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EIDORS: Electrical Impedance Tomography and Diffuse Optical Tomography Reconstruction Software

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Dual Models − Coarse model partially covers fine model

In many cases, the inverse model only covers part of the space. It is necessary to cover the entire space with the fine model to provide an accurate forward solution, but it is not possible to accurately calculate parameters for the low sensitivity parts of the space.

Coarse and Fine models

Create a fine model of a tank with 7 electrodes just near one side. The coarse model will only represent the conductivity near this region.
% Dual Partial $Id: dual_partial2d01.m 3504 2012-07-05 11:53:21Z aadler $

imdl= mk_common_model('d2c2',64);
fmdl= imdl.fwd_model;
% Only keep 11 electrodes on top
fmdl.electrode = fmdl.electrode([60:64,1:6]);
% New stimulation pattern with 11 electrodes
fmdl.stimulation = mk_stim_patterns(11,1,'{ad}','{ad}',{},1);
% Remove meas_select - it was created for 64 electrodes
fmdl = rmfield(fmdl,'meas_select'); 
subplot(121)
show_fem(fmdl); axis square

% Crop model
[cmdl,c2f_idx]= crop_model(fmdl, inline('(y-0.25)-abs(x)<0','x','y','z'));
subplot(122)
show_fem(cmdl);
axis(1.05*[-1 1 -1 1]); axis square;

print_convert dual_partial2d01a.png;


Figure: Fine (left) and dual (right) models

Simulated data

Data are simulated of a point rotating on a fine mesh (inverse crime in this case)
% Simulate object moving $Id: dual_partial2d02.m 2782 2011-07-14 21:14:12Z aadler $

% simulation model
imdl= mk_common_model('d2c2',64);
smdl= imdl.fwd_model;
% Only keep 11 electrodes on top 
smdl.electrode = smdl.electrode([60:64,1:6]);
smdl.stimulation = mk_stim_patterns(11,1,'{ad}','{ad}',{},1);

[vh,vi,xyr]= simulate_2d_movement(32, smdl, [0.75,0.05] );
% Only 12 to 2 O'clock
idx= 5:10;
vi= vi(:,idx); xyr= xyr(:,idx);

clf;subplot(121)
show_fem(smdl); axis square

% Show target positions
   theta= linspace(0,2*pi,50); xr= cos(theta); yr= sin(theta);
   hold on;
   for i=1:size(xyr,2)
       hh= plot(xyr(3,i)*xr+ xyr(1,i),xyr(3,i)*yr+ xyr(2,i));
       set(hh,'LineWidth',3,'Color',[0,0,0]);
   end
   hold off;

print_convert dual_partial2d02a.png;


Figure: Simulation data

Reconstruction models

Create fine mesh (fmdl_mdl) and dual mesh (dmdl_mdl) reconstruction models.
% Dual Partial $Id: dual_partial2d03.m 4839 2015-03-30 07:44:50Z aadler $

% base model
imdl.RtR_prior = @prior_noser;
imdl.hyperparameter.value = .2;

% Reconstruction model - only fine reconstruction
frec_mdl= imdl;
frec_mdl.fwd_model= fmdl; % fine model

% Reconstruction model - dual model
drec_mdl = imdl;
drec_mdl.fwd_model= fmdl; % fine model
% coarse to fine mapping
nf_els= size(fmdl.elems,1); nc_els= size(cmdl.elems,1);
drec_mdl.fwd_model.coarse2fine= sparse(c2f_idx, 1:nc_els, 1, nf_els, nc_els);

Reconstructed images (projected to fine mesh)

% Dual Partial $Id: dual_partial2d04.m 2749 2011-07-14 13:32:13Z aadler $

clf; levels= [inf,inf,0,1,1];

% reconstruct fine model
imgf= inv_solve(frec_mdl, vh, vi);

show_slices(imgf,levels);
print_convert dual_partial2d04a.png;

% reconstruct dual model
imgd= inv_solve(drec_mdl, vh, vi);

show_slices(imgd,levels);
print_convert dual_partial2d04b.png;




Figure: Fine (top) and dual (bottom) reconstructed images

Reconstructed images (projected to coarse mesh)

% Dual Partial $Id: dual_partial2d05.m 2749 2011-07-14 13:32:13Z aadler $

clf; levels= [inf,inf,0,1,1];

% reconstruct fine model params onto coarse model
imgf= inv_solve(frec_mdl, vh, vi);
imgf.fwd_model= cmdl;
imgf.elem_data= imgf.elem_data(c2f_idx,:);

show_slices(imgf,levels);
print_convert dual_partial2d05a.png;

% reconstruct dual model on coarse mesh
imgd= inv_solve(drec_mdl, vh, vi);
imgd.fwd_model= cmdl;

show_slices(imgd,levels);
print_convert dual_partial2d05b.png;




Figure: Fine (top) and dual (bottom) reconstructed images

Last Modified: $Date: 2017-02-28 13:12:08 -0500 (Tue, 28 Feb 2017) $ by $Author: aadler $