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pluto_svd.jl
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pluto_svd.jl
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### A Pluto.jl notebook ###
# v0.19.36
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
# ╔═╡ 8f3a2089-c44c-450d-8d10-d355c9503ed7
begin
using PlutoUI
using LaTeXStrings
using Plots
using Colors
using ImageFiltering
using Images
using ImageMagick
using TestImages
using ImageView
using ImageTransformations
using LinearAlgebra
using LowRankApprox
end
# ╔═╡ 6b1ce739-aca6-47d6-842a-701712b2cc61
md"""
# SVD and low rank approximation
"""
# ╔═╡ a1dc1236-a3d4-456a-ab2b-5a3f2bbac38a
md"""
## 1. Singular Value Decomposition
Let $\mathbb{K}$ denote $\mathbb{R}$ or $\mathbb{C}$, and let $A \in \mathbb{K}^{m{\times}n}$. Set $r=\operatorname{rank}(A)$ for short. We state well-known facts.
1. $\operatorname{rank}(A^\ast{A})=\operatorname{rank}(AA^\ast)=\operatorname{rank}(A).$
2. $A^\ast{A}\geqq0.$
3. There exist positive numbers $\sigma_1 \geqq \dotsc \geqq \sigma_r >0$ such that
$\det(zE_n-A^\ast{A})=
(z-\sigma_1^2)\dotsc(z-\sigma_r^2)z^{n-r}.$
4. There exist an orthonormal system $\{u_1,\dotsc,u_m\}$ of $\mathbb{K}^m$ and an orthonormal system $\{v_1,\dotsc,v_n\}$ of $\mathbb{K}^n$ such that
$A
=
U{\Sigma}V^\ast
=
\sum_{l=1}^r
\sigma_l u_l v_l^\ast.$
5. The above decomposition of a matrix $A$ is said to be the singular value decomposition of $A$, where $U$ and $V$ are unitary matrices and $\Sigma$ is a rectangular diagonal matrix of the form
$U=[u_1,\dotsc,u_m],
\quad
V=[v_1,\dotsc,v_n],
\quad
\Sigma
=
\begin{bmatrix}
\sigma_1 & & &
\\
& \ddots & &
\\
& & \sigma_r &
\\
& & &
\end{bmatrix}.$
"""
# ╔═╡ 1e72b8d9-b161-4f31-92e1-b90a25a02a35
md"""
## 2. Low rank approximation
Let $A \in \mathbb{K}^{m{\times}n}$, and let $r=\operatorname{rank}(A)$.
Suppose that the SVD of $A$ is given by
$A=\displaystyle\sum_{l=1}^r\sigma_l u_l v_l^\ast.$
Set
$A_k:=\sum_{l=1}^k\sigma_l u_l v_l^\ast,
\quad
k=1,\dotsc,r.$
Then $\operatorname{rank}(A_k)=k$. It is well-known that $A_k$ is the best approximation of $A$ among $m{\times}n$ matrices with rank $k$. More precisely we have the Eckart-Young inequality of the form
$\lVert{A-B}\rVert
\geqq
\lVert{A-A_k}\rVert,
\quad
B\in\mathbb{K}^{m{\times}n},
\
\operatorname{rank}(B)=k,$
where $\lVert\cdot\rVert$ is the spectral norm of matrices defined by
$\lVert{B}\rVert
:=
\max_{\substack{x \in \mathbb{K}^n \\ x\ne0}}
\frac{\lvert{Bx}\rvert}{\lvert{x}\rvert},$
and $\lvert\cdot\rvert$ is the standard norm of numeric vectors. The same inequlity holds if we replace the spectral norm by the Frobenius norm or the nuclear norm.
"""
# ╔═╡ 2b8cafe5-0efb-4892-957c-055c63bf23ed
begin
I=load("./material/char_kway_teow.jpg");
X=imresize(I, ratio=1/4);
(p,q)=size(X);
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
# ╔═╡ dd7e95e0-8d80-44f0-8ad6-5c9dece4d18f
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
# ╔═╡ f29dc05b-8209-42a1-8f37-4f608c216459
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
# ╔═╡ 33e36dbc-6682-4de6-a911-7eb2a0ec4f05
md"""
rank = $(@bind r Slider(1:rank, show_value=true))
"""
# ╔═╡ a9f38308-8483-4b3a-9160-f2622027c7e2
begin
p1=plot(colorview(RGB,W[:,:,:,r]),
title="Low Rank Approx 1-100/384",
xaxis=false,
xticks=false,
yaxis=false,
yticks=false,
grid=false);
p2=plot(colorview(RGB,X),
title="Original RGB Image",
xaxis=false,
xticks=false,
yaxis=false,
yticks=false,
grid=false);
plot(p1,p2)
end
# ╔═╡ b1e6cc97-a684-4cc2-8a53-5514544b453b
size(X)
# ╔═╡ 00000000-0000-0000-0000-000000000001
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PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
TestImages = "5e47fb64-e119-507b-a336-dd2b206d9990"
[compat]
Colors = "~0.12.10"
ImageFiltering = "~0.7.5"
ImageMagick = "~1.2.2"
ImageTransformations = "~0.9.5"
ImageView = "~0.11.6"
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LaTeXStrings = "~1.3.0"
LowRankApprox = "~0.5.3"
Plots = "~1.38.16"
PlutoUI = "~0.7.51"
TestImages = "~1.8.0"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
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