Tomography is the process of reconstructing a density distribution from given integrals over sections of the distribution. In our example, we will work with tomography on black and white images. Suppose $x$ be the vector of $n$ pixel densities, with $x_j$ denoting how white pixel $j$ is. Let $y$ be the vector of $m$ line integrals over the image, with $y_i$ denoting the integral for line $i$. We can define a matrix $A$ to describe the geometry of the lines. Entry $A_{ij}$ describes how much of pixel $j$ is intersected by line $i$. Assuming our measurements of the line integrals are perfect, we have the relationship that
$$y = Ax$$However, anytime we have measurements, there are usually small errors that occur. Therefore it makes sense to try to minimize
$$\|y - Ax\|_2^2.$$This is simply an unconstrained least squares problem; something we can readily solve!
In [2]:
using Convex, Gadfly, SCS
line_mat_x = readdlm("tux_sparse_x.txt")
line_mat_y = readdlm("tux_sparse_y.txt")
line_mat_val = readdlm("tux_sparse_val.txt")
line_vals = readdlm("tux_sparse_lines.txt")
# Form the sparse matrix from the data
# Image is 50 x 50
img_size = 50
# The number of pixels in the image
num_pixels = img_size * img_size
line_mat = spzeros(3300, num_pixels)
num_vals = length(line_mat_val)
for i in 1:num_vals
x = int(line_mat_x[i])
y = int(line_mat_y[i])
line_mat[x + 1, y + 1] = line_mat_val[i]
end
pixel_colors = Variable(num_pixels)
# line_mat * pixel_colors should be close to the line_integral_values
# to reflect that, we minimize a norm
objective = sumsquares(line_mat * pixel_colors - line_vals)
problem = minimize(objective)
solve!(problem, SCSSolver(verbose=0))
rows = zeros(img_size*img_size)
cols = zeros(img_size*img_size)
for i = 1:img_size
for j = 1:img_size
rows[(i-1)*img_size + j] = i
cols[(i-1)*img_size + j] = img_size + 1 - j
end
end
# Plot the image using the pixel values obtained!
p = plot(
x=rows, y=cols, color=reshape(evaluate(pixel_colors), img_size, img_size), Geom.rectbin,
Scale.ContinuousColorScale(Scale.lab_gradient(color("black"), color("white")))
)
Out[2]: