Transformations Between Images and Look-Up Tables

Figure 1.8: Direct and inverse transformations between images.

Since this is a rather delicate topic that could lead to some ambiguity, it is useful to devote a section to how transformations between images are actually applied.

Let $f$ be a generic bijective transformation

$$f : \mathbb{R}^{2} \to \mathbb{R}^{2}$$ (1.128)

that maps point $(u,v)^{\top}$ belonging to the source image to point $(u',v')^{\top}$ in the destination image, that is,

$$(u',v')^{\top} = f(u,v)$$ (1.129)

This transformation is called Forward Warping.

Since images are not continuous but quantized into pixels, transformation $f$ cannot be used directly in real applications because it could both leave holes in the second image and project the same point from the first image multiple times. For these reasons, when an image is processed, the inverse transformation $f^{-1}$ is always used; for each point in the destination image $(u',v')^{\top}$, it returns the point in the source image $(u,v)$ from which the color is sampled, namely:

$$(u,v)^{\top} = f^{-1}(u',v')$$ (1.130)

This transformation is called Inverse Warping.

Clearly, the source image is also composed of pixels, but knowing point $(u,v)^{\top}$ makes it straightforward to use techniques such as linear interpolation to determine the pixel value.

If function $f^{-1}$ is very complex and the same transformation is to be applied to multiple images, computational time can be saved by creating a Look-Up Table (LUT) of $(u,v)^{\top}$ elements, with the same size as the destination image, in which the result of transformation (1.130) is stored for each element.