├── MATLAB-Code ├── Z_data.mat ├── project_eit.asv └── project_eit.m ├── MATLAB-Outputs ├── equations.png ├── layers.gif ├── layers.mp4 ├── tumor-animation.gif └── tumorAnimation.mp4 └── README.md /MATLAB-Code/Z_data.mat: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/candicei/MATLAB-ElT/80cfe84dab2ed16c8ddeea37b93dc5868f7f6be5/MATLAB-Code/Z_data.mat -------------------------------------------------------------------------------- /MATLAB-Code/project_eit.asv: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/candicei/MATLAB-ElT/80cfe84dab2ed16c8ddeea37b93dc5868f7f6be5/MATLAB-Code/project_eit.asv -------------------------------------------------------------------------------- /MATLAB-Code/project_eit.m: -------------------------------------------------------------------------------- 1 | %% MATH 406 2 | % Electrical Impedance Tomography (EIT) 3 | % Problem to determine non-uniformity within a circular region (a tumor 4 | % in the region) based on boundary values 5 | % 6 | % Using the solution u(r,theta) = 7 | % [I*a/(2*pi*sigma)]*log[(a^2+r^2+2arcos(phi)/(a^2+r^2-2arcos(phi)] 8 | % where phi = theta-pi/2 9 | % 10 | % Z_data.txt is a file containing for measurement data of voltages 11 | % - size: 16x8 12 | % 13 | % Variables 14 | % [X,Y] = points on the boundary where measurements are taken 15 | % [x_mesh, y_mesh] = made by the mesh we want to determine our inner 16 | % points and will be mapped to... 17 | % [XX, YY] = the mesh points (line above) to the edges given by the 18 | % transformations mapX and mapY 19 | % % % % % 20 | 21 | clear;clc; 22 | % OPENING THE DATA FILE (delta v1) 23 | load('Z_data.mat'); 24 | 25 | % CONSTANTS 26 | a = 10; % radius 27 | sigma = 1 ; % conductance 28 | I = 1/a; % current 29 | k = 16; % boundary points 30 | n = 8; % number of excitation location pairs (of current) 31 | eps = 1e-9; % small number for the voltage boundary values to not have an effect on the data 32 | C1 = I*a/(2*sigma*pi); 33 | N = n*100; % number of radial points within the mesh 34 | 35 | % MAPPING FUNCTIONS 36 | mapX = @(x,y) 2.*a^2.*x./(y.^2+x.^2+a^2); 37 | mapY = @(X) (a^2-X.^2).^(1/2); 38 | 39 | % VARIABLES 40 | phi = 0:pi/n:(2*pi); 41 | u1 = zeros(k+1,n); 42 | r = a; % where we r 43 | 44 | % HOMOGENOUS BOUNDARY VOLTAGE VALUES 45 | u_foo(:,1) = C1*(log(a^2+a^2+2.*a.*a.*cos(phi))- log(a^2+a^2-2.*a.*a.*cos(phi))); 46 | u_foo(abs(u_foo) == inf) = eps; %checking for voltage values at infinity to be set as eps to have little effect on data 47 | % creating the homogenous matrix (of boundary voltages at the points) 48 | for i = 1:n 49 | u1(:,i) = u_foo; 50 | end 51 | 52 | % CREATING THE ZH MATRIX 53 | ZH = diff(u1); % in order to the the delta u1 matrix 54 | ZH([1,8,9,16],:) = eps; 55 | 56 | 57 | % CREATING THE DELTA SIGMA MATRIX 58 | del_sig = -sigma.*(Z./ZH - 1); 59 | del_sig(abs(del_sig)==inf) = eps; % replacing inf locations 60 | 61 | % CREATING THE MESH 62 | r_foo = linspace(0,a,N); 63 | phi_foo = linspace(0,2*pi,N); 64 | [r_mesh, phi_mesh] = meshgrid(r_foo, phi_foo); 65 | x_mesh = r_mesh.*cos(phi_mesh); 66 | y_mesh = r_mesh.*sin(phi_mesh); 67 | 68 | % these XX and YY transformations only go to the top half (+sqrt()) 69 | XX = mapX(x_mesh,y_mesh); 70 | YY = mapY(XX); 71 | 72 | % creating the actual measurment points X, Y on the boundary for only the 73 | % top half 74 | phi = 0:pi/n:(2*pi-pi/n); 75 | X = a.*cos(phi); edgesX = fliplr(X(1:length(X)/2+1)); 76 | Y = a.*sin(phi); edgesY = fliplr(Y(1:length(Y)/2+1)); 77 | 78 | % We check by regions to see if XX is within certain regions (separated by 79 | % the equipotential lines from the mapped x_mesh, y_mesh. This is compare 80 | % to values of our measurement points X and Y. The regions are specified 81 | % such that region 1 = region from X(pi) - X(15*pi/16), region 2 = 82 | % X(15*pi/16) - X(14*pi/16) and so on. Region 8 is the region from phi 83 | % pi/16 to 0. 84 | % This only checks the top curve. From this, 85 | % we determin a mapping (a look-up table) from regions 1, 2, ... 8, to 86 | % determine in which equipotential line we exist in. 87 | % We search down on columns first then to the next row. 88 | lookupX = discretize(XX,edgesX); 89 | 90 | 91 | % CREATING THE MULTIPLE EXCITATION POINT MAPS 92 | uuu = repmat(lookupX,1,1,k/2); 93 | shift = 1*floor(N/(k)); % number of points to "circshift" the data 94 | uu = lookupX; 95 | 96 | %regions 1->8 averaged with the bottom half 97 | avg_del_sig = del_sig(fliplr(1:k/2),:)+del_sig((k/2+1):k,:)/2; 98 | % obtaining the 3d graph for all points for excitation points at phi/16 99 | for ii = 1:n 100 | for i = 1:length(avg_del_sig) 101 | uu(lookupX == i) = avg_del_sig(i,ii); 102 | end 103 | uuu(:,:,ii) = circshift(uu,shift*(ii-1)); 104 | end 105 | 106 | % PLOTTING THE FIGURES 107 | titlename = ['Excitation at \phi = \pi/16']; 108 | figure(1) 109 | for i = 1:n 110 | subplot(2,4,i) 111 | surf(x_mesh,y_mesh,uuu(:,:,i)) 112 | view([0 90]) 113 | shading interp 114 | titlename(22) = num2str(i-1); 115 | title(titlename) 116 | xlabel('X'); 117 | ylabel('Y'); 118 | end 119 | 120 | figure(2) 121 | surf(x_mesh,y_mesh,sum(uuu,3)) 122 | title('Location of Tumor') 123 | zlabel('Conductivity Pertubation(\Delta\sigma)') 124 | xlabel('X') 125 | ylabel('Y') 126 | view([0 90]) 127 | colormap(parula) 128 | shading interp 129 | colorbar 130 | 131 | % To Visualize 132 | % surf(x_mesh,y_mesh,zeros(size(x_mesh))) 133 | % view([0,90]) 134 | % figure(2) 135 | % surf(XX,YY,zeros(size(XX))) 136 | % view([0,90]) 137 | 138 | -------------------------------------------------------------------------------- /MATLAB-Outputs/equations.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/candicei/MATLAB-ElT/80cfe84dab2ed16c8ddeea37b93dc5868f7f6be5/MATLAB-Outputs/equations.png -------------------------------------------------------------------------------- /MATLAB-Outputs/layers.gif: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/candicei/MATLAB-ElT/80cfe84dab2ed16c8ddeea37b93dc5868f7f6be5/MATLAB-Outputs/layers.gif -------------------------------------------------------------------------------- /MATLAB-Outputs/layers.mp4: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/candicei/MATLAB-ElT/80cfe84dab2ed16c8ddeea37b93dc5868f7f6be5/MATLAB-Outputs/layers.mp4 -------------------------------------------------------------------------------- /MATLAB-Outputs/tumor-animation.gif: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/candicei/MATLAB-ElT/80cfe84dab2ed16c8ddeea37b93dc5868f7f6be5/MATLAB-Outputs/tumor-animation.gif -------------------------------------------------------------------------------- /MATLAB-Outputs/tumorAnimation.mp4: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/candicei/MATLAB-ElT/80cfe84dab2ed16c8ddeea37b93dc5868f7f6be5/MATLAB-Outputs/tumorAnimation.mp4 -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Electrical Impedance Tomography (EIT) with Green's Functions 2 | This project is to learn an application of applying Green's Functions towards Electrical Impedance Tomography (EIT) to image a simulated tumour in a circular region using MATLAB. This is part of a project course from the University of British Columbia. 3 | 4 | 5 | **See the resulting EIT scan as a gif!** 6 | 7 |

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10 | 11 | Trouble viewing the GIF? Download the video here: [Video](MATLAB-Outputs/tumorAnimation.mp4). 12 | 13 | 14 | --- 15 | # Table of Contents 16 | 1. [Introduction to Electrical Impedance Tomography](#intro) 17 | 2. [Problem Description](#problem-description) 18 | 3. [Solution](#solution) 19 | 1. [Solution to the Boundary Value Problem](#bvp) 20 | 2. [Implementation](#implementation) 21 | 3. [Superimposing the Solution](#overlay) 22 | 4. [Impedance Solution](#final) 23 | 4. [Equations](#equations) 24 | 25 | --- 26 | 27 | ## Introduction to Electrical Impedance Tomography 28 | 29 | Electrical impedance tomography (EIT) is an imaging method which uses surface measurements to determine variations in electrical conductivity, permittivity, and impedance within an object to image that specific object. It is highly used in medical imaging as it is noninvasive and biological tissues and fluids have considerable variations from each other in conductivity and is therefore distinguishiable using EIT. 30 | 31 | In the regular set-up for an EIT system, surface electrodes are attached to the outer layer of the area being examined. Current is then applied to node pairs (located such that the examined area separates the two nodes) and the voltage across the nodes is measured such that conductivity can be measured. 32 | 33 | --- 34 | 35 | ## Problem Description 36 | To determine the changes of conductivity (noted by equation [(1)](#equations)) within a circular region from a finite number of voltage measurements on a boundry, a current is applied to parts of the boundary. This then becomes a partial differential equation subject to the boundary conditions as seen in equations [(2)](#equations) and [(3)](#equations). Altogether, this is a boundary value problem (BVP). 37 | 38 | This is called an inverse problem because the material properties are determined from measurements of field quantities and in this case, can lead to identifying anomolies such as tumours which are known to have much higher conductivities than normal tissues. Thus, because of this property of tumours, identifying areas of perturbed conductivities can image the tumour! 39 | 40 | ### The Data 41 | The data is of simulated voltage measurements along the boundary of a circular region. 42 | 43 | - [Z_data.mat](MATLAB-Code/Z_data.mat) 44 | 45 | Information about the file: 46 | * Contains 128 = 16 X 8 data points representing the voltage differences from 16 nodes. 47 | * The nodes are uniformly distributed along the boundary of the circle. 48 | * Due to symmetry, there are only 8 indepedent axially distributed nodes simulating the pairs k - k'. 49 | * The first of the 16 is along the x axis, aka the phi angle is 0. 50 | * Z1 is the voltage difference between the stimulating axis T1 and the next terminal marked T2. 51 | * Subsequent elements represent the voltage differences as seen as in equation [(4)](#equations). 52 | 53 | 54 | --- 55 | 56 | ## Solution 57 | 58 | ### Solution to the Boundary Value Problem 59 | The solution to the BVP noted by equations [(2)](#equations) and [(3)](#equations) is found to be as shown in equation [(5)](#equations). This is found by using Green's functions. 60 | 61 | 62 | ### Implementation 63 | In order to find the conductivity in the material, equation [(6)](#equations) is implemented using voltage measurements (v) from the Z_data.mat file and the (u) calculated from equation [(5)](#equations). 64 | 65 | 66 | ### Superimposing the Solutions 67 | The gif shows the results of the calculated impedances. To the left, it shows us the inidvidual responses from exciting one of the eight axial nodes. In each frame, the yellow shows a high intensity along a strip that the anomaly exists in. As we rotate around the region, we superimpose all these images to get the image on the right of the gif. The superimposed image provides the location of where the anomoly exists within the circular region (all determined by boundary values)! 68 | 69 |

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72 | 73 | Trouble viewing the GIF? Download the video here: [Video](MATLAB-Outputs/layers.mp4). 74 | 75 | 76 | 77 | ### Final EIT Scan 78 | The completed EIT scan is the superposition of the 8 axial excitations. The complete code written in MATLAB can be downloaded here: [project_eit.m](MATLAB-Code/project_eit.m). 79 | 80 | The final scan is seen as the [gif found at the top of the page](#gif)! 81 | 82 | --- 83 | 84 | ## Equations 85 | An image of the equations used are found in the image below: 86 | 87 |

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