EIDORS: Electrical Impedance Tomography and Diffuse Optical Tomography Reconstruction Software

Hosted by

3D-3D Dual Models

Application of 3D-3D dual models has many advantages. One can easily change inverse model resolution to fit number of unknowns to the power of his computer. It very natural to limit space where the image is reconstructed only to the area of interest, and maybe the most important "inverse crime" won't be ever commited.

This tutorial is based on classical tank problem, where forward problem is solved using fine mesh (Netgen generated) with 36,000 elements and inverse problem is based on very coarse mesh with 688 elements.

% Dual model 3d-3d image reconstruction demo
%
% (C) 2009, Bartosz Sawicki
% Licenced under the GPLv2 or later
% $Id: dual_3d3d.m 1835 2009-06-11 21:50:26Z sawickib $

%% Create forward, fine tank model
electrodes_per_plane = 16;
number_of_planes = 2;
tank_radius = 0.2;
tank_height = 0.5;

[fine_mdl,centres] = create_tank_mesh_ng( ...
   tank_radius, ...   
   tank_height, ...  
   'C', ... % Rect_or_Circ_electrode
   log2(electrodes_per_plane), ...  
   number_of_planes, ...  
   0.15, ... % first_plane_starts
   0.2, ... % height_between_centres
   0.01, ... % electrode_width
   0.05, ... % electrode_height, ...
   'tank', ... % fname, 
   0 ... % elec_mesh_density  
   );

% Determine stimulation paterns
stim_pat = mk_stim_patterns(electrodes_per_plane, number_of_planes, ...
              '{ad}','{ad}',{'meas_current'});

% Parameters for forward model
fine_mdl.stimulation= stim_pat;
fine_mdl.solve=      'aa_fwd_solve';
fine_mdl.system_mat= 'aa_calc_system_mat';
fine_mdl.jacobian=   'aa_calc_jacobian';
fine_mdl.normalize_measurements= 0;
fine_mdl.np_fwd_solve.perm_sym= '{n}';

          
%% Solve homogeneous model 
disp('Homogeneous model');

% Every cells has the same material
mat= ones( size(fine_mdl.elems,1) ,1);

homg_img= eidors_obj('image', 'homogeneous image', ...
                     'elem_data', mat, ...
                     'fwd_model', fine_mdl );

homg_data=fwd_solve( fine_mdl, homg_img);


%% Create inclusion and solve inhomogenous model
disp('Inhomogeneous model');

% Parameters of spherical inclusion
center = [0.10, 0, 0.2];
radius = 0.05;
inclusion_material = 10;

inhomg_img = create_inclusion(homg_img, center, radius, inclusion_material);

figure; show_fem( inhomg_img );

inhomg_data=fwd_solve( fine_mdl, inhomg_img);


%% Create coarse model for inverse problem

coarse_mdl_maxh = 0.07; % maximum element size 
coarse_mdl = create_cylinder_mesh_ng('cylinder', tank_radius, ...
                                     tank_height, coarse_mdl_maxh) ;


%% Create inverse model
disp('Inverse model')

inv3d.name=  'Dual 3d-3d EIT inverse';
inv3d.fwd_model= fine_mdl;

disp('Calculating coarse2fine mapping ...');
inv3d.fwd_model.coarse2fine = ...
       mk_coarse_fine_mapping( fine_mdl, coarse_mdl);
disp('   ... done');

% Parameters for inverse model
inv3d.solve= 'np_inv_solve';
inv3d.hyperparameter.value = 1e-4;
inv3d.R_prior= 'np_calc_image_prior';
inv3d.np_calc_image_prior.parameters= [3 1]; 
inv3d.jacobian_bkgnd.value= 1;
inv3d.reconst_type= 'difference';

inv3d= eidors_obj('inv_model', inv3d);

recon_img= inv_solve( inv3d, homg_data, inhomg_data);

figure; show_fem( recon_img );


%% Show results on coarse mesh

mat= recon_img.elem_data;
coarse_recon_img= eidors_obj('image', 'reconstructed coarse image', ...
                     'elem_data', mat, ...
                     'fwd_model', coarse_mdl );

figure; show_fem( coarse_recon_img );              

Figure: Netgen generated mesh for tank problem, with a 2×16 electrodes. Mesh consists of 36,000 elements. Single spherical inclusion was applied.

Figure: Results of reconstruction. On the left results shown on coarse mesh used in inverse problem. On the right solution projected into fine, forward mesh.