pluto_linear_algebra_j.jl

### A Pluto.jl notebook ###
# v0.19.43

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

# ╔═╡ bf5554d5-5c14-48bb-8dda-bf264cf660a9
begin
	using PlutoUI
    using LaTeXStrings
	using Formatting
	using Plots
	using Colors 
	using ImageFiltering
	using Images
	using ImageMagick
	using TestImages
	using ImageView
	using ImageTransformations
	using LinearAlgebra
	using LowRankApprox
	using Wavelets
	using DSP
	using Primes
    using DataFrames
end

# ╔═╡ 4fc5029e-1e90-408d-b5c6-c563918ae660
md""" #### 数ベクトル空間の線形代数
1. 行列と数ベクトル

2. 行列が表すデータ

3. 特異値分解と応用

４. 離散ウェーブレット分解と応用
"""

# ╔═╡ e8b6f53b-eb6a-4f9b-99e5-8ee87f2ceda2
md"""
#### 1. 行列と数ベクトル 
"""

# ╔═╡ 9aa96b35-d37d-46c2-90c9-509d06126fd5
md"""
##### 1-1. 行列と数ベクトルの定義

さて$m$と$n$を正整数とし$m{\times}n$個の(実)数 

$a_{ij}, \quad i=1,\dotsc,m, \quad j=1,\dotsc,n$ 

を行を $m$ 個で列が $n$ 個になるように並べて $[\quad]$ または $(\quad)$ で括ったもの

$A=[a_{ij}]
:=
\begin{bmatrix}
a_{11} & a_{12} & \dotsb & a_{1n}
\\
a_{21} & a_{22} & \dotsb & a_{2n}
\\
\vdots & \vdots &  & \vdots
\\
a_{m1} & a_{m2} & \dotsb & a_{mn}
\end{bmatrix}$

を $m{\times}n$ 行列という. 例えば

$\begin{bmatrix}
1 & 2 & 3
\\
4 & 5 & 6
\end{bmatrix},
\quad
\begin{bmatrix}
1 & 2 & 3
\\
4 & 5 & 6
\\
7 & 8 & 9
\end{bmatrix},
\quad
\begin{bmatrix}
1
\\
4
\\
7
\end{bmatrix},
\quad
\begin{bmatrix}
1 & 2 & 3
\end{bmatrix}$

は行列である. 特に列または行の数が1である場合

$\vec{a}
=
\begin{bmatrix}
a_1 \\ \vdots \\ a_m
\end{bmatrix},
\quad
\vec{b}
=
\begin{bmatrix}
b_1 & \dotsb & b_n
\end{bmatrix}$

はそれぞれ $m$ 次元列ベクトル, $n$ 次元行ベクトルとよぶ. 
"""

# ╔═╡ b4bcd9fb-1960-445d-bde1-5755fd5662e8
md"""
##### 1-2. 行列と数ベクトルの演算

- 行列 $A$ の転置行列 $A^T$: $m{\times}n$ 行列 $A=[a_{ij}]$ の転置行列 $A^T$ という $n{\times}m$ 行列を次のように定義する:
$A^T:=[a_{ji}], 
\quad 
\begin{bmatrix}
1 & 2 & 3
\\
4 & 5 & 6
\end{bmatrix}^T
=
\begin{bmatrix}
1 & 4
\\
2 & 5
\\
3 & 6
\end{bmatrix}.$

- 行列の和 $A+B$: $A=[a_{ij}]$ と $B=[b_{ij}]$ が同じ $m{\times}n$ 行列であるとき $A+B$ を次のように定義する:
$A+B:=[a_{ij}+b_{ij}],$
$\begin{bmatrix}
1 & 2 & 3
\\
4 & 5 & 6
\end{bmatrix}
+
\begin{bmatrix}
7 & 8 & 9
\\
10 & 11 & 12
\end{bmatrix}
=
\begin{bmatrix}
1+7 & 2+8 & 3+9
\\
4+10 & 5+11 & 6+12
\end{bmatrix}
=
\begin{bmatrix}
8 & 10 & 12
\\
14 & 16 & 18
\end{bmatrix}.$

- 行列のスカラー倍 $cA$: 数 $c$ と $A=[a_{ij}]$ に対して $A$ のスカラー倍 $cA$ を次のように定義する:
$cA:=[ca_{ij}],
\quad
5
\begin{bmatrix}
1 & 2 & 3
\\
4 & 5 & 6
\end{bmatrix}
=
\begin{bmatrix}
5 & 10 & 15
\\
20 & 25 & 30
\end{bmatrix}.$

- 行列と列ベクトルの積 $A\vec{x}$: $\vec{a}_1,\dotsc,\vec{a}_n$ を $m$ 次元列ベクトルとし, $m{\times}n$ 行列 $A$ を $A:=\begin{bmatrix}\vec{a}_1 & \dotsb & \vec{a}_n\end{bmatrix}$ とする. $n$ 次元列ベクトル $\vec{x}=\begin{bmatrix}x_1 \\ \vdots \\ x_n\end{bmatrix}$ に対して $m$ 次元列ベクトル $A\vec{x}$ を次のように定義する:
$A\vec{x}:=x_1\vec{a}_1+\dotsb+x_n\vec{a}_n,$
$\begin{bmatrix}
1 & 2
\\
3 & 4
\\
5 & 6
\end{bmatrix}
\begin{bmatrix}
100 \\ 10
\end{bmatrix}
=
100
\begin{bmatrix}
1
\\
3
\\
5
\end{bmatrix}
+
10
\begin{bmatrix}
2
\\
4
\\
6
\end{bmatrix}
=
\begin{bmatrix}
120
\\
340
\\
560
\end{bmatrix}.$

- 行列の積 $AB$: $m{\times}n$ 行列 $A$ と $n{\times}l$ 行列 $B:=\begin{bmatrix}\vec{b}_1 & \dotsb & \vec{b}_l\end{bmatrix}$ に対して $m{\times}l$ 行列 $AB$ を次のように定義する:
$AB:=\begin{bmatrix}A\vec{b}_1 & \dotsb & A\vec{b}_l\end{bmatrix},$
$\begin{bmatrix}
1 & 2
\\
3 & 4
\\
5 & 6
\end{bmatrix}
\begin{bmatrix}
10 & 10
\\ 
1 & 100
\end{bmatrix}
=
\begin{bmatrix}
12 & 210
\\
34 & 430
\\
56 & 650
\end{bmatrix}.$


"""

# ╔═╡ 642ba2ee-ba66-4601-9edb-677c23e70780
md"""
##### 1-3. 平面上の線形写像の例
さて $A$ を $m{\times}n$ 行列, $\vec{x}$ と $\vec{y}$ を $n$ 次元列ベクトル, $c$ と $d$ を数とする. 
次の対応 

$\vec{x} \mapsto A\vec{x}$

は $n$ 次元列ベクトルの全体 $\mathbb{R}^n$ から $m$ 次元列ベクトルの全体 $\mathbb{R}^m$ への写像であり, 列ベクトルの和とスカラー倍を保存する: 

$A(c\vec{x}+d\vec{y}) = c(A\vec{x}) + d(A\vec{y}).$

このような性質をもつ写像は線形写像とよばれる. 以下では $2\times2$ 行列が定義する平面 $\mathbb{R}^2$ から平面 $\mathbb{R}^2$ への線形写像の例を2つ見てみよう.

$\vec{x} \mapsto \vec{s}:=A\vec{x},$

すなわち

$\begin{bmatrix} x \\ y \end{bmatrix}
\mapsto 
\begin{bmatrix} s \\ t \end{bmatrix}
:=
\begin{bmatrix} a & b \\ c & d \end{bmatrix}
\begin{bmatrix} x \\ y \end{bmatrix}
=
x
\begin{bmatrix} a \\ c \end{bmatrix}
+
y
\begin{bmatrix} b \\ d \end{bmatrix}$

とする.
"""

# ╔═╡ b8264135-0b84-4395-8d3a-ad8f42265004
md"""
###### 1-3-1. 伸長
２つの定数 $\lambda\ne0$ かつ $\mu\ne0$ を導入する. 

$\begin{bmatrix} s \\ t \end{bmatrix}
:=
\begin{bmatrix} \lambda & 0 \\ 0 & \mu \end{bmatrix}
\begin{bmatrix} x \\ y \end{bmatrix}
=
\begin{bmatrix} \lambda{x} \\ \mu{y} \end{bmatrix}$

は $x$ 軸の尺度を $\lambda$ 倍に引き伸ばし, $y$ 軸の尺度を $\mu$ 倍に引き伸ばす写像である. 
$\lambda<0$ ならば $x$ 軸と $s$ 軸の向きは逆になる. 

原点を中心とする単位円の方程式 $x^2+y^2=1$ は楕円の方程式

$\frac{s^2}{\lambda^2}+\frac{t^2}{\mu^2}=1$

に移る. $\lambda=\sqrt{2}$, $\mu=1/\sqrt{2}$ の場合の図は以下のようになる.

"""

# ╔═╡ beeb450c-2191-4591-a289-d0d956e83e98
begin 
	ellipse=load("./image/ellipse.png");
end

# ╔═╡ df792cbc-3244-453d-9cb8-6bc359b881c5
md"""
###### 1-3-2. 回転
原点を中心に反時計回りに角度 $\theta \in [0,2\pi)$ の回転を与える行列 $P(\theta)$ は

$P(\theta)
=
\begin{bmatrix}
\cos\theta & -\sin\theta
\\
\sin\theta & \cos\theta
\end{bmatrix}$

である. 第1列と第2列の関係は以下のようになり, 直交していることがわかる:

$\begin{bmatrix}
-\sin\theta
\\
\cos\theta
\end{bmatrix}
=
\begin{bmatrix}
\cos(\theta+\pi/2)
\\
\sin(\theta+\pi/2)
\end{bmatrix}.$

角度が $\theta=\pi/6$ のときに $st$-平面に $x$-軸と $y$-軸を書き込んでみる:

$\begin{bmatrix} s \\ t \end{bmatrix}
=
\begin{bmatrix}
\cos(\pi/6) & -\sin(\pi/6)
\\
\sin(\pi/6) & \cos(\pi/6)
\end{bmatrix}
\begin{bmatrix} x \\ y \end{bmatrix}
=
x
\begin{bmatrix} \sqrt{3}/2 \\ 1/2 \end{bmatrix}
+
y
\begin{bmatrix} -1/2 \\ \sqrt{3}/2 \end{bmatrix}.$

"""

# ╔═╡ 8e88d7de-5b94-46dc-bb37-b2a89287372e
begin 
	rotation=load("./image/rotation.png");
end

# ╔═╡ 734f2151-1622-466c-8f64-0b86d81b32f2


# ╔═╡ 2cc48257-83e1-4fa4-8659-edc962d92bf2
begin
	xx=rand(1550:1950, 100)/10;
    XX=[ones(100) xx];
    yy=(xx.^2).*rand(160:400, 100)/100000;
	yy=round.(yy, digits=1);
    bb=XX\yy;
    tt=155:1:195;
    zz=[ones(length(tt)) tt]*bb;
	ZZ=[xx yy];
	for i=1:1
	end
end

# ╔═╡ 9ff98a24-2646-4009-9482-9d066faa1716
md"""
#### 2. 行列が表すデータ
"""

# ╔═╡ 56cc741b-f300-4558-a078-a161f58307a9
md"""
##### 2-1. データ行列
下記は100人の身長と体重のデータ一覧であるが, $100\times2$ の行列とみなすことができる. 一般に個体数 $N$ で各個体がそれぞれ $p$ 種類のデータをもっているとき, そのデータ一覧は $N{\times}p$ 行列として表すことができる. 
"""

# ╔═╡ dc3cb19c-c076-4dc4-9d38-90a5c4956f3b
begin
	df = DataFrame(height = xx, weight = yy)
end

# ╔═╡ c8263139-b11e-4404-a311-e5a345944168
md"""
身長を横軸に体重を縦軸にとった平面上に各個人のデータを点で表すことができる. これを散布図という. 線形代数の一般逆行列あるいは特異値分解を利用すると, 線形回帰(最小二乗法)による身長と体重の関係を表す直線を求めることができる.
"""

# ╔═╡ e5700f04-1645-4fa7-9fa9-b2cc608120b7
begin
	scatter(xx,yy, 
		grid=false,
		label="sample", 
		xlabel="height [cm]", 
		ylabel="weight [kg]", 
        title="Linear Regression between Height & Weight")
    plot!(tt,zz,linewidth=2, label="regression",color="magenta")
end

# ╔═╡ d412da3b-992f-450f-8c4f-989f033697e8
md"""
##### 2-2. 白黒画像 (grayscale image)
さて, 白黒画像 (grayscale image) は各小正方形(ピクセル)に $0$ から $255$ までの整数, あるいは, 
それらを $255$ で割った $0$ から $1$ までの実数を成分とする行列である. 
$0$ は黒を表し $255$ は白を表す. 数値は色の強さを表し大きいほど強い. 例えば次の行列

$\begin{bmatrix}
0 & 15  & 30  & 45 & 60 & 75
\\ 
90 & 105 & 120  & 135  & 150 & 165
\\ 
180 & 195  & 210  & 225 & 240 & 255
\end{bmatrix}$

の表す白黒画像は以下のとおりである。
"""

# ╔═╡ 9ff00640-ee11-4d18-9fbd-3d6ef5264ee5
begin
    Gsample=[0 15 30 45 60 75; 
		     90 105 120 135 150 165;
		     180 195 210 225 240 255]/255;
	plot(Gray.(Gsample),
		xaxis=false, 
        xticks=false, 
        yaxis=false, 
        yticks=false)
end

# ╔═╡ aa109280-a4ef-427a-8b75-850acd667d12
md"""
##### 2-3. RGB画像

RGB画像は同じサイズの白黒画像のデータの3つの行列をデータとし, 各行列を三原色の赤と緑と青でそれぞれ着色して合わせたものである. プログラミング言語の乱数機能で生成した3つの$16\times16$行列とその3つ組が表すRGB画像を示す.

"""

# ╔═╡ 34c4dc39-0b65-4abd-9997-c8bfaf02b529
begin
# read image, resize, decompose}
	L1=4;
	RII=rand(2^L1, 2^L1);
	GII=rand(2^L1, 2^L1);
	BII=rand(2^L1, 2^L1);
	RRR=zeros(3,2^L1, 2^L1);
	GGG=zeros(3,2^L1, 2^L1);
	BBB=zeros(3,2^L1, 2^L1);
	RGBII=zeros(3,2^L1, 2^L1);
	RRR[1,:,:]=RII[:,:];
	GGG[2,:,:]=GII[:,:];
	BBB[3,:,:]=BII[:,:];
	RGBII[1,:,:]=RII[:,:];
	RGBII[2,:,:]=GII[:,:];
	RGBII[3,:,:]=BII[:,:];

QQQ1=plot(colorview(RGB,RRR),
        title="Grayscale Image (Red)",
        xaxis=false, 
        xticks=false, 
        yaxis=false, 
        yticks=false);
QQQ2=plot(colorview(RGB,GGG),
        title="Grayscale Image (Green)",
        xaxis=false, 
        xticks=false, 
        yaxis=false, 
        yticks=false);
QQQ3=plot(colorview(RGB,BBB),
        title="Grayscale Image (Blue)",
        xaxis=false, 
        xticks=false, 
        yaxis=false, 
        yticks=false, 
        grid=false);
QQQ4=plot(colorview(RGB,RGBII),
        title="RGB Image",
        xaxis=false, 
        xticks=false, 
        yaxis=false, 
        yticks=false, 
        grid=false);
plot(QQQ1,QQQ2,QQQ3,QQQ4,size=(1000,1000),layout=(2,2))
end

# ╔═╡ 23591d92-e9d9-4565-94d0-6d3b37c8befc
md"""
#### 3. 特異値分解と応用
"""

# ╔═╡ a7744957-4472-4012-ad31-372cd48fa78c
md""" ##### 3-1. 特異値分解の例

行列 

$A
=
\frac{1}{8}
\begin{bmatrix} 
\sqrt{6}+4\sqrt{2} & \sqrt{6}-4\sqrt{2} 
\\ 
-4\sqrt{6}+\sqrt{2} & 4\sqrt{6}+\sqrt{2} 
\end{bmatrix}$ 

は次のように分解される:

$A
=
U \Sigma V^T
=
\begin{bmatrix} \cos(\pi/3) & \sin(\pi/3) \\ -\sin(\pi/3) & \cos(\pi/3) \end{bmatrix}
\begin{bmatrix} 2 & 0 \\ 0 & 1/2 \end{bmatrix}
\begin{bmatrix} \cos(\pi/4) & \sin(\pi/4) \\ -\sin(\pi/4) & \cos(\pi/4) \end{bmatrix}^T.$

さて $U$ は時計回りの $\pi/3$ の回転を与える行列であり, $V^T$ は時計回りの $\pi/4$ の回転を与える行列であり, $A$ は回転 ($V^T$), 伸長 ($\Sigma$), 回転 ($U$) の合成写像であることがわかる. 

$\begin{bmatrix} s \\ t \end{bmatrix}
=
A
\begin{bmatrix} x \\ y \end{bmatrix}
=
\frac{x}{8}
\begin{bmatrix} \sqrt{6}+4\sqrt{2} \\ -4\sqrt{6}+\sqrt{2} \end{bmatrix}
+
\frac{y}{8}
\begin{bmatrix} \sqrt{6}-4\sqrt{2} \\ 4\sqrt{6}+\sqrt{2}  \end{bmatrix}$

とすると, 単位円の方程式 $x^2+y^2=1$ は傾いた楕円の方程式 $49s^2+15\sqrt{3}st+19t^2=16$ に移る.  
"""

# ╔═╡ 8d8b6675-a71f-421b-bc62-c0a6fb9b3e9e
begin 
	svdexample=load("./image/svd.png");
end

# ╔═╡ 2e5e3122-1b2b-4dcf-88d4-f79b3e727bbf
md"""
$\sigma_1=2,
\quad
\sigma_2=\frac{1}{2},$

$\vec{v}_1
=
\frac{1}{\sqrt{2}}
\begin{bmatrix}
1 \\ -1 
\end{bmatrix},
\quad
\vec{v}_2
=
\frac{1}{\sqrt{2}}
\begin{bmatrix}
1 \\ 1 
\end{bmatrix},$

$\vec{u}_1
=
\frac{1}{2}
\begin{bmatrix}
1 \\ -\sqrt{3} 
\end{bmatrix},
\quad
\vec{u}_2
=
\frac{1}{2}
\begin{bmatrix}
\sqrt{3} \\ 1 
\end{bmatrix}$

とおくと, 


$A
=
\sum_{j=1}^2\sigma_j\vec{u}_j\vec{v}_j^T
=
\frac{1}{\sqrt{2}}
\begin{bmatrix}
1 \\ -\sqrt{3} 
\end{bmatrix}
\begin{bmatrix}
1 & -1 
\end{bmatrix}
+
\frac{1}{4\sqrt{2}}
\begin{bmatrix}
\sqrt{3} \\ 1 
\end{bmatrix}
\begin{bmatrix}
1 & 1 
\end{bmatrix},$

すなわち

$A
=
\frac{1}{8}
\begin{bmatrix} 
\sqrt{6}+4\sqrt{2} & \sqrt{6}-4\sqrt{2} 
\\ 
-4\sqrt{6}+\sqrt{2} & 4\sqrt{6}+\sqrt{2} 
\end{bmatrix}
=
\frac{1}{8}
\begin{bmatrix} 4\sqrt{2} & -4\sqrt{2} \\ -4\sqrt{6} & 4\sqrt{6} \end{bmatrix}
+
\frac{1}{8}
\begin{bmatrix} \sqrt{6} & \sqrt{6} \\ \sqrt{2} & \sqrt{2} \end{bmatrix}$

と表すことができる. 
"""

# ╔═╡ b4937ad5-c8c7-492c-86d6-981a14f274ca


# ╔═╡ 3f45852f-ad72-4ba2-b147-a14151e8ed97
md""" ##### 3-2. 一般の行列の特異値分解と低階最良近似
一般の $m{\times}n$ 行列 $A=\begin{bmatrix}\vec{a}_1 & \dotsb & \vec{a}_n \end{bmatrix}$ に対して $A$ の階数 (rank) とよばれる非負整数

$r
:=
\operatorname{dim}\bigl(
\{
A\vec{x}
=
x_1\vec{a}_1+\dotsb+x_n\vec{a}_n 
\ \vert \ 
\vec{x} \in \mathbb{R}^n
\}
\bigr)$ 

が定義される. $A$ の成分がすべて $0$ の場合を除くと $1 \leqq r \leqq \min\{m,n\}$ が成立する. 
"""

# ╔═╡ 52b8e90f-41d2-4791-9042-799053d5ec38
begin 
	strang=load("./image/strang.jpeg");
end

# ╔═╡ 3d656643-bf04-4ac7-99fc-6d98a571523a
md""" 
このとき 

$\sigma_1 \geqq \dotsb \geqq \sigma_r>0,
\quad
\vec{u}_1,\dotsc,\vec{u}_r \in \mathbb{R}^m,
\quad
\vec{v}_1,\dotsc,\vec{v}_r \in \mathbb{R}^n$

が存在して

$A=\sum_{j=1}^r\sigma_j\vec{u}_j\vec{v}_j^T$

と表すことができることが知られている. 

$A_k:=\sum_{j=1}^k\sigma_j\vec{u}_j\vec{v}_j^T, \quad k=1,\dotsc,r$

とすると, $A_k$ の階数は $k$ であり, 階数は $k$ のすべての行列のうち $A$ に最も近い行列, 
すなわち階数は $k$ のすべての行列の中で $A_k$ の最良の近似を与える行列であることが知られている. 
香港城市大学 (City University of Hong Kong) の食堂の焼きそばの画像 $384\times512\times3$ を低階最良近似してみる. 
"""

# ╔═╡ 789e78db-4f6a-4f7a-bba0-8ab119d3f1fb
begin
# read image, resize, decompose
	#I=load("./material/char_kway_teow.jpg");
	#X=imresize(I, ratio=1/4);
	I=load("./material/CityU.jpg");
	X=imresize(I, ratio=1/8);	
    (p,q)=size(X);
	L=7;
    A=channelview(X);
    R=Array{Float64}(A[1,:,:]);
	G=Array{Float64}(A[2,:,:]);
	B=Array{Float64}(A[3,:,:]);


	RU, RS, RV=psvd(R);
	GU, GS, GV=psvd(G);
	BU, BS, BV=psvd(B);
	
	rank=100;
	DR=zeros(p,q,rank);
	DG=zeros(p,q,rank);
	DB=zeros(p,q,rank);
    for r=1:rank
	    DR[:,:,r]=sum(RS[n]*RU[1:p,n]*(RV[1:q,n])' for n=1:r);
        DG[:,:,r]=sum(GS[n]*GU[1:p,n]*(GV[1:q,n])' for n=1:r);
        DB[:,:,r]=sum(BS[n]*BU[1:p,n]*(BV[1:q,n])' for n=1:r);
	end
end

# ╔═╡ 6e01a823-9696-41b7-96a3-cc1157bd472d
begin
    Y=zeros(3,p,q,rank+16);
	Z=zeros(p,q)
	for r=1:4
	    Y[1,:,:,r]=R;
        Y[2,:,:,r]=G;
        Y[3,:,:,r]=B;
	end
    for r=5:8
	    Y[1,:,:,r]=Z;
        Y[2,:,:,r]=Z;
        Y[3,:,:,r]=Z;
    end
	for r=9:rank+8
        Y[1,:,:,r]=DR[:,:,r-8];
        Y[2,:,:,r]=DG[:,:,r-8];
        Y[3,:,:,r]=DB[:,:,r-8];
	end
    for r=rank+9:rank+12
        Y[1,:,:,r]=Z;
        Y[2,:,:,r]=Z;
        Y[3,:,:,r]=Z;
	end
    for r=rank+13:rank+16
        Y[1,:,:,r]=R;
        Y[2,:,:,r]=G;
        Y[3,:,:,r]=B;
	end
end

# ╔═╡ 6f41a7c7-6c0a-447f-a9ff-6ef79d8c21b0
begin
    W=zeros(3,p,q,rank);
    for r=1:rank
	    W[1,:,:,r]=DR[:,:,r];
        W[2,:,:,r]=DG[:,:,r];
        W[3,:,:,r]=DB[:,:,r];
	end
end

# ╔═╡ bf7a523b-3783-4e45-a201-cbffaaf4859a
md"""
rank = $(@bind r Slider(1:rank, show_value=true))
"""

# ╔═╡ 780baf1a-b008-44dc-be56-75d0b5cf621e
begin
SVD1=plot(colorview(RGB,W[:,:,:,r]),
        title="approximation 1-100/384",
        xaxis=false, 
        xticks=false, 
        yaxis=false, 
        yticks=false, 
        grid=false);
SVD2=plot(colorview(RGB,X),
        title="original RGB image",
        xaxis=false, 
        xticks=false, 
        yaxis=false, 
        yticks=false, 
        grid=false);
plot(SVD1,SVD2)
end

# ╔═╡ 17c01101-a17c-43f6-9a08-260f13e17c13


# ╔═╡ c53a862a-0a1d-479e-a84e-f32d7bd7cfc8
md"""
#### ４. 離散ウェーブレット分解と応用
"""

# ╔═╡ 6b1ce739-aca6-47d6-842a-701712b2cc61
md"""
##### 4-1. 離散ハール・ウェーブレット
離散ウェーブレットとはある性質をみたす2つの数ベクトルの組でのことである. 
最も単純なのは次のハール・ウェーブレットである. 

$\vec{u}
=
\frac{1}{\sqrt{2}}
\begin{bmatrix}
1
\\
1
\\
0
\\
\vdots
\\
0 
\end{bmatrix}, 
\quad
\vec{v}
=
\frac{1}{\sqrt{2}}
\begin{bmatrix}
1
\\
-1
\\
0
\\
\vdots
\\
0 
\end{bmatrix}
\in\mathbb{R}^{N},$

これを用いると数ベクトルや行列の隣接する成分の平均をとるという操作を線形代数だけを用いて系統的に定義することができる. 2回の操作で

$\vec{a}_0
=
\begin{bmatrix}
1 \\ 3 \\ 5 \\ 7 
\end{bmatrix}
\mapsto 
\vec{a}_1
=
\begin{bmatrix}
2 \\ 2 \\ 6 \\ 6 
\end{bmatrix}
\mapsto 
\vec{a}_2
=
\begin{bmatrix}
4 \\ 4 \\ 4 \\ 4 
\end{bmatrix}$

$A_0
=
\begin{bmatrix}
0 & 2 & 4 & 6
\\ 
8 & 10 & 12 & 14 
\\ 
16 & 18 & 20 & 22
\\ 
24 & 26 & 28 & 30
\end{bmatrix}
\mapsto 
A_1
=
\begin{bmatrix}
5 & 5 & 9 & 9
\\ 
5 & 5 & 9 & 9 
\\ 
21 & 21 & 25 & 25
\\ 
21 & 21 & 25 & 25
\end{bmatrix}
\mapsto 
A_2
=
\begin{bmatrix}
15 & 15 & 15 & 15
\\ 
15 & 15 & 15 & 15 
\\ 
15 & 15 & 15 & 15
\\ 
15 & 15 & 15 & 15
\end{bmatrix}$

のようにな. 平均をとる操作を $\ell$ 回行って得られる $\vec{a}_\ell$ や $A_\ell$ をレベル $\ell$ の近似部分, 残りの $\vec{a} - \vec{a}_\ell$ や $A - A_\ell$ をレベル $\ell$ の詳細部分という. 

$\vec{a}_0 = \vec{a}_\ell + (\vec{a}_0-\vec{a}_\ell),
\quad
\vec{A}_0 = \vec{A}_\ell + (\vec{A}_0-\vec{A}_\ell),$

画像データのウェーブレット分解を観察してみよう.

""" 

# ╔═╡ de96f98e-7217-496f-9bf0-6c3e13706ec0
begin
	L0=3
	p0=8;
	q0=8;
    G0=[  0   4   8  12  16  20  24  28;
		  32  36  40  44  48  52  56  60;
		  64  68  72  76  80  84  88  92;
		  96 100 104 108 112 116 120 124;
		 128 132 136 140 144 148 152 156;
		 160 164 168 172 176 180 184 188; 
		 192 196 200 204 208 212 216 220;
		 224 228 232 236 240 244 248 252]/255; 

	XG0=zeros(p0,q0,L0);
	for l=1:L0
		XG0[:,:,l]=dwt(G0, wavelet(WT.haar), l);
	end

	XG0approx=zeros(p0,q0,L0);	
	for l=1:L0
	    XG0approx[1:Int(p0/2^l),1:Int(q0/2^l),l]=XG0[1:Int(p0/2^l),1:Int(q0/2^l),l];
	end
	XG0detail=XG0-XG0approx;

    YG0approx=zeros(p0,q0,L0);
	YG0detail=zeros(p0,q0,L0);
	for l=1:L0
		YG0approx[:,:,l]=idwt(XG0approx[:,:,l], wavelet(WT.haar), l);
		YG0detail[:,:,l]=idwt(XG0detail[:,:,l], wavelet(WT.haar), l);
	end
	
    ZG0approx=zeros(p0,q0,L0+1);
	ZG0detail=zeros(p0,q0,L0+1);
	ZG0approx[:,:,1]=G0;
	for l=2:L0+1
        ZG0approx[:,:,l]=YG0approx[:,:,l-1];
	    ZG0detail[:,:,l]=YG0detail[:,:,l-1];
	end
		
end

# ╔═╡ b9212600-820b-4edc-a265-ddf8334700c9
md""" 
##### 4-2. $8\times8$ 白黒画像

"""

# ╔═╡ b0b82a93-e29c-4c68-a2a7-45f181658f19
md"""
Level = $(@bind l0 Slider(0:L0, show_value=true))
"""

# ╔═╡ 3c393bc2-7da4-431b-a984-e8db6a97f5dc
begin
Z1=plot(Gray.(G0),
        #title="Original Image",
        xaxis=false, 
        xticks=false, 
        yaxis=false, 
        yticks=false);
Z2=plot(Gray.(ZG0approx[:,:,l0+1]),
        #title="Haar",
        xaxis=false, 
        xticks=false, 
        yaxis=false, 
        yticks=false);
Z3=plot(Gray.(ZG0detail[:,:,l0+1]),
        #title="Daubechies 2",
        xaxis=false, 
        xticks=false, 
        yaxis=false, 
        yticks=false);
plot(Z1,Z2,Z3,layout=(1,3))
end

# ╔═╡ fe147b2f-fdc8-4f6e-8efa-fb04862aa2dd
begin

# Decomposition Filter 
	RXII=zeros(2^L1,2^L1,L1);
	GXII=zeros(2^L1,2^L1,L1);
	BXII=zeros(2^L1,2^L1,L1);
	for l=1:L1
		RXII[:,:,l]=dwt(RII, wavelet(WT.haar), l);
		GXII[:,:,l]=dwt(GII, wavelet(WT.haar), l);
		BXII[:,:,l]=dwt(BII, wavelet(WT.haar), l);
	end

# Splitting 
	RXIIapprox=zeros(2^L1,2^L1,L1);
	GXIIapprox=zeros(2^L1,2^L1,L1);
	BXIIapprox=zeros(2^L1,2^L1,L1);
	for l=1:L1
	    RXIIapprox[1:2^(L1-l),1:2^(L1-l),l]=RXII[1:2^(L1-l),1:2^(L1-l),l];
		GXIIapprox[1:2^(L1-l),1:2^(L1-l),l]=GXII[1:2^(L1-l),1:2^(L1-l),l];
		BXIIapprox[1:2^(L1-l),1:2^(L1-l),l]=BXII[1:2^(L1-l),1:2^(L1-l),l];
	end
	RXIIdetail=RXII-RXIIapprox;
	GXIIdetail=GXII-GXIIapprox;
	BXIIdetail=BXII-BXIIapprox;

# Composition Filter
	RYIIapprox=zeros(2^L1,2^L1,L1);
	RYIIdetail=zeros(2^L1,2^L1,L1);
	GYIIapprox=zeros(2^L1,2^L1,L1);
	GYIIdetail=zeros(2^L1,2^L1,L1);
	BYIIapprox=zeros(2^L1,2^L1,L1);
	BYIIdetail=zeros(2^L1,2^L1,L1);
	for l=1:L1
		RYIIapprox[:,:,l]=idwt(RXIIapprox[:,:,l], wavelet(WT.haar), l);
		RYIIdetail[:,:,l]=idwt(RXIIdetail[:,:,l], wavelet(WT.haar), l);
		GYIIapprox[:,:,l]=idwt(GXIIapprox[:,:,l], wavelet(WT.haar), l);
		GYIIdetail[:,:,l]=idwt(GXIIdetail[:,:,l], wavelet(WT.haar), l);
		BYIIapprox[:,:,l]=idwt(BXIIapprox[:,:,l], wavelet(WT.haar), l);
		BYIIdetail[:,:,l]=idwt(BXIIdetail[:,:,l], wavelet(WT.haar), l);
	end

# RGB
	ZIIapprox=zeros(3,2^L1,2^L1,L1+1);
	ZIIdetail=zeros(3,2^L1,2^L1,L1+1);
	ZIIapprox[1,:,:,1]=RII;
	ZIIapprox[2,:,:,1]=GII;
	ZIIapprox[3,:,:,1]=BII;
	for l=2:L1+1
		ZIIapprox[1,:,:,l]=RYIIapprox[:,:,l-1];
		ZIIapprox[2,:,:,l]=GYIIapprox[:,:,l-1];
		ZIIapprox[3,:,:,l]=BYIIapprox[:,:,l-1];
		ZIIdetail[1,:,:,l]=RYIIdetail[:,:,l-1];
		ZIIdetail[2,:,:,l]=GYIIdetail[:,:,l-1];
		ZIIdetail[3,:,:,l]=BYIIdetail[:,:,l-1];
	end

end

# ╔═╡ 4d12cd8c-7d1d-4611-887f-bfdd53da48e9
md""" 
##### 4-3. 乱数で生成した $16\times16$ RGB画像

"""

# ╔═╡ 7aeb1dd0-19d3-4ab5-9767-b52f1d0d6fe7
md"""
Level = $(@bind l1 Slider(0:L1, show_value=true))
"""

# ╔═╡ 4a79dcb1-a2a8-4cb7-863c-9ca9e8f88e66
begin
Q0=plot(colorview(RGB,ZIIapprox[:,:,:,1]),
        #title="Original Image",
        xaxis=false, 
        xticks=false, 
        yaxis=false, 
        yticks=false);
Q1=plot(colorview(RGB,ZIIapprox[:,:,:,l1+1]),
        #title="Approximation",
        xaxis=false, 
        xticks=false, 
        yaxis=false, 
        yticks=false);
Q2=plot(colorview(RGB,ZIIdetail[:,:,:,l1+1]),
        #title="Detail",
        xaxis=false, 
        xticks=false, 
        yaxis=false, 
        yticks=false, 
        grid=false);
plot(Q0,Q1,Q2,layout=(1,3))
end

# ╔═╡ 2b8cafe5-0efb-4892-957c-055c63bf23ed
begin

# Decomposition Filter 
	XR=zeros(p,q,L);
	XG=zeros(p,q,L);
	XB=zeros(p,q,L);
	for l=1:L
		XR[:,:,l]=dwt(R, wavelet(WT.haar), l);
		XG[:,:,l]=dwt(G, wavelet(WT.haar), l);
		XB[:,:,l]=dwt(B, wavelet(WT.haar), l);
	end

# Splitting 
	XRapprox=zeros(p,q,L);
	XGapprox=zeros(p,q,L);
	XBapprox=zeros(p,q,L);	
	for l=1:L
	XRapprox[1:Int(p/2^l),1:Int(q/2^l),l]=XR[1:Int(p/2^l),1:Int(q/2^l),l];
    XGapprox[1:Int(p/2^l),1:Int(q/2^l),l]=XG[1:Int(p/2^l),1:Int(q/2^l),l];
	XBapprox[1:Int(p/2^l),1:Int(q/2^l),l]=XB[1:Int(p/2^l),1:Int(q/2^l),l];
	end
	XRdetail=XR-XRapprox;
	XGdetail=XG-XGapprox;
	XBdetail=XB-XBapprox;

# Composition Filter
	YRapprox=zeros(p,q,L);
	YGapprox=zeros(p,q,L);
	YBapprox=zeros(p,q,L);
	YRdetail=zeros(p,q,L);
	YGdetail=zeros(p,q,L);
	YBdetail=zeros(p,q,L);
	for l=1:L
		YRapprox[:,:,l]=idwt(XRapprox[:,:,l], wavelet(WT.haar), l);
		YGapprox[:,:,l]=idwt(XGapprox[:,:,l], wavelet(WT.haar), l);
		YBapprox[:,:,l]=idwt(XBapprox[:,:,l], wavelet(WT.haar), l);
		YRdetail[:,:,l]=idwt(XRdetail[:,:,l], wavelet(WT.haar), l);
		YGdetail[:,:,l]=idwt(XGdetail[:,:,l], wavelet(WT.haar), l);
		YBdetail[:,:,l]=idwt(XBdetail[:,:,l], wavelet(WT.haar), l);
	end

# RGB
	Wapprox=zeros(3,p,q,L+1);
	Wdetail=zeros(3,p,q,L+1);
	Wapprox[:,:,:,1]=A;
	for l=2:L+1
		Wapprox[1,:,:,l]=YRapprox[:,:,l-1];
		Wapprox[2,:,:,l]=YGapprox[:,:,l-1];
		Wapprox[3,:,:,l]=YBapprox[:,:,l-1];
		Wdetail[1,:,:,l]=YRdetail[:,:,l-1];
		Wdetail[2,:,:,l]=YGdetail[:,:,l-1];
		Wdetail[3,:,:,l]=YBdetail[:,:,l-1];
	end
end

# ╔═╡ 6468175c-7a79-4ab8-8879-eb442694c06d
md""" 
##### 4-4. $384\times512$ RGB画像
香港城市大学の食堂の焼きそば $384=2^7\times3$, $512=2^9$

"""

# ╔═╡ 33e36dbc-6682-4de6-a911-7eb2a0ec4f05
md"""
Level = $(@bind l Slider(0:L, show_value=true))
"""

# ╔═╡ a9f38308-8483-4b3a-9160-f2622027c7e2
begin
P1=plot(colorview(RGB,A[:,:,:]),
        #title="Approximation",
        xaxis=false, 
        xticks=false, 
        yaxis=false, 
        yticks=false, 
        grid=false);
P2=plot(colorview(RGB,Wapprox[:,:,:,l+1]),
        #title="Approximation",
        xaxis=false, 
        xticks=false, 
        yaxis=false, 
        yticks=false, 
        grid=false);
P3=plot(colorview(RGB,Wdetail[:,:,:,l+1]),
        #title="Detail",
        xaxis=false, 
        xticks=false, 
        yaxis=false, 
        yticks=false, 
        grid=false);
plot(P1,P2,P3,size=(1536,384),layout=(1,3))
end

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
Colors = "5ae59095-9a9b-59fe-a467-6f913c188581"
DSP = "717857b8-e6f2-59f4-9121-6e50c889abd2"
DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0"
Formatting = "59287772-0a20-5a39-b81b-1366585eb4c0"
ImageFiltering = "6a3955dd-da59-5b1f-98d4-e7296123deb5"
ImageMagick = "6218d12a-5da1-5696-b52f-db25d2ecc6d1"
ImageTransformations = "02fcd773-0e25-5acc-982a-7f6622650795"
ImageView = "86fae568-95e7-573e-a6b2-d8a6b900c9ef"
Images = "916415d5-f1e6-5110-898d-aaa5f9f070e0"
LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
LowRankApprox = "898213cb-b102-5a47-900c-97e73b919f73"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
Primes = "27ebfcd6-29c5-5fa9-bf4b-fb8fc14df3ae"
TestImages = "5e47fb64-e119-507b-a336-dd2b206d9990"
Wavelets = "29a6e085-ba6d-5f35-a997-948ac2efa89a"

[compat]
Colors = "~0.12.10"
DSP = "~0.7.9"
DataFrames = "~1.6.1"
Formatting = "~0.4.2"
ImageFiltering = "~0.7.8"
ImageMagick = "~1.3.0"
ImageTransformations = "~0.10.1"
ImageView = "~0.12.1"
Images = "~0.26.0"
LaTeXStrings = "~1.3.1"
LowRankApprox = "~0.5.5"
Plots = "~1.40.0"
PlutoUI = "~0.7.55"
Primes = "~0.5.6"
TestImages = "~1.8.0"
Wavelets = "~0.10.0"
"""

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

julia_version = "1.10.4"
manifest_format = "2.0"
project_hash = "e7a3735e3f7ab5d904662ad4b055b1324ae4d436"

[[deps.AbstractFFTs]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "d92ad398961a3ed262d8bf04a1a2b8340f915fef"
uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c"
version = "1.5.0"
weakdeps = ["ChainRulesCore", "Test"]

    [deps.AbstractFFTs.extensions]
    AbstractFFTsChainRulesCoreExt = "ChainRulesCore"
    AbstractFFTsTestExt = "Test"

[[deps.AbstractPlutoDingetjes]]
deps = ["Pkg"]
git-tree-sha1 = "c278dfab760520b8bb7e9511b968bf4ba38b7acc"
uuid = "6e696c72-6542-2067-7265-42206c756150"
version = "1.2.3"

[[deps.Adapt]]
deps = ["LinearAlgebra", "Requires"]
git-tree-sha1 = "0fb305e0253fd4e833d486914367a2ee2c2e78d0"
uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e"
version = "4.0.1"
weakdeps = ["StaticArrays"]

    [deps.Adapt.extensions]
    AdaptStaticArraysExt = "StaticArrays"

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

[[deps.ArnoldiMethod]]
deps = ["LinearAlgebra", "Random", "StaticArrays"]
git-tree-sha1 = "62e51b39331de8911e4a7ff6f5aaf38a5f4cc0ae"
uuid = "ec485272-7323-5ecc-a04f-4719b315124d"
version = "0.2.0"

[[deps.ArrayInterface]]
deps = ["Adapt", "LinearAlgebra", "Requires", "SparseArrays", "SuiteSparse"]
git-tree-sha1 = "bbec08a37f8722786d87bedf84eae19c020c4efa"
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version = "7.7.0"

    [deps.ArrayInterface.extensions]
    ArrayInterfaceBandedMatricesExt = "BandedMatrices"
    ArrayInterfaceBlockBandedMatricesExt = "BlockBandedMatrices"
    ArrayInterfaceCUDAExt = "CUDA"
    ArrayInterfaceGPUArraysCoreExt = "GPUArraysCore"
    ArrayInterfaceStaticArraysCoreExt = "StaticArraysCore"
    ArrayInterfaceTrackerExt = "Tracker"

    [deps.ArrayInterface.weakdeps]
    BandedMatrices = "aae01518-5342-5314-be14-df237901396f"
    BlockBandedMatrices = "ffab5731-97b5-5995-9138-79e8c1846df0"
    CUDA = "052768ef-5323-5732-b1bb-66c8b64840ba"
    GPUArraysCore = "46192b85-c4d5-4398-a991-12ede77f4527"
    StaticArraysCore = "1e83bf80-4336-4d27-bf5d-d5a4f845583c"
    Tracker = "9f7883ad-71c0-57eb-9f7f-b5c9e6d3789c"

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

[[deps.AxisAlgorithms]]
deps = ["LinearAlgebra", "Random", "SparseArrays", "WoodburyMatrices"]
git-tree-sha1 = "01b8ccb13d68535d73d2b0c23e39bd23155fb712"
uuid = "13072b0f-2c55-5437-9ae7-d433b7a33950"
version = "1.1.0"

[[deps.AxisArrays]]
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git-tree-sha1 = "16351be62963a67ac4083f748fdb3cca58bfd52f"
uuid = "39de3d68-74b9-583c-8d2d-e117c070f3a9"
version = "0.4.7"

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

[[deps.BitFlags]]
git-tree-sha1 = "2dc09997850d68179b69dafb58ae806167a32b1b"
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version = "0.1.8"

[[deps.BitTwiddlingConvenienceFunctions]]
deps = ["Static"]
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[[deps.Bzip2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
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version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "330f955bc41bb8f5270a369c473fc4a5a4e4d3cb"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.6+0"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "691634e5453ad362044e2ad653e79f3ee3bb98c3"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.39.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e92a1a012a10506618f10b7047e478403a046c77"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.5.0+0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+1"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "49ce682769cd5de6c72dcf1b94ed7790cd08974c"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.5+0"

[[deps.adwaita_icon_theme_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "hicolor_icon_theme_jll"]
git-tree-sha1 = "28401767f30e5743ef5e3b0be71417bc911d3952"
uuid = "b437f822-2cd6-5e08-a15c-8bac984d38ee"
version = "43.0.1+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "a68c9655fbe6dfcab3d972808f1aafec151ce3f8"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.43.0+0"

[[deps.gdk_pixbuf_jll]]
deps = ["Artifacts", "Glib_jll", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Libtiff_jll", "Pkg", "Xorg_libX11_jll", "libpng_jll"]
git-tree-sha1 = "e9190f9fb03f9c3b15b9fb0c380b0d57a3c8ea39"
uuid = "da03df04-f53b-5353-a52f-6a8b0620ced0"
version = "2.42.8+0"

[[deps.hicolor_icon_theme_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b458a6f6fc2b1a8ca74ed63852e4eaf43fb9f5ea"
uuid = "059c91fe-1bad-52ad-bddd-f7b78713c282"
version = "0.17.0+3"

[[deps.iso_codes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "d837a5d2a19d54243dafb6cc98d0f590a603dfa1"
uuid = "bf975903-5238-5d20-8243-bc370bc1e7e5"
version = "4.15.1+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"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.8.0+1"

[[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", "Zlib_jll"]
git-tree-sha1 = "93284c28274d9e75218a416c65ec49d0e0fcdf3d"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.40+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 = "1.52.0+1"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"

[[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 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+1"
"""

# ╔═╡ Cell order:
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