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