\(\newcommand{L}[1]{\| #1 \|}\newcommand{VL}[1]{\L{ \vec{#1} }}\newcommand{R}[1]{\operatorname{Re}\,(#1)}\newcommand{I}[1]{\operatorname{Im}\, (#1)}\)

Extra transform exercise

>>> #: standard imports
>>> import numpy as np
>>> import numpy.linalg as npl
>>> import matplotlib.pyplot as plt
>>> # print arrays to 4 decimal places
>>> np.set_printoptions(precision=4, suppress=True)
>>> import nibabel as nib

>>> #: gray colormap and nearest neighbor interpolation by default
>>> plt.rcParams['image.cmap'] = 'gray'
>>> plt.rcParams['image.interpolation'] = 'nearest'

What transforms did I add?

I took this image

and I made a new copy, in img, hdr format (see The image header and affine):

I then modified the affine in ds114_sub009_highres_moved.hdr to add an extra rotation and translation.

What extra translation and rotation did I add?

Hint: remember the rotation formulae from Rotations and rotation matrices:

\[\begin{split}\begin{alignat}{1} R_x(\theta) &= \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\[3pt] 0 & \sin \theta & \cos \theta \\[3pt] \end{bmatrix} \\[6pt] R_y(\theta) &= \begin{bmatrix} \cos \theta & 0 & \sin \theta \\[3pt] 0 & 1 & 0 \\[3pt] -\sin \theta & 0 & \cos \theta \\ \end{bmatrix} \\[6pt] R_z(\theta) &= \begin{bmatrix} \cos \theta & -\sin \theta & 0 \\[3pt] \sin \theta & \cos \theta & 0\\[3pt] 0 & 0 & 1\\ \end{bmatrix} \end{alignat}\end{split}\]

You might also want to compare against results from the rotations.py module:

>>> #: functions to make rotation matrices
>>> from rotations import x_rotmat, y_rotmat, z_rotmat

Final hint:

  • inverse sine, cosine are np.arcsin, np.arccos.
>>> #: Get the original image affine
>>> import nibabel as nib
>>> orig_img = nib.load('ds114_sub009_highres.nii')
>>> print(orig_img.affine)
[[   0.9989   -0.0605    0.0109 -129.8257]
 [   0.0427    1.263     0.2336 -119.0906]
 [  -0.0215   -0.3028    0.9723 -143.4178]
 [   0.        0.        0.        1.    ]]

>>> #: Get the new image affine
>>> moved_img = nib.load('ds114_sub009_highres_moved.img')
>>> moved_img.affine
array([[   0.9416,   -0.4311,   -0.0586,  -98.8336],
       [   0.336 ,    1.1887,    0.2264, -164.1377],
       [  -0.0215,   -0.3028,    0.9723, -158.4178],
       [   0.    ,    0.    ,    0.    ,    1.    ]])

>>> #- Work out what transform has been added in the new affine

>>> #- What rotation and translation has been applied?