3.3. Image Interpolation

Interpolation

Interpolation estimates pixel values at non-integer coordinates when transforming or resizing images. It is essential in inverse warping, where the transformed coordinate $(x,y)$ usually doesn’t align with integer pixel positions.

In general, interpolation defines a function $\ell (x,y)$ that maps continuous coordinates to pixel intensities using the discrete image as its parameter set.

• Nearest neighbor interpolation assigns to the new location the value of the pixel at the nearest integer (rounded), such that
  ${\ell }_{\text{NN}}(x,y)=\ell (\lfloor x+0.5\rfloor ,\lfloor y+0.5\rfloor ).$
  It is the fastest and simplest interpolation method, but often produces blocky or jagged edges due to intensity jumps.
  ⠀
• Bilinear interpolation adds smoothing by using the four nearest pixels surrounding the non-integer coordinate $(x,y)$. Let $x_0=\lfloor x\rfloor$, $y_0=\lfloor y\rfloor$, $\Delta x=x-x_0$, $\Delta y=y-y_0$. Then,
  $\ell (x,y{)}_{\text{Bil}}=(1-\Delta x)(1-\Delta y)\ell (x_0,y_0)\Delta x(1-\Delta y)\ell (x_0+1,y_0)(1-\Delta x)\Delta y\ell (x_0,y_0+1)\Delta x\Delta y\ell (x_0+1,y_0+1)$