Interpolation filters

For interpolating data, Pandora2d uses its own interpolation filters. Those filters are used in the refinement step.

Bicubic

This filter uses the following kernel in 1D:

$W(x) = \begin{cases} (\alpha + 2)|x|^3 - (\alpha + 3) |x|^2 + 1 & \text{if } |x| \leq 1 \\ \alpha |x|^3 - 5\alpha |x|^2 + 8\alpha |x| - 4\alpha & \text{if } 1 < |x| < 2 \end{cases}$

Where

$\alpha = -0.5$

and

$x = a - \text{fractional\_shift}$

In the following image, two lines of an image are sketched with on each a point to interpolate at indices 3.5 and 5.5:

For both points to interpolate, the fractional_shift is 0.5. Thus, the coefficients to apply to pixels’ values are the same for both points.

$x$ values for which kernel is not null are available for $a$ ranging from $-1$ to $2$ .

Thus, for an image in 2D, the filter is applied on an array of shape $4 \times 4$ where coefficients are applied on columns then lines.

Sinc

This filter use a cardinal sine of the form:

$\frac{\sin(x\ \pi)}{x\ \pi}$

Where $x$ is the fractional shift.

Computed coefficients are windowed by a Gaussian of form:

$\exp\left(\frac{-2\ \pi\ x^2}{\sigma^2}\right)$

Where $\sigma$ correspond to the size of the filter (half width of the window).