Density Images

Figure 2

Figure 3

Figure 4

Figure 5
Matlab Code - Isolines
%Set up system (R = density image)
sz = 64; %pixels
[L, R1, R2] = Experiment1(sz); %[g/cm^3]
%Forward project density images
X1 = L*R1(:); %[g/cm^2]
X2 = L*R2(:);
%Mass Attenuation Coefficients (NIST)
M1 = 0.240183*MU(6) + 0.759818*MU(9); %teflon [cm^2/g]
M2 = .90*MU(22) + .06*MU(13) + .04*MU(23); %titanium
%Energy Fluence
Psi = [1 1 1 1.5 2 3.5 5 7.5 10 10 10 15 20 35 50]'; %bin width
Psi = Psi/sum(Psi); %normalize to 1
Psi = Psi/sum(Psi);
AU1 = X1.*M1';
AU2 = X2.*M2';
d = exp(-AU1-AU2)*Psi;
%M1 approximation
len = length(d);
x = zeros(len,1);
F = @(x) exp(-x*M1')*Psi;
for i = 1:len
x(i) = fzero(@(x) F(x)-d(i), [0 100]);
end
IM = L\x;
IM2 = reshape(IM,sz,sz); %recover density image
IM2(IM2 < 0)=0;
figure
imagesc(IM2);
colormap gray
Sino = reshape(x,sz,[]);
figure
imagesc(Sino);
colormap gray
%isoline
i = 2465;
exp(-x(i)*M1')*Psi
d(i)
exp(-X1(i)*M1'-X2(i)*M2')*Psi
x_iso = linspace(0,x(i),100);
y_iso = zeros(size(x_iso));
ISO = @(x,y) exp(-x*M1'-y*M2')*Psi;
for j = 1:length(x_iso)
y_iso(j) = fzero(@(y) ISO(x_iso(j),y)-d(i), [-1 100]);
end
figure
plot(x_iso,y_iso);
hold on
plot(X1(i),X2(i),'x');
function [L, R1, R2] = Experiment1(sz)
%Create Density Images
[im1, im2, im3] = MakeCircles(sz); %(100 px diameter)
R1 = 2.20*im1; %Teflon [g/cm^3]
R2 = 4.43*(im2+im3); %Titanium [g/cm^3]
%image dimensions
Sz = size(im1);
vxls = prod(Sz);
%detector coordinates
px = .5:Sz(1);
p = [px; zeros(size(px))];
dets = length(px);
%views
views = dets+101;
%Create proj matrix
L = SparseProj(Sz, views, dets, vxls, p);
%convert pixel width to cm
width = 5; %[cm]
D = Sz(1);
L = L*width/D;
end
function L = SparseProj(Sz, views, dets, vxls, p)
%angles
t = (0:views-1)*pi*2/views; %angle
%rotation matrix
R =@(t) [cos(t) -sin(t); sin(t) cos(t)];
%proj direction vector
r = [0; 1];
r = r/norm(r); %normalize to 1
%Isocenter
Iso = Sz/2; %center of image
%initialize lengths matrix
L = cell(views,dets);
for j=1:views
t_j = t(j); %angle of proj
r_j = R(t_j)*r; %rotate proj direction vector
for i=1:dets
%rotate p about Iso
p_i = R(t_j)*(p(:,i)-Iso')+Iso';
%project along r
[inds, lens] = Proj(Sz,p_i,r_j);
%store lengths
L{j,i} = {inds, lens};
end
end
elem = 0; %sparse element count
for j=1:views
for i=1:dets
len = length(L{j,i}{1});
elem = elem+len;
end
end
I = zeros(elem,1);%sparse indices
J = zeros(elem,1);
V = zeros(elem,1);
ptr = 1;
for j=1:views
for i=1:dets
inds = L{j,i}{1};
lens = L{j,i}{2};
len = length(inds);
I(ptr:ptr+len-1) = i+(j-1)*dets;
J(ptr:ptr+len-1) = inds;
V(ptr:ptr+len-1) = lens;
ptr = ptr+len;
end
end
L = sparse(I,J,V,views*dets,vxls);
end
function [inds, lens] = Proj(Sz,p,r)
%voxel array size
Sx = Sz(1);
Sy = Sz(2);
%intersection lines
x_i = 0:Sz(1);
y_i = 0:Sz(2);
%intersection parameter (t)
t_x = (x_i-p(1))/r(1);
t_y = (y_i-p(2))/r(2);
t = [t_x t_y];
t = unique(sort(t));
%intersection coordinates (x,y)
x = p(1) + r(1)*t;
y = p(2) + r(2)*t;
%valid region
mask = (x >= 0) & (y >= 0) & (x <= Sx) & (y <= Sy);
t = t(mask);
%find midpoints
t_mid = convn(t,[.5 .5],'valid');
%find lengths
lens = diff(t);
%round to pixel coordinates
x_mid = p(1) + r(1)*t_mid;
y_mid = p(2) + r(2)*t_mid;
x_co = ceil(x_mid);
y_co = ceil(y_mid);
inds = sub2ind(Sz,x_co,y_co);
end
function [Z, Z1, Z2] = MakeCircles(sz)
[X,Y] = ndgrid(1:sz,1:sz);
cx = round(.5*sz);
cy = round(.5*sz);
r1 = round(.125*sz); %radius of little circles
r2 = round(.437*sz); %radius of big circle
r3 = round(.219*sz); %radius of circle of little circles
circle =@(cx,cy,r) sqrt((X-cx).^2+(Y-cy).^2) < r;
t1 = 0;
t2 = 2*pi/2;
cx1 = cx-r3*cos(t1);
cy1 = cy-r3*sin(t1);
cx2 = cx-r3*cos(t2);
cy2 = cy-r3*sin(t2);
Z = circle(cx,cy,r2);
Z1 = circle(cx1,cy1,r1);
Z2 = circle(cx2,cy2,r1);
Z = Z -Z1 -Z2;
end
function mu1 = MU(i)
%Source:
%https://physics.nist.gov/PhysRefData/XrayMassCoef/tab3.html
mu_cell = {
[7.217e+00, 2.148e+00, 1.059e+00, 5.612e-01, 4.546e-01, 4.193e-01, 4.042e-01, 3.914e-01, 3.854e-01, 3.764e-01, 3.695e-01, 3.570e-01, 3.458e-01, 3.355e-01, 3.260e-01, 3.091e-01, 2.944e-01, 2.651e-01, 2.429e-01, 2.112e-01, 1.893e-01, 1.729e-01, 1.599e-01, 1.405e-01, 1.263e-01, 1.129e-01, 1.027e-01, 8.769e-02, 6.921e-02, 5.806e-02, 5.049e-02, 4.498e-02, 3.746e-02, 3.254e-02, 2.539e-02, 2.153e-02]
[6.084e+01, 1.676e+01, 6.863e+00, 2.007e+00, 9.329e-01, 5.766e-01, 4.195e-01, 2.933e-01, 2.476e-01, 2.092e-01, 1.960e-01, 1.838e-01, 1.763e-01, 1.703e-01, 1.651e-01, 1.562e-01, 1.486e-01, 1.336e-01, 1.224e-01, 1.064e-01, 9.535e-02, 8.707e-02, 8.054e-02, 7.076e-02, 6.362e-02, 5.688e-02, 5.173e-02, 4.422e-02, 3.503e-02, 2.949e-02, 2.577e-02, 2.307e-02, 1.940e-02, 1.703e-02, 1.363e-02, 1.183e-02]
[2.339e+02, 6.668e+01, 2.707e+01, 7.549e+00, 3.114e+00, 1.619e+00, 9.875e-01, 5.054e-01, 3.395e-01, 2.176e-01, 1.856e-01, 1.644e-01, 1.551e-01, 1.488e-01, 1.438e-01, 1.356e-01, 1.289e-01, 1.158e-01, 1.060e-01, 9.210e-02, 8.249e-02, 7.532e-02, 6.968e-02, 6.121e-02, 5.503e-02, 4.921e-02, 4.476e-02, 3.830e-02, 3.043e-02, 2.572e-02, 2.257e-02, 2.030e-02, 1.725e-02, 1.529e-02, 1.252e-02, 1.109e-02]
[6.041e+02, 1.797e+02, 7.469e+01, 2.127e+01, 8.685e+00, 4.369e+00, 2.527e+00, 1.124e+00, 6.466e-01, 3.070e-01, 2.251e-01, 1.792e-01, 1.640e-01, 1.554e-01, 1.493e-01, 1.401e-01, 1.328e-01, 1.190e-01, 1.089e-01, 9.463e-02, 8.471e-02, 7.739e-02, 7.155e-02, 6.286e-02, 5.652e-02, 5.054e-02, 4.597e-02, 3.938e-02, 3.138e-02, 2.664e-02, 2.347e-02, 2.121e-02, 1.819e-02, 1.627e-02, 1.361e-02, 1.227e-02]
[1.229e+03, 3.766e+02, 1.597e+02, 4.667e+01, 1.927e+01, 9.683e+00, 5.538e+00, 2.346e+00, 1.255e+00, 4.827e-01, 3.014e-01, 2.063e-01, 1.793e-01, 1.665e-01, 1.583e-01, 1.472e-01, 1.391e-01, 1.243e-01, 1.136e-01, 9.862e-02, 8.834e-02, 8.065e-02, 7.460e-02, 6.549e-02, 5.890e-02, 5.266e-02, 4.791e-02, 4.108e-02, 3.284e-02, 2.798e-02, 2.476e-02, 2.248e-02, 1.945e-02, 1.755e-02, 1.495e-02, 1.368e-02]
[2.211e+03, 7.002e+02, 3.026e+02, 9.033e+01, 3.778e+01, 1.912e+01, 1.095e+01, 4.576e+00, 2.373e+00, 8.071e-01, 4.420e-01, 2.562e-01, 2.076e-01, 1.871e-01, 1.753e-01, 1.610e-01, 1.514e-01, 1.347e-01, 1.229e-01, 1.066e-01, 9.546e-02, 8.715e-02, 8.058e-02, 7.076e-02, 6.361e-02, 5.690e-02, 5.179e-02, 4.442e-02, 3.562e-02, 3.047e-02, 2.708e-02, 2.469e-02, 2.154e-02, 1.959e-02, 1.698e-02, 1.575e-02]
[3.311e+03, 1.083e+03, 4.769e+02, 1.456e+02, 6.166e+01, 3.144e+01, 1.809e+01, 7.562e+00, 3.879e+00, 1.236e+00, 6.178e-01, 3.066e-01, 2.288e-01, 1.980e-01, 1.817e-01, 1.639e-01, 1.529e-01, 1.353e-01, 1.233e-01, 1.068e-01, 9.557e-02, 8.719e-02, 8.063e-02, 7.081e-02, 6.364e-02, 5.693e-02, 5.180e-02, 4.450e-02, 3.579e-02, 3.073e-02, 2.742e-02, 2.511e-02, 2.209e-02, 2.024e-02, 1.782e-02, 1.673e-02]
[4.590e+03, 1.549e+03, 6.949e+02, 2.171e+02, 9.315e+01, 4.790e+01, 2.770e+01, 1.163e+01, 5.952e+00, 1.836e+00, 8.651e-01, 3.779e-01, 2.585e-01, 2.132e-01, 1.907e-01, 1.678e-01, 1.551e-01, 1.361e-01, 1.237e-01, 1.070e-01, 9.566e-02, 8.729e-02, 8.070e-02, 7.087e-02, 6.372e-02, 5.697e-02, 5.185e-02, 4.459e-02, 3.597e-02, 3.100e-02, 2.777e-02, 2.552e-02, 2.263e-02, 2.089e-02, 1.866e-02, 1.770e-02]
[5.649e+03, 1.979e+03, 9.047e+02, 2.888e+02, 1.256e+02, 6.514e+01, 3.789e+01, 1.602e+01, 8.205e+00, 2.492e+00, 1.133e+00, 4.487e-01, 2.828e-01, 2.214e-01, 1.920e-01, 1.639e-01, 1.496e-01, 1.298e-01, 1.176e-01, 1.015e-01, 9.073e-02, 8.274e-02, 7.649e-02, 6.717e-02, 6.037e-02, 5.399e-02, 4.915e-02, 4.228e-02, 3.422e-02, 2.960e-02, 2.663e-02, 2.457e-02, 2.195e-02, 2.039e-02, 1.846e-02, 1.769e-02]
[7.409e+03, 2.666e+03, 1.243e+03, 4.051e+02, 1.785e+02, 9.339e+01, 5.467e+01, 2.328e+01, 1.197e+01, 3.613e+00, 1.606e+00, 5.923e-01, 3.473e-01, 2.579e-01, 2.161e-01, 1.781e-01, 1.600e-01, 1.370e-01, 1.236e-01, 1.064e-01, 9.502e-02, 8.664e-02, 8.006e-02, 7.029e-02, 6.316e-02, 5.646e-02, 5.145e-02, 4.430e-02, 3.594e-02, 3.122e-02, 2.818e-02, 2.610e-02, 2.348e-02, 2.197e-02, 2.013e-02, 1.946e-02]
[6.542e+02, 5.960e+02, 5.429e+02, 6.435e+03, 3.194e+03, 1.521e+03, 5.070e+02, 2.261e+02, 1.194e+02, 7.030e+01, 3.018e+01, 1.557e+01, 4.694e+00, 2.057e+00, 7.197e-01, 3.969e-01, 2.804e-01, 2.268e-01, 1.796e-01, 1.585e-01, 1.335e-01, 1.199e-01, 1.029e-01, 9.185e-02, 8.372e-02, 7.736e-02, 6.788e-02, 6.100e-02, 5.454e-02, 4.968e-02, 4.282e-02, 3.487e-02, 3.037e-02, 2.753e-02, 2.559e-02, 2.319e-02, 2.181e-02, 2.023e-02, 1.970e-02]
[9.225e+02, 6.474e+02, 4.530e+02, 5.444e+03, 4.004e+03, 1.932e+03, 6.585e+02, 2.974e+02, 1.583e+02, 9.381e+01, 4.061e+01, 2.105e+01, 6.358e+00, 2.763e+00, 9.306e-01, 4.881e-01, 3.292e-01, 2.570e-01, 1.951e-01, 1.686e-01, 1.394e-01, 1.245e-01, 1.065e-01, 9.492e-02, 8.647e-02, 7.988e-02, 7.008e-02, 6.296e-02, 5.629e-02, 5.129e-02, 4.426e-02, 3.613e-02, 3.159e-02, 2.873e-02, 2.681e-02, 2.445e-02, 2.313e-02, 2.168e-02, 2.127e-02]
[1.185e+03, 4.022e+02, 3.621e+02, 3.957e+03, 2.263e+03, 7.880e+02, 3.605e+02, 1.934e+02, 1.153e+02, 5.033e+01, 2.623e+01, 7.955e+00, 3.441e+00, 1.128e+00, 5.685e-01, 3.681e-01, 2.778e-01, 2.018e-01, 1.704e-01, 1.378e-01, 1.223e-01, 1.042e-01, 9.276e-02, 8.445e-02, 7.802e-02, 6.841e-02, 6.146e-02, 5.496e-02, 5.006e-02, 4.324e-02, 3.541e-02, 3.106e-02, 2.836e-02, 2.655e-02, 2.437e-02, 2.318e-02, 2.195e-02, 2.168e-02]
[1.570e+03, 5.355e+02, 3.092e+02, 3.192e+03, 2.777e+03, 9.784e+02, 4.529e+02, 2.450e+02, 1.470e+02, 6.468e+01, 3.389e+01, 1.034e+01, 4.464e+00, 1.436e+00, 7.012e-01, 4.385e-01, 3.207e-01, 2.228e-01, 1.835e-01, 1.448e-01, 1.275e-01, 1.082e-01, 9.614e-02, 8.748e-02, 8.077e-02, 7.082e-02, 6.361e-02, 5.688e-02, 5.183e-02, 4.480e-02, 3.678e-02, 3.240e-02, 2.967e-02, 2.788e-02, 2.574e-02, 2.462e-02, 2.352e-02, 2.338e-02]
[1.913e+03, 6.547e+02, 3.018e+02, 2.494e+02, 2.473e+03, 1.118e+03, 5.242e+02, 2.860e+02, 1.726e+02, 7.660e+01, 4.035e+01, 1.239e+01, 5.352e+00, 1.700e+00, 8.096e-01, 4.916e-01, 3.494e-01, 2.324e-01, 1.865e-01, 1.432e-01, 1.250e-01, 1.055e-01, 9.359e-02, 8.511e-02, 7.854e-02, 6.884e-02, 6.182e-02, 5.526e-02, 5.039e-02, 4.358e-02, 3.590e-02, 3.172e-02, 2.915e-02, 2.747e-02, 2.552e-02, 2.452e-02, 2.364e-02, 2.363e-02]
[2.429e+03, 8.342e+02, 3.853e+02, 2.168e+02, 2.070e+03, 1.339e+03, 6.338e+02, 3.487e+02, 2.116e+02, 9.465e+01, 5.012e+01, 1.550e+01, 6.708e+00, 2.113e+00, 9.872e-01, 5.849e-01, 4.053e-01, 2.585e-01, 2.020e-01, 1.506e-01, 1.302e-01, 1.091e-01, 9.665e-02, 8.781e-02, 8.102e-02, 7.098e-02, 6.373e-02, 5.697e-02, 5.193e-02, 4.498e-02, 3.715e-02, 3.293e-02, 3.036e-02, 2.872e-02, 2.682e-02, 2.589e-02, 2.517e-02, 2.529e-02]
[2.832e+03, 9.771e+02, 4.520e+02, 1.774e+02, 1.637e+03, 1.473e+03, 7.037e+02, 3.901e+02, 2.384e+02, 1.075e+02, 5.725e+01, 1.784e+01, 7.739e+00, 2.425e+00, 1.117e+00, 6.483e-01, 4.395e-01, 2.696e-01, 2.050e-01, 1.480e-01, 1.266e-01, 1.054e-01, 9.311e-02, 8.453e-02, 7.795e-02, 6.826e-02, 6.128e-02, 5.478e-02, 4.994e-02, 4.328e-02, 3.585e-02, 3.188e-02, 2.950e-02, 2.798e-02, 2.628e-02, 2.549e-02, 2.496e-02, 2.520e-02]
[3.184e+03, 1.105e+03, 5.120e+02, 1.703e+02, 1.424e+02, 1.275e+03, 7.572e+02, 4.225e+02, 2.593e+02, 1.180e+02, 6.316e+01, 1.983e+01, 8.629e+00, 2.697e+00, 1.228e+00, 7.012e-01, 4.664e-01, 2.760e-01, 2.043e-01, 1.427e-01, 1.205e-01, 9.953e-02, 8.776e-02, 7.958e-02, 7.335e-02, 6.419e-02, 5.762e-02, 5.150e-02, 4.695e-02, 4.074e-02, 3.384e-02, 3.019e-02, 2.802e-02, 2.667e-02, 2.517e-02, 2.451e-02, 2.418e-02, 2.453e-02]
[4.058e+03, 1.418e+03, 6.592e+02, 2.198e+02, 1.327e+02, 1.201e+03, 9.256e+02, 5.189e+02, 3.205e+02, 1.469e+02, 7.907e+01, 2.503e+01, 1.093e+01, 3.413e+00, 1.541e+00, 8.679e-01, 5.678e-01, 3.251e-01, 2.345e-01, 1.582e-01, 1.319e-01, 1.080e-01, 9.495e-02, 8.600e-02, 7.922e-02, 6.929e-02, 6.216e-02, 5.556e-02, 5.068e-02, 4.399e-02, 3.666e-02, 3.282e-02, 3.054e-02, 2.915e-02, 2.766e-02, 2.704e-02, 2.687e-02, 2.737e-02]
[4.867e+03, 1.714e+03, 7.999e+02, 2.676e+02, 1.218e+02, 1.187e+02, 1.023e+03, 6.026e+02, 3.731e+02, 1.726e+02, 9.341e+01, 2.979e+01, 1.306e+01, 4.080e+00, 1.830e+00, 1.019e+00, 6.578e-01, 3.656e-01, 2.571e-01, 1.674e-01, 1.376e-01, 1.116e-01, 9.783e-02, 8.851e-02, 8.148e-02, 7.122e-02, 6.388e-02, 5.709e-02, 5.207e-02, 4.524e-02, 3.780e-02, 3.395e-02, 3.170e-02, 3.035e-02, 2.892e-02, 2.839e-02, 2.838e-02, 2.903e-02]
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[1.00000e-03, 1.08088e-03, 1.16830e-03, 1.16830e-03, 1.24630e-03, 1.32950e-03, 1.32950e-03, 1.50000e-03, 2.00000e-03, 3.00000e-03, 3.33200e-03, 3.33200e-03, 3.41048e-03, 3.49080e-03, 3.49080e-03, 4.00000e-03, 4.04610e-03, 4.04610e-03, 4.42089e-03, 4.83040e-03, 4.83040e-03, 5.00000e-03, 5.18230e-03, 5.18230e-03, 6.00000e-03, 8.00000e-03, 1.00000e-02, 1.50000e-02, 1.63003e-02, 1.63003e-02, 1.79166e-02, 1.96932e-02, 1.96932e-02, 2.00000e-02, 2.04721e-02, 2.04721e-02, 3.00000e-02, 4.00000e-02, 5.00000e-02, 6.00000e-02, 8.00000e-02, 1.00000e-01, 1.09651e-01, 1.09651e-01, 1.50000e-01, 2.00000e-01, 3.00000e-01, 4.00000e-01, 5.00000e-01, 6.00000e-01, 8.00000e-01, 1.00000e+00, 1.25000e+00, 1.50000e+00, 2.00000e+00, 3.00000e+00, 4.00000e+00, 5.00000e+00, 6.00000e+00, 8.00000e+00, 1.00000e+01, 1.50000e+01, 2.00000e+01]
[1.00000e-03, 1.00334e-03, 1.00670e-03, 1.00670e-03, 1.11018e-03, 1.22430e-03, 1.22430e-03, 1.30316e-03, 1.38710e-03, 1.38710e-03, 1.50000e-03, 2.00000e-03, 3.00000e-03, 3.44180e-03, 3.44180e-03, 3.52548e-03, 3.61120e-03, 3.61120e-03, 4.00000e-03, 4.17380e-03, 4.17380e-03, 5.00000e-03, 5.00090e-03, 5.00090e-03, 5.18067e-03, 5.36690e-03, 5.36690e-03, 6.00000e-03, 8.00000e-03, 1.00000e-02, 1.50000e-02, 1.67331e-02, 1.67331e-02, 2.00000e-02, 2.03137e-02, 2.03137e-02, 2.07054e-02, 2.11046e-02, 2.11046e-02, 3.00000e-02, 4.00000e-02, 5.00000e-02, 6.00000e-02, 8.00000e-02, 1.00000e-01, 1.12601e-01, 1.12601e-01, 1.50000e-01, 2.00000e-01, 3.00000e-01, 4.00000e-01, 5.00000e-01, 6.00000e-01, 8.00000e-01, 1.00000e+00, 1.25000e+00, 1.50000e+00, 2.00000e+00, 3.00000e+00, 4.00000e+00, 5.00000e+00, 6.00000e+00, 8.00000e+00, 1.00000e+01, 1.50000e+01, 2.00000e+01]
[1.00000e-03, 1.02220e-03, 1.04490e-03, 1.04490e-03, 1.15314e-03, 1.27260e-03, 1.27260e-03, 1.35409e-03, 1.44080e-03, 1.44080e-03, 1.50000e-03, 2.00000e-03, 3.00000e-03, 3.55170e-03, 3.55170e-03, 3.63859e-03, 3.72760e-03, 3.72760e-03, 4.00000e-03, 4.30340e-03, 4.30340e-03, 5.00000e-03, 5.18220e-03, 5.18220e-03, 5.36198e-03, 5.54800e-03, 5.54800e-03, 6.00000e-03, 8.00000e-03, 1.00000e-02, 1.50000e-02, 1.71663e-02, 1.71663e-02, 2.00000e-02, 2.09476e-02, 2.09476e-02, 2.13487e-02, 2.17574e-02, 2.17574e-02, 3.00000e-02, 4.00000e-02, 5.00000e-02, 6.00000e-02, 8.00000e-02, 1.00000e-01, 1.15606e-01, 1.15606e-01, 1.50000e-01, 2.00000e-01, 3.00000e-01, 4.00000e-01, 5.00000e-01, 6.00000e-01, 8.00000e-01, 1.00000e+00, 1.25000e+00, 1.50000e+00, 2.00000e+00, 3.00000e+00, 4.00000e+00, 5.00000e+00, 6.00000e+00, 8.00000e+00, 1.00000e+01, 1.50000e+01, 2.00000e+01]
};
%load E,mu
E = E_cell{i}';
mu = mu_cell{i}';
%remove discontinuity
[~,i] = unique(E);
j = setdiff( 1:numel(E), i );
E(j) = E(j) + 1e-12;
%NIST X-ray beam energy levels [keV]
E1 = [3 4 5 6 8 10 15 20 30 40 50 60 80 100 150]'; %15 energy levels
%interpolate
mu1 = interp1(E,mu,E1*1e-3);
end