e i d o r s - ejw - about i d o r s electrical impedance ... – damped gauss-newton ... part i. e i...

E I D O R S Electrical Impedance Tomography and Diffuse Optical Tomography Reconstruction Software

Sujin Ahn

2011. 11. 24

GIST

Kyung Hee University, 2011 1

Contents

• Introduction

• Part I. EIDORS

– 2D model

– 3D model

• Part II. NETGEN

• Exercise


• Let Ω be an imaging object with boundary 𝜕Ω.

• Forward problem

– Given current injection sources and sinks on 𝜕Ω, and admittivity 𝛾 over Ω, find the boundary electric potential 𝑉 on 𝜕Ω.

• Inverse problem

– For a given set of boundary voltage 𝑉 and injected currents 𝐼 at the boundary 𝜕Ω, recover the admittivity 𝛾 inside Ω.


EIT problem

Admittivity Boundary Voltage

Forward

Inverse

Given model, injected current

• Assumptions for imaging object Ω – No interior sources of charges – Quasi-static fields

𝛻 ⋅ 𝛾 𝐫 𝛻𝑢𝑗 𝐫 = 0 in Ω

𝑢𝑗 + 𝑧𝑘𝛾𝜕𝑢𝑗

𝜕𝐧 𝜀𝑘

= 𝑉𝑗,𝑘 for 𝑘 = 1, … , 𝐿

𝛾𝜕𝑢𝑗

𝜕𝐧= 0 on 𝜕Ω\ 휀𝑘

𝐿

𝑘=1

𝛾𝜕𝑢𝑗

𝜕𝐧𝜀𝑘

𝑑𝑠 = 0 if 𝑘 ∈ 1, … , 𝐿 \*𝑗 − 1, 𝑗+

𝛾𝜕𝑢𝑗

𝜕𝐧𝜀𝑗

𝑑𝑠 = 𝐼 = − 𝛾𝜕𝑢𝑗

𝜕𝐧𝜀𝑗−1

𝑑𝑠


FEM formulation FEM formulation

Complete Electrode Model

• Assembled Complete System

• 𝑨𝑴 𝑖, 𝑗 = 𝛾𝛻𝑢𝑖 ⋅ 𝛻𝑢𝑗𝑑𝐫 , Ω

𝑨𝒛 𝑖, 𝑗 = 1

𝑧𝑘𝑢𝑖𝑢𝑗𝜀𝑘

𝑑𝑠𝐿𝑘=1

• 𝑨𝑽 𝑖, 𝑘 = − 1

𝑧𝑘𝑢𝑖𝜀𝑘

𝑑𝑠, 𝑨𝑫 𝑠, 𝑘 =

1

𝑧𝑘𝑒𝑘 𝑖𝑓 𝑠 = 𝑘

0 𝑖𝑓 𝑠 ≠ 𝑘

1 ≤ 𝑖, 𝑗 ≤ 𝑛𝑣 # of vertices , 1 ≤ 𝑠, 𝑘 ≤ 𝐿 # of electrodes

↔ Ax=b


FEM formulation FEM formulation (forward)

𝑢

v

𝐴𝑀 + 𝐴𝑧 𝐴𝑉

𝐴𝑉𝑇 𝐴𝐷

L

L

𝒏𝑽

𝒏𝑽

× =

0

I

u : nodal potential v : boundary voltage

• How to solve Ax=b

– Conjugate Gradient (CG)

– General Minimal Residual (GMRes)

– Bi-Conjugated Gradients

(BiCG, CGS, BiCGStab)

– Quasi Minimal Residual

(QMR, Symmetric QMR)

,etc.


FEM formulation FEM formulation (forward)

• The admittivity 𝛾 is formulated by a least-square minimum of the residual error 𝐹 𝛾 − 𝑉𝑚 which is

𝛾∗ = argmin𝛾 1

2𝐹 𝛾 − 𝑉𝑚

2

where 𝐹: 𝐶𝑛𝐸 → 𝐶𝐿 (forward operator)


FEM formulation FEM formulation (inverse)

• Let D 𝛾 = 𝐹 𝛾 − 𝑉𝑚.

• D 𝛾 + 𝛿𝛾 ≈ 𝐷 𝛾 + 𝐷′ 𝛾 𝛿𝛾 +1

2𝐷′′ 𝛾 𝛿𝛾2 + 𝑂 𝛿𝛾3

(Taylor expansion)

• 𝐷 𝛾 + 𝛿𝛾 ≈ 0 for appropriate 𝛿𝛾

Neglecting 2nd order terms and above

𝐷 𝛾 = −𝐷′ 𝛾 𝛿𝛾

𝐷′ 𝛾 =𝜕𝐹 𝛾 𝑑𝑚

𝜕𝜎𝑘= − 𝛻𝑢 𝐼𝑑 𝛻u I𝑚 𝑑𝑉

𝑣𝑜𝑥𝑒𝑙 𝑘

where 𝑢(𝐼𝑑) is the potential for the d-th drive pattern, 𝑢 𝐼𝑚 is the hypothetical measurement potential.

Our goal is to find a proper 𝛅𝛄 to get 𝜸∗ by iterative or single step!


FEM formulation FEM formulation (linearization)

• For difference imaging, let 𝐽 = 𝐷′ 𝛾 , 𝐹 𝛾 ≈ 𝑉0 where 𝑉0 is the voltage when Ω is homogeneous.

𝐽 𝛿𝛾 = 𝑉𝑚 − 𝑉0

𝛿𝛾 = 𝐽†(𝑉𝑚 − 𝑉0) ,

where 𝐽† is a pseudo-inverse matrix of 𝐽.

1. When 𝐽 (with size nr x nc) is full rank, 𝐽† can be

directly calculated as follows;

– nr < nc 𝐽† = 𝐽𝑇 𝐽 𝐽𝑇 −1

– nr > nc 𝐽† = 𝐽𝑇𝐽 −1𝐽𝑇

2. When 𝐽 is not full rank, 𝐽† is generally computed

by Singular Value Decomposition (SVD)


FEM formulation FEM formulation (linearization)

• Let 𝐽 ∈ 𝑅𝑟 × 𝑐.

∃ 𝑈 ∈ 𝑅𝑟×𝑟 , 𝑉 ∈ 𝑅𝑐×𝑐 s. t. J = U Σ 𝑉𝑇

where U, V are orthogonal matrices and Σ ∈ 𝑅𝑟×𝑐 is a diagonal matrix whose diagonal consists of the singular values *𝜎𝑖+ of 𝐽.

𝐽† = V Σ† 𝑈𝑇

where Σ† is Σ𝑇 with nonzero 𝜎𝑖 replaced by 1/𝜎𝑖.


FEM formulation FEM formulation (SVD)

• In our 2D model, singular values after 50th value are almost zero.

Replace 𝜎51, … 𝜎208 by zero before taking inversion.

It is important to find the singular value at which the normalized value suddenly drops!


FEM formulation FEM formulation (Truncated SVD)

𝐽 = 𝑈Σ𝑉𝑇

• How to solve 𝛾∗ = argmin𝛾 1

2𝐹 𝛾 − 𝑉𝑚

2

– Truncated SVD

– Steepest Descent

– Newton-Raphson

– Damped Gauss-Newton

– Levenberg-Marquardt

– Quasi-Newton

, etc.


FEM formulation FEM formulation (inversion methods)

Part I. E I D O R S


• Free software algorithms for forward and inverse modeling for EIT and Diffusion based Optical tomography

• Requirement

– Matlab (≥ 7.0) or Octave (≥ 3.4)

– Netgen Mesher

• http://eidors3d.sourceforge.net


EIDORS

http://eidors3d.sourceforge.net/

To try the EIDORS software, follow these steps: 1. Download the software

− Release Version: EIDORS 3.5 (14 Jul 2011)

2. Unzip the software in a directory /path/to/eidors or C:/path/to/eidors

3. Start Matlab

4. type: >>run /path/to/eidors/startup.m (windows >>run C:/path/to/eidors/startup.m)

5. Execute one of the sample programs in the /path/to/eidors/examples directory (such as demo_real.m)


Getting Started

http://prdownloads.sf.net/eidors3d/eidors-v3.5.zip


Flow chart for numerical simulation

Inverse computation

Visualization

Model generation

Forward computation

Dense mesh

• mk_common_model • ng_mk_cyl_models

• fwd_solve

• Recover admittivity

• inv_solve • calc_jacobian

Model generation

Forward computation

Coarse mesh

• fwd_solve

Synthetic data

𝐽 𝛿𝛾 = 𝑉𝑚 − 𝑉0

• show_fem • show_slices

Startup;

% Generate 2D mesh

imb= mk_common_model('c2c',16);

e= size(imb.fwd_model.elems,1); bkgnd= 1;

% Solve Homogeneous model

img= eidors_obj('image','');

img.elem_data= bkgnd * ones(e,1);

img.fwd_model= imb.fwd_model;

vh= fwd_solve( img );

% Add Two triangular elements

img.elem_data([25, … ,121:124])=bkgnd * 2;

img.elem_data([95, … , 22])=bkgnd * 2;

vi= fwd_solve( img );

% Add -12dB SNR

vi_n= vi;

nampl= std(vi.meas - vh.meas)*10^(-18/20);

vi_n.meas = vi.meas + nampl *randn(size(vi.meas));

show_fem(img); axis square; axis off


EIDORS (2D model, forward)

mk_common_model (mdlstr, n_elec) • Utility function to create common

EIT 2D/3D FEM models • Several templates for models e.g. mk_common_model('c2C',16) - 2D circ model with 576 elems)

% Generate 2D mesh












% Add -12dB SNR

vi_n= vi;






% Generate 2D mesh












% Add -12dB SNR

vi_n= vi;






eidors_obj (type, name, varargin) • Constructor to create a eidors

structure • USAGE: as a constructor

- type: obj type: fwd_model, inv_model, data, image

- name: text string identifier for model (may be '')

- varargin: properites and values for object

% Generate 2D mesh












% Add -12dB SNR

vi_n= vi;






fwd_solve(img) • Calculate data from a fwd_model

object and an image

% Generate 2D mesh












% Add -12dB SNR

vi_n= vi;






Show_fem (img, options) • Show the EIDORS finite element

model • options specifies a set of options - options(1): show colorbar - options(2): show numbering on electrodes - options(3): number elements (=1) or nodes (=2);

% Create Inverse Model

inv2d= eidors_obj('inv_model', 'EIT inverse');

inv2d.reconst_type= 'difference';

inv2d.jacobian_bkgnd.value= 1;

% This is not an inverse crime;

(inv_mdl ≠ fwd_mdl)

imb= mk_common_model('b2c',16);

inv2d.fwd_model= imb.fwd_model;

inv2d.fwd_model.np_fwd_solve.perm_sym= '{y}';

% Guass-Newton solvers

inv2d.solve= @np_inv_solve;

% Tikhonov prior

inv2d.hyperparameter.value = 1e-3;

inv2d.RtR_prior= @tikhonov_image_prior;

inv2d.saveon=0;

imgr= inv_solve( inv2d, vh, vi);


EIDORS (2D model, inverse)

Several options for computing prior matrix @tikhonov_image_prior; @noser_image_prior, @laplace_image_prior @gaussian_HPF_prior @ab_calc_tv_prior (total variation)



inv2d.reconst_type= 'difference';

inv2d.jacobian_bkgnd.value= 1;

% This is not an inverse crime;

(inv_mdl ≠ fwd_mdl)

imb= mk_common_model('b2c',16);

inv2d.fwd_model= imb.fwd_model;

inv2d.fwd_model.np_fwd_solve.perm_sym= '{y}';

% Guass-Newton solvers


% Tikhonov prior


inv2d.RtR_prior= @tikhonov_image_prior;

inv2d.saveon=0;



EIDORS (2D model, inverse)

inv_solve (inv_model, data1, data2) • Calculate ‘difference’ image from

an inv_model and data • data1 : reference data

(homogeneous or @low freq) • data2 : perturbed data (inhomogeneous or @high freq) inv_solve (inv_model, data) • Calculate ‘static’ image from an

inv_model and data


3D model I

% Generate 3d model imb= mk_common_model('n3r2',16); e= size(imb.fwd_model.elems,1); bkgnd= 1; % Homogenous Data img= eidors_obj('image','Demo Image'); img.elem_data= bkgnd*ones(e,1); img.fwd_model= imb.fwd_model; vh= fwd_solve( img ); % Inhomogenous Data load datacom A B; img.elem_data(A)= bkgnd*1.2; img.elem_data(B)= bkgnd*0.8; clear A B; vi= fwd_solve( img ); % Add 12dB noise vi_n= vi; vi_n.meas = vi.meas + std(vi.meas - vh.meas)/10^(12/20) ... *randn(size(vi.meas)); sig= sqrt(norm(vi.meas - vh.meas));



inv3d.reconst_type= 'difference'; inv3d.jacobian_bkgnd.value = 1;

inv3d.fwd_model= imb.fwd_model; inv3d.fwd_model.np_fwd_solve.perm_sym= '{y}';

inv3d.saveon=0;

% Gauss-Newton Solver



% Tikhonov prior

inv3d.R_prior= @tikhonov_image_prior;


Forward Inverse


3D model II

% Generate cylinder model

fmdl =ng_mk_cyl_models([2,1,0.1],[16,1],[0.05]); fmdl.stimulation = mk_stim_patterns(16,1,[0,1],[0,1],{},1);

% Homogeneous object

imgh= mk_image( fmdl, 1 );

% Fwd Simulation

vh = fwd_solve( imgh);

% Target Object

select_fcn = inline('(x-0.2).^2 + (y-0.5).^2 + (z-1).^2<0.1^2','x','y','z');

memb_frac = elem_select( imgh.fwd_model, select_fcn);

imgt= mk_image( fmdl, 1 + memb_frac );

% Fwd Simulation

vt = fwd_solve( imgt);

% Add SNR 2.0 noise

vn = add_noise(4, vt, vh);

% Create Inverse Model fmdlt=ng_mk_cyl_models([2,1,0.2],[16,1], 0.05); fmdlt.stimulation = mk_stim_patterns(16,1,[0,1],[0,1],{},1); img= mk_image( fmdlt, 1 ); inv3d= eidors_obj('inv_model', 'EIT inverse'); inv3d.reconst_type= 'difference'; inv3d.jacobian_bkgnd.value = 1; inv3d.fwd_model= img.fwd_model; inv3d.fwd_model.np_fwd_solve.perm_sym= '{y}'; inv3d.saveon=0; %% Gauss-Newton Solver inv3d.hyperparameter.value = 1e-3; inv3d.solve= @np_inv_solve; %% Tikhonov prior inv3d.R_prior= @tikhonov_image_prior; imgr= inv_solve( inv3d, vh, vn);

Forward Inverse

Part II. Mesh generation


• NETGEN is an automatic 3d tetrahedral mesh generator, developed by Joachim Schöberl

• http://sourceforge.net/projects/netgen-mesher/

• Getting started 1. Go to the NETGEN path

2. Run netgen.exe


FEM formulation NETGEN

http://sourceforge.net/projects/netgen-mesher/




FEM formulation NETGEN (boxcyl.geo) ## Two cylinders on a box algebraic3d #define box: solid box = plane (0, 0, 0.5; -1, 0, 0) and plane (0, 0, 0.5; 0, -1, 0) and plane (0, 0, 0.5; 0, 0, -1) and plane (2, 1.5, 1; 1, 0, 0) and plane (2, 1.5, 1; 0, 1, 0) and plane (2, 1.5, 1; 0, 0, 1); #define cylinders: solid cyls = (cylinder (0.5, 0.75, 0; 0.5, 0.75, 2; 0.3) or cylinder (1.5, 0.75, 0; 1.5, 0.75, 2; 0.3) ) and plane (0, 0, 0.7; 0, 0, -1) and plane (0, 0, 1.5; 0, 0, 1); #combine both: solid main = box or cyls; #define sub-domains: tlo main; singular edge box cyls;

Every .geo file must contain the keyword algebraic3d before any non-comment line.



The keyword solid defines a named solid, here the solid box is defined.



The definition tlo main declares the solid main as a top-level-object, what is necessary for meshing.


FEM formulation NETGEN (Available primitives)

1. A half space, i.e., a plane and everything on one side of it.

where p=(px, py, pz) on the plane with the outer normal n=(nx, ny, nz).

2. A cylinder of infinite length, given by a=(ax, ay, az) and b=(bx, by, bz)

on the central axis and the radius r

3. A sphere with the center c=(cx, cy, cz) and the radius r

4. A orthobrick is a brick parallel to the coordinate axis. It is specified by

two opposite corner points a=(ax, ay, az) and b=(bx, by, bz)

plane(px, py, pz; nx, ny, nz)

cylinder(ax, ay, az; bx, by, bz; r)

sphere(cx, cy, cz; r)

orthobrick(ax, ay, az; bx, by, bz)


FEM formulation Exercise

1. Generate the following model

(1) cylinder with radius 2, height 3, and maximum size 0.7

(2) 16 electrodes with radius 0.5 at the half of height.

(Use ng_mk_cyl_models)

2. Make a spherical object centered at (1,0,0) with radius 1.

3. Generate homogeneous data (vh.meas) and inhomogeneous

data (vi.meas) under the following conditions.

(1) Adjacent injection and adjacent measurement

(2) The conductivity is 1 in the background and 2 in the

anomaly.

4. Find the difference image using vh.meas and vi.meas using

the inverse operator in the slide 23.


FEM formulation Exercise

NETGEN

Generate a sphere with raidus 5 and put one circular area with radius 0.5 on the spherical surface .


FEM formulation Homework

1. Generate the following model by using NETGEN.

(1) sphere with radius 5.

(2) equi-distanced16 electrodes on the equator with radius 0.5.

(3) a cubic object centered at (2.5,0,0) with the edge length 2.

2. Generate homogeneous data (vh.meas) and inhomogeneous

data (vi.meas) on the model from problem 1 under the following

conditions.

(1) Adjacent injection and adjacent measurement

(2) The conductivity is 1 in the background and 2 in the anomaly.

3. Find the difference image using the pseudo inverse of the

sensitivity matrix by truncated SVD.

e i d o r s - ejw - about i d o r s electrical impedance ... – damped gauss-newton ... part i. e i...

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