Variational and PDE Models for Image Interpolation

The nonlinear PDE-based image reconstruction domain is approached in this chapter. The state of the art inpainting approaches based on differential models are described in the first section. The interpolation techniques based on variational models are described first and the PDE-based inpainting methods that do not follow variational principles are discussed next. Then, our main contributions in this image processing field are presented in the following sections. The structural inpainting techniques developed by us in variational or PDE form are based on second and fourth order nonlinear diffusion models. Our variational reconstruction algorithms are described in the second section, while the nonlinear PDE-based image interpolation solutions developed by us are detailed in the last section of this chapter.

[1]  Tony F. Chan,et al.  Morphologically invariant PDE inpaintings , 2001 .

[2]  Patrick Pérez,et al.  Region filling and object removal by exemplar-based image inpainting , 2004, IEEE Transactions on Image Processing.

[3]  Pascal Getreuer,et al.  Total Variation Inpainting using Split Bregman , 2012, Image Process. Line.

[4]  Tudor Barbu,et al.  A novel variational framework for structural image completion , 2017, 2017 International Conference on Optimization of Electrical and Electronic Equipment (OPTIM) & 2017 Intl Aegean Conference on Electrical Machines and Power Electronics (ACEMP).

[5]  Tudor Barbu,et al.  Variational image inpainting technique based on nonlinear second-order diffusions , 2016, Comput. Electr. Eng..

[6]  Lin He,et al.  Cahn--Hilliard Inpainting and a Generalization for Grayvalue Images , 2009, SIAM J. Imaging Sci..

[7]  Manya V. Afonso,et al.  Blind Inpainting Using $\ell _{0}$ and Total Variation Regularization , 2015, IEEE Transactions on Image Processing.

[8]  Tudor Barbu,et al.  Hybrid image interpolation technique based on nonlinear second and fourth-order diffusions , 2017, 2017 International Symposium on Signals, Circuits and Systems (ISSCS).

[9]  Karl Kunisch,et al.  Total Generalized Variation , 2010, SIAM J. Imaging Sci..

[10]  Evelyn Lunasin,et al.  The Navier–Stokes–Voight model for image inpainting , 2009, 0901.4548.

[11]  Marrick Neri,et al.  Total variation-based image inpainting and denoising using a primal-dual active set method , 2014 .

[12]  Sos S. Agaian,et al.  Quantifying image similarity using measure of enhancement by entropy , 2007, SPIE Defense + Commercial Sensing.

[13]  A. Bertozzi,et al.  Unconditionally stable schemes for higher order inpainting , 2011 .

[14]  Andrew P. Witkin,et al.  Scale-Space Filtering , 1983, IJCAI.

[15]  Guillermo Sapiro,et al.  Image inpainting , 2000, SIGGRAPH.

[16]  Firas A. Jassim.  Image Inpainting by Kriging Interpolation Technique , 2013, ArXiv.

[17]  Stanley Osher,et al.  Image Decomposition and Restoration Using Total Variation Minimization and the H1 , 2003, Multiscale Model. Simul..

[18]  Pascal Getreuer,et al.  Rudin-Osher-Fatemi Total Variation Denoising using Split Bregman , 2012, Image Process. Line.

[19]  Michael K. Ng,et al.  Multiplicative Noise Removal with Spatially Varying Regularization Parameters , 2010, SIAM J. Imaging Sci..

[20]  Carola-Bibiane Schönlieb,et al.  A Combined First and Second Order Variational Approach for Image Reconstruction , 2012, Journal of Mathematical Imaging and Vision.

[21]  Kristian Bredies,et al.  A TGV-Based Framework for Variational Image Decompression, Zooming, and Reconstruction. Part I: Analytics , 2015, SIAM J. Imaging Sci..

[22]  Carola-Bibiane Schönlieb,et al.  Combined First and Second Order Total Variation Inpainting using Split Bregman , 2013, Image Process. Line.

[23]  Jean-Michel Morel,et al.  Level lines based disocclusion , 1998, Proceedings 1998 International Conference on Image Processing. ICIP98 (Cat. No.98CB36269).

[24]  George B. Arfken.  7 – FUNCTIONS OF A COMPLEX VARIABLE II: CALCULUS OF RESIDUES , 1985 .

[25]  Viorel Barbu,et al.  Nonlinear Differential Equations of Monotone Types in Banach Spaces , 2010 .

[26]  Tudor Barbu,et al.  Nonlinear anisotropic diffusion-based structural inpainting framework , 2017, 2017 13th International Conference on Advanced Technologies, Systems and Services in Telecommunications (TELSIKS).

[27]  L. Ambrosio,et al.  Approximation of functional depending on jumps by elliptic functional via t-convergence , 1990 .

[28]  Alexei A. Efros,et al.  Texture synthesis by non-parametric sampling , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[29]  Tony F. Chan,et al.  Euler's Elastica and Curvature-Based Inpainting , 2003, SIAM J. Appl. Math..

[30]  Tony F. Chan,et al.  Nontexture Inpainting by Curvature-Driven Diffusions , 2001, J. Vis. Commun. Image Represent..

[31]  Tony F. Chan,et al.  Non-texture inpainting by curvature-driven diffusions (CDD) , 2001 .

[32]  Guillermo Sapiro,et al.  Navier-stokes, fluid dynamics, and image and video inpainting , 2001, Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001.

[33]  V. Caselles,et al.  Exemplar-based Image Inpainting and Applications , 2011 .

[34]  Carola-Bibiane Schönlieb,et al.  Partial Differential Equation Methods for Image Inpainting , 2015, Cambridge monographs on applied and computational mathematics.

[35]  Tudor Barbu,et al.  Structural image interpolation using a nonlinear second-order hyperbolic PDE-based model , 2017, 2017 E-Health and Bioengineering Conference (EHB).

[36]  Lucas Pereira,et al.  Image replacement through texture synthesis , 1997, Proceedings of International Conference on Image Processing.

[37]  Jianhong Shen,et al.  Digital inpainting based on the Mumford–Shah–Euler image model , 2002, European Journal of Applied Mathematics.