Image Reconstruction and Image Priors

Image Reconstruction and Image Priors

Vadim Soloviev,Josias Elisee,

Tim Rudge,Simon Arridge

Munich

April 24, 2009

P4 University College London – Computer Science (UCL)

UCL has an annual turnover of £500M, academic and research staff totaling 4,000, and over 3,000 PhD research students. The department of Computer Science has over 50 academic staff with specialist groups involved in Imaging Science, Computer Graphics, BioInformatics.Intelligent Systems Networking, and Software Systems Engineering.

CMICIn 2005, the Centre for Medical Imaging (CMIC) was formed jointly between Computer Science and the department of Medical Physics & BioEngineering to create a world class grouping combining excellence in medical imaging sciences withinnovative computational methodology, finding application in biomedical research and in healthcare. The research of the group focuses on detailed structural and functional analysis in neurosciences, imaging to guide interventions, image analysis in drug discovery, imaging in cardiology and imaging in oncology with a strong emphasis on e-science technologies. The Centre has very close links with the Faculty of Clinical Sciences, the Faculty of Life Sciences and associated Clinical Institutes, in particular the Institute of Neurology,the Institute of Child Health and the Centre for Neuroimaging Techniques (CNT),

• Main tasks attributed to the organisation: The main tasks for P4 UCL are WP4 with some input into WP3, WP6 and WP7. We will contribute mathematical and computational techniques for the development of forward and inverse modeling in optical tomography in the diffuse regime particularly using priors, and to the simulation of new imaging devices and the analysis of clinical data.

Objective 4.1 : To develop FMT inversion utilizing XCT image priors without strong anatomy function correlations.

Progress:• Developed structured priors orientating the reconstructed FMT images

to have level sets parallel to those of XCT image• Developed information theoretic priors orientating the reconstructed

FMT images to have maximum joint entropy with the XCT image• Initial tests on simulated 3D images of mouse from a realistic atlas

Significant Results• Reconstructions in 3D depend only linearly on total number of pixels in

reconstructed image and independent of the number of pixels in data.

Deviations from Annex 1• None

Failure to meet critical objectives• Not Applicable

Use of resources• No deviation from work

Structural priors

Choice of Prior))(exp()( xx Consider

)(xLx

drxx D )|(|)( where

xDxx TD :||and D a symmetric tensor

kDxL T:)(By variation

D

D

x

xk

||

)|(|'

where

Lagged_Diffusivity Gauss-Newton Method

Choice of Prior (2)

kss)(

TID ˆˆ

Examples: 1st order Tikhonov

sT

Tk

TT

s

Ts

)(

Total Variation

Now choose Where is normal to level set of another image xref

x xref

)/|exp(| Txref

Target object: cylinder with embedded inhomogeneitiesRadius: 25 mm, height: 50 mmBackground: a=0.01 mm-1, s=1 mm-1

Red: Inclusions with increased absorptionBlue: Inclusions with increased scattering

Measurements: 80 source locations, 80 detector locations, arranged in 5 rings at elevations -20, -10, 0, +10, +20

Data: log amplitude and phase for source modulated at 100MHzMultiplicative Gaussian noise 0.5%

FEM mesh: 83142 nodes, 444278 4-noded tetrahedraReconstruction grid: 80x80x80

Cross sections through target for planes z=16, z=60 and y=40

a

s

Example: Cylinder with inclusions

Reconstruction: Nonlinear conjugate gradients (50 iterations) with line searchPrior: TV with hyperparameter = 10-4 and smoothing parameter = 0.1

a

s

Reconstruction with flat TV prior

Iso-surfaces Cross sections

a

s

Reconstruction with TV prior using correct structural information

Edge prior


Reconstruction

a

s


a

s

TV prior using undifferentiated structural information

Edge prior


Reconstruction

a

s


a

s

TV prior using partial structural information

Edge prior


Reconstruction

a

s


Results using 3D Edge-Weighted Priors

Digimouse: Slices (z=0mm) through solutions with no prior-based regulari-sation (top), and regularisation ¿ = 10¡ 3 (bottom).

Results using 3D Edge-Weighted Priors (2)

Digimouse: Slice (z=0mm) through regularisation edgeweighting °.Digimouse: Solution on a line through the target at various values of ¿.

Information Theoretic Priors

² marginal entropy measures the uncertainty of a single random vector ofrandomvariables

² joint entropy measures theentropy of a joint system, in this context con-sisting of a pair of random vectors denoting the reconstructed and refer-ence image.

Consider the use of joint entropy (J E) H (¢;¢) or mutual information (MI)M I (¢;¢) as regularization Functionals will be introduced in the reconstructionprocess

ª J E (x;xref) = H(x;xref) and

ª M I (x;xref) = ¡ M I (x;xref)

leading to theminimization of J E and maximization of MI.

Marginal and Joint Entropies

Target Distributions and Reference Images

Target distributions and the 5 reference images pairs, incommensuratelyrelated to thetarget gray values. Ref 1displays full correspondencebetween itsfeatures and the ones in the target distributions. Ref. 2 contains features notexisting in the target space. Ref. 3 is missing features. Thegradient in Ref. 4risesby centeringa2D Gaussian (¾: 50pixels) on top of Ref. 2andmultiplyingthepixels values underneath. Wealso add 5%Gaussian multiplicativenoise

Reconstructions

¹ a and ¹ 0s reconstructions by introducing the available reference image pairswith joint entropy or mutual information. The converged TK1 reconstructionsareprovided for comparison alongwith the initialization guess

a

’s

Objective 4.2 : To incorporate XCT image segmentation into the FMT

Progress:

• Developed segmentation of XCT based on anisotropic diffusion (Perona-Malik algorithm)

• Developed hexahedral adaptive mesh generation from XCT images

• Incorporated mesh reduction methods using public-domain software ISO2MESH

• Developed Boundary Element (BEM) and hybrid Boundary-Finite Element (BEM-FEM) methods

Significant Results

• Reconstructions using FEM only for internal organs are much faster than using a complete FEM mesh

Deviations from Annex 1

• None

Failure to meet critical objectives

• Not Applicable

Use of resources

• No deviation from work

Segmentation Requirements

• construction of meshes for numerical modelling • construction of priors as required in Objective 4.1• post-reconstruction object labelling and analysis

Anisotropic Diffusion Based Segmentation

Left: sliceof original CT image. Right: segmented image.

Mesh Generation

(a) Rendered rat's head. (b) Rat's hexahedral mesh. Mesh is re ned morealongboundaris of theobject. (c) Rendered rat's skeleton. (d) Extracted skele-ton mesh.

BEM and BEM-FEM approach

Mouse and liver meshes (3234 nodes, 1616 elements and 4582 nodes, 2290elements respectively) - thevolumemesh is 8649nodesand 4659elements large

BEM-FEM results

Energy density ¯eld on themouse surfaceand central slicewithin the liver

Objective 4.3 : To calculate spatially varying optical attenuation in tissues in-vivo.

Progress:• Developed non-linear reconstruction method for attenuation making use

of Louiville transformation from diffusion to Schrodinger equation.

Significant Results• Reconstruction of attenuation from steady-state data is dependent on

good estimates of spatially varying scatter.




Excitation

Figure1: Digimouse: Simulated excitationdata fromtarget °uorophoreconcen-tration radius 5mm, positioned at (5,0,0)mm. (Images individually log scaled)

Fluorescence

Figure 1: Digimouse: Simulated °uorescence data from target °uorophoreconcentration radius 5mm, positioned at (5,0,0)mm. (Images individually logscaled)

Reconstruction

Digimouse: Target °uorophore concentrationDigimouse: Iso-surface through solution at 0.35, and slice through solution atz=0mm.

Objective 4.4 :To develop FMT inversion based on simultaneous XCT segmentation and classification

Progress:• Developed combined reconstruction/segmentation method combining

Gauss-Newton image reconstruction with fuzzy-kmeans image classification.

• Developed fully hierarchical Bayesian framework

Significant Results• Classification error less than 5% for simulated noisy data.




x

x,Cx

y Cy

Data Noise Statistics

Image

Image Statistics Class Statistics

ReconstructionStep

EstimationStep

Prior UpdateStep

Combined Reconstruction Classification

Heirarchical Bayesian MethodIn thereconstruction stepweaimtooptimizetheoptical parametersx given themeasurements y and the mixtures of Gaussians prior p(xj³ ;¸;µ) using Bayes'rule. Unfortunately, this is a non-convex optimization problem. Hence, weapproximate theprior distribution by taking themaximuma posteriori (MAP)estimateof the indicator variables at each pixel so that

p(xj³ ;¸;µ) =Y

i

Y

`

p(xi jµ )³ i ` ; (1)

where we have also approximated class variances to be a constant C` =°¡ 1=2I . After this operation, both the prior and likelihood terms are Gaus-sian and ¯nding the MAP estimate of the optical parameters x is a convexoptimization problem.

To solve this problem, we assume that the precision C ¡ 1y of the measure-

ment noise can be factored as C ¡ 1y = LTy L y using the Cholesky decomposi-

tion. Theprior distribution can bewritten asmultivariatenormal distributionN (¹x;°¡ 1=2I ), where ¹xi = m` if pixel i belongs to class . After these changeswecan transform to aminimisation problem

xk+1 = argminx

kL y(y ¡ f (x))k2+°kx ¡ ¹xk2: (2)

Deliverables

4.1 Inversion algorithms• Deliverable 4.1 was created as an inversion code. Two versions were

developed : • compiled C++ code using UCL Toast Libraries and OpenGL graphics• Matlab program using Mex version of TOAST libraries and Matlab

Graphics• Individual installations on partner systems will be provided at the next

project meeting.

Conclusions• FEM and BEM based solvers• Linear and non-linear reconstruction• Large Data Sets using Matrix-Free approach• Structural Priors incorporating image information, not dependent on

segmentation• Statistical Priors based on information theory• Matlab based code available on web

http://web4.cs.ucl.ac.uk/research/vis/toast/index.html

Documents

Image Reconstruction and Image Priors