A New Nonlocal H1 Model for Image Denoising

Following ideas of Kindermann et al. (Multiscale Model. Simul. 4(4):1091–1115, 2005) and Gilboa and Osher (Multiscale Model. Simul. 7:1005–1028, 2008) we introduce new nonlocal operators to interpret the nonlocal means filter (NLM) as a regularization of the corresponding Dirichlet functional. Then we use these nonlocal operators to propose a new nonlocal H1 model, which is (slightly) different from the nonlocal H1 model of Gilboa and Osher (Multiscale Model. Simul. 6(2):595–630, 2007; Proc. SPIE 6498:64980U, 2007). The key point is that both the fidelity and the smoothing term are derived from the same geometric principle. We compare this model with the nonlocal H1 model of Gilboa and Osher and the nonlocal means filter, both theoretically and in computer experiments. The experiments show that this new nonlocal H1 model also provides good results in image denoising and closer to the nonlocal means filter than the H1 model of Gilboa and Osher. This means that the new nonlocal operators yield a better interpretation of the nonlocal means filter than the nonlocal operators given in Gilboa and Osher (Multiscale Model. Simul. 7:1005–1028, 2008).

[1]  Yan Jin,et al.  A Nonlocal Version of the Osher-Solé-Vese Model , 2011, Journal of Mathematical Imaging and Vision.

[2]  Guy Gilboa,et al.  Nonlocal evolutions for image regularization , 2007, Electronic Imaging.

[3]  Alessandro Foi,et al.  Image Denoising by Sparse 3-D Transform-Domain Collaborative Filtering , 2007, IEEE Transactions on Image Processing.

[4]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[5]  Xavier Bresson,et al.  Bregmanized Nonlocal Regularization for Deconvolution and Sparse Reconstruction , 2010, SIAM J. Imaging Sci..

[6]  Jitendra Malik,et al.  Scale-Space and Edge Detection Using Anisotropic Diffusion , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[7]  L. Shires Image , 2018, Victorian Literature and Culture.

[8]  Roberto Manduchi,et al.  Bilateral filtering for gray and color images , 1998, Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271).

[9]  L. Modica,et al.  Partial Differential Equations and the Calculus of Variations , 1989 .

[10]  L. P. I︠A︡roslavskiĭ Digital picture processing : an introduction , 1985 .

[11]  A. Bruckstein,et al.  K-SVD : An Algorithm for Designing of Overcomplete Dictionaries for Sparse Representation , 2005 .

[12]  J. Craggs Applied Mathematical Sciences , 1973 .

[13]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[14]  M. Elad,et al.  $rm K$-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation , 2006, IEEE Transactions on Signal Processing.

[15]  Guy Gilboa,et al.  Nonlocal Operators with Applications to Image Processing , 2008, Multiscale Model. Simul..

[16]  Yves Meyer,et al.  Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures , 2001 .

[17]  Bernhard Schölkopf,et al.  Regularization on Discrete Spaces , 2005, DAGM-Symposium.

[18]  T. Chan NON-LOCAL UNSUPERVISED VARIATIONAL IMAGE SEGMENTATION MODELS , 2008 .

[19]  B. Schölkopf,et al.  A Regularization Framework for Learning from Graph Data , 2004, ICML 2004.

[20]  Jaakko Astola,et al.  From Local Kernel to Nonlocal Multiple-Model Image Denoising , 2009, International Journal of Computer Vision.

[21]  Ron Kimmel,et al.  A general framework for low level vision , 1998, IEEE Trans. Image Process..

[22]  G. Sapiro,et al.  Geometric partial differential equations and image analysis [Book Reviews] , 2001, IEEE Transactions on Medical Imaging.

[23]  Karen O. Egiazarian,et al.  Pointwise Shape-Adaptive DCT for High-Quality Denoising and Deblocking of Grayscale and Color Images , 2007, IEEE Transactions on Image Processing.

[24]  Stanley Osher,et al.  Deblurring and Denoising of Images by Nonlocal Functionals , 2005, Multiscale Model. Simul..

[25]  Jean-Michel Morel,et al.  A Review of Image Denoising Algorithms, with a New One , 2005, Multiscale Model. Simul..

[26]  Gabriel Peyré,et al.  Image Processing with Nonlocal Spectral Bases , 2008, Multiscale Model. Simul..

[27]  Tony F. Chan,et al.  Image processing and analysis - variational, PDE, wavelet, and stochastic methods , 2005 .

[28]  T. Chan,et al.  WAVELET INPAINTING BY NONLOCAL TOTAL VARIATION , 2010 .

[29]  Luminita A. Vese,et al.  Nonlocal Variational Image Deblurring Models in the Presence of Gaussian or Impulse Noise , 2009, SSVM.

[30]  G. Aubert,et al.  Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations (Applied Mathematical Sciences) , 2006 .

[31]  Guy Gilboa,et al.  Nonlocal Linear Image Regularization and Supervised Segmentation , 2007, Multiscale Model. Simul..

[32]  Gjlles Aubert,et al.  Mathematical problems in image processing , 2001 .

[33]  Leonid P. Yaroslavsky,et al.  Digital Picture Processing , 1985 .

[34]  Pierre Kornprobst,et al.  Mathematical problems in image processing - partial differential equations and the calculus of variations , 2010, Applied mathematical sciences.

[35]  Bin Dong,et al.  Level Set Based Nonlocal Surface Restoration , 2008, Multiscale Model. Simul..

[36]  Karen O. Egiazarian,et al.  Image restoration by sparse 3D transform-domain collaborative filtering , 2008, Electronic Imaging.

[37]  Stanley Osher,et al.  Image Recovery via Nonlocal Operators , 2010, J. Sci. Comput..

[38]  Joachim Weickert,et al.  Anisotropic diffusion in image processing , 1996 .

[39]  Laurent D. Cohen,et al.  Non-local Regularization of Inverse Problems , 2008, ECCV.

[40]  Charles-Alban Deledalle,et al.  Non-local Methods with Shape-Adaptive Patches (NLM-SAP) , 2012, Journal of Mathematical Imaging and Vision.

[41]  Tony F. Chan,et al.  Image processing and analysis , 2005 .