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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.

Source code: dual_3d3d.m
% Dual model 3d-3d image reconstruction demo
%
% (C) 2009, Bartosz Sawicki
% Licenced under the GPLv2 or later
% $Id: dual_3d3d.m 4839 2015-03-30 07:44:50Z aadler $

%% Create forward, fine tank model
electrodes_per_plane = 16;
number_of_planes = 2;
tank_radius = 0.2;
tank_height = 0.5;
fine_mdl = ng_mk_cyl_models([tank_height,tank_radius],...
    [electrodes_per_plane,0.15,0.35],[0.01]);

% 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=      'fwd_solve_1st_order';
fine_mdl.system_mat= 'system_mat_1st_order';
fine_mdl.jacobian=   'jacobian_adjoint';
fine_mdl = mdl_normalize(fine_mdl,0);

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

% Every cells has the same material
homg_img= mk_image(fine_mdl, 1 );

homg_data=fwd_solve( 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);

show_fem( inhomg_img );
print_convert dual_3d3d01a.png '-density 80'

inhomg_data=fwd_solve( inhomg_img);


%% Create coarse model for inverse problem

coarse_mdl_maxh = 0.07; % maximum element size 
coarse_mdl = ng_mk_cyl_models([tank_height,tank_radius,coarse_mdl_maxh],[0],[]);


%% 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= @eidors_default;
inv3d.hyperparameter.value = .01;
inv3d.R_prior= @prior_laplace;
inv3d.jacobian_bkgnd.value= 1;
inv3d.reconst_type= 'difference';

inv3d= eidors_obj('inv_model', inv3d);

recon_img= inv_solve( inv3d, homg_data, inhomg_data);

show_fem( recon_img );
print_convert dual_3d3d02a.png '-density 80'


%% Show results on coarse mesh

coarse_recon_img= mk_image(coarse_mdl, recon_img.elem_data);

show_fem( coarse_recon_img );              
print_convert dual_3d3d02b.png '-density 80'


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.

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