Image Restoration
Image Restoration Diffusion Denoising Deconvolution Super-resolution Tomographic Reconstruction
Diffusion Term Consider only the regularization term E-L equation: (Laplace equation) Steepest Descent:
Evolution of Laplace's Equation 0 10 100 1000 10000 t
Evolution of TV Equation 0 10 100 1000 10000 t
Isotropic & Anisotropic Diffusion
Acquisition model with noise noise + original image u(x) acquired images + n(x) = z(x)
Denoising Acquisition model Minimization problem Data term Weighting parameter Regularization term
Equivalent Formulations Constraint minimization Maximum A Posteriori (MAP) estimate Bayes' Theorem:
Denoising functional: E-L equation: Discrete solution: Set of linear equations
Original image u noisy image z
Regularization Weight λ x x x
Regularization L2 norm Tichonov L1 norm Total Variation Nonconvex
Regularization
Noisy input image
noisy original
Anisotropic Denoising noisy BEEM image TV + histogram equalization wavelet-based denoising TV-based denoising
Anisotropic Denoising noisy BEEM image TV + histogram equalization wavelet-based denoising TV-based denoising
Acquisition model with blur noise channel + original image [uu(x) * acquired image h](x) + n(x) = z(x)
Motion Blur * =
Out-of-focus Blur * =
Inverse Filter Add noise equivalent
Wiener Filter Replace by a constant Rato of power spectra equivalent
Deblurring (Deconvolution) Acquisition mode Minimization problem Data term Weighting parameter Regularization term
E-L equation: Discrete solution: Set of linear equations
Long-time Exposure
Multi-Channel Acquisition Model noise channel K channel 2 + channel 1 original image [uu(x) * acquired images hk](x) + nk(x) = zk(x)
Blind Deconvolution Acquisition model Minimization problem Data term Image regularization term Blur Regularization term
Blur Regularization Term u z1 = u z1 * h2 = * h1 u * h2 h1 * h2 * u = h1 * h2 * u = 0 z2 h1 * z2
Alternating Minimization Minimization of F(u,{hk}) over u and hk alternates. Input: Blurred images and estimation of the blur size Output: Reconstructed image and the blurs
Astronomical Imaging Degraded images Reconstructed image Blur estimation
Multichannel Deconvolution Super-resolution
Super-resolution
Super-resolution Sub-pixel shifts Interpolation on a high-resolution grid
Superresolution Acquisition model Minimization problem Data term Image regularization term Blur Regularization term
Superresolution rough registration Optical zoom (ground truth) Superresolved image (2x)
SR limits
Superresolution of Video Interpolated video Super-resolved video (2x)
Tomographic Reconstruction CT SPECT MRI PET X-rays gamma rays electromagnetic waves positron-electron annihilation
Tomography Principle 1D projections of 2D objects
Sinogram Projections (sinogram) = Radon Transform Reconstruction Inverse Radon (Filtered Back Projection)
Variational Reconstruction R operator performing projections z sinogram Our optimization problem is
original Back Projection 15 projections (in Fourier domain) Variational Reconstruction
End
Between-Image Shift original image PSF degraded image [ u h k ] τ k x,y +nk x,y =z k x,y [ u g k ] x,y +nk x,y =z k x,y
Long-time exposure II The Poor Fisherman, Paul Gauguin, 1896