Arrige.ppt [Lecture seule]

Image Reconstruction in Optical Tomography Simon R Arridge1, 1.- Centre of Medical Image Computing, Dept. Computer Science, University College London, UK

Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Introduction •

Physical models

•

X-Ray CT vs OT

•

The Forward Problem

•

Inverse Scattering Approach

•

Bayesian Approach and Optimisation

•

Numerical Methods

•

Approximation Errors

•

Shape Based Methods

•

Transport Equation Approach

•

Dynamic Imaging


Physical Models


Light Propagation Maxwell’ Maxwell’s Equations Bethe-Salpeter Equation

Boltzmann Equation

Multigroup Boltzmann Equation

Variable Speed Boltzmann

Pn Equations

Fokker-Plank Equation

P1 Equations Telegrapher’ Telegrapher’s Equations

Diffusion Equation

Advection-Diffusion Equation


Radiative Transport Equation (RTE) RTE in frequency domain (modulation amplitude ω):

attenuation coefficient

radiance scattering phase coefficient function

source term

Change of radiance Ι at position r into direction θ ε S2, given absorption and scattering coefficients, and scattering phase function Θ. Boundary Condition :

Phase Function :


Diffusion transport model (frequency) Diffusion equation results from the assumption that radiance only varies linearly with angular direction (plus a few others)

Field Φ ( photon density): Diffusion Coefficient : Boundary condition: Measurement operator: Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Diffuse light transport in tissue • Typical optical parameters in tissue: µa = 0.01 … 0.1 mm-1 µs' = 1 … 10 mm-1 • Scatter-dominated; unscattered component is negligible in all practical applications

s

• Light propagates as a density wave • Low spatial information content of boundary measurements • Linear reconstruction techniques are in general not applicable

Simulation of stationary photon density in a circle (radius 25 mm) with inhomogeneities, given a boundary source µa = 0.025 mm-1, µs = 2 mm-1


X-Ray CT vs OT


X-Ray CT : A well-posed Imaging Problem

The Forward Problem (Radon Transform) can be written

g(l,θ) I l

f(x,y) I0

Where the forward operator is a linear integral operator with kernel K(l,θ;x,y)=δ(x-lSinθ+sCosθ,y-lCosθ-sSinθ)

θ Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Backprojection in X-Ray CT

The Adjoint Operator (Backprojection) has the same kernel

f(x,y)


The Inversion Formula in X-Ray CT Using forward and adjoint operators we get the inversion formula

Backprojection

Filter

• Reconstruction is only exact with complete data • In practice reconstruction is implemented in Fourier domain where (Ρ*Ρ)-1 is simple


Projection measurements X-ray CT: Scattering negligible: projections contain shadows of internal objects

OT: High scattering: photon density wave with low spatial information content surface projections

inclusion

→ direct linear reconstruction (Radon transform) from pencil beam transmission

→ iterative nonlinear model-based reconstruction (optimisation problem) from diffuse photon density wave


Regions of measurement sensitivity Multiple scattering: photons propagate by random walk through tissue → uncertainty of path between source and detector Photon measurement density functions (PMDF): areas of influence for a measurement for a given source-detector pair ("banana") random photon path

PMDF: homogeneous

PMDF: with inclusion

Shape of PMDF is affected by internal parameter distribution → backprojection of measurements is nonlinear! Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

The Forward Problem


OT : An ill-posed Imaging Problem

g+(ρm,θm) Due to multiple scattering, the initial “ray” in direction gives rise to multiple rays at all points on the output surface

µa(x,y,z) µs(x,y,z) Outgoing

Albedo operator

Incoming

g-(ρs,θs) Surface Transport Green’s function Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

The Diffusion Approximation in OT

If scattering is very extensive, the outgoing radiation is Lambertian

Incoming radiation is replaced by an equivalent diffuse source

g-(θs)

JnRobin source Source

q0

or

Isotropic


The Forward Problem (1)

The Diffusion Approximation equivalent to the albedo operator is the linear Robin-to-Neumann operator

where GδΩ is Green’s function for inhomogeneous b.c. case


Green’s Operators

For a given Green’s function G define the corresponding Green’s operator as the integral transform with kernel G

Define the measurable via the boundary derivative operator

Simplifying notation:


Inverse Scattering Approach


Linearisation (1)

Assume we know a reference state x0 = (µa0,D0)T with corresponding wave Φ0, and we want to find the scattered wave Φδ due to a change in state xδ =(α,β)T. We have µa0 = µa + α D= D0+ β Φ=Φ(0)+ Φδ (the initial state is not necessarily homogeneous). We get


Linearisation (2)

Formal solution using Green’s operator for the reference state Where V(a,b) is the “potential” operator. Let G0 be the Green’s function for the reference state. Then

Apply divergence theorem and assume β|δΩ=0 leads to

The Lippman Schwinger Equation Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Born Approximation

Lippman-Schwinger equation is formally solved in a Neumann Series

The Born Approximation simply truncates at second term :

Solution (in principle): Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Data Transformation

Dynamic range in the data is very large and the Born Approximation doesn’t work. Apply a functional transformation to the data. A simple choice is Logarithm => Rytov approximation

Linearised system becomes Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

ReΦδ µa

ImΦδ

Scattered fields

Re(logΦ)δ Im(logΦ)δ D Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Discrete System


Discrete Measurements

Fields Φj indexed by the incoming waves ηj.

Measurement at detector i is a weighted integral on the boundary Change in measurement due to (α,β) is given by

Where Κ*j is adjoint operator Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Discrete Solution Basis

Since the expected resolution is low, we represent the solution in a low resolution, smooth basis such as cubic spline pixels


Discrete System

We arrive at a discretised linear system


Adjoint formulation

Let G*δΩ be the Green’s function of the adjoint problem

Then the following holds: “The measured flux at ρm due to an isotropic source at r’ is equal to the complex conjugate of the photon density at r’ due to an adjoint Neumann source at ρm” Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Sensitivity functions

This means that the kernel of the Frechet derivative Κ can be represented as the product of forward and adjoint Green’s functions.

Which serves as the definition of the sensitivity functions


Sensitivity - mua

Real

Amplitude

Imag

Phase


Sensitivity - D Real

Amplitude

Imag

Phase


Bayesian Approach and Optimisation


The Forward Problem (2)

For some incoming radiation η

Combine all possible incoming radiation patterns to define “complete” non-linear mapping


Non-linear Reconstruction

From the Forward Mapping F(µa,D; ρm, ρm) derive discrete model Define the (Least Squares) data functional Together with a penalty term Minimisation of Equivalent to maximising Bayesian a posteriori probability


Newton Method

Iteratively solve Large value of τ ~Steepest Descent – when far from minimum Small value of τ ~Guass-Newton – when near to minimum. Include line search If run to convergence, terminates when the gradient is zero


Adjoint construction of gradient

Direct Field Φ

µa gradient Re(ΦΨ)

Cumulative µa gradient Total µa gradient

Adjoint Field Ψ

D gradient Re(gradΦ.gradΨ)

Cumulative D gradientTotal D gradient


Filtration µa gradient

D gradient

Filtered µa gradient

Filtered D gradient

µa reconstruction

D reconstruction


Iteration

D update

µa update

µa reconstruction

D reconstruction


Numerical Methods

•

Numerical Methods

•

- Finite Element Methods - Basis Selection - Newton-Krylov Reconstruction Method - Regularistion


TOAST forward model: Diffusion/FEM

Partition domain Ω into L nonoverlapping elements, joined at D vertex nodes. Approximate solution Φ to the diffusion equation by piecewise polynomial and continuous function Φh:

given the vector of nodal solutions Φj, and basis functions ψj(r) with limited support. Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

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TOAST element types (2-D)

linear

quadratic

cubic

isoparametric


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Finite element model

Substituting Φh into the DE leads to residual R:

Galerkin method: Weighted average of R vanishes over domain Ω when weighting function is chosen to be u:


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Integration by parts:


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In matrix notation:

with system matrices


TOAST matrix optimisers

no optimisation

minimum bandwidth

Tinney scheme 2

minimum degree


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Time-resolved data generation

Full temporal solution for Γ(t): Approximate ∂Φ/∂t with finite differences, e.g. Crank-Nicholson

Simulation of Γ(t) across a circle (r=25), nsteps = 400 homog. absorption 0.005, 0.01 and 0.02 Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

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Direct calculation of the moments of Γ(t)

Instead of calculating integral transforms from G(t) they can be calculated directly: Calculate 0th moment (m0) using stationary equation: Higher moments mn can be generated iteratively:


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Forward data generation in TOAST

femmesh Mesh generator

meshmod Mesh optimiser

makeqm Source/detector definition

Definition file femdata Forward solver

Data Data files files


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1. Generate a mesh from boundary information (optode positions, MRI image etc.) Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

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femmesh


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2. Edit and optimise the mesh with meshmod Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

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meshmod


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3. Define boundary source and detector positions and source-detector connectivity Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

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QM file format

s1 m1

makeqm

m0 s0

s2

m5 s5

m2

m4 s3

m3

s4

QM file Dimension 2

Header

SourceList 8 24.5 0 22.0 9.3 ...

Source coordinates

MeasurementList 8 24.5 4.9 20.8 13.9 ...

Detector coordinates

LinkList 6: 0 1 2 5 6 7 6: 0 1 2 3 6 7

Connectivity


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4. Specify options and parameters for the forward solver in a definition file. Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

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Definition file [FILES] ROOTNAME MESHFILE QMFILE LOGFILE

= = = =

gendata demo.msh demo.qm gendata.log

←Base file name ←Mesh file name ←Source-detector spec file name ←Log file name

[FORWARD_SOLVER] ALGORITHM = CHOLESKY BOUNDARY_CONDITION = ROBIN SOURCE_TYPE = NEUMANN SOURCE_PROFILE = GAUSSIAN SOURCE_WIDTH = 2.0

←Use direct system matrix solver ←Use Robin boundary condition ←Specify source as boundary flux ←Gaussian source profile ←1/e width of Gaussian

[INIT_PARAM] RESET_MUA = MESH RESET_P2 = HOMOG 1 MUS RESET_N = MESH

←Take absorption parameters stored with mesh ←Reset scatter parameters to homogeneous ←Refractive index

[DTYPE_0] TYPE = Moment NMOM = 1 NORMALISED = TRUE ENORM = NORMALISED FILE = demo_m1.fem

←First data set ←Type is Mellin transform (moment) ←Order is 1 ←Normalise with integral of TPSF ←Use data as standard deviation estimate ←Output file name

[END] . . Optical . Current Issues in Functional Imaging with Devices, Montreal, 11-12 May 2006

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5. Run the forward solver to produce boundary data files. Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

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femdata


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Reconstruction in TOAST

Data Data files files MONSTIR/ femdata

Definition file

QM file

toast Inverse solver

µa image

Outline generator

Mesh file

toastim viewer

µs image


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toastim NIM image viewer


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toastim NIM image viewer


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2-D reconstruction from simulated data 0.009

µa 0.06 0.8

µs 5

Target Unstructured mesh Bicubic pixel Simultaneous reconstruction from 16x16 phase+modulation data CG solver, 50 iterations, object radius 25 mm Forward basis: cubic (10-noded triangle) Inverse basis: unstructured and regular bicubic

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3-D reconstruction from simulated data

3D baby head model * • Dimensions: 100 x 90 x 90 mm • Discretisation: 35293 tetrahedra (10-noded, quadratic) * Courtesy B. Kaan Karamete, Rensselaer Polytechnic Institute

Finite Element Method

Matrix System :


Mesh Generation


Development of 3D FEM Head Models These need to incorporate as much a priori knowledge of tissue structure and optical properties as possible.


Generate individual surface meshes Original generic surface

Measured optode positions

New, individual surface



CT Scan Surface

Photogramme tric surface

Comparison between surfaces


Boundary Element Method •Second Greens theorem -> Integral Representation

•Boundary Integral Representation


Discretisation •Discretisation of the boundary into quadratic triangles: •Parametric Surfaces ->

-> Surface mesh

•Integration over elements:

•Gauss quadrature •Singularities Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Building linear system matrix


Forward problem BEM solution

3-layer model of head Source

Diffuse photon propagation in multilayered geometries Jan Sikora et al 2006 Phys. Med. Biol. 51 497-516


The inverse basis

We require a basis expansion to express the parameters µ and D in the context of the inverse problem:

An obvious choice is bi=ui, i.e. re-using the basis of the forward model, but it is generally advantageous to decouple the inverse and forward bases, and to use an independent basis expansion for the inverse problem. Desirable properties of the inverse basis are: • • • •

regular grid problem-dependent resolution (regularisation) local normalised:

(A) Piecewise polynomial basis (1)

Linear regular basis Given a regular grid defined by lattice points

with grid spacings δ1, δ2, δ3 we define a local piecewise tri-linear basis b by

(A) Piecewise polynomial basis (2)

Cubic regular basis Linear basis can be extended to higher order, e.g. tri-cubic grid defined by cubic spline interpolants

(B) Blob basis (1)

"Blob basis": radially symmetric volume elements arranged on a regular grid:

Various choices for the profile function B exist. B is chosen to have a limited support radius a. Blob bases are commonly used in image reconstruction due to inherent smoothness properties.

(B) Blob basis (2)

Linear ramp

with scaling parameter s. Gaussian with truncated support

with scaling parameter s and width parameter σ.

(B) Blob basis (3)

Hanning function

Kaiser-Bessel window function

where Im is the modified Bessel function of order m, a is the support radius and α is a shape parameter. Using m=2 ensures a continuous derivative at the blob boundary.

(B) Blob basis (4)

Cubic B-spline

where

(B) Blob basis (5)

Radial blob profiles

(B) Blob basis (6)

Mapping error: homogeneous image

(B) Blob basis (7)

Mapping error: inhomogeneous test image

2D test problem

µa

µs

Target

Object: 2D circle radius 25 mm background parameters: µa=0.025mm-1, µs=2mm-1 with embedded absorption and scattering features Forward model: unstructured mesh with 33000 nodes, 7300 elements (10-noded triangles using 3rd order polynomial basis functions) Data: 32 source positions, 32 detector positions, 30 detector positions used for each source (fanbeam geometry) Measurements: phase shift and modulation amplitude for 100MHz modulated input signal.

Reconstruction results (1)

Polynomial basis FEM model for inverse problem: 3529 nodes, 768 elements (10noded triangles) Solver: nonlinear CG

µa

µs

Unstructured piecewise linear (FEM)

Regular piecewise cubic basis (20x20)


Blob basis (1)

µa

µs

Ramp basis

Hanning

Spline 20x20

Bessel (10)


Blob basis (2)

µa

µs

Gauss (σ=0.7)

Gauss (σ=1.0)

20x20

Basis Selection : Conclusions

• In OT, the unstructured mesh of the FEM forward model is not an adequate basis for the inverse problem • An independent low-resolution and regular basis improves the results of the reconstruction. • Polynomial and blob basis functions show similar performance, with polynomial bases having the advantage of being normalised. • Blob basis types require tuning of support radius and shape parameters to reduce shape-dependent artefacts.

Newton-Krylov Reconstruction Scheme


Gauss-Newton framework Consider a model f(x) = y where nonlinear operator f: RN→RM maps a finite-dimensional (discretised) parameter distribution x into a finite-dimensional set of data y. Image x is sampled from a continuous distribution x(r) over domain Ω by means of some basis expansion. Image reconstruction - consider the regularised optimisation problem: Given a set of measurements y, find

where ||.|| is Euclidian norm, ψ: RN→R is regularisation operator acting on image x with hyperparameter τ.


Define objective function Ψ : RN→R:

Assume Ψ twice differentiable. Quadratic approximation to Ψ Ψ (xk+δ) for a step δ from current estimate xk is

If δ is a minimiser of Q, it satisfies

which leads to Newton step


By substituting Ψ′ and Ψ′′ and neglecting the second derivative in f (Gauss-Newton approximation) we get

where J = {∂fi/∂xj} ∈ RM×N is the Jacobian of the forward operator. GN avoids computation of Hessian f′′(x), at the cost of potential loss of local quadratic convergence of Newton method. Approaches to restore local convergence: (i) Levenberg-Marquardt trust region approach (LM): (ii) damped Gauss-Newton approach (DGN):


Notation: Hessian term: Gradient term: DGN update: LM update:

How do we store H (NxN dense!) to solve the linear step?


Krylov linear step Krylov methods are a class of iterative solvers for linear problems Ax = b where the iterates at step k lie in the Krylov space spanned by the orthogonal sequence

GMRES

with r0 = b - Ax0. Examples: conjugate gradients (CG), bi-conjugate gradients (BiCG), biconjugate gradients stabilised (BiCGSTAB), generalised minimal residuals (GMRES) Using a Krylov solver for the normal equation for LM or DGN avoids the explicit formation of Hk: Hk is accessed only in terms of matrix-vector multiplications

H can therefore be represented implicitly by its components


Normalisation

We want JTJ + τϑ′′ to have diagonal 1, by applying diagonal rescaling M:

where diagonal Mk ∈ RN×N is given by


Data space transformation

Transforming data to a dimensionless space is important where data are composed of different components, and where the components have different physical dimensions (and magnitudes) Apply transformation matrix T ∈ RM×M: Rescaled forward model: Rescaled Jacobian:


Parameter space transformation

By the same argument, apply transformation g to the parameter space: leading to transformed minimisation problem

with rescaled Jacobian:

represented by diagonal scaling matrix S: with


Regularisation Tikhonov prior of the form

Test case: 2-D circular object, 32x32 measurements with added Gaussian-distributed random noise. Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

2-D reconstruction: circular object Geometry: Circle radius 25 mm Background: µa = 0.025 mm-1, µs = 2 mm-1 + embedded objects

Forward data generation: FEM: 32971 nodes, 7261 10-noded triangles, 3rd order polynomial basis functions

Reconstruction: FEM: 3511 nodes, 6840 3-noded triangles, linear basis functions Solution basis expansion: 20x20 bicubic pixel basis

Measurements: 32 sources, 32 detectors, 1024 measurements Source modulation: 50 MHz Each measurement consists of phase and modulation amplitude data. Noise: 0-2% Gaussian random noise


Initial homogeneous estimate Global ("1-pixel") optimisation of homogeneous initial estimate

Robustness of reconstruction vs. initial estimate

objective function µa

global fit

0.0307

background

2

L1 image residuals

0.025 2.16

initial estimate µs Parameters: Data noise: 1% Gaussian noise Solver: DGN Regularisation: Tikhonov, τ=7e-5

final iterate


Stability of convergence in the presence of noise L1 image residuals of final iteration as a function of homogeneous initial parameter estimate Regularisation parameter: Noise level 0% 1% 2%

t 2⋅10-5 7⋅10-5 3⋅10-4

absorption image

scatter image


2-D reconstruction results Target

DGN

LM 0.05

µa

0.0125 4

µs

1

Convergence rate DGN and LM: 2-D test case

objective function

Objective function vs. iteration count

LM

DGN

DGN with inexact line search converges significantly faster than LM Despite higher cost of DGN iteration (line search) it provides higher performance Similar results for wide range of initial homogeneous estimates


Convergence rate vs. GMRES stopping criterion convergence vs runtime

objective function

objective function

convergence vs iteration

GMRES stopping criterion is not critical for GN convergence Moderate stopping criterion provides convergence rate equivalent to direct solution of explicit Hessian Newton-GMRES is converges significantly faster in terms of runtime (This is 2D! Much more severe in 3D)


2-D test case

absorption

Radius 25 0.0125 Background: µa=0.025, µs=2

scatter

0.05 1

Target

Reconstruction

4

Forward mesh: 32971 nodes 7261 triangles 3rd order Lagrangian shape functions Reconstruction: 20x20 bicubic pixel basis Data: 32Qx32M phase shift, modulation amplitude Solver: nonlinear CG (50 iterations)


3-D reconstruction: cylinder object Geometry: Cylinder radius 25 mm, height 50 mm Background: µa = 0.01 mm-1, µs = 1 mm-1 + embedded objects Measurements: Source and detector sites arranged in 5 rings on the cylinder mantle, 8 sources and 8 detectors per ring → 1600 measurements Source modulation: 50 MHz Each measurement consists of phase and modulation amplitude data. FEM models: Data generation: 83142 nodes, 444278 tetrahedra Reconstruction: 9043 nodes, 45438 tetrahedra

Reconstruction: DGN data scaling: average per data types parameter transformation: log


3-D reconstruction: cylinder object Absorption

Target

DGN

0.0068

Scatter

Target

DGN

0.02

0.6

4


3D Reconstructions 0.0125

Absorption

0.05 1

Scatter Target

Reconstruction


4

Newton-Krylov : Conclusions • Krylov solution of Gauss-Newton linear step avoids explicit formation of Hessian → applicable to large-scale 3-D problems • Implicit Hessian can incorporate both data and parameter space transformations • For the test cases considered here, variable step length (line search, DGN) converges faster than adjustment of LM parameter with fixed step length 1. • Next step: We have assumed that the formation of the Jacobian is feasible, while the formation of the Hessian is not (M < N). If this doesn't hold, implicit form of the Jacobian can be considered.


Regularisation Methods


Diffusion regularisation methods In the regularity term, we consider The energy becomes:

The gradient is:

The behaviour of the function influences the local direction of the regularization. A specific estimation of this function in the minimization problem is crucial. Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Decomposition of the gradient Decompose the gradient of the regularity term by using the local object structures:

The diffusion process depends on the choice of this function. Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Choice of the function

Existence, unicity and stability of the solution: •

twice continuously differentiable.

• •

is increasing function for all x in R+.

•

is convex for all x in R+.

•

is concave for all x in R+.

1


Choice of the function Restoration by controlling the diffusion in the regularity term : •In homogeneous regions: the diffusion must be isotropic. •In a neighbourhood of an edge: the diffusion must be anisotropic. it must act in the direction of the orthogonal of the normal N and not across it.

1


Choice of the function


Choice of the function The gradient of the regularity term becomes:

is chosen by the Perona and Malik’s diffusion function:

The choice estimator.

will be estimated by using the robust statistic


Choice of the function ! 0.15

"! ( #x ) !x="

0.1

"! ' ( #x )

1

0.8

0.6

0.05

0.4

!x="

0.2 0 0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

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0.8

1

"! ' ( #x )

1

0.8

0

0.2

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0.7

0.8

0.9

1

"! " ( #x )

1

0.8

#x

0.6

0.6

0.4

0.4

!x="

0.2

0.1

!x="

0.2

0

0 0

0.1

0.2

0.3

0.4

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0.7

0.8

0.9

1

0

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0.7

0.8


0.9

1

2D Comparison study: Modified Perona Malik vs Tikhonov Scattering coefficients

Absorption coefficients

Original 2D-object Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006


Modified PM

Tikhonov

Scattering coefficients



Diffusion Modified PM

Diffusion Tikhonov

Scattering coefficients


Shape-Based Methods


Shape Based Methods


Explicit Shape Reconstruction • Suppose domain Ω consists of set of piecewise continuous regions with boundaries Ci and parameters {µj, κj}

• Define basis set for Ci • Define forward mapping F:{γ,µ,κ}−>y • Inverse problem : find {γ,µ,κ} from y V.Kolehmainen et.al., Inverse Problems, 1999; PMB, 2000


Shape Inverse Problem • Start from guess for the boundary coefficients • Create measurements • Jacobian (linearisation) • From • Update • Iterate until min


Boundary Recovery


Boundary Recovery


Parametric Surface • A closed surface extracted from a voluminous object without holes is topologically equivalent to a sphere’s surface. • A homeomorphism exists mapping the closed surface to the unit sphere. • Two parameters θ and φ can be assigned for each point on the surface. • a nice smooth parametric description of the surface is then possible using weighted sum of basis functions expressed on the sphere such as spherical harmonics.


Parametric Description

φ

z y x

θ Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Steps for mapping on the sphere

Extraction of the surface’s net of nodes.

Initial mapping

=>Not limited to star shaped objects

Optimisation for the mapping

Voxel faces -> Spherical squares


Spherical Harmonic Representation

3rd degree

7th degree

11th degree


Boundary Element Method •Second Greens theorem -> Integral Representation

•Boundary Integral Representation


Discrete linear system


Results : movie

blue: exact inhomogeneous region for simulated data red: initial guess for inhomogenity Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

3D BEM Explicit Shape reconstruction


Level set formulation In a two-level shape inverse problem we assume that parameters ml, l∈{1,2} are of the form

where for each parameter domain Ω is subdivided into disjoint zones, in each of which parameter ml can only assume either an interior or exterior constant value.

S1 Ω\S1 m1(x)

Ω\S2 S2 m2(x)


Level set function For each parameter l∈{1,2}, define a smooth level-set function ψl so that

such that the boundary ∂Sl of Sl is defined by the zero contour of ψl The iterative reconstruction of ∂Sl is then realised as a time evolution approach, where ∂Sl and ψl are represented as functions of an artificial evolution time t:


Inverse problem We want to solve the problem Given the outer shape ∂Ω and external parameters ml,e: Find the level set functions ψl and interior parameter values ml,i that minimises the least-squares cost functional

where Formulate force terms f1(x,t), f2(x,t), g1(t), g2(t) that define a descent flow for J:

shape evolution

contrast evolution


Test problem: inclusions in noisy background Reconstructions of absorption and diffusion inclusions in circular domain. Radius: 25mm Mesh: 6840 elements 3511 nodes

Parameters: external: absorption: m1,e = 0.01mm-1 diffusion: m2,e = 0.33mm internal:

absorption: m1,i = 0.02mm-1 diffusion: m2,i = 0.165mm

initial:

absorption: m1,i = 0.015mm-1 diffusion: m2,i = 0.22mm

Measurements: 32 sources x 32 detectors, log amplitude + phase, sources modulated at 100MHz

target absorption (m1)

target diffusion (m2)

Noise: 5% parameter noise in background 1% data noise


Reconstruction: time evolution


Reconstruction results Comparison of level set results with Gauss-Newton pixel-based reconstruction level set reconstruction

pixel-based reconstruction

diffusion

absorption

target


Modelling Errors


Approximation Errors Consider the inverse problem x is a distributed parameter (i.e. function). y is sampled finite dimensional (i.e. a vector). A is linear or non linear mapping For computational implementation represent x in a finite dimensional space

Forward problem converges to accurate model as Maybe computationally infeasible to use accurate model. Instead consider accurate model

ε(x) is modelling error. Use Bayesian techniques


Approximation Errors Assume that the continuous model

Can be approximated by

Then the disretized “exact” model within measurement accuracy is

Model Reduction. Choose

Model reduction map “Exact” reduced model

Given the probability density of (xδ,e), the model reduction operator P and the forward models Aδ, and Ah, derive a computational model for the posterior density π(xh|y) Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Implementation Let π(xδ) be the prior probability density in RN for accurate model. Samples :

Samples of additive noise model approximate π(n)

Mean Covariance

In Gaussian case, assuming xh and e are mutually independent, posterior becomes


Mesh Set up


Prior Model •

Samples


Estimation results •

Approximation Errors


Estimation results Relative approximation errors vs mesh density

Forward Model

Inverse Model


Estimation results •

Expected Errors vs Measurement Noise


Reconstructions


Reconstruction Error Estimates Marginal Densities


Transport Equation Approach


RTE vs Diffusion

(a)

(b)

The cross-sectional p hoton d ensity calculated using the P1 to P1 5 eq uations at ( a) 0 , 2 , 5 and 2 5 m m from the source and ( b ) as a function of the ang ular variab le ( θ) on the centre line at 2 5 m m from the source ( µS 1 m m -1 , µa = 0 . 0 1 m m -1 and g = 0 ) Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Radiative Transport Equation (RTE) RTE in frequency domain (modulation amplitude ω):

attenuation coefficient Boundary Condition :

radiance scattering phase coefficient function

source term

Phase Function :


Finite Element Method for RTE : Galerkin Test Space and model Space are the same, e.g piecewise linear

Matrix System :


Stream Line Diffusion Modification •Classical FEM method leads to oscillatory solutions in some cases, especially in voids (Ray Effect). •Ray effect is mitigated by using large number of directions, or by using smoother basis functions in angular variable •Stream line diffusion method adds a smoothing term in the directional derivative term by making test space basis functions of the form

•Model space and test space are different (Petrov-Galerkin method) •Parameter δ is dependent on local scattering and absorption •See Richling et.al. 2001


Finite Element Method for RTE with SDM


Diffusion Approximation DA represents radiance as linear variation over Sn-1

Diffusion Equation :

Boundary condition :


Finite Element Method for DA : Galerkin

Matrix System :


Hybrid RTE-DA ΩRTE Domains ΩRTE and ΩDA separated by interface Γ Γ

ΩDA

Matrix System :

Related work by Bal and Maday 2002 Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Example

Optical Properties


Results : gap


Results : gap


Space and time complexity


Results : hole


Results : hole


Head mesh example

RTE-DA

DA only


PN method •Change angular basis to spherical harmonics •Integral operator is diagonal (isotropic scattering) •Outscatter and loss terms are diagonal •Transport term has off diagonal terms (tridiagonal in 1D – cf rotated frame method) Even-parity representation of radiance, source and phase function:

leads to second-order formulation of RTE of the form:

where C+ and D are operators constructed from the even-parity quantities. Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

System matrix assembly Given a L-dimensional spatial basis, and a M-dimensional angular basis, solution vector φ is organised as where each M-dimensional block φi represents the angular expansion for a single spatial basis term. Likewise, system matrix K consists of L x L blocks, each of dimension M x M. For PN model of order N, the dimension of the angular basis in this formulation is given by Kij blocksize: PN order

1

blocksize 1

1x1

3

6x6

5

15 x 15

7

28 x 28

9

45 x 45

11

66 x 66

P1

P3

1

P5 6

6 15

15


Variable order PN implementation

Local adaptation of PN order by elimination of rows and columns of K on a per-node basis. Efficient implementation by sparse matrix structure. Diagonals of eliminated blocks substituted with identity.

M(P7)

M(P1)

M(P1)

eliminate

M(P7)

M(P7)

eliminate

M(P1) eliminate

1

M(P1) eliminate

1

M(P7)

i-1

node i

i+1

i+2


2D test case with low-scattering ring

4 mm

µs=2 mm-1

µs=0.01 mm-1

Scatter distribution

P1

6 mm

P7

variable order case 1 (VO1)

source

P1

P7

variable order case 2 (VO2)

Radius: 25 mm. Ring: width 3 mm (20 ≤ r ≤ 23) Parameters: µa = 0.025mm-1, µs = 2mm-1 (background), µs = 0.01mm-1 (ring) Refractive index: n = 1 Anisotropy factor: g = 0.5 Mesh: 4278 3-noded triangles, 2209 nodes Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

source

phase

log amplitude

Complex radiance field distributions

P1

P3

P7

P1+7 (VO2)

Marked difference in field ln φ(r) = ln A(r) + iϕ(r) between P1 and P7 solutions → diffusion approximation fails. Mixed order solution restores P7 results throughout domain Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Fields for uniform-order models

ring

ring ring source

log amplitude

source

phase

Cross sections of complex photon density fields (log amplitude and phase) for uniform order PN models P1, P3, P5, P7 along central axis y = 0. In this case, P1 and P3 break down in the low-scattering layer and in proximity to the source. From P5, the solutions have essentially converged. Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Fields for mixed-order models

ring

ring ring source

log amplitude

source

phase

Cross sections of complex photon density fields (log amplitude and phase) for mixed order PN models VO1 and VO2 (P1 + P7) along central axis y = 0. Both mixed-order models show very good agreement with the uniform P7 solution throughout the medium. Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Radiance field differences

log amplitude

phase

P 7 - P1

P7 - P1+7


Boundary data for uniform-order models

Measurement of complex boundary exitance data (log amplitude and phase projected in normal direction) at 32 locations with equal angular spacing along the circumference. Low PN order solutions underestimate the magnitude of amplitude and phase in the presence of the clear layer. Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Boundary data for mixed-order models

Mixed-order PN solutions agree well with uniform P7 solution. For phase, there is a significant improvement of extending the area high-order solution to the boundary (VO2), instead of restricting to the clear layer (VO1). Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

PN solver performance

mixed

uniform

PN order

Memory [MByte] MByte]

runtime [sec] total lin. lin. solver

GMRES iterations

1

7

1.2

0.7

18

3

20

14

12

25

5

59

65

58

27

7

129

223

196

38

9

235

491

408

42

1+7 (VO1)

66

106

79

36

1+7 (VO2)

87

124

98

32

Performance comparison between uniform and mixed PN order solutions. VO1: variable order case 1 (narrow P7 ring) VO2: variable order case 2 (wide P7 ring) Current Issues in Functional Imaging with Optical Devices, Montreal, 11-12 May 2006

Transport Optical Tomography : results 1 Oliver Dorn, Inverse Problems 8 cm 1998

µs = 1 1/cm µa = 0.1 1/cm g = 0.5


Transport Optical Tomography : results 2

8 cm

µs = 5 1/cm µa = 0.1 1/cm g = 0.5



8 cm

µs = 10 1/cm µa = 0.1 1/cm g = 0.9



8 cm

µs = 10 1/cm µa = 0.1 1/cm g = 0.5



8 cm

µs = 10 1/cm µa = 0.1 1/cm g = 0.0



8 cm

µs = 100 1/cm µa = 0.1 1/cm g = 0.9


Transport Approach : Conclusions •Hybrid RTE-diffusion model •Applied to low scattering and void problems •1st order method with explicit inteface conditions •Stream line diffusion model mitigates ray effects •PN method with Dirichlet interface condition •Memory and computational saving


Dynamic Imaging


Softfield Tomography – DOT

Figure 1. A fibre holder helmet on the head of an infant during an imaging scan

Figure 2. Ultrasound image of infant with haemorrhage in left ventricle.


Neonatal Brain Imaging – functional imaging of the motor cortex


Tomographic mapping of functional activation of the motor cortex in the neonate

Passive movement of Left Arm

Passive movement of Right Arm


Conclusions

•

Optical Imaging is usually based on the Diffusion Equation

•

Approaches make use of inverse scattering theory (linear, small perturbations) or optimisation theory (non-linear, “absolute” imaging)

•

Efficient methods using adjoint fields which are analogous to “back-projection”

•

Model based approach using finite elements


Arrige.ppt [Lecture seule]

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