A bunch of note and code on the topic of Quaternion: applications in graphics and vision

A quaternion is a ring of number which can be written as $$\mathbf{H} = \{ x + y\mathbf{i} + z\mathbf{j} + w\mathbf{k} | x,y,z,w \in \mathbf{R} \},$$ where $\mathbf{i}, \mathbf{j}, \mathbf{k}$ are complex roots in terms of production, which safisfies $$\mathbf{i}^2=\mathbf{j}^2=\mathbf{k}^2=-1, \mathbf{i}\mathbf{j}=\mathbf{k}, \mathbf{j}\mathbf{k}=\mathbf{i},\mathbf{k}\mathbf{i}=\mathbf{j}.$$

The quaternion is a vector sapce in terms of adding, scalaring and norming, which means, $$(x_1 + y_1\mathbf{i} + z_1\mathbf{j} + w_1\mathbf{k}) + (x_2 + y_2\mathbf{i} + z_2\mathbf{j} + w_2\mathbf{k}) = (x_1 + x_2) + (y_1+y_2)\mathbf{i} + (z_1+z_2)\mathbf{j} + (w_1+w_2)\mathbf{k},$$ $$\lambda(x + y\mathbf{i} + z\mathbf{j} + w\mathbf{k}) = \lambda x + \lambda y\mathbf{i} + \lambda z\mathbf{j} + \lambda w\mathbf{k}\lambda \in \mathbf{R}.$$ $$||x + y\mathbf{i} + z\mathbf{j} + w\mathbf{k}|| = \sqrt{x^2+y^2+z^2+w^2}.$$ We have production of two quaternions by the linearity and production rule of $\{\mathbf{i},\mathbf{j},\mathbf{k}\}$, by caculation,

$$ \begin{align} (x + y\mathbf{i} + z\mathbf{j} + w\mathbf{k})(a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k}) = & (xa - yb - zc - wd) + (xb + ya + zd - wc) \mathbf{i} \\ + & (xc + wa + wb - yd)\mathbf{j} + (xd + wa + yc - zb) \mathbf{k}. \end{align} $$

So that we can proof, $$\forall p, q\in\mathbf{H}, ||pq|| = ||p||\cdot||q||.$$ And especially, a quaternion has its conjugation, which is $*(x + y\mathbf{i} + z\mathbf{j} + w\mathbf{k})=x - y\mathbf{i} - z\mathbf{j} - w\mathbf{k}$. We can prove that, $$*p\times p = p\times*p = ||p||^2$$ So that, a quaternion can obtain its inverse apparently. $$\frac{*p}{||p||^2} \times p = p \times \frac{*p}{||p||^2} = 1, \Rightarrow p^{-1} = \frac{*p}{||p||^2}.$$

We denote a quaternion $q = x + y\mathbf{i} + z\mathbf{j} + w\mathbf{k} = x + \mathbf{v}$, where $\mathbf{v}$ can be viewed as a correspondence of $\mathbf{R}^3$. The initial benifit is the production of quaternions,

$$ \begin{align} (x + y\mathbf{i} + z\mathbf{j} + w\mathbf{k})(a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k}) = & (x + \mathbf{t})(a + \mathbf{r}) \\ = & (xa - \langle \mathbf{t}, \mathbf{r} \rangle) + x \mathbf{r} + a \mathbf{t} + \mathbf{t} \wedge \mathbf{r}. \end{align} $$

And naturally, a vector from $\mathbf{R}^3$ can be embedded into $\mathbf{H}$, that $\forall (x,y,z) \in \mathbf{R}^3, H(x, y, z) = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \in \mathbf{H}$.

Actually, every $q \in \mathbf{H}$ satisfies $||q||=1$ can represent a rotation map in $\mathbf{R}^3$ by given,

$$\forall \mathbf{x} \in \mathbf{R}^3, \mbox{Rot}[q](\mathbf{x}) = q H(\mathbf{x}) q^{-1}.$$

Where the true background mean of that can be represented as,

$$\forall q \in \mathbf{H}, ||q||=1 \Rightarrow q = \cos \frac{\theta}{2} + \sin \frac{\theta}{2} \mathbf{t}, \mathbf{t}\in \mathbf{R}^3, ||\mathbf{t}||=1.$$

The meaning behind it is, $\mathbf{t}$ is the unchanged vector in $\mbox{Rot}[q]$, $\theta$ is the angle rotated by $\mathbf{t}$ by the rule of right hand orientation.

And it is natual to to think of rotation matrices, here we explore the relation ship between the two objects.

Suppose $q = \cos \frac{\theta}{2} + \sin \frac{\theta}{2} \mathbf{v}$, then $\forall \mathbf{x} = (x, y, z)^T \in \mathbf{R}^3$, here,

$$ \begin{align} \mbox{Rot}[q](x) = & (\cos \frac{\theta}{2} + \sin \frac{\theta}{2} \mathbf{v}) \mathbf{x} (\cos \frac{\theta}{2} - \sin \frac{\theta}{2} \mathbf{v}) \\ = & (\cos \frac{\theta}{2} + \sin \frac{\theta}{2} \mathbf{v}) (\sin \frac{\theta}{2} \langle \mathbf{x}, \mathbf{v} \rangle + \cos \frac{\theta}{2} \mathbf{x} + \sin \frac{\theta}{2} \mathbf{v} \wedge \mathbf{x}) \\ = & \cos \frac{\theta}{2} \sin \frac{\theta}{2} \langle \mathbf{x}, \mathbf{v} \rangle - \langle \sin \frac{\theta}{2} \mathbf{v}, \cos \frac{\theta}{2} \mathbf{x} + \sin \frac{\theta}{2} \mathbf{v} \wedge \mathbf{x} \rangle + \\ & \sin^2 \frac{\theta}{2} \langle \mathbf{x}, \mathbf{v} \rangle \mathbf{v} + \cos^2 \frac{\theta}{2} \mathbf{x} + \cos \frac{\theta}{2} \sin \frac{\theta}{2} \mathbf{v} \wedge \mathbf{x} + \sin \frac{\theta}{2} \cos \frac{\theta}{2} \mathbf{v} \wedge \mathbf{x} + \sin^2 \frac{\theta}{2} \mathbf{v} \wedge (\mathbf{v} \wedge \mathbf{x}) \\ = & \sin \theta \mathbf{v} \wedge \mathbf{x} + \cos \theta \mathbf{x} + (1 - \cos\theta)\mathbf{v}\mathbf{v}^T\mathbf{x}. \end{align} $$

So that, $$\mbox{Rot}[q] = \sin \theta [\mathbf{v}]_{\times}+(I-\mathbf{v}\mathbf{v}^T)\cos\theta + \mathbf{v}\mathbf{v}^T.$$

Note that, $\mathbf{v}$ is a eigen value correspond to $\mbox{Rot}[q]$, so that $\forall \mathbf{R}\in \mathbf{O}(3)$, $\mathbf{v}$ can be computed by the null space of $\mbox{Rot}[q] - I$.

Since then, there equation can be viewed as a linear equation to $\sin \theta, \cos \theta$.

[Environment]. We support c++, which depend on eigen3` (but which can be replaced, for the main usage is the structure of vector, matrix and implementation of linear solver and eigen solver).

The enironment is initially set on an archlinux machine, the procedure of setting environment could be,

sudo pacman -S eigen3 g++ python3 

And for other systems (e.g. Ubuntu, MacOS, Windows), the environmet could be broken (solving currently), the eigen3 in submodule is not working...

For c++ compilation, you can use

make compile

To compile the binary file, and we support clean for remove, all for build and run (as test).

The python bnding is depended on pybind11, you can install it like,

python -m pip install  pybind11          ## install it
python -m pip install 'pybind11[global]' ## install it system wide

And to run the installation, you should go to python directory by cd python. and go with

python -m pip install . --verbose ## of course if you want more details

Currently, this is quite troblesome...

[Under developing]. This could be super cool!

• Wikipedia: Quaternions, Quaternions and saptial rotation.

• Courses: CMU15.462: Computer Graphics, GAMES105.

• Blogs: .

This is a result of main.test().

q1 = (1,2i,3j,4k), norm = 5.47723
q2 = (0.866025,0.288675i,0.288675j,0.288675k), norm = 1
q1 + q2 = (1.86603,2.28868i,3.28868j,4.28868k)
12 * q1 = (12,24i,36j,48k)
q1.normalize()(0.182574,0.365148i,0.547723j,0.730297k)
q2.to_mat(): m = 
 0.666667 -0.333333  0.666667
 0.666667  0.666667 -0.333333
-0.333333  0.666667  0.666667

m.T * m = 
           1 -2.77556e-17 -2.77556e-17
-2.77556e-17            1            0
-2.77556e-17            0            1

mat_to_quat(m): (0.866025,0.288675i,0.288675j,0.288675k)