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I've come across the Iterative Closest Point algorithm using quaternions (as described in "A Method for Registration of 3-D Shapes" by Besl and McKay) and I'm wondering, why it works. To me it seems like some magical algorithms, because I have no idea why the eigenvector corresponding to the maximum of the eigenvalue of the matrix Q turns out to be the optimal rotation. Also, I don't know why the matrix Q has to look like it does.

I'd appreciate any explanation as well as any links to sources that derive or proof the algorithm.

Thanks in advance!

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That paper uses the Horn method for computing the best rotation (unit quaternion) aligning two sets or corresponding 3D vectors.

Horn method is described in detail in the paper: "Closed-form solution of the absolute orientation using unit quaternions".

In his paper Horn is looking for the unit quaternion maximizing the following functional:

$arg \max_q E(q) = \sum_i E_i(q)$

S.t. $q q^* = 1$

Where $q$ is the unit quaternion to find and $E_i(q)$ is:

$E_i(q) = (q x_i q^*) \cdot y_i$

Given the sets of corresponding pure quaternion vectors $\{x_i\}$ and $\{y_i\}$.

The first thing that Horn does is to switch from quaternion algebra to linear algebra by expressing the quaternion products in terms of matrix products:

$arg \max_q E(q) = \sum_i E_i(q)$

$E_i(q) = q^T N_i q$

S.t $q^T q = 1$

This time $q$ is a $4 \times 1$ column vector and $N_i$ is a $4 \times 4$ matrix formed using only $x_i$ and $y_i$.

Departing from quaternion expression: $ (q x_i q^*) \cdot y_i = (q x_i) \cdot (y_i q)$

That can be transformed to linear algebra language as: $(R_x q)^T (R_y q) = q^T ( R_x^T R_y) q = q^T N_i q$

The maximization can be expressed as:

$arg \max_q E(q) = q^T (\sum_i N_i) q$

S.t $q^T q = 1$

Since $N = \sum_i N_i$ is symmetric (because is a sum of symmetric matrices), according to the spectral theorem, the quadratic form $q^T N q$ subject to constraint $q^T q = 1$ has a max value when $q$ is one of the eigenvectors of $N$, specifically the one with the largest eigenvalue.

Proof:

Since $N$ is real symmetric matrix the spectral theorem guarantee that $N$ can be decomposed as $N = V^T A V$. Where $V$ is orthonormal matrix which columns are eigenvectors of $N$ and $A$ is diagonal matrix with eigenvalues of $N$.

$q^T (V^T A V) q = p^T A p$

$p^T A p = \sum_i {a_i p_i^2}$

It turns out that:

$\sum_i {a_i p_i^2} < a_0 \sum_i { p_i^2}$ being $a_0$ the largest eigenvalue, since $\sum_i { p_i^2} = 1$.

$\sum_i {a_i p_i^2} < a_0$

So we can pick $q$ to be the eigenvector (column of $V$) corresponding to the largest eigenvalue $a_0$ to achieve the largest posible value of the quadratic form.

That concludes the proof.

You can find it more detailed at: Horn paper Appendix 3 or here:

http://sfb649.wiwi.hu-berlin.de/fedc_homepage/xplore/tutorials/mvahtmlnode16.html

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