Inverse Perspective Mapping

The previous sections presented examples of inverse perspective: the possibility of recovering a 3D point from a 2D image point and knowledge of a constraint in the world on whose surface the point lies. It is always possible to define a system involving the optical ray (9.26) and a variety in $\mathbb{R}^3$:

$$\left\{ \begin{array}{l} \mathbf{x} = \lambda \mathbf{V} \mathbf{p} + \mathbf{t} \\ f(\mathbf{x}) = 0 \\ \end{array}\right.$$ (9.47)

where $\mathbf{V}=\mathbf{R}^{-1}\mathbf{K}$ has been defined. The same formulation is used in computer graphics to denote RayTracing techniques. Several cases are generalized in this section.

Intersection of an Optical Ray and a Plane

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

$$\hat{\mathbf{n}} \cdot \mathbf{x} + q = 0$$ (9.48)

provides a constraint that allows the intersection of the optical ray (9.26) with plane (9.48). System (9.47) is linear and can be solved for $\lambda$; substituting $\lambda$ into the first equation then determines the 3D point. It is also possible to define a linear mapping associated with the line-plane intersection in the form $\mathbf{x} \equiv \mathbf{A}_{4 \times 3} \mathbf{p}$, having defined

$$\mathbf{A}_{4 \times 3} = \begin{bmatrix} \left( \hat{\math... ...{\top} \\ \hat{\mathbf{n}}^{\top} \end{bmatrix} \mathbf{V}$$ (9.49)

Intersection of an Optical Ray and a Sphere

The variety has equation

$$\left\Vert \mathbf{x} - \mathbf{x}_0 \right\Vert^2 = r^2$$ (9.50)

which, together with system (9.47), yields

$$\lambda^2 \left\Vert \mathbf{V} \mathbf{p} \right\Vert^2 + 2... ...ht) + \left\Vert \mathbf{t} - \mathbf{x}_0 \right\Vert^2 = r^2$$ (9.51)

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