational approach for image reconstruction, J. Math. Imaging Vis. 48 (2014), pp. 308–338. doi: 10.1007/s10851-013-0445-4
- P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intel. 12 (1990), pp. 629–639. doi: 10.1109/34.56205
- L.I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992), pp. 259–268. doi: 10.1016/0167-2789(92)90242-F
- R. Srivastava and S. Srivastava, Restoration of Poisson noise corrupted digital images with nonlinear PDE based filters along with the choice of regularization parameter estimation, Pattern Recogn. Lett.34 (2013), pp. 1175–1185. doi: 10.1016/j.patrec.2013.03.026
- J. Weickert, Anisotropic Diffusion in Image Processing, B.G. Teubner, Stuttgart, 1998.
- Y.-L. You and M. Kaveh, Fourth-order partial differential equation for noise removal, IEEE Trans. Image Process. 9 (2000), pp. 1723–1730. doi: 10.1109/83.869184
- W.L. Zeng, X.B. Lu, and X.H. Tan, Non-linear fourth-order telegraph-diffusion equation for noise removal, IET Image Process. 7 (2013), pp. 335–342. doi: 10.1049/iet-ipr.2012.0155