1. 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

4. NOISE MODELING
For a positive coefficient:
For a negative coefficient
Given a threshold t, if P > t, the coefficient could be due to the noise.
On the other habd, if P < t, the coefficient cannot be due to the noise,
and a significant coefficient is detected.

13. NGC2997 MULTIRESOLUTION SUPPORT

40. DECONVOLUTION SIMULATION
LUCY
PIXON
Wavelet

44. 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.

45. Continuous Ridgelet Transform
Ridgelet Transform (Candes, 1998):
Ridgelet function:
The function is constant along lines.
Transverse to these ridges, it is a wavelet.

49. Digital Ridgelet Transform :

50. Example application of Ridgelets

51. SNR = 0.1

53. Undecimated Wavelet Filtering (3 sigma)

54. Ridgelet Filtering (5sigma)

55. Line detection by the ridgelet transform

56. NEWTON/XMM Image
of the supernovae SN1604
Ridgelet Filtering

58. 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 :

59. The Curvelet Transform
Wavelet
Curvelet
Width = Length^2

60. The Curvelet Transform
J.L. Starck, E. Candès and D. Donoho,
"Astronomical Image Representation by the Curvelet Transform,
Astronomy and Astrophysics, 398, , 2003.

61. CURVELET FILTERING NOISE MODELING For a positive coefficient:
For a negative coefficient
Given a threshold t:
if P > t, the coefficient could be due to the noise.
if P < t, the coefficient cannot be due to the noise,
and a significant coefficient is detected.
Hard Thresholding:

62. Curvelet
Undecimated WT
Lena + Gaussian Noise

64. FILTERING

68. Algorithm :

69. a) Simulated image (gaussians+lines) b) Simulated image + noise c) A trous algorithm
d) Curvelet transform e) coaddition c+d f) residual = e-b

70. a) A370
b) a trous
c) Ridgelet + Curvelet
Coaddition b+c

71. a) NGC2997
b) atrous
c) Ridgelet
d) Coaddition b+c

72. Galaxy SBS
Ridgelet
Curvelet
A trous WT

73. = + = + Galaxy SBS 0335-052 10 micron GEMINI-OSCIR CTM Curvelet
A trous WT
Ridgelet +
Residual
CTM =
Rid+Curvelet +
Clean Data

74. Noise Standard Deviation
PSNR
Lena
Curvelet
Decimated wavelet
Undecimated wavelet
Noise Standard Deviation

77. Barbara
Curvelet
Decimated wavelet
Undecimated wavelet

78. Curvelet
Wavelet
Curvelet

79. 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 :

80. We propose solving the following optimization problem:
min
Complexity_penalty , subject to
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.

86. DECONVOLUTION: We propose solving the following optimization problem:
min
Complexity_penalty , subject to
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.

89. 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

91. The Curvelet Transform
Wavelet
Curvelet
Width = Length^2

92. The Curvelet Transform
Wavelet
Curvelet
Width = Length^2