Lecture 4 (Feb 15)

Relationship to image inpainting
Taking the mean, mode and median of the neighboring intensities:
$E_2(g) =  \sum_{i=1}^N \left| f_i-g \right|^2$, $E_0(g) =  \sum_{i=1}^N \left| f_i-g \right|^0$ and $E_1(g) =  \sum_{i=1}^N \left| f_i-g \right|^1$.

Intensity transformation [$s=T(r)$]: most are application/image dependent.

Reading Materials

Lecture notes: p.31-35

Slides: [click here]

Lecture 3 (Feb 11)

Image Interpolation

Properties of the piecewise linear interpolation:
1. Will not create new global extrema, i.e. $\max_{x\in[x_1,x_M]} p(x) = \max_i f_i$ and  $\min_{x\in[x_1,x_M]} p(x) = \min_i f_i$.
2. $TV(\mathbf{g})=TV(\mathbf{f})$ where $TV(\cdot)$ is the total variation of the signal.
3. It is a linear transformation from $\mathbb{R}^M$ to $\mathbb{R}^{2M-1}$.

Two dimensional (image) interpolation: bilinear interpolation.

Relationship to image inpainting
Taking the mean, mode and median of the neighboring intensities:
$E_2(g) =  \sum_{i=1}^N \left| f_i-g \right|^2$, $E_0(g) =  \sum_{i=1}^N \left| f_i-g \right|^0$ and $E_1(g) =  \sum_{i=1}^N \left| f_i-g \right|^1$.

Reading materials

Lecture notes: p.29-31

HW1

Problems:
#5.9, 10, 11, 14, 15

Problems: [click here]
Due: Feb 25 (Monday)
Submit: To the TA in the tutorial

Lecture 2 (Feb 8)

Problems about the simple imaging model

Sampling: $\Omega \Rightarrow \Omega'=\{(x,y),x=1,2,\cdots,M, y=1,2,\cdots,N\}$.

Quantization: $[0,L-1] \Rightarrow \{0,1,2,\cdots,L-1\}$. MATLAB data type: uint8 and uint16.

Image interpolation

Given a signal $(x_i,f(x_i))$ for $i=1,2,\cdots,M$, we want to determine an approximation to $f(x^*)$ for $x^* \ne x_i$. Using
(i) High order polynomial interpolation: $p(x) \in P^{M-1}$ such that $p(x_i)=f(x_i) \Rightarrow$ one needs to invert a BIG matrix. This can be partially fixed by better basis for interpolation, such as Lagrange Interpolating Polynomials or Newton's Divided Difference.
(ii) Piecewise polynomial interpolation: determine $p_i(x) \in P^N$ for some $N \ll M-1$ such that the polynomial defined only piecewisely for $x \in [x_i,x_{i+1}]$ for $i=1,\cdots,M-1$. Then
$$
p(x) = \left\{
\begin{array}{cc}
p_1(x) & \mbox{ if $x\in[x_1,x_2]$} \\
p_2(x) & \mbox{ if $x\in[x_2,x_3]$} \\
\vdots & \\
p_{M-1}(x) & \mbox{ if $x\in[x_{M-1},x_M]$.}
\end{array}
\right.
$$
In particular, if $N=1$, the interpolation is called a piecewise linear interpolation.

Some properties of the piecewise linear interpolation

Properties of the piecewise linear interpolation:
1. Will not create new global extrema, i.e. $\max_{x\in[x_1,x_M]} p(x) = \max_i f_i$ and  $\min_{x\in[x_1,x_M]} p(x) = \min_i f_i$.
2. $TV(\mathbf{g})=TV(\mathbf{f})$ where $TV(\cdot)$ is the total variation of the signal.
3. It is a linear transformation from $\mathbb{R}^M$ to $\mathbb{R}^{2M-1}$.

Reading materials

Lecture notes: p.21-22,25-29
Presentation file: [click here]

Lecture 1 (Feb 1)

Overview

Introduction

A simple imaging model.
$$
f: \Omega \rightarrow [0,L]
$$
where $\Omega$ is a bounded domain in $\mathbb{R}^2$.

However, such a simple model has two limitations:
(1) Sampling;
(2) Quantization.

Reading materials

Lecture notes: p.20-21
Presentation file: [click here]

Lecture Hours

Week 1		1-Feb	Lecture Hours	Tutorial Hours
		L: Tim	1.5	0
Week 1	4-Feb	8-Feb
	T: Cancel
	L: Cancel	L: Tim	1.5	0
Week 2	11-Feb	15-Feb
	T:Alan
	L: Tim	L: Tim	3	1
Week 3	18-Feb	22-Feb
	T: Tim
	L: Tim	L: Tim	4	0
Week 4	25-Feb	1-Mar
	T: Alan
	L: Alan	L: Alan	0	4
Week 5	4-Mar	8-Mar
	T: Alan
	L: Tim	L: Tim	3	1
Week 6	11-Mar	15-Mar
	T: Tim
	L: Tim	L: Tim	4	0
Week 7	18-Mar	22-Mar
	T: Alan
	L: Tim	L: Tim	3	1
Week 8	25-Mar	29-Mar
	T: Alan
	L: Tim	L: Tim	3	1
Week 9	1-Apr
	T: Tim
	L: Tim		2.5	0
Week 10	8-Apr	12-Apr
	T: Alan
	L: Tim	L: Tim	3	1
Week 11	15-Apr
	T: Alan
	L: Tim		1.5	1
Week 11		26-Apr
		L: Tim	1.5	0
Week 12	29-Apr	3-May
	T: Tim
	L: Tim	L: Alan	2.5	1.5
Week 13	6-May
	T: Tim
	L: Tim		2.5	0
			36.5	11.5

Syllabus

[click here]