Affine Transformation, 2D, 4-Degrees of Freedom

(a variation of the previous theme).

## 1. Intro

What I did last time was about fitting of affine transformation of full 6-DoF. 4-DoF this time. By 4-DoF, I mean uniform scaling, rotation and x, y translations.

Say scale is , rotation angle is and translations are . Then the 4-DoF transformation from to can be written as:

.

Letting and , the transform above can be rewritten as:

.

So again my problem is to choose nice values of that minimise this error function:

.

## 2. by Calculus

Again, in order for the error function to have minimum value at some , all partials must be 0s. With some hand calculation (a bit tedious), this requirement leads to a system of equations as follows:

,

,

,

.

This system looks quite cumbersome, but is solvable. 1stly for :

,

.

And once are obtained, are too:

,

.

sweat sweat sweat. And this is ugly.

## 2. by Linear Algebra

Pretending I am already given nice , I can write:

.

Tweaking and rearranging the above non-equation to factor out (a bit of puzzle), I can rewrite it as follows:

.

Then I would rely on Householder Transform again to solve for . Hum. In this way, I can successfully have my computers work instead of me working hard. Relieved.

lll