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Image reconstruction and Image Priors

Image reconstruction and Image Priors. Tim Rudge Simon Arridge, Vadim Soloviev Josias Elisee, Christos Panagiotou Petri Hiltunen (Helsinki University of Technology). Fast reconstruction algorithm Edge-based image priors Joint entropy image priors Gaussian-mixture classification priors.

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Image reconstruction and Image Priors

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  1. Image reconstruction and Image Priors Tim Rudge Simon Arridge, Vadim Soloviev Josias Elisee, Christos Panagiotou Petri Hiltunen (Helsinki University of Technology)

  2. Fast reconstruction algorithm • Edge-based image priors • Joint entropy image priors • Gaussian-mixture classification priors

  3. 1. Fast reconstruction • Image compression method • Reduce matrix size • Explicit fast inversion • Optics Letters, Vol. 35, Issue 5, pp. 763-765 (2010)

  4. Measurement setup

  5. Forward operator • Size of matrix A = (nx* ny* ns* nθ) x nrecon = very big

  6. i,j = source, detector • w = pixel detector profile • P = projection to image • S = diag(1/ye) = normalisation • Gf / Gf* = Green's operator / adjoint operator (fluorescent λ)‏ • Ue = excitation field

  7. Compress each image • Where rows of Z: ...are basis functions in image • E.g. Wavelets, Fourier (sine/cosine)‏

  8. Form compressed system By replacing window functions w, with basis functions z in: Size of matrix = (nz* ns* nθ) x nrecon = more reasonable

  9. Solve compressed system • Matrix is (nz* ns * nθ) x (nz* ns * nθ)‏ • Small enough to store and solve explicitly • Typically solves in < 10s

  10. Some results

  11. Redundancy in data

  12. 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

  13. Huber edge prior (region), simulated data 2% noise

  14. 3. Joint entropy priors

  15. 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

  16. Combined Reconstruction Classification y Cy Data Noise Statistics x Reconstruction Step Estimation Step Image x,Cx l,q Class Statistics Image Statistics Prior Update Step

  17. Anim2d.mov

  18. 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

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