Image restoration, noise models, detection, deconvolution

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Image restoration, noise models, detection,
deconvolution

• Outline
• Image formation model
• Noise models
• Inverse problems in image processing
• Bayesian approaches
• Wiener filtering
• Maximum-entropy methode
• Shrinkage, Sparsity
• Applications os multiscale representations

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NOISE MODELING
For a positive coefficient
For a negative coefficient
Given a threshold t, if P gt t, the coefficient
could be due to the noise. On the other habd, if
P lt t, the coefficient cannot be due to the
noise, and a significant coefficient is detected.
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NGC2997 MULTIRESOLUTION SUPPORT
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DECONVOLUTION SIMULATION
LUCY
PIXON
Wavelet
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Problems related to the WT

• 1) Edges representation
• if the WT performs better than the FFT to
• represent edges in an image, it is still not
  optimal.
• 2) There is only a fixed number of directional
  elements
• independent of scales.
• 3) Limitation of existing scale concepts
• there is no highly anisotropic elements.

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Continuous Ridgelet Transform
Ridgelet Transform (Candes, 1998)
Ridgelet function
Transverse to these ridges, it is a wavelet.
The function is constant along lines.
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Digital Ridgelet Transform
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Example application of Ridgelets
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SNR 0.1
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Undecimated Wavelet Filtering (3 sigma)
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Ridgelet Filtering (5sigma)
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Line detection by the ridgelet transform
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NEWTON/XMM Image of the supernovae SN1604
Ridgelet Filtering
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The Curvelet Transform
The curvelet transform opens us the possibility
to analyse an image with different block sizes,
but with a single transform. The idea is to
first decompose the image into a set of wavelet
bands, and to analyze each band by a ridgelet
transform. The block size can be changed at each
scale level.

• Algorithm

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The Curvelet Transform
Wavelet
Curvelet
Width Length2
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The Curvelet Transform
J.L. Starck, E. Candès and D. Donoho, "Astronomica
l Image Representation by the Curvelet
Transform, Astronomy and Astrophysics, 398,
785--800, 2003.
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CURVELET FILTERING
NOISE MODELING
For a positive coefficient
For a negative coefficient
Given a threshold t if P gt t, the coefficient
could be due to the noise. if P lt t, the
coefficient cannot be due to the noise, and a
significant coefficient is detected.
Hard Thresholding
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FILTERING
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• Algorithm

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a) Simulated image (gaussianslines) b)
Simulated image noise c) A
trous algorithm
d) Curvelet transform
e) coaddition cd
f) residual e-b
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a) A370
b) a trous
c) Ridgelet Curvelet
Coaddition bc
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a) NGC2997
b) atrous
d) Coaddition bc
c) Ridgelet
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Galaxy SBS 0335-052
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Galaxy SBS 0335-052 10 micron GEMINI-OSCIR
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PSNR
Lena
Curvelet
Decimated wavelet
Undecimated wavelet
Noise Standard Deviation
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Barbara
Curvelet
Decimated wavelet
Undecimated wavelet
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Curvelet
Curvelet
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RESTORATION HOW TO COMBINE SEVERAL MULTISCALE
TRANSFORMS ?
The problem we need to solve for image
restoration is to make sure that our
reconstruction will incorporate information
judged as significant by any of our
representations.

• Very High Quality Image Restoration, in Signal
  and Image Processing IX, San Diego, 1-4 August,
  2001,
• Eds Laine, Andrew F. Unser, Michael A.
  Aldroubi, Akram, Vol. 4478, pp 9-19, 2001.

Notations
Consider K linear transforms
and the coefficients of x after applying
.
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We propose solving the following optimization
problem
Where C is the set of vectors which obey the
linear constraints
positivity constraint
is significant
The second constraint guarantees that the
reconstruction will take into account any
pattern which is detected by any of the K
transforms.
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DECONVOLUTION
We propose solving the following optimization
problem
Where C is the set of vectors which obey the
linear constraints
positivity constraint
is significant
The second constraint guarantees that the
reconstruction will take into account any
pattern which is detected by any of the K
transforms.
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Multiscale Transforms
Critical Sampling
Redundant Transforms Pyramidal
decomposition (Burt and Adelson) (bi-)
Orthogonal WT
Undecimated Wavelet Transform Lifting scheme
construction Isotropic
Undecimated Wavelet Transform Wavelet Packets
Complex
Wavelet Transform Mirror Basis
Steerable Wavelet
Transform
Dyadic Wavelet
Transform
Nonlinear Pyramidal
decomposition (Median)
New Multiscale Construction
Contourlet
Ridgelet Bandelet
Curvelet (Several
implementations) Finite Ridgelet
Transform Platelet (W-)Edgelet
Adaptive Wavelet

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The Curvelet Transform
Wavelet
Curvelet
Width Length2
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The Curvelet Transform
Wavelet
Curvelet
Width Length2
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