medicinal-imaging/exercise/3. Computed Tomography/Part2_inverse problem - exercise.jl

### A Pluto.jl notebook ###
# v0.19.12

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

# ╔═╡ b573185c-25aa-11eb-14ee-4f35f39e4f60
using DelimitedFiles, Plots, PlutoUI, FFTW; gr(size = (640, 360)); PlutoUI.TableOfContents(depth=4)

# ╔═╡ 00c34a30-2834-11eb-331c-95de2567ce32
using Interpolations

# ╔═╡ 9c0deb3c-25a8-11eb-239d-b739f0934042
md"""
# Numerical Methods of Medical Imaging - Exercise 3

[Institute for Biomedical Imaging](https://www.tuhh.de/ibi/home.html), Hamburg University of Technology

* 👨‍🏫 Prof. Dr. Tobias Knopp
* 👨‍🏫 Dr. Martin Möddel
"""

# ╔═╡ ed576dc2-25a8-11eb-12c3-35fcb09a7d44
md"""
# Computed Tomography - Part II

* 📅 Due date: 22.11.2022, 12 a.m.
* You can earn 1 point for each task (7 points in total).
* 💣 Part II contains also a small coding challenge for those interested (1 bonus point).

The inverse problem in computed tomography is to determine the attenuation given some Radon data. Here, two different inversion strategies are discussed.
"""

# ╔═╡ 499a2a82-25ad-11eb-1cc4-91e69d30a6e6
md"""
## Shepp Logan Phantom

To start off we are going to load and visualize the Shepp Logan phantom Radon data.
"""

# ╔═╡ 9cfc0e64-25aa-11eb-19c4-d78e0f6b03e8
begin
	p = readdlm("sheppLogan.dat")
	M,N = size(p)
	ξs=range(-1/sqrt(2)*(1-1/M),step=2/sqrt(2)/M,length=M)
	γs = range(0.0,step=π/N,length=N)
end

# ╔═╡ d64e93ec-25aa-11eb-129c-bd82a7f07ffa
heatmap(γs,ξs,p,
	c = :viridis,
	xlabel = "γ",
	ylabel = "ξ"
)

# ╔═╡ 12d9ae66-25ad-11eb-102b-7ddeebec9526
md"""
## 4. Filtered Backprojection

In practice Radon data $p(\xi,\gamma)$ is inverted using the filtered backprojection, which consists of three steps:

* The first step consists of a 1D Fourier transformation along the $\xi$-dimension

$P(q,\gamma) = \int_{\mathbb R} p(\xi,\gamma)e^{-2\pi i \xi q} d\xi.$

* Next, a ramp filter is applied and the result is transformed back into Radon space

$h(\xi,\gamma) = \int_{\mathbb R} P(q,\gamma)\vert q \vert e^{2\pi i\xi q}dq.$

* Last, the attanuation coefficient is obtained by integrating $h(\xi,\gamma)$ along the path $\xi(\gamma) = x\cos \gamma +y\sin\gamma$

$\mu(x,y) = \int_0^\pi h(\xi(\gamma),\gamma)d\gamma.$
"""

# ╔═╡ 92d01a86-25bc-11eb-3736-7f409c153fd7
md"""
#### 🎓 Task

Calculate `P` from the discrete Shepp Logan phantom Radon data `p` by 1D Fourier transformation along the $\xi$-dimension. Exploit the fact that `p` is a real array and use the `rfft` instead of the `fft`.
"""

# ╔═╡ 32c477c8-25b3-11eb-20f8-fb55bc0566b5
P = p;

# ╔═╡ ead8c092-25b3-11eb-28ec-7514cf5ffb06
heatmap(abs.(P),
	c = :viridis,
	xlabel = "γ",
	ylabel = "ξ"
)

# ╔═╡ 08158066-25bc-11eb-0820-c361b2249fc8
md"""
#### 🎓 Task
Calculate `h` by filtering and back transformation of `P`. Use the `irfft` function for efficient inverse Fourier transformation.
"""

# ╔═╡ 61dbb53c-25b9-11eb-387c-fdc757db0931
h = p;

# ╔═╡ 9d07c95c-25b9-11eb-1945-b72a11da9806
heatmap(γs,ξs,h,
	c = :viridis,
	xlabel = "γ",
	ylabel = "ξ"
)

# ╔═╡ c1db14a0-25b9-11eb-109f-41ec55c0146c
md"""
#### 🎓 Task

Write a function `μ(h,x,y)` to approximate

$\mu(x,y) = \int_0^\pi h(\xi(\gamma),\gamma)d\gamma$

from `h`. Approximate this integral by taking the discretization of your data into account. Make use of every angle available along the $\gamma$-direction of the discrete data `h` for approximation using a simple Riemann sum.

Along the $\xi$-direction $\xi(\gamma)$ will likely not fall onto the discretization grid, in which case the exact value $h(\xi(\gamma),\gamma)$ is unknown. To this end, search for the nearest data point in `h` (nearest-neighbor interpolation) along the $\xi$-direction and use it as an estimate for $h(\xi(\gamma),\gamma)$.
"""

# ╔═╡ c07167ce-25c1-11eb-2290-cd8871136b7a
function μ(h,x,y)
	M,N = size(h)
	ξs = range(-1/sqrt(2)*(1-1/M),step=2/sqrt(2)/M,length=M)
	γs = range(0.0,step=π/N,length=N)
	out = zero(eltype(h))
	
	# do stuff
	
	return out
end

# ╔═╡ d2a7da5d-2290-4d78-9567-6b0a24ca0156
md"Adapt the grid size:"

# ╔═╡ 7edc23ac-2815-11eb-24db-27b9485ebf22
@bind L Slider(1:150, default=25, show_value=true)

# ╔═╡ 63135366-2815-11eb-3c25-cdfc9853385e
locations = range(-1/2*(1-1/L),step=1/L,length=L)

# ╔═╡ 693919f0-281b-11eb-1f01-87295724208e
md"""
## 5. Fourier Based Reconstruction in Computed Tomography

* 💣 This is no regular exercise but a small coding challenge for those interested.

![](https://imgs.xkcd.com/comics/frequency/car_germany.gif)

"Frequency" taken from [xkcd](https://xkcd.com/1331/) is licensed under [CC](https://en.wikipedia.org/wiki/en:Creative_Commons) [BY-NC 2.5](https://creativecommons.org/licenses/by-nc/2.5/) license. Check out the link for more!

The Radon transformation is given by

$p(\xi,\gamma) = \int_{-\infty}^\infty \mu(\xi \cos\gamma - \eta \sin\gamma, \xi\sin \gamma +\eta \cos\gamma)\text{d}\eta,$

the 1D-Fourier transform of $p(\xi,\gamma)$ along the first dimension by

$P(q,\gamma) = \int_{-\infty}^\infty p(\xi,\gamma) e^{-2\pi i q\xi} \text{d}\xi$

the 2D-Fourier transform of the attenuation coefficients $\mu(x,y)$ by

$M(u,v) = \int_{\mathbb R^2} \mu(x,y) e^{-2\pi i (xu+yv)} \text{d}x\text{d}y.$

The **Fourier-Slice-Theorem** states that

$P(q,\gamma) = M(q\cos\gamma,q\sin\gamma)$

which can be used for a reconstruction of our Radon data

$\mu(x,y) = \int_{\mathbb R^2} M(u,v) e^{2\pi i (xu+yv)} \text{d}u\text{d}v.$

The steps neccessary are:
* First, a 1D discrete Fourier transform of $p$ along the $\xi$-direction.
* Second, an interpolation from polar to Cartesian grid. I.e. a direct application of the theorem.
* Last, an inverse 2D Fourier transform of $M$.

#### 🎓💣 Task
Implement a function returning $\mu$ given the radon data `p` and final grid size `K` using the Fourier based reconstruction method just described.

Hints:
* Consider a vector `v`. Use the functions [`fftshift`, `ifftshift`](https://juliamath.github.io/AbstractFFTs.jl/stable/api/#AbstractFFTs.fftshift) for a circular shift of the periodic signal so that it becomes centered at index 1. Its discrete Fourier transform `fftshift(fft(ifftshift(v)))` then has frequencies 
```julia
M = length(fftshift(fft(ifftshift(v))))
if iseven(M)
    frequencies = -div(M,2):div(M,2)-1
else
    frequencies = -div(M-1,2):div(M-1,2)
end
```

* In this particular case you only have angles $0\le\gamma<\pi$ available. Hence, you will need to make use of $(q\cos(\gamma),q\sin(\gamma)) = -(q\cos(\gamma-\pi),q\sin(\gamma-\pi))$ for polar-to-Cartesian transformation.
* For interpolation (and extrapolation if you need it) you can use the `Interpolations` package. Of special interest will be [this](https://juliamath.github.io/Interpolations.jl/latest/control/#Gridded-interpolation-1) segment in the documentation.
"""

# ╔═╡ 6d65d24c-2821-11eb-304e-4f59c903950e
function μ(p, K=100)
	
	# do some cool stuff
	
	return zeros(eltype(p),K,K)
end

# ╔═╡ d575d7c2-2814-11eb-32ae-1991037b7658
phantom = [μ(h,x,y) for x in locations, y in locations];

# ╔═╡ 16c687b2-2815-11eb-1535-d95a9253aeb7
heatmap(locations,locations,phantom,
	c = :viridis,
	xlims = (-0.5,0.5),
	ylims = (-0.5,0.5),
	aspect_ratio = 1,
	xlabel = "x",
	ylabel = "y"
)

# ╔═╡ cf191765-4d82-4f3e-b4f6-788bdd4f7d8c
md"Adapt the grid size `K`:"

# ╔═╡ 231e5ff4-283c-11eb-091b-ff43ad805431
@bind K Slider(1:1:150, default=41, show_value=true)

# ╔═╡ d90abf40-2834-11eb-3861-eda6ee7a1000
heatmap(abs.(μ(p,K)),
	c = :viridis,
	aspect_ratio = 1,
	xlabel = "x",
	ylabel = "y"
)

# ╔═╡ b7469c30-0b02-425f-b064-149ef2f86d4b
begin
	hint(text) = Markdown.MD(Markdown.Admonition("hint", "Hint", [text]))
	not_defined(variable_name) = Markdown.MD(Markdown.Admonition("danger", "Oh, oh! 😱", [md"Variable **$(Markdown.Code(string(variable_name)))** is not defined. You should probably do something about this."]))
	failed_test(text=md"Some tests failed. Keep working on it!") = Markdown.MD(Markdown.Admonition("danger", "Oh, oh! 😱", [text]))
	still_missing(text=md"Replace `missing` with your solution.") = Markdown.MD(Markdown.Admonition("warning", "Let's go!", [text]))
	keep_working(text=md"The answer is not quite right.") = Markdown.MD(Markdown.Admonition("danger", "Keep working on it!", [text]));
	yays = [md"Great! 🥳", md"Correct! 👏", md"Tada! 🎉"]
	correct(text=md"$(rand(yays)) Let's move on to the next task.") = Markdown.MD(Markdown.Admonition("correct", "Got it!", [text]))
end;

# ╔═╡ d649e96f-f54d-450e-9fb8-b1d306f3de4a
if size(P) != (184,180)
	keep_working(md"The size of `P` is not quite right.")
elseif !(isapprox(8044.0 + 0.0im,P[1,1],atol=1e-5)) || !(isapprox(-3.1296370420222672 - 13.524313951361393im,P[100,100],atol=1e-5))
	keep_working(md"`P` is not quite right.")
else
	correct()
end

# ╔═╡ 47db0f4a-dd92-400a-86f5-99c8a27778f4
let x = zeros(5,5); x[3,3] = 1.0; μtest(r) = norm(r,2)<=0.5 ? 1.0 : 0.0
	if size(h) != (367,180)
		keep_working(md"The size of `h` is not quite right.")
	elseif !(isapprox(-13.520243,h[1,1],atol=1e-5)) || !(isapprox(55.643539,h[100,100],atol=1e-5))
		keep_working(md"`h` is not quite right.")
	else
		correct()
	end
end

# ╔═╡ af59d647-cfe7-4da5-ae93-8876eb11dee3
let x = [-0.134134  -22.1585  -13.1535  -14.7317   -18.8289   -22.5123  -21.4767   -26.1091
  -8.60523   -16.9745  -23.3918   62.7502    50.3356   -16.8203   -6.31704   -2.70192
 -23.227      56.7793   61.2494   67.9982    -8.25005   70.4288   52.8148   -29.4127
 335.286      65.2589  -13.2265   -4.30947  -19.1694    89.4779   59.0391   346.347
 333.685      49.3326   55.06    -14.7453    63.2481   101.868    61.9505   328.903
 -13.1561     67.9968   69.4684   67.1149    65.531     49.8706   65.5727   -20.2306
 -20.0441    -32.0541   55.7314   66.39      61.8352    74.1205   -9.05851   -9.72341
 -14.7304    -13.8669  -16.8306  -10.9165   -22.3654   -23.9891  -33.9383   -24.2068]
	locations = range(-1/2*(1-1/8),step=1/8,length=8)
	M = [μ(h,xx,yy) for xx in locations, yy in locations]
	if !(isapprox(x,M,atol=1e-3))
		keep_working(md"Your version of `μ` is not quite right.")
	else
		correct()
	end
end

# ╔═╡ 2fee5db2-4b63-4c57-8e5c-778fdc4d7153
let locations = range(-1/2*(1-1/8),step=1/8,length=8)
	M = abs.(μ(p,20))
	if !(isapprox(M[1,1],19.67317,atol=1e-3)) || !(isapprox(M[10,10],6.75829,atol=1e-3))
		keep_working(md"Your version of `μ` is not quite right.")
	else
		correct()
	end
end

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
DelimitedFiles = "8bb1440f-4735-579b-a4ab-409b98df4dab"
FFTW = "7a1cc6ca-52ef-59f5-83cd-3a7055c09341"
Interpolations = "a98d9a8b-a2ab-59e6-89dd-64a1c18fca59"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"

[compat]
FFTW = "~1.5.0"
Interpolations = "~0.14.6"
Plots = "~1.35.3"
PlutoUI = "~0.7.43"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised

[[AbstractFFTs]]
deps = ["ChainRulesCore", "LinearAlgebra"]
git-tree-sha1 = "69f7020bd72f069c219b5e8c236c1fa90d2cb409"
uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c"
version = "1.2.1"

[[AbstractPlutoDingetjes]]
deps = ["Pkg"]
git-tree-sha1 = "8eaf9f1b4921132a4cff3f36a1d9ba923b14a481"
uuid = "6e696c72-6542-2067-7265-42206c756150"
version = "1.1.4"

[[Adapt]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "195c5505521008abea5aee4f96930717958eac6f"
uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e"
version = "3.4.0"

[[ArgTools]]
uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f"
version = "1.1.1"

[[Artifacts]]
uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33"

[[AxisAlgorithms]]
deps = ["LinearAlgebra", "Random", "SparseArrays", "WoodburyMatrices"]
git-tree-sha1 = "66771c8d21c8ff5e3a93379480a2307ac36863f7"
uuid = "13072b0f-2c55-5437-9ae7-d433b7a33950"
version = "1.0.1"

[[Base64]]
uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f"

[[BitFlags]]
git-tree-sha1 = "84259bb6172806304b9101094a7cc4bc6f56dbc6"
uuid = "d1d4a3ce-64b1-5f1a-9ba4-7e7e69966f35"
version = "0.1.5"

[[Bzip2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "19a35467a82e236ff51bc17a3a44b69ef35185a2"
uuid = "6e34b625-4abd-537c-b88f-471c36dfa7a0"
version = "1.0.8+0"

[[Cairo_jll]]
deps = ["Artifacts", "Bzip2_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "JLLWrappers", "LZO_jll", "Libdl", "Pixman_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll", "Zlib_jll", "libpng_jll"]
git-tree-sha1 = "4b859a208b2397a7a623a03449e4636bdb17bcf2"
uuid = "83423d85-b0ee-5818-9007-b63ccbeb887a"
version = "1.16.1+1"

[[ChainRulesCore]]
deps = ["Compat", "LinearAlgebra", "SparseArrays"]
git-tree-sha1 = "e7ff6cadf743c098e08fca25c91103ee4303c9bb"
uuid = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
version = "1.15.6"

[[ChangesOfVariables]]
deps = ["ChainRulesCore", "LinearAlgebra", "Test"]
git-tree-sha1 = "38f7a08f19d8810338d4f5085211c7dfa5d5bdd8"
uuid = "9e997f8a-9a97-42d5-a9f1-ce6bfc15e2c0"
version = "0.1.4"

[[CodecZlib]]
deps = ["TranscodingStreams", "Zlib_jll"]
git-tree-sha1 = "ded953804d019afa9a3f98981d99b33e3db7b6da"
uuid = "944b1d66-785c-5afd-91f1-9de20f533193"
version = "0.7.0"

[[ColorSchemes]]
deps = ["ColorTypes", "ColorVectorSpace", "Colors", "FixedPointNumbers", "Random"]
git-tree-sha1 = "1fd869cc3875b57347f7027521f561cf46d1fcd8"
uuid = "35d6a980-a343-548e-a6ea-1d62b119f2f4"
version = "3.19.0"

[[ColorTypes]]
deps = ["FixedPointNumbers", "Random"]
git-tree-sha1 = "eb7f0f8307f71fac7c606984ea5fb2817275d6e4"
uuid = "3da002f7-5984-5a60-b8a6-cbb66c0b333f"
version = "0.11.4"

[[ColorVectorSpace]]
deps = ["ColorTypes", "FixedPointNumbers", "LinearAlgebra", "SpecialFunctions", "Statistics", "TensorCore"]
git-tree-sha1 = "d08c20eef1f2cbc6e60fd3612ac4340b89fea322"
uuid = "c3611d14-8923-5661-9e6a-0046d554d3a4"
version = "0.9.9"

[[Colors]]
deps = ["ColorTypes", "FixedPointNumbers", "Reexport"]
git-tree-sha1 = "417b0ed7b8b838aa6ca0a87aadf1bb9eb111ce40"
uuid = "5ae59095-9a9b-59fe-a467-6f913c188581"
version = "0.12.8"

[[Compat]]
deps = ["Dates", "LinearAlgebra", "UUIDs"]
git-tree-sha1 = "3ca828fe1b75fa84b021a7860bd039eaea84d2f2"
uuid = "34da2185-b29b-5c13-b0c7-acf172513d20"
version = "4.3.0"

[[CompilerSupportLibraries_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "e66e0078-7015-5450-92f7-15fbd957f2ae"
version = "0.5.2+0"

[[Contour]]
git-tree-sha1 = "d05d9e7b7aedff4e5b51a029dced05cfb6125781"
uuid = "d38c429a-6771-53c6-b99e-75d170b6e991"
version = "0.6.2"

[[DataAPI]]
git-tree-sha1 = "46d2680e618f8abd007bce0c3026cb0c4a8f2032"
uuid = "9a962f9c-6df0-11e9-0e5d-c546b8b5ee8a"
version = "1.12.0"

[[DataStructures]]
deps = ["Compat", "InteractiveUtils", "OrderedCollections"]
git-tree-sha1 = "d1fff3a548102f48987a52a2e0d114fa97d730f0"
uuid = "864edb3b-99cc-5e75-8d2d-829cb0a9cfe8"
version = "0.18.13"

[[Dates]]
deps = ["Printf"]
uuid = "ade2ca70-3891-5945-98fb-dc099432e06a"

[[DelimitedFiles]]
deps = ["Mmap"]
uuid = "8bb1440f-4735-579b-a4ab-409b98df4dab"

[[Distributed]]
deps = ["Random", "Serialization", "Sockets"]
uuid = "8ba89e20-285c-5b6f-9357-94700520ee1b"

[[DocStringExtensions]]
deps = ["LibGit2"]
git-tree-sha1 = "5158c2b41018c5f7eb1470d558127ac274eca0c9"
uuid = "ffbed154-4ef7-542d-bbb7-c09d3a79fcae"
version = "0.9.1"

[[Downloads]]
deps = ["ArgTools", "FileWatching", "LibCURL", "NetworkOptions"]
uuid = "f43a241f-c20a-4ad4-852c-f6b1247861c6"
version = "1.6.0"

[[Expat_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "bad72f730e9e91c08d9427d5e8db95478a3c323d"
uuid = "2e619515-83b5-522b-bb60-26c02a35a201"
version = "2.4.8+0"

[[FFMPEG]]
deps = ["FFMPEG_jll"]
git-tree-sha1 = "b57e3acbe22f8484b4b5ff66a7499717fe1a9cc8"
uuid = "c87230d0-a227-11e9-1b43-d7ebe4e7570a"
version = "0.4.1"

[[FFMPEG_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "JLLWrappers", "LAME_jll", "Libdl", "Ogg_jll", "OpenSSL_jll", "Opus_jll", "PCRE2_jll", "Pkg", "Zlib_jll", "libaom_jll", "libass_jll", "libfdk_aac_jll", "libvorbis_jll", "x264_jll", "x265_jll"]
git-tree-sha1 = "74faea50c1d007c85837327f6775bea60b5492dd"
uuid = "b22a6f82-2f65-5046-a5b2-351ab43fb4e5"
version = "4.4.2+2"

[[FFTW]]
deps = ["AbstractFFTs", "FFTW_jll", "LinearAlgebra", "MKL_jll", "Preferences", "Reexport"]
git-tree-sha1 = "90630efff0894f8142308e334473eba54c433549"
uuid = "7a1cc6ca-52ef-59f5-83cd-3a7055c09341"
version = "1.5.0"

[[FFTW_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "c6033cc3892d0ef5bb9cd29b7f2f0331ea5184ea"
uuid = "f5851436-0d7a-5f13-b9de-f02708fd171a"
version = "3.3.10+0"

[[FileWatching]]
uuid = "7b1f6079-737a-58dc-b8bc-7a2ca5c1b5ee"

[[FixedPointNumbers]]
deps = ["Statistics"]
git-tree-sha1 = "335bfdceacc84c5cdf16aadc768aa5ddfc5383cc"
uuid = "53c48c17-4a7d-5ca2-90c5-79b7896eea93"
version = "0.8.4"

[[Fontconfig_jll]]
deps = ["Artifacts", "Bzip2_jll", "Expat_jll", "FreeType2_jll", "JLLWrappers", "Libdl", "Libuuid_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "21efd19106a55620a188615da6d3d06cd7f6ee03"
uuid = "a3f928ae-7b40-5064-980b-68af3947d34b"
version = "2.13.93+0"

[[Formatting]]
deps = ["Printf"]
git-tree-sha1 = "8339d61043228fdd3eb658d86c926cb282ae72a8"
uuid = "59287772-0a20-5a39-b81b-1366585eb4c0"
version = "0.4.2"

[[FreeType2_jll]]
deps = ["Artifacts", "Bzip2_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "87eb71354d8ec1a96d4a7636bd57a7347dde3ef9"
uuid = "d7e528f0-a631-5988-bf34-fe36492bcfd7"
version = "2.10.4+0"

[[FriBidi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "aa31987c2ba8704e23c6c8ba8a4f769d5d7e4f91"
uuid = "559328eb-81f9-559d-9380-de523a88c83c"
version = "1.0.10+0"

[[GLFW_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pkg", "Xorg_libXcursor_jll", "Xorg_libXi_jll", "Xorg_libXinerama_jll", "Xorg_libXrandr_jll"]
git-tree-sha1 = "d972031d28c8c8d9d7b41a536ad7bb0c2579caca"
uuid = "0656b61e-2033-5cc2-a64a-77c0f6c09b89"
version = "3.3.8+0"

[[GR]]
deps = ["Base64", "DelimitedFiles", "GR_jll", "HTTP", "JSON", "Libdl", "LinearAlgebra", "Pkg", "Preferences", "Printf", "Random", "Serialization", "Sockets", "Test", "UUIDs"]
git-tree-sha1 = "cf7bf90e483228f6c988e474b420064e5351b892"
uuid = "28b8d3ca-fb5f-59d9-8090-bfdbd6d07a71"
version = "0.69.4"

[[GR_jll]]
deps = ["Artifacts", "Bzip2_jll", "Cairo_jll", "FFMPEG_jll", "Fontconfig_jll", "GLFW_jll", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Libtiff_jll", "Pixman_jll", "Pkg", "Qt5Base_jll", "Zlib_jll", "libpng_jll"]
git-tree-sha1 = "bc9f7725571ddb4ab2c4bc74fa397c1c5ad08943"
uuid = "d2c73de3-f751-5644-a686-071e5b155ba9"
version = "0.69.1+0"

[[Gettext_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "9b02998aba7bf074d14de89f9d37ca24a1a0b046"
uuid = "78b55507-aeef-58d4-861c-77aaff3498b1"
version = "0.21.0+0"

[[Glib_jll]]
deps = ["Artifacts", "Gettext_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Libiconv_jll", "Libmount_jll", "PCRE2_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "fb83fbe02fe57f2c068013aa94bcdf6760d3a7a7"
uuid = "7746bdde-850d-59dc-9ae8-88ece973131d"
version = "2.74.0+1"

[[Graphite2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "344bf40dcab1073aca04aa0df4fb092f920e4011"
uuid = "3b182d85-2403-5c21-9c21-1e1f0cc25472"
version = "1.3.14+0"

[[Grisu]]
git-tree-sha1 = "53bb909d1151e57e2484c3d1b53e19552b887fb2"
uuid = "42e2da0e-8278-4e71-bc24-59509adca0fe"
version = "1.0.2"

[[HTTP]]
deps = ["Base64", "CodecZlib", "Dates", "IniFile", "Logging", "LoggingExtras", "MbedTLS", "NetworkOptions", "OpenSSL", "Random", "SimpleBufferStream", "Sockets", "URIs", "UUIDs"]
git-tree-sha1 = "4abede886fcba15cd5fd041fef776b230d004cee"
uuid = "cd3eb016-35fb-5094-929b-558a96fad6f3"
version = "1.4.0"

[[HarfBuzz_jll]]
deps = ["Artifacts", "Cairo_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "Graphite2_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg"]
git-tree-sha1 = "129acf094d168394e80ee1dc4bc06ec835e510a3"
uuid = "2e76f6c2-a576-52d4-95c1-20adfe4de566"
version = "2.8.1+1"

[[Hyperscript]]
deps = ["Test"]
git-tree-sha1 = "8d511d5b81240fc8e6802386302675bdf47737b9"
uuid = "47d2ed2b-36de-50cf-bf87-49c2cf4b8b91"
version = "0.0.4"

[[HypertextLiteral]]
deps = ["Tricks"]
git-tree-sha1 = "c47c5fa4c5308f27ccaac35504858d8914e102f9"
uuid = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2"
version = "0.9.4"

[[IOCapture]]
deps = ["Logging", "Random"]
git-tree-sha1 = "f7be53659ab06ddc986428d3a9dcc95f6fa6705a"
uuid = "b5f81e59-6552-4d32-b1f0-c071b021bf89"
version = "0.2.2"

[[IniFile]]
git-tree-sha1 = "f550e6e32074c939295eb5ea6de31849ac2c9625"
uuid = "83e8ac13-25f8-5344-8a64-a9f2b223428f"
version = "0.5.1"

[[IntelOpenMP_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "d979e54b71da82f3a65b62553da4fc3d18c9004c"
uuid = "1d5cc7b8-4909-519e-a0f8-d0f5ad9712d0"
version = "2018.0.3+2"

[[InteractiveUtils]]
deps = ["Markdown"]
uuid = "b77e0a4c-d291-57a0-90e8-8db25a27a240"

[[Interpolations]]
deps = ["Adapt", "AxisAlgorithms", "ChainRulesCore", "LinearAlgebra", "OffsetArrays", "Random", "Ratios", "Requires", "SharedArrays", "SparseArrays", "StaticArrays", "WoodburyMatrices"]
git-tree-sha1 = "842dd89a6cb75e02e85fdd75c760cdc43f5d6863"
uuid = "a98d9a8b-a2ab-59e6-89dd-64a1c18fca59"
version = "0.14.6"

[[InverseFunctions]]
deps = ["Test"]
git-tree-sha1 = "49510dfcb407e572524ba94aeae2fced1f3feb0f"
uuid = "3587e190-3f89-42d0-90ee-14403ec27112"
version = "0.1.8"

[[IrrationalConstants]]
git-tree-sha1 = "7fd44fd4ff43fc60815f8e764c0f352b83c49151"
uuid = "92d709cd-6900-40b7-9082-c6be49f344b6"
version = "0.1.1"

[[JLFzf]]
deps = ["Pipe", "REPL", "Random", "fzf_jll"]
git-tree-sha1 = "f377670cda23b6b7c1c0b3893e37451c5c1a2185"
uuid = "1019f520-868f-41f5-a6de-eb00f4b6a39c"
version = "0.1.5"

[[JLLWrappers]]
deps = ["Preferences"]
git-tree-sha1 = "abc9885a7ca2052a736a600f7fa66209f96506e1"
uuid = "692b3bcd-3c85-4b1f-b108-f13ce0eb3210"
version = "1.4.1"

[[JSON]]
deps = ["Dates", "Mmap", "Parsers", "Unicode"]
git-tree-sha1 = "3c837543ddb02250ef42f4738347454f95079d4e"
uuid = "682c06a0-de6a-54ab-a142-c8b1cf79cde6"
version = "0.21.3"

[[JpegTurbo_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b53380851c6e6664204efb2e62cd24fa5c47e4ba"
uuid = "aacddb02-875f-59d6-b918-886e6ef4fbf8"
version = "2.1.2+0"

[[LAME_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "f6250b16881adf048549549fba48b1161acdac8c"
uuid = "c1c5ebd0-6772-5130-a774-d5fcae4a789d"
version = "3.100.1+0"

[[LERC_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "bf36f528eec6634efc60d7ec062008f171071434"
uuid = "88015f11-f218-50d7-93a8-a6af411a945d"
version = "3.0.0+1"

[[LZO_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e5b909bcf985c5e2605737d2ce278ed791b89be6"
uuid = "dd4b983a-f0e5-5f8d-a1b7-129d4a5fb1ac"
version = "2.10.1+0"

[[LaTeXStrings]]
git-tree-sha1 = "f2355693d6778a178ade15952b7ac47a4ff97996"
uuid = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f"
version = "1.3.0"

[[Latexify]]
deps = ["Formatting", "InteractiveUtils", "LaTeXStrings", "MacroTools", "Markdown", "OrderedCollections", "Printf", "Requires"]
git-tree-sha1 = "ab9aa169d2160129beb241cb2750ca499b4e90e9"
uuid = "23fbe1c1-3f47-55db-b15f-69d7ec21a316"
version = "0.15.17"

[[LazyArtifacts]]
deps = ["Artifacts", "Pkg"]
uuid = "4af54fe1-eca0-43a8-85a7-787d91b784e3"

[[LibCURL]]
deps = ["LibCURL_jll", "MozillaCACerts_jll"]
uuid = "b27032c2-a3e7-50c8-80cd-2d36dbcbfd21"
version = "0.6.3"

[[LibCURL_jll]]
deps = ["Artifacts", "LibSSH2_jll", "Libdl", "MbedTLS_jll", "Zlib_jll", "nghttp2_jll"]
uuid = "deac9b47-8bc7-5906-a0fe-35ac56dc84c0"
version = "7.84.0+0"

[[LibGit2]]
deps = ["Base64", "NetworkOptions", "Printf", "SHA"]
uuid = "76f85450-5226-5b5a-8eaa-529ad045b433"

[[LibSSH2_jll]]
deps = ["Artifacts", "Libdl", "MbedTLS_jll"]
uuid = "29816b5a-b9ab-546f-933c-edad1886dfa8"
version = "1.10.2+0"

[[Libdl]]
uuid = "8f399da3-3557-5675-b5ff-fb832c97cbdb"

[[Libffi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "0b4a5d71f3e5200a7dff793393e09dfc2d874290"
uuid = "e9f186c6-92d2-5b65-8a66-fee21dc1b490"
version = "3.2.2+1"

[[Libgcrypt_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgpg_error_jll", "Pkg"]
git-tree-sha1 = "64613c82a59c120435c067c2b809fc61cf5166ae"
uuid = "d4300ac3-e22c-5743-9152-c294e39db1e4"
version = "1.8.7+0"

[[Libglvnd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll", "Xorg_libXext_jll"]
git-tree-sha1 = "7739f837d6447403596a75d19ed01fd08d6f56bf"
uuid = "7e76a0d4-f3c7-5321-8279-8d96eeed0f29"
version = "1.3.0+3"

[[Libgpg_error_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "c333716e46366857753e273ce6a69ee0945a6db9"
uuid = "7add5ba3-2f88-524e-9cd5-f83b8a55f7b8"
version = "1.42.0+0"

[[Libiconv_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "42b62845d70a619f063a7da093d995ec8e15e778"
uuid = "94ce4f54-9a6c-5748-9c1c-f9c7231a4531"
version = "1.16.1+1"

[[Libmount_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "9c30530bf0effd46e15e0fdcf2b8636e78cbbd73"
uuid = "4b2f31a3-9ecc-558c-b454-b3730dcb73e9"
version = "2.35.0+0"

[[Libtiff_jll]]
deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "LERC_jll", "Libdl", "Pkg", "Zlib_jll", "Zstd_jll"]
git-tree-sha1 = "3eb79b0ca5764d4799c06699573fd8f533259713"
uuid = "89763e89-9b03-5906-acba-b20f662cd828"
version = "4.4.0+0"

[[Libuuid_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7f3efec06033682db852f8b3bc3c1d2b0a0ab066"
uuid = "38a345b3-de98-5d2b-a5d3-14cd9215e700"
version = "2.36.0+0"

[[LinearAlgebra]]
deps = ["Libdl", "libblastrampoline_jll"]
uuid = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"

[[LogExpFunctions]]
deps = ["ChainRulesCore", "ChangesOfVariables", "DocStringExtensions", "InverseFunctions", "IrrationalConstants", "LinearAlgebra"]
git-tree-sha1 = "94d9c52ca447e23eac0c0f074effbcd38830deb5"
uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688"
version = "0.3.18"

[[Logging]]
uuid = "56ddb016-857b-54e1-b83d-db4d58db5568"

[[LoggingExtras]]
deps = ["Dates", "Logging"]
git-tree-sha1 = "5d4d2d9904227b8bd66386c1138cf4d5ffa826bf"
uuid = "e6f89c97-d47a-5376-807f-9c37f3926c36"
version = "0.4.9"

[[MKL_jll]]
deps = ["Artifacts", "IntelOpenMP_jll", "JLLWrappers", "LazyArtifacts", "Libdl", "Pkg"]
git-tree-sha1 = "41d162ae9c868218b1f3fe78cba878aa348c2d26"
uuid = "856f044c-d86e-5d09-b602-aeab76dc8ba7"
version = "2022.1.0+0"

[[MacroTools]]
deps = ["Markdown", "Random"]
git-tree-sha1 = "42324d08725e200c23d4dfb549e0d5d89dede2d2"
uuid = "1914dd2f-81c6-5fcd-8719-6d5c9610ff09"
version = "0.5.10"

[[Markdown]]
deps = ["Base64"]
uuid = "d6f4376e-aef5-505a-96c1-9c027394607a"

[[MbedTLS]]
deps = ["Dates", "MbedTLS_jll", "MozillaCACerts_jll", "Random", "Sockets"]
git-tree-sha1 = "6872f9594ff273da6d13c7c1a1545d5a8c7d0c1c"
uuid = "739be429-bea8-5141-9913-cc70e7f3736d"
version = "1.1.6"

[[MbedTLS_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "c8ffd9c3-330d-5841-b78e-0817d7145fa1"
version = "2.28.0+0"

[[Measures]]
git-tree-sha1 = "e498ddeee6f9fdb4551ce855a46f54dbd900245f"
uuid = "442fdcdd-2543-5da2-b0f3-8c86c306513e"
version = "0.3.1"

[[Missings]]
deps = ["DataAPI"]
git-tree-sha1 = "bf210ce90b6c9eed32d25dbcae1ebc565df2687f"
uuid = "e1d29d7a-bbdc-5cf2-9ac0-f12de2c33e28"
version = "1.0.2"

[[Mmap]]
uuid = "a63ad114-7e13-5084-954f-fe012c677804"

[[MozillaCACerts_jll]]
uuid = "14a3606d-f60d-562e-9121-12d972cd8159"
version = "2022.2.1"

[[NaNMath]]
deps = ["OpenLibm_jll"]
git-tree-sha1 = "a7c3d1da1189a1c2fe843a3bfa04d18d20eb3211"
uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
version = "1.0.1"

[[NetworkOptions]]
uuid = "ca575930-c2e3-43a9-ace4-1e988b2c1908"
version = "1.2.0"

[[OffsetArrays]]
deps = ["Adapt"]
git-tree-sha1 = "1ea784113a6aa054c5ebd95945fa5e52c2f378e7"
uuid = "6fe1bfb0-de20-5000-8ca7-80f57d26f881"
version = "1.12.7"

[[Ogg_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "887579a3eb005446d514ab7aeac5d1d027658b8f"
uuid = "e7412a2a-1a6e-54c0-be00-318e2571c051"
version = "1.3.5+1"

[[OpenBLAS_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Libdl"]
uuid = "4536629a-c528-5b80-bd46-f80d51c5b363"
version = "0.3.20+0"

[[OpenLibm_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "05823500-19ac-5b8b-9628-191a04bc5112"
version = "0.8.1+0"

[[OpenSSL]]
deps = ["BitFlags", "Dates", "MozillaCACerts_jll", "OpenSSL_jll", "Sockets"]
git-tree-sha1 = "ebe81469e9d7b471d7ddb611d9e147ea16de0add"
uuid = "4d8831e6-92b7-49fb-bdf8-b643e874388c"
version = "1.2.1"

[[OpenSSL_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e60321e3f2616584ff98f0a4f18d98ae6f89bbb3"
uuid = "458c3c95-2e84-50aa-8efc-19380b2a3a95"
version = "1.1.17+0"

[[OpenSpecFun_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "13652491f6856acfd2db29360e1bbcd4565d04f1"
uuid = "efe28fd5-8261-553b-a9e1-b2916fc3738e"
version = "0.5.5+0"

[[Opus_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "51a08fb14ec28da2ec7a927c4337e4332c2a4720"
uuid = "91d4177d-7536-5919-b921-800302f37372"
version = "1.3.2+0"

[[OrderedCollections]]
git-tree-sha1 = "85f8e6578bf1f9ee0d11e7bb1b1456435479d47c"
uuid = "bac558e1-5e72-5ebc-8fee-abe8a469f55d"
version = "1.4.1"

[[PCRE2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "efcefdf7-47ab-520b-bdef-62a2eaa19f15"
version = "10.40.0+0"

[[Parsers]]
deps = ["Dates"]
git-tree-sha1 = "595c0b811cf2bab8b0849a70d9bd6379cc1cfb52"
uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0"
version = "2.4.1"

[[Pipe]]
git-tree-sha1 = "6842804e7867b115ca9de748a0cf6b364523c16d"
uuid = "b98c9c47-44ae-5843-9183-064241ee97a0"
version = "1.3.0"

[[Pixman_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b4f5d02549a10e20780a24fce72bea96b6329e29"
uuid = "30392449-352a-5448-841d-b1acce4e97dc"
version = "0.40.1+0"

[[Pkg]]
deps = ["Artifacts", "Dates", "Downloads", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"]
uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"
version = "1.8.0"

[[PlotThemes]]
deps = ["PlotUtils", "Statistics"]
git-tree-sha1 = "8162b2f8547bc23876edd0c5181b27702ae58dce"
uuid = "ccf2f8ad-2431-5c83-bf29-c5338b663b6a"
version = "3.0.0"

[[PlotUtils]]
deps = ["ColorSchemes", "Colors", "Dates", "Printf", "Random", "Reexport", "SnoopPrecompile", "Statistics"]
git-tree-sha1 = "21303256d239f6b484977314674aef4bb1fe4420"
uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043"
version = "1.3.1"

[[Plots]]
deps = ["Base64", "Contour", "Dates", "Downloads", "FFMPEG", "FixedPointNumbers", "GR", "JLFzf", "JSON", "LaTeXStrings", "Latexify", "LinearAlgebra", "Measures", "NaNMath", "Pkg", "PlotThemes", "PlotUtils", "Printf", "REPL", "Random", "RecipesBase", "RecipesPipeline", "Reexport", "RelocatableFolders", "Requires", "Scratch", "Showoff", "SnoopPrecompile", "SparseArrays", "Statistics", "StatsBase", "UUIDs", "UnicodeFun", "Unzip"]
git-tree-sha1 = "524d9ff1b2f4473fef59678c06f9f77160a204b1"
uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
version = "1.35.3"

[[PlutoUI]]
deps = ["AbstractPlutoDingetjes", "Base64", "ColorTypes", "Dates", "Hyperscript", "HypertextLiteral", "IOCapture", "InteractiveUtils", "JSON", "Logging", "Markdown", "Random", "Reexport", "UUIDs"]
git-tree-sha1 = "2777a5c2c91b3145f5aa75b61bb4c2eb38797136"
uuid = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
version = "0.7.43"

[[Preferences]]
deps = ["TOML"]
git-tree-sha1 = "47e5f437cc0e7ef2ce8406ce1e7e24d44915f88d"
uuid = "21216c6a-2e73-6563-6e65-726566657250"
version = "1.3.0"

[[Printf]]
deps = ["Unicode"]
uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"

[[Qt5Base_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Fontconfig_jll", "Glib_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "OpenSSL_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libxcb_jll", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_keysyms_jll", "Xorg_xcb_util_renderutil_jll", "Xorg_xcb_util_wm_jll", "Zlib_jll", "xkbcommon_jll"]
git-tree-sha1 = "c6c0f690d0cc7caddb74cef7aa847b824a16b256"
uuid = "ea2cea3b-5b76-57ae-a6ef-0a8af62496e1"
version = "5.15.3+1"

[[REPL]]
deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"

[[Random]]
deps = ["SHA", "Serialization"]
uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

[[Ratios]]
deps = ["Requires"]
git-tree-sha1 = "dc84268fe0e3335a62e315a3a7cf2afa7178a734"
uuid = "c84ed2f1-dad5-54f0-aa8e-dbefe2724439"
version = "0.4.3"

[[RecipesBase]]
deps = ["SnoopPrecompile"]
git-tree-sha1 = "612a4d76ad98e9722c8ba387614539155a59e30c"
uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
version = "1.3.0"

[[RecipesPipeline]]
deps = ["Dates", "NaNMath", "PlotUtils", "RecipesBase", "SnoopPrecompile"]
git-tree-sha1 = "9b1c0c8e9188950e66fc28f40bfe0f8aac311fe0"
uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c"
version = "0.6.7"

[[Reexport]]
git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b"
uuid = "189a3867-3050-52da-a836-e630ba90ab69"
version = "1.2.2"

[[RelocatableFolders]]
deps = ["SHA", "Scratch"]
git-tree-sha1 = "90bc7a7c96410424509e4263e277e43250c05691"
uuid = "05181044-ff0b-4ac5-8273-598c1e38db00"
version = "1.0.0"

[[Requires]]
deps = ["UUIDs"]
git-tree-sha1 = "838a3a4188e2ded87a4f9f184b4b0d78a1e91cb7"
uuid = "ae029012-a4dd-5104-9daa-d747884805df"
version = "1.3.0"

[[SHA]]
uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"
version = "0.7.0"

[[Scratch]]
deps = ["Dates"]
git-tree-sha1 = "f94f779c94e58bf9ea243e77a37e16d9de9126bd"
uuid = "6c6a2e73-6563-6170-7368-637461726353"
version = "1.1.1"

[[Serialization]]
uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b"

[[SharedArrays]]
deps = ["Distributed", "Mmap", "Random", "Serialization"]
uuid = "1a1011a3-84de-559e-8e89-a11a2f7dc383"

[[Showoff]]
deps = ["Dates", "Grisu"]
git-tree-sha1 = "91eddf657aca81df9ae6ceb20b959ae5653ad1de"
uuid = "992d4aef-0814-514b-bc4d-f2e9a6c4116f"
version = "1.0.3"

[[SimpleBufferStream]]
git-tree-sha1 = "874e8867b33a00e784c8a7e4b60afe9e037b74e1"
uuid = "777ac1f9-54b0-4bf8-805c-2214025038e7"
version = "1.1.0"

[[SnoopPrecompile]]
git-tree-sha1 = "f604441450a3c0569830946e5b33b78c928e1a85"
uuid = "66db9d55-30c0-4569-8b51-7e840670fc0c"
version = "1.0.1"

[[Sockets]]
uuid = "6462fe0b-24de-5631-8697-dd941f90decc"

[[SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "b3363d7460f7d098ca0912c69b082f75625d7508"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.0.1"

[[SparseArrays]]
deps = ["LinearAlgebra", "Random"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

[[SpecialFunctions]]
deps = ["ChainRulesCore", "IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
git-tree-sha1 = "d75bda01f8c31ebb72df80a46c88b25d1c79c56d"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
version = "2.1.7"

[[StaticArrays]]
deps = ["LinearAlgebra", "Random", "StaticArraysCore", "Statistics"]
git-tree-sha1 = "f86b3a049e5d05227b10e15dbb315c5b90f14988"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.5.9"

[[StaticArraysCore]]
git-tree-sha1 = "6b7ba252635a5eff6a0b0664a41ee140a1c9e72a"
uuid = "1e83bf80-4336-4d27-bf5d-d5a4f845583c"
version = "1.4.0"

[[Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

[[StatsAPI]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "f9af7f195fb13589dd2e2d57fdb401717d2eb1f6"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.5.0"

[[StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "d1bf48bfcc554a3761a133fe3a9bb01488e06916"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.21"

[[TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"
version = "1.0.0"

[[Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"
version = "1.10.1"

[[TensorCore]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "1feb45f88d133a655e001435632f019a9a1bcdb6"
uuid = "62fd8b95-f654-4bbd-a8a5-9c27f68ccd50"
version = "0.1.1"

[[Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[TranscodingStreams]]
deps = ["Random", "Test"]
git-tree-sha1 = "8a75929dcd3c38611db2f8d08546decb514fcadf"
uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
version = "0.9.9"

[[Tricks]]
git-tree-sha1 = "6bac775f2d42a611cdfcd1fb217ee719630c4175"
uuid = "410a4b4d-49e4-4fbc-ab6d-cb71b17b3775"
version = "0.1.6"

[[URIs]]
git-tree-sha1 = "e59ecc5a41b000fa94423a578d29290c7266fc10"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.4.0"

[[UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[Unzip]]
git-tree-sha1 = "ca0969166a028236229f63514992fc073799bb78"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.2.0"

[[Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.19.0+0"

[[Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4528479aa01ee1b3b4cd0e6faef0e04cf16466da"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.25.0+0"

[[WoodburyMatrices]]
deps = ["LinearAlgebra", "SparseArrays"]
git-tree-sha1 = "de67fa59e33ad156a590055375a30b23c40299d3"
uuid = "efce3f68-66dc-5838-9240-27a6d6f5f9b6"
version = "0.5.5"

[[XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "58443b63fb7e465a8a7210828c91c08b92132dff"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.14+0"

[[XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.12+3"

[[Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "868e669ccb12ba16eaf50cb2957ee2ff61261c56"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.29.0+0"

[[libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+0"

[[libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.1.1+0"

[[libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.48.0+0"

[[p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"

[[x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+0"
"""

# ╔═╡ Cell order:
# ╟─9c0deb3c-25a8-11eb-239d-b739f0934042
# ╠═b573185c-25aa-11eb-14ee-4f35f39e4f60
# ╟─ed576dc2-25a8-11eb-12c3-35fcb09a7d44
# ╟─499a2a82-25ad-11eb-1cc4-91e69d30a6e6
# ╟─9cfc0e64-25aa-11eb-19c4-d78e0f6b03e8
# ╠═d64e93ec-25aa-11eb-129c-bd82a7f07ffa
# ╟─12d9ae66-25ad-11eb-102b-7ddeebec9526
# ╟─92d01a86-25bc-11eb-3736-7f409c153fd7
# ╠═32c477c8-25b3-11eb-20f8-fb55bc0566b5
# ╟─ead8c092-25b3-11eb-28ec-7514cf5ffb06
# ╟─d649e96f-f54d-450e-9fb8-b1d306f3de4a
# ╟─08158066-25bc-11eb-0820-c361b2249fc8
# ╠═61dbb53c-25b9-11eb-387c-fdc757db0931
# ╟─9d07c95c-25b9-11eb-1945-b72a11da9806
# ╟─47db0f4a-dd92-400a-86f5-99c8a27778f4
# ╟─c1db14a0-25b9-11eb-109f-41ec55c0146c
# ╠═c07167ce-25c1-11eb-2290-cd8871136b7a
# ╟─af59d647-cfe7-4da5-ae93-8876eb11dee3
# ╟─d2a7da5d-2290-4d78-9567-6b0a24ca0156
# ╟─7edc23ac-2815-11eb-24db-27b9485ebf22
# ╟─63135366-2815-11eb-3c25-cdfc9853385e
# ╟─d575d7c2-2814-11eb-32ae-1991037b7658
# ╟─16c687b2-2815-11eb-1535-d95a9253aeb7
# ╟─693919f0-281b-11eb-1f01-87295724208e
# ╠═00c34a30-2834-11eb-331c-95de2567ce32
# ╠═6d65d24c-2821-11eb-304e-4f59c903950e
# ╟─2fee5db2-4b63-4c57-8e5c-778fdc4d7153
# ╟─cf191765-4d82-4f3e-b4f6-788bdd4f7d8c
# ╟─231e5ff4-283c-11eb-091b-ff43ad805431
# ╟─d90abf40-2834-11eb-3861-eda6ee7a1000
# ╟─b7469c30-0b02-425f-b064-149ef2f86d4b
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002