Image reconstruction and Image Priors
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Image reconstruction and Image Priors
Tim Rudge
Simon Arridge, Vadim Soloviev
Josias Elisee, Christos Panagiotou
Petri Hiltunen (Helsinki University of Technology)
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1. Fast reconstruction algorithm
2. Edge-based image priors
3. Joint entropy image priors
4. Gaussian-mixture classification priors
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1. Fast reconstruction
Image compression method Reduce matrix size Explicit fast inversion Optics Letters, Vol. 35, Issue 5, pp. 763-765
(2010)
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Measurement setup
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Forward operator
•Size of matrix A = (nx* n
y* n
s* n
θ) x n
recon = very big
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i,j = source, detectorw = pixel detector profileP = projection to imageS = diag(1/ye) = normalisationGf / Gf* = Green's operator / adjoint operator (fluorescent λ)Ue = excitation field
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Compress each image
Where rows of Z:
...are basis functions in image
E.g. Wavelets, Fourier (sine/cosine)
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Form compressed system
By replacing window functions w, with basis functions z in:
Size of matrix = (nz* n
s* n
θ) x n
recon = more reasonable
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Solve compressed system
Matrix is (nz* n
s * n
θ) x (n
z* n
s * n
θ)
Small enough to store and solve explicitlyTypically solves in < 10s
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Some results
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Redundancy in data
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2. Edge priors
•Smoothing operator
•Spatially varying width
•Edge in prior image low smoothing
•Smoothing max. ║ to edge
•Prior image flat max. Smoothing
•No segmentation needed
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Huber edge prior (region), simulated data 2% noise
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3. Joint entropy priors
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4. Gaussian-mixture priors
•Tikhonov 0 == single Gaussian
•Use mixture of k Gaussians
•Iteratively:
•K-means cluster class statistics
•Construct inv. covariance Cx-1, mean μx
•Reconstruct with prior Cx-1, μx
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x
x,Cx
y Cy
Data Noise Statistics
Image
Image Statistics Class Statistics
ReconstructionStep
EstimationStep
Prior UpdateStep
Combined Reconstruction Classification
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Anim2d.mov
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People / papers
Petri Hiltunen (Helsinki) – Gaussian-mixture priors Phys. Med. Biol. 54, pp. 6457–6476, (2009)
Christos Panagiotou – Joint entropy priors J. Opt. Soc. Am., Vol. 26, Issue 5, pp. 1277-1290, (2009)
Wavelet method: Optics Letters, Vol. 35, Issue 5, (2010) pp. 763-765, (2010)
Martin Schweiger TOAST FEM code, other programming
Josias Elisee BEM method
Vadim Soloviev, Thanasis Zaccharopolous, Simon Arridge