affines
¶
Utility routines for working with points and affine transforms
Errors in calculating or using affines 


Add diagonal elements steps and translations starts to affine 

Apply affine matrix aff to points pts 

Apply numpy dot product function from right to left on arrays 

Combine a matrix and vector into an homogeneous affine 

Split a transform into its matrix and vector components. 

Return voxel size for each input axis given affine 
append_diag¶

nibabel.affines.
append_diag
(aff, steps, starts=())¶ Add diagonal elements steps and translations starts to affine
Typical use is in expanding 4x4 affines to larger dimensions. Nipy is the main consumer because it uses NxM affines, whereas we generally only use 4x4 affines; the routine is here for convenience.
 Parameters
 aff2D array
N by M affine matrix
 stepsscalar or sequence
diagonal elements to append.
 startsscalar or sequence
elements to append to last column of aff, representing translations corresponding to the steps. If empty, expands to a vector of zeros of the same length as steps
 Returns
 aff_plus2D array
Now P by Q where L =
len(steps)
and P == N+L, Q=N+L
Examples
>>> aff = np.eye(4) >>> aff[:3,:3] = np.arange(9).reshape((3,3)) >>> append_diag(aff, [9, 10], [99,100]) array([[ 0., 1., 2., 0., 0., 0.], [ 3., 4., 5., 0., 0., 0.], [ 6., 7., 8., 0., 0., 0.], [ 0., 0., 0., 9., 0., 99.], [ 0., 0., 0., 0., 10., 100.], [ 0., 0., 0., 0., 0., 1.]])
apply_affine¶

nibabel.affines.
apply_affine
(aff, pts)¶ Apply affine matrix aff to points pts
Returns result of application of aff to the right of pts. The coordinate dimension of pts should be the last.
For the 3D case, aff will be shape (4,4) and pts will have final axis length 3  maybe it will just be N by 3. The return value is the transformed points, in this case:
res = np.dot(aff[:3,:3], pts.T) + aff[:3,3:4] transformed_pts = res.T
This routine is more general than 3D, in that aff can have any shape (N,N), and pts can have any shape, as long as the last dimension is for the coordinates, and is therefore length N1.
 Parameters
 aff(N, N) arraylike
Homogenous affine, for 3D points, will be 4 by 4. Contrary to first appearance, the affine will be applied on the left of pts.
 pts(…, N1) arraylike
Points, where the last dimension contains the coordinates of each point. For 3D, the last dimension will be length 3.
 Returns
 transformed_pts(…, N1) array
transformed points
Examples
>>> aff = np.array([[0,2,0,10],[3,0,0,11],[0,0,4,12],[0,0,0,1]]) >>> pts = np.array([[1,2,3],[2,3,4],[4,5,6],[6,7,8]]) >>> apply_affine(aff, pts) #doctest: +ELLIPSIS array([[14, 14, 24], [16, 17, 28], [20, 23, 36], [24, 29, 44]]...)
Just to show that in the simple 3D case, it is equivalent to:
>>> (np.dot(aff[:3,:3], pts.T) + aff[:3,3:4]).T #doctest: +ELLIPSIS array([[14, 14, 24], [16, 17, 28], [20, 23, 36], [24, 29, 44]]...)
But pts can be a more complicated shape:
>>> pts = pts.reshape((2,2,3)) >>> apply_affine(aff, pts) #doctest: +ELLIPSIS array([[[14, 14, 24], [16, 17, 28]], <BLANKLINE> [[20, 23, 36], [24, 29, 44]]]...)
dot_reduce¶

nibabel.affines.
dot_reduce
(*args)¶ Apply numpy dot product function from right to left on arrays
For passed arrays \(A, B, C, ... Z\) returns \(A \dot B \dot C ... \dot Z\) where “.” is the numpy array dot product.
 Parameters
 **argsarrays
Arrays that can be passed to numpy
dot
function
 Returns
 dot_productarray
If there are N arguments, result of
arg[0].dot(arg[1].dot(arg[2].dot ... arg[N2].dot(arg[N1])))...
from_matvec¶

nibabel.affines.
from_matvec
(matrix, vector=None)¶ Combine a matrix and vector into an homogeneous affine
Combine a rotation / scaling / shearing matrix and translation vector into a transform in homogeneous coordinates.
 Parameters
 matrixarraylike
An NxM array representing the the linear part of the transform. A transform from an Mdimensional space to an Ndimensional space.
 vectorNone or arraylike, optional
None or an (N,) array representing the translation. None corresponds to an (N,) array of zeros.
 Returns
 xformarray
An (N+1, M+1) homogenous transform matrix.
See also
Examples
>>> from_matvec(np.diag([2, 3, 4]), [9, 10, 11]) array([[ 2, 0, 0, 9], [ 0, 3, 0, 10], [ 0, 0, 4, 11], [ 0, 0, 0, 1]])
The vector argument is optional:
>>> from_matvec(np.diag([2, 3, 4])) array([[2, 0, 0, 0], [0, 3, 0, 0], [0, 0, 4, 0], [0, 0, 0, 1]])
to_matvec¶

nibabel.affines.
to_matvec
(transform)¶ Split a transform into its matrix and vector components.
The tranformation must be represented in homogeneous coordinates and is split into its rotation matrix and translation vector components.
 Parameters
 transformarraylike
NxM transform matrix in homogeneous coordinates representing an affine transformation from an (N1)dimensional space to an (M1)dimensional space. An example is a 4x4 transform representing rotations and translations in 3 dimensions. A 4x3 matrix can represent a 2dimensional plane embedded in 3 dimensional space.
 Returns
 matrix(N1, M1) array
Matrix component of transform
 vector(M1,) array
Vector compoent of transform
See also
Examples
>>> aff = np.diag([2, 3, 4, 1]) >>> aff[:3,3] = [9, 10, 11] >>> to_matvec(aff) (array([[2, 0, 0], [0, 3, 0], [0, 0, 4]]), array([ 9, 10, 11]))
voxel_sizes¶

nibabel.affines.
voxel_sizes
(affine)¶ Return voxel size for each input axis given affine
The affine is the mapping between array (voxel) coordinates and mm (world) coordinates.
The voxel size for the first voxel (array) axis is the distance moved in world coordinates when moving one unit along the first voxel (array) axis. This is the distance between the world coordinate of voxel (0, 0, 0) and the world coordinate of voxel (1, 0, 0). The world coordinate vector of voxel coordinate vector (0, 0, 0) is given by
v0 = affine.dot((0, 0, 0, 1)[:3]
. The world coordinate vector of voxel vector (1, 0, 0) isv1_ax1 = affine.dot((1, 0, 0, 1))[:3]
. The final 1 in the voxel vectors and the[:3]
at the end are because the affine works on homogenous coodinates. The translations part of the affine istrans = affine[:3, 3]
, and the rotations, zooms and shearing part of the affine isrzs = affine[:3, :3]
. Because of the final 1 in the input voxel vector,v0 == rzs.dot((0, 0, 0)) + trans
, andv1_ax1 == rzs.dot((1, 0, 0)) + trans
, and the difference vector isrzs.dot((0, 0, 0))  rzs.dot((1, 0, 0)) == rzs.dot((1, 0, 0)) == rzs[:, 0]
. The distance vectors in world coordinates between (0, 0, 0) and (1, 0, 0), (0, 1, 0), (0, 0, 1) are given byrzs.dot(np.eye(3)) = rzs
. The voxel sizes are the Euclidean lengths of the distance vectors. So, the voxel sizes are the Euclidean lengths of the columns of the affine (excluding the last row and column of the affine). Parameters
 affine2D arraylike
Affine transformation array. Usually shape (4, 4), but can be any 2D array.
 Returns
 vox_sizes1D array
Voxel sizes for each input axis of affine. Usually 1D array length 3, but in general has length (N1) where input affine is shape (M, N).