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)

1. Fast reconstruction algorithm

2. Edge-based image priors

3. Joint entropy image priors

4. Gaussian-mixture classification priors

1. Fast reconstruction

Image compression method Reduce matrix size Explicit fast inversion Optics Letters, Vol. 35, Issue 5, pp. 763-765

(2010)

Measurement setup

Forward operator

•Size of matrix A = (nx* n

y* n

s* n

θ) x n

recon = very big

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

Compress each image

Where rows of Z:

...are basis functions in image

E.g. Wavelets, Fourier (sine/cosine)

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

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

Some results

Redundancy in data

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

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

3. Joint entropy priors

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

x

x,Cx

y Cy

Data Noise Statistics

Image

Image Statistics Class Statistics

ReconstructionStep

EstimationStep

Prior UpdateStep

Combined Reconstruction Classification

Anim2d.mov

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