Representation of a Three-Dimensional Moving Scene

Reference: An Invitation to 3-D vision, Chapter 2.

CheatSheet

2.1 Three-dimensional Euclidean space

A point in the Euclidean space:

A vector in the Euclidean space

Inner product of two vectors:

Cross product of two vectors:

For fixed $ u $, the cross product can be represented by a map from $ \mathbb{R}^3 $ to $ \mathbb{R}^3: u \mapsto u \times v $. We define this mapping matrix by

such that $ u \times v = \widehat{u}v $. Note that $ \widehat{u} $ is a skew-symmetric matrix, i.e. $ \widehat{u}^T = -\widehat{u} $.

The cross product defines a (one-to-one) map between a vector u and a $3 \times 3$ skew-symmetric matrix $ \widehat{u} $. The map is defined by a hat-operator

2.2 Rigid-body motion

Rigid-body: the distance between any pair of points $ (p,q) $ on it does not change over time as the object moves.

Euclidean transformation: a map that preserves the Euclidean distance between every pair of points. The set of all Euclidean transformation in 3-D space is denoted by $ E(3) $.

\[g: \mathbb{R}^3 \to \mathbb{R}^3; X \mapsto g(X)\]

Suppose a vector $ v=\mathbf{Y} - \mathbf{X} $. After the transformation of $g$, we obtain a new vector \(u = g_*(v) \doteq g(\mathbf{Y}) - g(\mathbf{X})\) where \(\begin{Vmatrix} g_*(v) \end{Vmatrix} = \begin{Vmatrix} v \end{Vmatrix}.\)

Some Euclidean transformations are not physically realizable. For example the map

\(f: \begin{bmatrix} X_1, X_2, X_3 \end{bmatrix} \mapsto \begin{bmatrix} X_1, X_2, -X_3 \end{bmatrix}\) preserves distances but not orientations. (reflection)

The map or transformation induced by a rigid-body motion is called a special Euclidean transformation. The word “special” indicates the fact that a transformation is orientation-preserving.

Rigid-body motion or special Euclidean transformation: A map that preserves the norm and the cross product of any two vectors

1. norm: \(\begin{Vmatrix} g_*(v) \end{Vmatrix} = \begin{Vmatrix} v \end{Vmatrix}, \forall v \in \mathbb{R}^3\)
2. cross product: \(g_*(u) \times g_*(v) = g_*(u \times v), \forall u, v \in \mathbb{R}^3\)
3. (consequently) angle or inner product: \(\left\langle u,v \right\rangle = \left\langle g_*(u), g_*(v) \right\rangle\)

Let us define a Cartesian coordinate frame, with its principal axes given by three orthonormal vectors $ e_1, e_2, e_3 \in \mathbb{R}^3 $ ($e_1 \times e_2 = e_3 $, right-handed frame)

After a rigid-body motion $ g $,

The resulting vectors still form a right-handed frame.

When a camera moves, the camera frame also moves along with the camera. The configuration of the camera is then determined by two components:

1. Translational part $ T $ : the vector between the origin of the world frame and that of the camera frame.
2. Rotational part $ R $ : the relative orientation of the camera frame $ C $, with coordinate axes $ (x,y,z) $, relative to the fixed world frame $ W $ with coordinate axes $ (X,Y,Z) $

2.3 Rotational motion and its representations

2.3.1 Orthogonal matrix representation of rotations

The frame C relative to the frame W is determined by the three orthonormal vectors:

Then the configuration of the rotating object is determined by the $ 3 \times 3 $ matrix $ R_{wc} \doteq \begin{bmatrix} r_1, r_2, r_3 \end{bmatrix} \in \mathbb{R}^3 $

Since $ r_1, r_2, r_3 $ form an orthonormal frame,

Since we only deal with a right-handed frame, $ det(R_{wc}) = +1 $. Hence $ R_{wc} $ is a special orthogonal matrix, which is denoted by

\[SO(3) \doteq \left\{ R \in \mathbb{R}^{3\times3} \middle|\ R^TR=I,\ det(R)=+1 \right\}\]

The space $ SO(3) $ is referred as the special orthogonal group of $ \mathbb{R}^3 $ or simply rotation group.

Suppose that for a given point $ p $, its coordinates in the world frame $ W $ is $ X_w = \left[ X_{1w}, X_{2w}, X_{3w} \right] $.

Since $ r_1, r_2, r_3 $ form a basis for $ \mathbb{R}^3 $, $ X_w $ can also be expressed as

where $ X_c = \left[ X_{1c}, X_{2c}, X_{3c} \right] $ is the coordinates of the same point $ p $ in the frame $ C $ relative to the frame $ W $.

$ R_{wc} $ transforms the coordinates $ X_c $ in the frame $ C $ to its coordinates $ X_w $ in the frame $ W $

2.3.2 Canonical exponential coordinates for rotations

Given a trajectory $ R(t) $ that describes a continuous rotational motion, the rotation must satisfy $ R(t)^TR(t) = I$.

By computing the derivative with respect to time $ t $,

The resulting equation shows that $ \dot{R}(t) R^T(t) \in \mathbb{R}^{3 \times 3} $ is a skew-symmetric matrix. There must exist a vector $ w(t) \in \mathbb{R}^3 $ such that

The space of all skew-symmetric matrices is denoted by

\[so(3) \doteq \left\{ \widehat{w} \in \mathbb{R}^{3 \times 3} \mid w\in \mathbb{R}^3 \right\}.\]

Note that $ so(3) $ depends only on three parameters.

Since the above equation is a linear ordinary differential equation (ODE), $ R(t) $ can be interpreted as the state transition matrix. Assuming $ R(0) = I $, the solution is

The matrix exponential indeed defines a map from the space $ so(3) $ to $ SO(3) $, the so-called exponential map

\[exp: so(3) \to SO(3); \quad \widehat{w} \mapsto e^{\widehat{w}}\]

We can verify the matrix $ e^{\widehat{w}t} $ is a rotation matrix.

Hence $ \left( e^{\widehat{w}t} \right)^T e^{\widehat{w}t} = I $.

The physical interpretation of $ R(t) = e^{\widehat{w}t} $ is that if $ \begin{Vmatrix} w \end{Vmatrix} = 1 $, $ R(t) = e^{\widehat{w}t} $ is simply a rotation around the axis $ w \in \mathbb{R}^3 $ by an angle $ t $ radians.

Theorem 2.8 (Logarithm of $ SO(3) $). For any $ R \in SO(3) $, there exists a (not necessarily unique) $ w \in \mathbb{R}^3 $ such that $ R = exp( \widehat{w} ) $. We denote the inverse of the exponential map by $ \widehat{w} = log(R) $.

Theorem 2.9 (Rodrigues’ formula for a rotation matrix). Given $ w \in \mathbb{R}^3$, the matrix exponential $ R = e^{\widehat{w}} $ is given by

\[e^{\widehat{w}} = I + {\widehat{w} \over \begin{Vmatrix} w \end{Vmatrix}} sin(\begin{Vmatrix} w \end{Vmatrix}) + {\widehat{w}^2 \over \begin{Vmatrix} w \end{Vmatrix}^2} (1 - cos(\begin{Vmatrix} w \end{Vmatrix})\]

if $ \begin{Vmatrix} w \end{Vmatrix} = 1 $, $ {\widehat{w} \over \begin{Vmatrix} w \end{Vmatrix}} = \widehat{w}, \begin{Vmatrix} wt \end{Vmatrix} = t $, so that

Note that the exponential map is not commutative.

2.4 Rigid-body motion and its representations

The figure illustrates a moving rigid object with a coordinate frame $ C $. The coordinates of the point $ p $ is either $ X_w $ in the world frame or $ X_c $ in the object frame. The relationship between them is given by

where $ R_{wc} \in SO(3) $ is the relative rotation between two frames and $ T_{wc} \in \mathbb{R}^3 $. The vector from the origin of the frame $ C $ to the point $ p $ in the world frame $ W $ is $ R_{wc}X_c $.

We denote the full rigid-body motion by $ g_{wc} = \left( R_{wc}, T_{wc} \right ) $, or simply $ g = \left( R, T \right ) $. In compact form, we rewrite

The set of all possible configurations of a rigid body can then be described by the space of rigid-body motions or special Euclidean transformations

\[SE(3) \doteq \left\{ g = (R, T) \mid R \in SO(3), T \in \mathbb{R}^3 \right\}\]

2.4.1 Homogeneous representation

Rigid body motion, \( X_w = R_{wc}X_c + T_{wc} \), is not linear but affine. An affine transformation can be converted to linear on by homogeneous coordinates. In homogeneous coordinates, a point $ p \in \mathbb{R}^3 $ and a vector $ v \in \mathbb{R}^3 $ are represented as

Using the homogeneous notation, the rigid-body motion can then be rewritten in a linear form.

$ \bar{g}_{wc} \in \mathbb{R}^{4 \times 4} $ is called the homogeneous representation of the rigid-body motion $ g_{wc} = \left( R_{wc}, T_{wc} \right) \in SE(3) $.

We can represent a rigid-body transformation of coordinates by a linear matrix multiplication. A matrix representation of the special Euclidean transformation is given by

\[SE(3) \doteq \left\{ \bar{g} = \begin{bmatrix} R & T \\ 0 & 1 \end{bmatrix} \middle| R \in SO(3), T \in \mathbb{R}^3 \right\} \in \mathbb{R}^{4 \times 4}\]

From the above definition, $ SE(3) $ forms a group.

Note that rigid motions act differently on points (rotation and translation) and vectors (only rotation)

2.4.2 Canonical exponential coordinates for rigid-body motions

From now on, for simplicity let us omit “-“ to indicate homogeneous representation and simply use $ g $.

Consider a moving rigid body described by $ SE(3): g(t) = \left( R(t), T(t) \right) $ or in the homogeneous representation

We know that $ \dot{R}(t) R^T(t) $ is a skew-symmetric matrix: there exists $ \widehat{w}(t) \in so(3) $ such that $ \widehat{w} = \dot{R}(t) R^T(t) $.

By defining a vector $ v(t) = \dot{T} - \widehat{w}(t)T(t) $, the above equation becomes a matrix, called a twist,

Then we have \(\dot{g}(t) = \dot{g}(t) g^{-1}(t) g(t) = \widehat{\xi}(t) g(t)\).

The set of all twists is denoted by

\[se(3) \doteq \left\{ \widehat{\xi} = \begin{bmatrix} \widehat{w} & v \\ 0 & 0 \end{bmatrix} \middle|\ \widehat{w} \in so(3), v \in \mathbb{R}^3 \right\} \subset \mathbb{R}^{4 \times 4}\]

We also define two operators, $\wedge$ and $\vee$, to convert between a twist $ \widehat{\xi} \in se(3) $ and its twist coordinates $ \xi \in \mathbb{R}^6 $ as follows:

Consider a case that $ \widehat{\xi} $ is a constant matrix: \(\dot{g}(t) = \widehat{\xi} g(t)\). This is a time-invariant linear ordinary differential equation. Assuming $ g(0) = I $, the solution is

By Rodrigues’ formula and properties of matrix exponential,

The exponential map defines a transformation from the space $se(3)$ to $SE(3)$,

The twist $ \widehat{\xi} \in se(3) $ is also called the exponential coordinates for $SE(3)$

Theorem 2.11 (Logarithm of $SE(3)$). For any $ g \in SE(3) $, there exist (not necessarily unique) twist coordinates $ \xi \in (v, w) $ such that $ g = exp(\hat{\xi}) $. We denote the inverse to the exponential map by $ \hat{\xi} = log(g) $