Inverse Perspective Mapping

In the previous sections, we have seen examples of inverse perspective: the ability to derive the 3D point given a 2D image point and the knowledge of a constraint in the world on whose surface the point lies. It is always possible to create a system between the optical ray (8.16) and a manifold in $\mathbb{R}^3$:

$$\left\{ \begin{array}{l} \mathbf{x} = \lambda \mathbf{V} \mathbf{p} + \mathbf{t} \\ f(\mathbf{x}) = 0 \\ \end{array}\right.$$ (8.38)

denoting it as $\mathbf{V}=\mathbf{R}^{-1}\mathbf{K}$. This same formulation is used in computer graphics to refer to RayTracing techniques. In this section, we will generalize various cases.

Intersection of Optical Ray and Plane

A generic plane in $\mathbb{R}^3$ expressed in the form

$$\hat{\mathbf{n}} \cdot \mathbf{x} + q = 0$$ (8.39)

serves as a constraint to allow the intersection between the optical ray (8.16) and the plane (8.39). The system (8.38) is linear and can be solved for $\lambda$ and, by substituting $\lambda$ into the first equation, determine the 3D point. It is also possible to create a linear application associated with the intersection of a plane and a line in the form $\mathbf{x} \equiv \mathbf{A}_{4 \times 3} \mathbf{p}$, having defined

$$\mathbf{A}_{4 \times 3} = \begin{bmatrix} \left( \hat{\math... ...{\top} \\ \hat{\mathbf{n}}^{\top} \end{bmatrix} \mathbf{V}$$ (8.40)

.

Intersection of an Optical Ray and a Sphere

The manifold has the equation

$$\left\Vert \mathbf{x} - \mathbf{x}_0 \right\Vert^2 = r^2$$ (8.41)

which, when combined with the system (8.38), allows us to obtain

$$\lambda^2 \left\Vert \mathbf{V} \mathbf{p} \right\Vert^2 + 2... ...ht) + \left\Vert \mathbf{t} - \mathbf{x}_0 \right\Vert^2 = r^2$$ (8.42)

The solution of the quadratic equation can therefore have 0 (no intersection), 1 (the optical ray is tangent to the sphere), or 2 roots (the optical ray intersects the sphere).