2019-06-03-denoising

Wiki

• white noise
  • white refers to the way signal power is distributed (independently) over time or along frequencies
  • white noise vector if each component has probability distribution with mean 0 and a finite variance, and are statistically independent
• gaussian white noise
  • white noise vector where each component assumes a gaussian distribution
    • the vector is a multivariate gaussian distribution
• nonlocal means: https://en.wikipedia.org/wiki/Non-local_means
  • weighted sum of all pixels’ local mean filter
• bilateral filter: https://en.wikipedia.org/wiki/Bilateral_filter
• total variation: https://en.wikipedia.org/wiki/Total_variation_denoising
  • denoising but edge perserving
  • 2D problem harder to solve (primal dual: https://link.springer.com/content/pdf/10.1023%2FB%3AJMIV.0000011325.36760.1e.pdf)

Denoising

• 1992_nonlinear_total_variantion_based_noise_removal_algorithms
  • abstract
    • denoising by minimizing total variations, constraint using Lagrange multipliers
    • solved by gradient-projection
  • intro
    • total variation norm
      • L1 norms of derivatives
      • nonlinear, computationall complex (compared to L2)
    • problem formulation
      • constraint optimization as time dependent nonlinear parabolic pde
        • derive the euler-lagrange equation for the optimization
      • solved with with a time-steping algorithm
  • results
    • denoise in flat regions AND preserves edge details
  • some follow-ups
    • primal-dual method for minimizing total variation
      • https://www.uni-muenster.de/AMM/num/Vorlesungen/MathemBV_SS16/literature/Chambolle2004.pdf
• 1998_bilateral_filtering_for_gray_and_color_images
  • bilateral filtering
• 2008_nonlocal_image_and_movie_denoising
  • nonlocal mean filtering
• 2010_fast_image_recovery_using_variable_splitting_and_constraint_optimization
  • abstract
    • using ADMM for solving unconstrained problem where objective includes
      • L2 data-fidelity term
      • non-smooth regularizer
    • variable splitting
      • equivalent to constrained optimization, addressed with augmented Lagrangian method
  • problem formulation
    • synthesis approach (i.e. with wavelets)
      • x = W\beta where
        • W are elements of wavelet frame
        • \beta are parameter to be estimated
    • analysis approach
      • x sampled randomly
      • based on regularizers that analyzes the image itself rather than representation in wavelet domain
        • i.e total variation regularizer
    • unified view
      • min_x 1/2 || Ax - y ||_2^2 + \tau \phi(x)
        • A = BW where A=B for analysis approach
      • solvers
        • iterative shrinkage / thresholding (IST)
          • relies on denoising function \Psi(y)
          • iteration: x_{k+1} = \Psi ( x_t - 1/\gamma A^T(Ax - y))
          • problem: slow
  • proposed algo: split augmented Lagrangian shrinkage algorithm (SALSA)
    • a variant of ADMM
• 2014_progressive_image_denoising_through_hybrid_graph_laplacian_regularization
  • abstract
    • laplacian regularized image denoising
    • semisupervised learning
  • intro
    • variational problem
      • minimize data fidelity term and regularization term
    • priors
      • locally smooth (nearby pixels more likely to have same/similar intensity values)
      • non-local self-similarity (pixels on same structure likely to have same or similar intensity)
  • graph laplacian regularized regression
    • graph laplacian regularizer
      • R(f) = \sum_{i,j} (f(x_i) - f(x_j))^2 w_{ij}
      • w_ij is edge weight which reflects affinity between two vertices x_i and x_j
        • want to design filters that is edge-preserving (bilateral filtering)

Plug and Play P3 Prior

• 2013_plug_and_play_priors_for_model_based_reconstruction
• 2016_algorithm_induced_prior_for_image_restoration

• 2016_plug_and_play_admm_for_image_restoration_fixed_point_convergence_and_applications

• 2017_regularization_by_denoising
  • intro
    • plug-and-play (P3)
      • use implicit priors for regularizing general inverse problems
      • problems
        • no clear objective function
        • parameter tuning ADMM
    • use of denoising engine for regularization of inverse problems
      • rho(x) = 1/2 x^T (x - f(x))
        • is proved to be convex (guaranteed convergence)
      • the goodness
        • explicit objective
        • gradient manageable
        • any inverse problem handled by calling denoising engine iteratively
      • applications
        • single-image superresolution
        • blurring

Learning denoising prior for inverse problems

• 2017_learning_deep_cnn_denoiser_prior_for_image_restoration
  • abstract
    • use CNN to learn powerful denoiser priors, then plug into ADMM, HQS (half quadratic splittings)
• 2017_learning_proximal_operators_using_denoising_networks_for_regularizing_inverse_image_problems
  • abstract
    • use CNN to learn proximal operators
    • did not perform better than flexISP for demosaiking

Review

• 2010_review_image_denoising_algorithm_with_a_new_one

• 2018_review_modern_regularization_methods_for_inverse_problems
  • linear/nonlinear regularization for inverse problems
  • pretty involved discussion

• 2018_review_cnn_for_inverse_problems_in_imaging