An interesting application of KLT in image processing is rotating a binary $($i$.$e$.$ black$-$and$-$white$)$ image

along a horizontal $($or vertical$)$ direction. For example, if you want to identify screwdrivers and

wrenches that may travel on a manufacturing belt using a robot hand with a camera, you can make

shape$-$wise template matching. But, to make template matching, the shape directions must be

aligned; you cannot correlate the shape of a $45$ degrees rotated wrench with another one rotated by

$-10$ degrees. We can use KLT to rotate all shapes to horizontal direction, and then apply a successful

template matching. The rotation requires identification of the direction of the tool $($may it be a

screwdriver or a wrench$).$ We do it as follows:

Assume that the tool is with black pixel over white background. We make a list of

coordinates $(x_i,y_i)$ of the black pixels.

We find the average coordinate $($corresponding to the center of the tool$)$ and subtract this

from all coordinates $($making them zero mean$).$

We construct a $22$ correlation matrix by finding $R=\left[\begin{array}{cc}E\{x_i^2\}& E\{x_i{y}_i\}\\ E\{y_i{x}_i\}& E\{y_i^2\}\end{array}\right]$

The eigenvectors of this matrix show the "major" and the "minor" axis of the tool, which

correspond to the "direction" of the tool. So$!..$ All we need to do is to rotate it to make it

horizontal. Therefore, we apply the KLT not to the pixel values, but to the pixel coordinates!

Question: $($programming$)$ Draw a black$-$white wrench image $($or find one from the internet$)$

I:$1...$

Write a small code to give you an array of the coordinates corresponding to the black pixels.

Average them and subtract this mean to normalize coordinates to the center of the wrench.

Then, find the $22$ autocorrelation of the coordinates and determine the eigenvectors. The

eigenvector with the larger eigenvalue must correspond to the vector in the direction of the

wrench, and the other eigenvector must be perpendicular, meaning the width of the

wrench. Using the angles of these vector, rotate$-$back the wrench and make it horizontal.

Show outputs of each step in your Matlab code.