I am carrying out 3D imaging alignment using (as an overview) the following coding scheme:
ref = imref3d(size(img(:,:,:,1)),[0 32] ,[0 32] ,[0 20] );
[optimizer, metric] = imregconfig( 'monomodal');
tform=imregtform(img(:,:,:,2), ref, img(:,:,:,1), ref, 'rigid', optimizer, metric);
For clarification, the variable `img` is a time series of 3D images (where each 3D image is a stack of 2D cross sections). At different time points, the object I am imaging can move. Thus, the purpose of the above code is to generate a geometric transformation that optimally aligns the second time point of the 3D volume ( `img(:,:,:,2)` ) to a reference image, which is the first time point of the 3D volume ( `img(:,:,:,1)`). The geometric transformation that is finally chosen by the optimization algorithm is output to the object `tform`. `tform` contains a 4x4 matrix as one of its properties (`T`); it is this 4x4 matrix that encodes the translation and rotation information. As you can see from my above code, I stored this 4x4 matrix in the variable `transform_mat`.
After reading through a lot of scattered mathworks documentation, I determined that this `transform_mat` variable (representing an affine **rigid body** transformation matrix) is in a "*post-multiplied*" form, which, as I understand it, just means that it is the transposed version of what one would traditionally see in a linear algebra text book.
For my purposes, **I am interested in extracting specific rotation information** from `transform_mat`. Here is what I have done so far:
rot_postmultiply=transform_mat(1:3,1:3);
rot_premultiply=rot_postmultply';
Just to quickly interject, I created the premultiplied version of the rotation matrix because I believe that many of the functions that **act on** rotation matrices assume that it is in its premultiplied form.
From this 3x3 rotation matrix, **I want to extract the rotations** (in radians) **about the STATIC x-axis, y-axis, and z-axis of the reference image**. My initial attempt at carrying this out is as follows:
eul = rotm2eul(rot_premultiply);
The `rotm2eul` function provides me the 3 **Euler Angles** associated with this 3x3 rotation matrix. `eul` is a 1x3 vector and, according to documentation, the "default order for Euler angle rotations is 'ZYX' ". However, I am not certain that *Euler Angles* are actually describing the information I want to extract (i.e. rotations about the static x, y, z axes of the reference image).
I do not have a strong linear algebra/geometric transformation background, but my understanding of Euler Angles is that each rotation that takes place **changes the coordinate system**. For example, after the rotation about the Z axis (the first value in the `eul` vector), we now have **new** X and Y axes (call them X' and Y'). Then the "Y-axis rotation" (the second value in `eul`) is actually the rotation about Y'...*not* Y. Repeat this argument for the final "X-rotation" (which would *really* be about an X'' axis).
If anyone could offer some insight about how to proceed (and if my concerns about Euler Angles are correct), I would greatly appreciate it!
Also, sorry if the word "**static**" is the incorrect terminology. Hopefully, I have provided sufficient context so that no confusion arises.
