7.fft.jl

### A Pluto.jl notebook ###
# v0.19.22

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

# ╔═╡ 129914e6-66cb-4ab9-a29e-8e317c0c9bc3
using PlutoUI

# ╔═╡ 8beb3f48-af68-4bc0-9c3e-12ae0713e455
using Test, SparseArrays, LinearAlgebra

# ╔═╡ ab3525e4-c4bb-11ed-2c7e-ab007b9c9196
using FFTW

# ╔═╡ 485cba37-5ada-4678-81ed-977216d29088
using Polynomials

# ╔═╡ 0f81530f-449b-4a32-a0f1-cc35378661ac
using Images

# ╔═╡ c3c583e6-79fa-4a8e-92ca-fa8a95a4982c
using Plots

# ╔═╡ 869f12fb-3ddb-4962-9243-9e0ce1f43e1d
md"""
### Fourier transformation

Given a function $f(x)$, the Fourier transformation is defined as

```math
\hat f(u) = \int_{-\infty}^{\infty} e^{-2\pi iux} f(x) dx.
```

Its inverse process, or the inverse Fourier transformation is defined as

```math
f(x) = \int_{-\infty}^{\infty} e^{2\pi iux} \hat f(u) dk
```
"""

# ╔═╡ 0796609b-286f-48a3-bd2d-faffd6b3073d
md"""
Similarly, given a two variable function $f(x, y)$, the two dimensional Fourier transformation is
```math
\hat f(u, v) = \int_{-\infty}^{\infty}dy\int_{-\infty}^\infty e^{-2\pi i(ux+vy)} f(x, y) dx.
```
The two dimensional inverse Fourier transformation is
```math
f(x, y) = \int_{-\infty}^{\infty}du\int_{-\infty}^\infty e^{2\pi i(ux+vy)} \hat f(u, v) dv.
```
"""

# ╔═╡ e9d47916-62c7-4918-b3cb-51655d9f1757
md"""
Fourier transformation can be used in
1. Image and audio compression,
1. Solving solid state system with translational invariance,
2. Understanding quantum Fourier transformation,
3. Understanding the Fourier optics.
"""

# ╔═╡ bc4696ad-a113-4817-867e-aa18dfe701cf
md"""
## The definition of Descrete Fourier Transformation (DFT)
"""

# ╔═╡ 162db6e3-1a4b-4b55-9583-b71a28205342
md"""
A $n$ dimensional quantum Fourier transformation.
```math
y_{i}=\sum _{n=0}^{n-1}x_{j}\cdot e^{-{\frac {i2\pi }{n}}ij}
```
"""

# ╔═╡ 2a166f85-8940-439c-b27e-3e2beb6ba691
md"""
This transformation is linear, which can be represented as the DFT matrix
"""

# ╔═╡ 44cf16d8-61bb-4c0e-b80d-3ec11876eab0
md"""
```math
F_n = \left(
\begin{matrix}
1 & 1 & 1 & \ldots & 1\\
1 & \omega & \omega^2 & \ldots & \omega^{n-1}\\
1 & \omega^2 & \omega^4 & \ldots & \omega^{2n-2}\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
1 & \omega^{n-1} & \omega^{2n-2} & \ldots & \omega^{(n-1)^2}\\
\end{matrix}
\right)
```
"""

# ╔═╡ 441d306f-ee12-4a31-8240-a516f7c7c5d6
md"""
where $\omega = e^{ik}$
This transformation is also reversible, which can be represented as $F_n^\dagger/n$.
"""

# ╔═╡ 509ffd4f-12de-44d2-aa8d-e2d1b334143c
function dft_matrix(n::Int)
	ω = exp(-2π*im/n)
	return [ω^((i-1)*(j-1)) for i=1:n, j=1:n]
end

# ╔═╡ 215237cd-f214-430a-9ee4-111b2577dbc8
@bind fourier_n NumberField(1:20, default=4)

# ╔═╡ 6b61a16f-3a9a-417b-a337-c85de9834e63
Fn = dft_matrix(fourier_n)

# ╔═╡ f8a52ce2-ef5f-4234-9646-e4f9a9f463f2
# dft matrix is unitary upto constant.
dft_matrix(fourier_n) * dft_matrix(fourier_n)' ./ fourier_n

# ╔═╡ 63a1c5a0-0544-4676-878b-2359ce3b215a
md"## The Cooley–Tukey's Fast Fourier transformation (FFT)"

# ╔═╡ a8d6e254-7a4d-40bd-a1ff-cb9166d5d373
md"We have a recursive algorithm to compute the DFT."

# ╔═╡ c8a87a74-e73b-4b40-ad13-0b8add2c9e72
md"""
```math
F_n x = \left(\begin{matrix}I_{n/2} & D_{n/2}\\I_{n/2} & -D_{n/2} \end{matrix}\right)\left(\begin{matrix} F_{n/2} & 0 \\ 0 & F_{n/2}\end{matrix}\right)\left(\begin{matrix}x_{\rm odd}\\x_{\rm even}\end{matrix}\right)
```
where $D_n = {\rm diag}(1, \omega, \omega^2, \ldots, \omega^{n-1})$.
"""

# ╔═╡ 85bbc1e0-e763-45c3-a3d8-7737f9c282df
md"""
Quiz: What is the computing time of a $F_n x$?

Hint: $T(n) = 2 T(n/2) + O(n)$.
"""

# ╔═╡ db53fca4-e20b-4140-9ff5-57fea662c3f6
@testset "fft decomposition" begin
	n = 4
	Fn = dft_matrix(n)
	F2n = dft_matrix(2n)

	# the permutation matrix to permute elements at 1:2:n (odd) to 1:n÷2 (top half)
	pm = sparse([iseven(j) ? (j÷2+n) : (j+1)÷2 for j=1:2n], 1:2n, ones(2n), 2n, 2n)

	# construct the D matrix
	ω = exp(-π*im/n)
	d1 = Diagonal([ω^(i-1) for i=1:n])

	# construct F_{2n} from F_n
	F2n_ = [Fn d1 * Fn; Fn -d1 * Fn]
	@test F2n * pm' ≈ F2n_
end

# ╔═╡ 14234cf7-cf73-409b-9369-7f4dc47a9868
md"## The Julia implementation"

# ╔═╡ 255631d5-f768-41c2-859e-15d8d10619d6
md"""
We implement the $O(n\log(n))$ time Cooley-Tukey FFT algorithm.
"""

# ╔═╡ 1bd59263-1bc5-4f32-96f2-1bab54a96a92
function fft!(x::AbstractVector{T}) where T
    N = length(x)
    @inbounds if N <= 1
        return x
    end
 
    # divide
    odd  = x[1:2:N]
    even = x[2:2:N]
 
    # conquer
    fft!(odd)
    fft!(even)
 
    # combine
    @inbounds for i=1:N÷2
       t = exp(T(-2im*π*(i-1)/N)) * even[i]
       oi = odd[i]
       x[i]     = oi + t
       x[i+N÷2] = oi - t
    end
    return x
end

# ╔═╡ b263a34f-21b8-4f7a-b211-c47e8590f840
@testset "fft" begin
	x = randn(ComplexF64, 8)
	@test fft!(copy(x)) ≈ dft_matrix(8) * x
end

# ╔═╡ 564c79b9-b79b-4d7a-9003-1350210957e1
md"The Julia package `FFTW.jl` contains a super fast FFT implementation."

# ╔═╡ 8a959693-0cfb-49c6-832d-307c90902986
@testset "fft" begin
	x = randn(ComplexF64, 8)
	@test fft(copy(x)) ≈ dft_matrix(8) * x
end

# ╔═╡ ad11f25c-7a1f-40c4-b1d9-c773e7e6baf2
md"""
## Application 1: Fast polynomial multiplication
"""

# ╔═╡ c699de9c-a02b-4810-9d53-eb1ced318470
md"Given two polynomials $p(x)$ and $q(x)$"

# ╔═╡ fc2f9537-6fb8-4218-8931-d8d91c87cb86
md"""
```math
p(x) = \sum_{k=0}^{n-1} a_k x^k
```
```math
q(x) = \sum_{k=0}^{n-1} b_k x^k
```
"""

# ╔═╡ 933b383c-2b96-4b15-b4fd-913e18c9a219
md"""
The multiplication of them is defined as
"""

# ╔═╡ eceda83a-749a-41a1-87fd-6930cb800dca
md"""
```math
p(x)q(x) = \sum_{k=0}^{2n-2} c_k x^{k}
```
"""

# ╔═╡ f00f373f-c7f8-4543-ad88-e7f01b6f8663
md"""
1. Evaluate $p(x)$ and $q(x)$ at $2n$ points $ω^0, \ldots , ω^{2n−1}$ using DFT. This step takes time $O(n \log n)$.

2. Obtain the values of $p(x)q(x)$ at these 2n points through pointwise multiplication
```math
\begin{align}
(p \circ q)(ω^0) &= p(ω^0) q(ω^0), \\
(p \circ q)(ω^1) &= p(ω^1) q(ω^1),\\
&\vdots\\
(p \circ q)(ω^{2n−1}) &= p(ω^{2n−1}) q(ω^{2n−1}).
\end{align}
```
This step takes time $O(n)$.

3. Interpolate the polynomial $p \circ q$ at the product values using inverse DFT to obtain coefficients $c_0, c_1, \ldots, c_{2n−2}$. This last step requires time $O(n \log n)$.
We can also use FFT to compute the convolution of two vectors $a = (a_0,\ldots , a_{n−1})$ and $b = (b_0, \ldots , b_{n−1})$, which is defined as a vector $c = (c_0, \ldots , c_{n−1})$ where
```math
c_j = \sum^j_{k=0} a_kb_{j−k}, ~~~~~~ j = 0,\ldots, n − 1.
```
The running time is again $O(n \log n)$.
"""

# ╔═╡ 2c711287-6adf-4b5f-9d29-8f13757795f0
p = Polynomial([1, 3, 2, 5, 6])

# ╔═╡ 53088523-c399-4715-b56c-a9cf4aead6f2
q = Polynomial([3, 1, 6, 2, 2])

# ╔═╡ a0ab3b3f-520f-46e6-8190-460ec08d2d2b
md"Step 1: evaluate $p(x)$ at $2n-1$ different points."

# ╔═╡ d7542d39-a07f-45ab-8c73-82195b921c32
pvals = fft(vcat(p.coeffs, zeros(4)))

# ╔═╡ bd0f5890-6d2f-4f01-ad01-f6bcf38d22c9
md"which is equivalent to computing:"

# ╔═╡ c66b1585-3375-4c43-978a-ce17b68480a6
let
	n = 5
	ω = exp(-2π*im/(2n-1))
	map(k->p(ω^k), 0:(2n-1))
end

# ╔═╡ 51fd341d-83dc-477a-80a0-0d7ca082b0c6
md"The same for $q(x)$."

# ╔═╡ 5ec492fd-15e1-43ba-984e-d053b7ef25c8
qvals = fft(vcat(q.coeffs, zeros(4)))

# ╔═╡ 13ddd827-ddf3-4326-bc4f-357343492d7a
md"Step 2: Compute $p(x) q(x)$ at $2n-1$ points."

# ╔═╡ dd124023-c34e-4f8b-ae8e-cdc0afba84d3
pqvals = pvals .* qvals

# ╔═╡ b92865b0-e88b-40da-a3b7-f6f0a07a2f5f
md"Step 3: Using the $2n-1$ point to fit the target polynomial."

# ╔═╡ 63cbb20c-dcaf-4492-a7f6-4e2840823a02
ifft(pqvals)

# ╔═╡ ed52d277-0b08-4e5a-a669-e5fe4df8798d
md"Summarize:"

# ╔═╡ f6a21eb8-2677-4b41-90be-4b0928025122
function fast_polymul(p::AbstractVector, q::AbstractVector)
	pvals = fft(vcat(p, zeros(length(q)-1)))
	qvals = fft(vcat(q, zeros(length(p)-1)))
	pqvals = pvals .* qvals
	return real.(ifft(pqvals))
end

# ╔═╡ 162439f7-c129-4ac2-b452-94cc956c7fae
function fast_polymul(p::Polynomial, q::Polynomial)
	Polynomial(fast_polymul(p.coeffs, q.coeffs))
end

# ╔═╡ 10d0bce2-4f23-4d7a-aa4a-aabb956f9b67
md"A similar algorithm has already been implemented in package `Polynomials`. One can easily verify the correctness."

# ╔═╡ 002ca7e5-062e-4e65-867f-3359e4b32e64
p * q

# ╔═╡ 59fd3d9c-e205-4020-b2bc-69eb22371f29
fast_polymul(p, q)

# ╔═╡ 08eb22af-3904-4aed-82c1-7cd17d24d036
md"## Application 2: Image compression"

# ╔═╡ 0fd767f8-6b13-44b1-b458-a9e92aada885
md"""
If you google the logo of the Hong Kong University of Science and Technology, you will probably find the following png of size ``2000 \times 3000``.
"""

# ╔═╡ c7e40dc5-ffad-4e92-98da-83a67947f6bd
img = Images.load("images/hkust-gz.png")

# ╔═╡ 75978d9f-2338-4dcf-a108-90131c7464d8
md"""
It is too large! We can compress it with the Fourier transformation algorithm.
To simplify the discussion, let us using the gray scale image.
"""

# ╔═╡ af294d84-118b-4efd-9f11-4c1a1854a270
gray_image = Gray.(img)

# ╔═╡ 490bd6fc-288b-46f3-9d43-2dce386cc633
# The gray scale image uses 8-bit fixed point numbers as the pixel storage type.
typeof(gray_image)

# ╔═╡ 7817368c-de8f-4896-a4c3-9ece3c629b99
img_data = Float32.(gray_image)

# ╔═╡ 55408e59-cdb4-451c-857d-c9ff52c69a7c
img_data_k = fftshift(fft(img_data))

# ╔═╡ 820f7c11-92e5-4123-b778-8b79c7f54677
# it is sparse!
Gray.(abs2.(img_data_k) ./ length(img_data_k))

# ╔═╡ c92f121f-bc9c-4663-9967-fd58750f80f1
md"""
We can store it in the sparse matrix format.
"""

# ╔═╡ 9ae65235-48f1-4197-8577-34cf2ef4b58c
@bind tolerence Slider(1:1000; default=100, show_value=true)

# ╔═╡ 0fb6ea2f-569a-44f7-9640-9343bc09d0be
sparse_img = let
	# let us discard all variables smaller than 1e-5
	img_data_k[abs.(img_data_k) .< tolerence] .= 0
	sparse(img_data_k)
end

# ╔═╡ 50997084-774d-4eba-912c-6a311ce56aca
compression_ratio = nnz(sparse_img) / (2000 * 3000)

# ╔═╡ 0919b332-c38e-4de4-9dad-e32cffa9ef7a
recovered_img = ifft(fftshift(Matrix(sparse_img)))

# ╔═╡ 71dea051-882c-41d2-84ad-2f5f8eb0ad04
Gray.(abs.(recovered_img))

# ╔═╡ 98fad9ce-ff15-4a5f-93f8-cf67f299cc5e
md"""
# Assignment
Watch this YouTube video: [https://youtu.be/jnxqHcObNK4](https://youtu.be/jnxqHcObNK4)

Use what you have learnt to solve the analyse the following sequential data.
"""

# ╔═╡ 6a21eb36-156d-4f04-a399-667e323ff08b
N = 5000

# ╔═╡ 308582bb-065a-4078-871d-0f36d16a329f
brain_signal = sin.(LinRange(0, 1000, N) ./ 10) .+ rand(N)

# ╔═╡ a1c74149-dd97-48b8-96d6-d12f000a09bb
plot(brain_signal)

# ╔═╡ 7fe5a514-a9c8-4ae3-8318-729bff933299
md"""
Here, we use the Ricker wavelet to analyse the above wave function.
```math
A  \left(1 - \left(\frac{x}{a}\right)^2\right) e^{-\frac{x^2}{2a^2}},
```
where $A = \frac{8}{\sqrt{3a}\pi}$.
"""

# ╔═╡ 3c567563-72fb-461d-a6b7-4153c28a44c6
function ricker(x, a)
	A = 8/π/sqrt(3a)
	return A * (1 - (x/a)^2) * exp(-x^2/a^2/2)
end

# ╔═╡ 9a72cda0-ee52-46d5-8dcf-c7b9fcc66798
let
	x = -10:0.01:10
	y = ricker.(x, 0.6)
	plot(x, y)
end

# ╔═╡ cb26291d-d6d6-4533-a37c-286cba0048b6
md"### Tasks"

# ╔═╡ 07153cbc-35e4-4cb2-b46c-e413ecfd6641
md"""
Please help me fix the following code to let the output be what we want. You need to implement the wavelet transformation `z = wavelet_transformation(x, y)`, such that
```math
z_i = \sum_j x_{j-i} y_j
```
You are supposed to implement the fast wavelet transformation that having time complexity $O(n \log(n))$ where $n$ is the size of $x$ and $y$.
"""

# ╔═╡ 289b1e96-9aa0-4c74-b545-4c7ebd0b37f2
function wavelet_transformation(signal::AbstractVector{T}, fw) where T
	# TODO: please remove the following line and add your own implementation!
	resulting_vector = randn(length(signal) + length(fw)-1)
	return resulting_vector
end

# ╔═╡ fc149a86-70cd-41a2-b122-719765449784
# this is the test program
let
	# the width parameter `a` in the Ricker wavelet is 1..500
	widths = 1:N÷10
	res = []
	for (j, a) in enumerate(widths)
		fw = ricker.(-1000:1000, a)   # the descretized wavelet of width `a`
		res_a = wavelet_transformation(brain_signal, fw)
		push!(res, res_a)
	end
	heatmap(hcat(res...); ylabel="time", xlabel="widths")
end

# ╔═╡ e8475529-9257-46ef-a28f-d041e371353c
md"""
In the submission (pull request), the following contents should be included
1. The correct implementation of the `wavelet_transformation` function.
2. The output image created by the above code block,
3. An interpretation of the output image.
"""

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git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

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

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

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "c6edfe154ad7b313c01aceca188c05c835c67360"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.4+0"

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

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

[[deps.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"

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

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

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

[[deps.libsixel_jll]]
deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Pkg", "libpng_jll"]
git-tree-sha1 = "d4f63314c8aa1e48cd22aa0c17ed76cd1ae48c3c"
uuid = "075b6546-f08a-558a-be8f-8157d0f608a5"
version = "1.10.3+0"

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

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

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

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

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

[[deps.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"
"""

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