Homologous Points in Alternative Image Spaces

Through an appropriate Warp-Table, it is possible to transform the input image into an alternative form that still preserves the ability to reconstruct the scene in three dimensions. To also maintain the concept of disparity, meaning having homologous points along the same vertical coordinate between the two images of the stereoscopic pair, some additional constraints are necessary: the horizontal image coordinate must be a function of the $x$ camera coordinate, while the vertical image coordinate must not be a function of the $x$. The function of the horizontal coordinate must be monotonic (typically increasing) in $x$, while the function of the vertical coordinate must be monotonically increasing in $y$.

A widespread parameterization is the polar one, where, to avoid confusion between the axes, the equation has been chosen as

$$\begin{array}{l} x' = \operatorname{atan} \frac{x}{\sqrt{y... ...^2}} \\ y' = \operatorname{atan} \frac{y}{z} \\ \end{array}$$ (9.83)

which projects a camera point $(x,y,z)$ onto an image point $(x',y')$ (for comparison, I recall that the perspective equation has equations $x'= x/z$, $y' = y/z$). Given this projection, it is possible to write the inverse equation as

$$\begin{array}{l} x = \sin(x') \\ y = \cos(x') \sin(y') \\ z = \cos(x') \cos(y') \end{array}$$ (9.84)

This parametrization allows for the projection of all coordinates of a hemisphere (up to 180 degrees) into an image space, which the pin-hole model does not permit. Therefore, this parametrization is convenient for remapping Fish-Eye cameras. Through this parametrization, it is possible to write an equation similar to that in (9.21) to triangulate two image points.