Metric stratum

The metric stratum corresponds to the group of similarities. These transformations correspond to Euclidean transformations (i.e. orthonormal transformation + translation) complemented with a scaling. When no absolute yardstick is available, this is the highest level of geometric structure that can be retrieved from images. This property is crucial for special effects since it enables the possibility to use scale models in movies.

A metric transformation can be represented as follows:

$$\left[\begin{array}{c} X' \\ Y' \\ Z' \end{array}\right] = \... ...begin{array}{c} t_{14} \\ t_{24} \\ t_{34} \end{array} \right]$$ (B27)

with $r_{ij}$ the coefficients of an orthonormal matrix. The coefficients $r_{ij}$ are related by 6 independent constraints $\sum^3_{k=1} r_{ik}r_{jk}=\delta_{ij}, ( 1 \leq i \leq j; 1 \leq j \leq 3)$ with $\delta_{ij}$ the Kronecker delta^B1. This corresponds to the matrix relation that ${\bf R}^\top{\bf R} = {\bf RR}^\top = {\bf I}$ and thus ${\bf R}^{-1} = {\bf R}^\top$. Recall that ${\bf R}$ is a rotation matrix if and only if ${\bf RR}^\top = {\bf I}$ and det${\bf R}=1$. In particular, an orthonormal matrix only has 3 degrees of freedom. Using homogeneous coordinates, (2.27) can be rewritten as ${\tt M}' \sim {\bf T}_{M} {\tt M}$, with

$${\bf T}_{M} \sim \left[\begin{array}{cccc} \sigma r_{11} &... ...32} & \sigma r_{33} & t_Z \\ 0 & 0 & 0 & 1\end{array} \right]$$ (B28)

A metric transformation therefore counts 7 independent degrees of freedom, 3 for orientation, 3 for translation and 1 for scale.

In this case there are two important new invariant properties: relative lengths and angles. Similar to the affine case, these new invariant properties are related to an invariant geometric entity. Besides leaving the plane at infinity unchanged similarity transformations also transform a specific conic into itself, i.e. the absolute conic. This geometric concept is more abstract than the plane at infinity. It could be seen as an imaginary circle located in the plane at infinity. In this text the absolute conic is denoted by $\Omega$. It will be seen that it is often more practical to represent this entity in 3D space by its dual entity $\Omega ^*$. When only the plane at infinity is under consideration, $\omega _\infty$ and $\omega^*_\infty$ are used to represent the absolute conic and the dual absolute conic (these are 2D entities). Figure 2.2 and Figure 2.3 illustrate these concepts.

Figure 2.2: The absolute conic $\Omega$ and the absolute dual quadric $\Omega ^*$ in 3D space.

Figure 2.3: The absolute conic $\omega _\infty$ and dual absolute conic $\omega _\infty ^*$ represented in the purely imaginary part of the plane at infinity ${\tt \Pi}_\infty$

The canonical form for the absolute conic $\Omega$ is:

$$\Omega : X^2 + Y^2 + Z^2 = 0 \mbox{ and } W=0$$ (B29)

Note that two equations are needed to represent this entity. The associated dual entity, the absolute dual quadric $\Omega ^*$, however, can be represented as a single quadric. The canonical form is:

$$\Omega^* \sim \left[\begin{array}{cccc} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&0 \end{array}\right] \enspace .$$ (B30)

Note that ${\tt\Pi}_\infty = [0 \,0 \,0 \,1]^\top$ is the null space of $\Omega ^*$. Let ${\tt M}_\infty \sim [X \, Y\, Z\, 0]^\top$ be a point of the plane at infinity, then that point in the plane at infinity is easily parameterized as ${\tt m}_\infty \sim [X\,\,Y\,Z]^\top$. In this case the absolute conic can be represented as a 2D conic:

$$\omega_\infty \sim \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 &... ...& 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right] \enspace .$$ (B31)

According to (2.28), applying a similarity transformation to ${\tt M}_\infty$ results in ${\tt m}_\infty \mapsto {\tt m'}_\infty \sim \sigma {\bf R} {\tt m}_\infty$. Using equations (2.14),(2.15) and (2.20), it can now be verified that a similarity transformation leaves the absolute conic and its associated entities unchanged:

$$\left[\begin{array}{cc} {\bf I}_{3 \times 3} & 0_3 \\ 0_3^\t... ...a {\bf R} & {\tt t} \\ 0_3^\top & 1 \end{array} \right]^\top$$ (B32)

and

$${\bf I}_{3 \times 3}\sim \sigma^{-1}{\bf R}^{-\top} {\bf I}_... ...} \sim \sigma {\bf R} {\bf I}_{3 \times 3} {\bf R}^\top \sigma$$ (B33)

Inversely, it is easy to prove that the projective transformations which leave the absolute quadric unchanged form the group of similarity transformations (the same could be done for the absolute conic and the plane at infinity):

$$\left[\begin{array}{cc} {\bf I}_{3 \times 3} & {\tt0}_3 \\ ... ...}^\top {\bf A}^\top & {\tt c}^\top {\tt c} \end{array} \right]$$

Therefore ${\bf A}{\bf A}^\top \sim {\bf I}_{3 \times 3}$ and ${\tt c} = {\tt0}_3$ which are exactly the constraints for a similarity transformation.

Angles can be measured using Laguerre's formula (see for example [172]). Assume two directions are characterized by their vanishing points ${\tt v}$ and ${\tt v}'$ in the plane at infinity (i.e. the intersection of a line with the plane at infinity indicating the direction). Compute the intersection points ${\tt j}$ and ${\tt j}'$ between the absolute conic and the line through the two vanishing points. The following formula based on the cross-ratio then gives the angle (with $i=\sqrt{-1}$):

$$\alpha = \frac{1}{2i} \log \{ {\tt v}_1 , {\tt v}_2 ; {\tt j}, {\tt j}' \}$$ (B34)