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pluto_ct.jl
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### A Pluto.jl notebook ###
# v0.19.46
using Markdown
using InteractiveUtils
# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
quote
local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
local el = $(esc(element))
global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
el
end
end
# ╔═╡ 5ffc5ec0-f91f-11ec-0552-37ef5f25102d
begin
using PlutoUI
using Plots
using Symbolics
using ForwardDiff
using LaTeXStrings
using Images
using TestImages
using ImageView
using LinearAlgebra
# using FFTW
# using DSP
nbsp = html" "
end
# ╔═╡ ab1c3d69-0499-4fad-b2a8-5360ed7339e1
begin
include("xray.jl")
segment(A, B; color="black", kwargs...) = plot([A[1], B[1]], [A[2], B[2]]; color=color, kwargs...)
segment!(A, B; color="black", kwargs...) = plot!([A[1], B[1]], [A[2], B[2]]; color=color, kwargs...)
segment!(p, A, B; color="black", kwargs...) = plot!(p, [A[1], B[1]], [A[2], B[2]]; color=color, kwargs...)
I=load("./material/stomach.png");
z=Gray.(I);
w,h = size(z);
Nr_div2 = floor(sqrt(w*w+h*h)/2);
Nr = Int64(Nr_div2*2+1);
angles = (0:1:179)/180*pi; # [rad]
sinogram = radon(z,angles);
for i=1:1
end
end
# ╔═╡ 2dc43448-3cf4-475f-8b61-d16c3594d9e3
md"""
## X-ray transform on $\mathbb{R}^2$
A planar line $\gamma$ is parametrized by $(\theta,t)\in[0,\pi]\times\mathbb{R}$ as
$
\begin{aligned}
\gamma
=
\gamma(\theta,t)
& =
\{(x,y)\in\mathbb{R}^2\ : \ x\cos\theta+y\sin\theta=t\}
\\
& =
\{(x,y)\in\mathbb{R}^2\ : \ \cos\theta(x-t\cos\theta)+\sin\theta(y-t\sin\theta)=0\}
\\
& =
\{\vec{x}\in\mathbb{R}^2\ : \ \vec{\omega}\cdot(\vec{x}-t\vec{\omega})=0\}
\\
& =
\{(t\cos\theta-s\sin\theta,t\sin\theta+s\cos\theta)\ : \ s\in\mathbb{R}\},
\\
\vec{\omega}
& =
(\cos\theta,\sin\theta)\in\mathbb{S}^1.
\end{aligned}$
The X-ray transform of a function $f$ of $(x,y)\in\mathbb{R}^2$ is defined by
$\mathcal{R}_1f(\gamma)=\mathcal{R}_1f(\theta,t)
:=\int_\gamma f=\int_{-\infty}^\infty f(t\cos\theta-s\sin\theta,t\sin\theta+s\cos\theta)ds.$
We have $\mathcal{R}_1f(\theta+\pi,-t)=\mathcal{R}_1f(\theta,t)$ for $(\theta,t)\in[0,\pi]\times\mathbb{R}$.
"""
# ╔═╡ a2a26295-531b-424d-9fd7-b3974987623b
begin
plot([1,101],[-50/sqrt(3)-12.5,50/sqrt(3)-12.5],label="\$t\$",
arrow=3,linewidth=2,linecolor=:lightgreen,
title="Parametrization of the X-ray transform on the plane",
xlim=(0,102),
ylim=(-51,51),
xlabel="\$x\$",
ylabel="\$y\$",
xticks=([21],[0]),
yticks=([-30],[0]),
aspect_ratio=1.0,
size=(600,600),
annotations = [(35,-26, "\$θ\$", 15)])
plot!([79,25],[-sqrt(3)*29,sqrt(3)*25],label="\$s\$",
arrow=3,linewidth=2,linecolor=:magenta)
plot!([1,101],[-30,-30],label=false,
arrow=3,linewidth=1,linecolor=:black)
plot!([21,21],[-50,50],label=false,
arrow=3,linewidth=1,linecolor=:black)
scatter!((56,-10),markersize = 5,label=false, color=:blue)
end
# ╔═╡ 76cb2ec2-fc76-4ad5-8d57-777395755f7f
md"""
#### Example of X-ray transform on $\mathbb{R}^2$
"""
# ╔═╡ 092ba07e-4866-40c6-ade8-25d9c2218011
md"""
θ = $(@bind a1 Slider(0:1:179, show_value=true, default=0))
$nbsp $nbsp $nbsp
t = $(@bind b1 Slider(-500:1:500, show_value=true, default=0))
"""
# ╔═╡ 8ed1312b-f5a3-4220-a865-9a12ed09a691
begin
if a1==90
P=(456,1);
Q=(456,657);
R=(1,328-b1);
S=(914,328-b1);
else
P=(1,328+456*tan(pi*a1/180));
Q=(914,328-456*tan(pi*a1/180));
R=(457+b1/cos(pi*a1/180)-328*tan(pi*a1/180),1);
S=(457+b1/cos(pi*a1/180)+328*tan(pi*a1/180),657);
end
p=plot(Gray.(z),
title="Original Grayscale Image",
size=(600,400),
xlim=(1,914),
ylim=(1,657),
xticks = ([1 h/2 h],[-floor(h/2),0,floor(h/2)]),
xlabel="x",
yticks = ([1 w/2 w],[floor(w/2),0,-floor(w/2)]),
ylabel="y");
segment!(P,Q,color=:lightgreen,width=2,legend=false);
segment!(R,S,color=:magenta,width=4,legend=false);
for i=1:1
end
end
# ╔═╡ 7769d576-c24f-4bd3-b130-e55faa113e55
begin
A=zeros(1);
B=zeros(1);
A[1]=a1+1;
B[1]=b1+563;
q=plot(sinogram,
size=(600,400),
title="Sinogram (X-ray transform)",
st=:heatmap,
color=:jet1,
xaxis="θ [radian]",
xticks = ([1 90 180], [0,"π/2","π"]),
yaxis="t",
yticks = ([1 Nr/2 Nr], [-floor(Nr/2),0,floor(Nr/2)]));
scatter!(A,B,markersize = 10,label = false, color=:magenta);
for i=1:1
end
end
# ╔═╡ edc9bd70-2c59-4140-b992-28c53f4c6137
begin
plot(p, q,
layout=2,
size=(1600,500),
left_margin=Plots.Measures.Length(:mm, 5.0),
right_margin=Plots.Measures.Length(:mm, 15.0),
top_margin=Plots.Measures.Length(:mm, 5.0),
bottom_margin=Plots.Measures.Length(:mm, 5.0))
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
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# ╔═╡ 00000000-0000-0000-0000-000000000002
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