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pluto_ct2.jl
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### A Pluto.jl notebook ###
# v0.19.45
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/stomachsq.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 $(\theta,t)\in[0,\pi]\times\mathbb{R}$.
"""
# ╔═╡ 76cb2ec2-fc76-4ad5-8d57-777395755f7f
md"""
#### Example of X-ray transform on $\mathbb{R}^2$
"""
# ╔═╡ 0536974e-02cb-4905-9c95-32be3adfeaca
md"""
θ = $(@bind a2 Slider(0:1:179, show_value=true, default=0))
"""
# ╔═╡ 1c501345-e920-46ec-8503-6ca424a44270
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(-200:1:200, show_value=true, default=0))
"""
# ╔═╡ 8ed1312b-f5a3-4220-a865-9a12ed09a691
begin
if a1==90
P=(220,1);
Q=(220,440);
R=(1,220-b1);
S=(440,220-b1);
else
P=(1,220+220*tan(pi*a1/180));
Q=(440,220-220*tan(pi*a1/180));
R=(220+b1/cos(pi*a1/180)-220*tan(pi*a1/180),1);
S=(220+b1/cos(pi*a1/180)+220*tan(pi*a1/180),440);
end
p=plot(Gray.(z),
title="Original Grayscale Image",
size=(600,400),
xlim=(1,440),
ylim=(1,440),
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=4,legend=false);
segment!(R,S,color=:magenta,width=4,legend=false);
for i=1:1
end
end
# ╔═╡ c961036d-a14f-47b8-9bd3-ca89c1e5ef32
begin
c0=-200;
c1=-100;
c2=0;
c3=100;
c4=200;
if a2==90
R20=(1,220-c0);
S20=(440,220-c0);
R21=(1,220-c1);
S21=(440,220-c1);
R22=(1,220-c2);
S22=(440,220-c2);
R23=(1,220-c3);
S23=(440,220-c3);
R24=(1,220-c4);
S24=(440,220-c4);
else
R20=(220+c0/cos(pi*a2/180)-220*tan(pi*a2/180),1);
S20=(220+c0/cos(pi*a2/180)+220*tan(pi*a2/180),440);
R21=(220+c1/cos(pi*a2/180)-220*tan(pi*a2/180),1);
S21=(220+c1/cos(pi*a2/180)+220*tan(pi*a2/180),440);
R22=(220+c2/cos(pi*a2/180)-220*tan(pi*a2/180),1);
S22=(220+c2/cos(pi*a2/180)+220*tan(pi*a2/180),440);
R23=(220+c3/cos(pi*a2/180)-220*tan(pi*a2/180),1);
S23=(220+c3/cos(pi*a2/180)+220*tan(pi*a2/180),440);
R24=(220+c4/cos(pi*a2/180)-220*tan(pi*a2/180),1);
S24=(220+c4/cos(pi*a2/180)+220*tan(pi*a2/180),440);
end
p2=plot(Gray.(z),
title="Original Grayscale Image",
xlim=(1,440),
ylim=(1,440),
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!((220-210*cos(pi*a2/180),220+210*sin(pi*a2/180)),(220+210*cos(pi*a2/180),220-210*sin(pi*a2/180)),color="lightgreen",width=4,legend=false);
segment!(R20,S20,color=:magenta,width=2,legend=false);
segment!(R21,S21,color=:magenta,width=2,legend=false);
segment!(R22,S22,color=:magenta,width=2,legend=false);
segment!(R23,S23,color=:magenta,width=2,legend=false);
segment!(R24,S24,color=:magenta,width=2,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+312;
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=(1400,400),
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
# ╔═╡ 3b12c09a-1d91-4f41-bcec-75bc837c1cf8
begin
q2=plot(sinogram[:,a2+1],
size=(600,400),
title="Slice of Sinogram (X-ray transform)",
xaxis="t",
xticks = ([112 212 312 412 512], [-200,-100,0,100,200]),
yaxis="R_1f(θ,*)",
legend=:false,
color=:skyblue,
width=4)
for i=1:1
end
end
# ╔═╡ bd325028-8239-41cb-8a85-7c02b1db76d8
begin
plot(p2, q2,
layout=2,
size=(1400,400),
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
# ╔═╡ 51e4a92a-b166-4b67-87bf-ee1c3f337500
begin
size(sinogram)
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
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ImageView = "86fae568-95e7-573e-a6b2-d8a6b900c9ef"
Images = "916415d5-f1e6-5110-898d-aaa5f9f070e0"
LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
Symbolics = "0c5d862f-8b57-4792-8d23-62f2024744c7"
TestImages = "5e47fb64-e119-507b-a336-dd2b206d9990"
[compat]
ForwardDiff = "~0.10.35"
ImageView = "~0.11.5"
Images = "~0.25.3"
LaTeXStrings = "~1.3.0"
Plots = "~1.38.15"
PlutoUI = "~0.7.51"
Symbolics = "~5.5.0"
TestImages = "~1.7.1"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
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