pluto_linear_algebra.jl

### A Pluto.jl notebook ###
# v0.19.46

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""" #### Linear Algebra of Numeric Vectors
1. Matrices and numeric vectors

2. Data described by matrices

3. Singular value decomposition and low rank approximation

４. Wavelet decomposition and image processing
"""

# ╔═╡ e8b6f53b-eb6a-4f9b-99e5-8ee87f2ceda2
md"""
#### 1. Matrices and numeric vectors 
"""

# ╔═╡ 9aa96b35-d37d-46c2-90c9-509d06126fd5
md"""
##### 1-1. Definition of matrices and numeric vectors

Let $m$ and $n$ be positive integers. Given $A=[a_{ij}]$ is said to be an 
$m{\times}n$ matrix if $mn$ (real) numbers 

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

are given, arranged in the form of $m$ rows and $n$ columns, and enclosed by $[\quad]$ or $(\quad)$: 

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

For example

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

are matrices. If $m$ or $n$ is equal to one, 

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

are said to be $m$-dimensional column vector and $n$-dimensional row vector respectively. If $m=n=1$, $[a_{11}]$ is idenfified as the scalar $a_{11}$.  
"""

# ╔═╡ b4bcd9fb-1960-445d-bde1-5755fd5662e8
md"""
##### 1-2. Operations of matrices and numeric vectors

- For an $m{\times}n$ matrix $A=[a_{ij}]$, we define its transpose $A^T$, which is an  $n{\times}m$ matrix, by:
$A^T:=[a_{ji}], 
\quad 
\begin{bmatrix}
1 & 2 & 3
\\
4 & 5 & 6
\end{bmatrix}^T
=
\begin{bmatrix}
1 & 4
\\
2 & 5
\\
3 & 6
\end{bmatrix}.$

- For $m{\times}n$ matrices $A=[a_{ij}]$ and $B=[b_{ij}]$, we define their sum $A+B$, which is also an $m{\times}n$ matrix, by:
$A+B:=[a_{ij}+b_{ij}],$
$\begin{bmatrix}
1 & 2 & 3
\\
4 & 5 & 6
\end{bmatrix}
+
\begin{bmatrix}
7 & 8 & 9
\\
10 & 11 & 12
\end{bmatrix}
=
\begin{bmatrix}
1+7 & 2+8 & 3+9
\\
4+10 & 5+11 & 6+12
\end{bmatrix}
=
\begin{bmatrix}
8 & 10 & 12
\\
14 & 16 & 18
\end{bmatrix}.$

- Foa a matrix $A=[a_{ij}]$ and a scalar $cA$, we define a scalar product $cA$ by:
$cA:=[ca_{ij}],
\quad
5
\begin{bmatrix}
1 & 2 & 3
\\
4 & 5 & 6
\end{bmatrix}
=
\begin{bmatrix}
5 & 10 & 15
\\
20 & 25 & 30
\end{bmatrix}.$

- Let $\vec{a}_1,\dotsc,\vec{a}_n$ be $m$-dimensional column vectors, and 
$A$ be an $m{\times}n$ matrix defined by $A:=\begin{bmatrix}\vec{a}_1 & \dotsb & \vec{a}_n\end{bmatrix}$. For an $n$-dimensional column vector $\vec{x}=\begin{bmatrix}x_1 \\ \vdots \\ x_n\end{bmatrix}$, we define the $m$-dimensional column vector $A\vec{x}$ by:

$A\vec{x}:=x_1\vec{a}_1+\dotsb+x_n\vec{a}_n,$
$\begin{bmatrix}
1 & 2
\\
3 & 4
\\
5 & 6
\end{bmatrix}
\begin{bmatrix}
100 \\ 10
\end{bmatrix}
=
100
\begin{bmatrix}
1
\\
3
\\
5
\end{bmatrix}
+
10
\begin{bmatrix}
2
\\
4
\\
6
\end{bmatrix}
=
\begin{bmatrix}
120
\\
340
\\
560
\end{bmatrix}.$

- Let $A$ be an $m{\times}n$ matrix, and let $B$ be an $n{\times}l$ matrix of the form $B:=\begin{bmatrix}\vec{b}_1 & \dotsb & \vec{b}_l\end{bmatrix}$. We define the product $AB$, which is an $m{\times}l$ matrix, by:
$AB:=\begin{bmatrix}A\vec{b}_1 & \dotsb & A\vec{b}_l\end{bmatrix},$
$\begin{bmatrix}
1 & 2
\\
3 & 4
\\
5 & 6
\end{bmatrix}
\begin{bmatrix}
10 & 10
\\ 
1 & 100
\end{bmatrix}
=
\begin{bmatrix}
12 & 210
\\
34 & 430
\\
56 & 650
\end{bmatrix}.$


"""

# ╔═╡ 642ba2ee-ba66-4601-9edb-677c23e70780
md"""
##### 1-3. Examples of linear transforms on the plane.
Let $A$ be an $m{\times}n$ matrix, let $\vec{x}$ and $\vec{y}$ be $n$-dimensional column vectors respectively, and let $c$ and $d$ be constants. 
The correspondence 

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

is a mapping of the set $\mathbb{R}^n$ of all the $n$-dimensional column vectors to $\mathbb{R}^m$, and preserves addition and scalar multiplication: 

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

Such a mapping is said to be a linear mapping, and in particular a linear transform in case that $m=n$. Let us show two examples of linear transform on $\mathbb{R}^2$ defined by $2\times2$ matrices:

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

that is, 

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

"""

# ╔═╡ b8264135-0b84-4395-8d3a-ad8f42265004
md"""
###### 1-3-1. Dilation
Let $\lambda\ne0$ and $\mu\ne0$ be constants. The linear transform  

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

makes the scale of $x$-axis to be $\lambda$, and makes the scale of $y$-axis to be $\mu$. If $\lambda<0$, then the orientation of $x$-axis is opposite to that of $u$-axis. 

Under this linear transformation, the equation $x^2+y^2=1$ of the unit circle with the center at the origin becomes 

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

The following figure describes the case of $\lambda=\sqrt{2}$ and $\mu=1/\sqrt{2}$.

"""

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

# ╔═╡ df792cbc-3244-453d-9cb8-6bc359b881c5
md"""
###### 1-3-2. Rotation
Let $\theta \in [0,2\pi)$ be a parameter. The matrix $P(\theta)$ defined by
$P(\theta)
=
\begin{bmatrix}
\cos\theta & -\sin\theta
\\
\sin\theta & \cos\theta
\end{bmatrix},$

gives the linear transform describing the anti-clockwise rotation of $\theta$ with the center at the origin. We remark that the first and second columns are unit vectors, and that the second column can be written as 

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

which is orthogonal to the first column.

When $\theta=\pi/6$, we draw $x$-axis and $y$-axis in the $uv$-plane:

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

"""

# ╔═╡ 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. Data described by matrices
"""

# ╔═╡ 56cc741b-f300-4558-a078-a161f58307a9
md"""
##### 2-1. Data matrices

Below is a list of height and weight data for 100 people, which can be considered as a $100\times2$ matrix. In general, when the number of individuals is $N$ and each individual has $p$ types of data, the data list can be expressed as a matrix.

"""

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

# ╔═╡ c8263139-b11e-4404-a311-e5a345944168
md"""
Each individual's data can be plotted as a point on a plane with height on the horizontal axis and weight on the vertical axis. This is called a scatter plot. By using pseudoinverse or singular value decomposition in linear algebra, it is possible to obtain the best line that shows the relationship between height and weight using linear regression (the least squares method).
"""

# ╔═╡ 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. Grayscale images
A grayscale image is a matrix in which each small square (pixel) has an integer between $0$ and $255$, or a real number between $0$ and $1$ obtained by dividing the integer by $255$. The integer $0$ represents black and the integer $255$ represents white. The number represents the intensity of the color, and the higher the number, the stronger the color. For example, the matrix

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

becomes the following grayscale image:
"""

# ╔═╡ 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 images

An RGB image is created by combining three matrices of grayscale image data of the same size, each of which is colored with the three primary colors red, green, and blue. The following shows three $16\times16$ matrices generated by the random numbers created by a programming language, and the RGB image by combining the triplet.

"""

# ╔═╡ 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. Singular value decomposition and low rank approximation
"""

# ╔═╡ a7744957-4472-4012-ad31-372cd48fa78c
md""" ##### 3-1. An example of singular value decomposition

The $2\times2$ matrix

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

is decomposed as 

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

$U$ is a matrix of the clockwise rotation of $\pi/3$, $V^T$ is a matrix of the anti-clockwise rotation of $\pi/4$, and one can understand that $A$ defines a composition mapping of the rotation ($V^T$), the dilation ($\Sigma$), and rotation ($U$). 

If we set 

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

then the equation $x^2+y^2=1$ of the unit circle becomes a slanted ellipse  $49s^2+15\sqrt{3}st+19t^2=16$. The clockwise rotation gives the directions of the major and minor axes of the ellipse. 
"""

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

# ╔═╡ 2e5e3122-1b2b-4dcf-88d4-f79b3e727bbf
md"""
If we set 

$\sigma_1=2,
\quad
\sigma_2=\frac{1}{2},$

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

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

then

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

that is, 

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

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


# ╔═╡ 3f45852f-ad72-4ba2-b147-a14151e8ed97
md""" ##### 3-2. Singular value decomposition and low rank approximation of matrices
In general an $m{\times}n$ matrix $A=\begin{bmatrix}\vec{a}_1 & \dotsb & \vec{a}_n \end{bmatrix}$ has a specific nonnegative integer which is called as the rank of  $A$ defined by 

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

It follows that $1 \leqq r \leqq \min\{m,n\}$ unless all the entries of $A$ are zero.  
"""

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

# ╔═╡ 3d656643-bf04-4ac7-99fc-6d98a571523a
md""" 
It is known that there exist  

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

such that we have 

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

Set 

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

It is known that the rank of $A_k$ is $k$ and closest to $A$ among all the $m{\times}n$ matrices of rank $k$, that is, $A_k$ is the best approximation of $A$ among all the $m{\times}n$ matrices of rank $k$.   

We compute $384\times512\times3$ RGB image of fried noodle at a canteen, City University of Hong Kong: 
"""

# ╔═╡ 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. Wavelet decomposition and image processing
"""

# ╔═╡ 6b1ce739-aca6-47d6-842a-701712b2cc61
md"""
##### 4-1. The Haar wavelet
The discrete wavelet is a pair of orthonomal two vectors $\vec{u}$ and $\vec{v}$ satisfying some condition, and the most typical example of the discrete wavelets is the Haar wavelet. They are very simple vectors: 

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

Roughly speaking the discrete wavelet transform by the Haar wavelet is to take averages of neighboring elements. The elements of a vector are transformed  to the average of neighboring two elements, and the elements of a matrix are converted to the average of neighboring 2 by 2 elements. The following examples show the outcomes of repeating the discere wavelet tranform twice: 

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

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


The output such as $\vec{a}_\ell$ and $A_\ell$ of $\ell$ times operations is said to be the approximation part of level $\ell$, and the remainder term such as $\vec{a}_0-\vec{a}_\ell$ and $A_0-A_\ell$ is called the detail part of level $\ell$. 

We shall observe the wavelet decomposition of image data using the Haar wavelet.

""" 

# ╔═╡ 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""" $8\times8$ grayscale image

"""

# ╔═╡ 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. $16\times16$ RGB image created by random numbers

"""

# ╔═╡ 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 image
fried noodle at a canteen, City University of Hong Kong

 $384=2^7\times3$, $512=2^9$

"""

# ╔═╡ 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 = "1.10.5"
manifest_format = "2.0"
project_hash = "e7a3735e3f7ab5d904662ad4b055b1324ae4d436"

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version = "1.5.0"
weakdeps = ["ChainRulesCore", "Test"]

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

[[deps.AbstractPlutoDingetjes]]
deps = ["Pkg"]
git-tree-sha1 = "c278dfab760520b8bb7e9511b968bf4ba38b7acc"
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    [deps.ArrayInterface.weakdeps]
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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 = "5.11.0+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", "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"
"""

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