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