Image Restoration. Diffusion Denoising Deconvolution Super-resolution Tomographic Reconstruction

Transcription

1 Image Restoration

2 Image Restoration Diffusion Denoising Deconvolution Super-resolution Tomographic Reconstruction

3 Diffusion Term Consider only the regularization term E-L equation: (Laplace equation) Steepest Descent:

4 Evolution of Laplace's Equation t

5 Evolution of TV Equation t

6 Isotropic & Anisotropic Diffusion

7 Acquisition model with noise noise + original image u(x) acquired images + n(x) = z(x)

8 Denoising Acquisition model Minimization problem Data term Weighting parameter Regularization term

9 Equivalent Formulations Constraint minimization Maximum A Posteriori (MAP) estimate Bayes' Theorem:

10 Denoising functional: E-L equation: Discrete solution: Set of linear equations

11 Original image u noisy image z

12 Regularization Weight λ x x x

13 Regularization L2 norm Tichonov L1 norm Total Variation Nonconvex

14 Regularization

15 Noisy input image

16 noisy original

17 Anisotropic Denoising noisy BEEM image TV + histogram equalization wavelet-based denoising TV-based denoising

18 Anisotropic Denoising noisy BEEM image TV + histogram equalization wavelet-based denoising TV-based denoising

19 Acquisition model with blur noise channel + original image [uu(x) * acquired image h](x) + n(x) = z(x)

20 Motion Blur * =

21 Out-of-focus Blur * =

22 Inverse Filter Add noise equivalent

23 Wiener Filter Replace by a constant Rato of power spectra equivalent

24 Deblurring (Deconvolution) Acquisition mode Minimization problem Data term Weighting parameter Regularization term

25 E-L equation: Discrete solution: Set of linear equations

26 Long-time Exposure

27 Multi-Channel Acquisition Model noise channel K channel 2 + channel 1 original image [uu(x) * acquired images hk](x) + nk(x) = zk(x)

28 Blind Deconvolution Acquisition model Minimization problem Data term Image regularization term Blur Regularization term

29 Blur Regularization Term u z1 = u z1 * h2 = * h1 u * h2 h1 * h2 * u = h1 * h2 * u = 0 z2 h1 * z2

30 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

31 Astronomical Imaging Degraded images Reconstructed image Blur estimation

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36 Multichannel Deconvolution Super-resolution

37 Super-resolution

38 Super-resolution Sub-pixel shifts Interpolation on a high-resolution grid

39 Superresolution Acquisition model Minimization problem Data term Image regularization term Blur Regularization term

40 Superresolution rough registration Optical zoom (ground truth) Superresolved image (2x)

41 SR limits

42 Superresolution of Video Interpolated video Super-resolved video (2x)

43 Tomographic Reconstruction CT SPECT MRI PET X-rays gamma rays electromagnetic waves positron-electron annihilation

44 Tomography Principle 1D projections of 2D objects

45 Sinogram Projections (sinogram) = Radon Transform Reconstruction Inverse Radon (Filtered Back Projection)

46 Variational Reconstruction R operator performing projections z sinogram Our optimization problem is

47 original Back Projection 15 projections (in Fourier domain) Variational Reconstruction

48 End

49 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

50 Long-time exposure II The Poor Fisherman, Paul Gauguin, 1896