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pluto_ct.jl
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pluto_ct.jl
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### A Pluto.jl notebook ###
# v0.19.16
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
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("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);
z3=zeros(1);
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}\},
\end{aligned}$
where $\vec{\omega}=(\cos\theta,\sin\theta)\in\mathbb{S}^1$. The X-ray transform of a function $f$ of $(x,y)=\mathbb{R}^2$ is defined by
$\mathcal{R}_1f(\gamma)=\mathcal{R}_1f(\theta,t)
:=\int_\gamma f=\int_{-\infty}^\infty f(t\cos\theta-s\sin\theta,t\sin\theta+s\cos\theta)ds,$
Note that $\mathcal{R}_1f(\theta+\pi,-t)=\mathcal{R}_1f(\theta,t)$.
"""
# ╔═╡ 76cb2ec2-fc76-4ad5-8d57-777395755f7f
md"""
## Example
"""
# ╔═╡ 38ea56dd-c2f9-4a67-af20-c01234fbee5b
begin
nbsp = html" "
end
# ╔═╡ 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);
z1=zeros(1);
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);
z2=zeros(1);
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
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
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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 = "~0.10.30"
ImageView = "~0.11.1"
Images = "~0.25.2"
LaTeXStrings = "~1.3.0"
Plots = "~1.31.1"
PlutoUI = "~0.7.39"
Symbolics = "~4.8.3"
TestImages = "~1.7.1"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
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