3.2. Image Warping

Warping

Image warping is the process of geometrically transforming an image so that pixels move to new locations according to a mapping function or transformation matrix. Warping changes the spatial arrangement of pixels, not their intensity values.

• In forward warping, each pixel in the source image $f$ is mapped to its new position in the destination image $g$ using $p^{\prime }=Tp$, such that $g(Tp)=f(p)$. Although conceptually simple, forward warping is rarely used in practice because of misalignment, which may yield holes in the output image or a single pixel in $g$ receiving multiple values from $f$.
  ⠀
• Instead of pushing source pixels forward, inverse warping pulls values $p$ from the source to fill every pixel $p^{\prime }$ in the destination image, such that $p=T^{-1}{p}^{\prime }$, and therefore $g(p^{\prime })=f(T^{-1}{p}^{\prime })$. However, since $p$ typically lies between discrete pixels, its value must be interpolated from nearby samples.

Because forward warping maps pixels to irregular, non-grid positions in $g$ (some overlap while others leave holes), there’s no consistent neighborhood to sample from for interpolation. Inverse warping fixes this by sampling from the regular grid of the source image, where interpolation is well-defined.