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]