quaternions
¶
Functions to operate on, or return, quaternions
The module also includes functions for the closely related angle, axis pair as a specification for rotation.
Quaternions here consist of 4 values w, x, y, z
, where w
is the
real (scalar) part, and x, y, z
are the complex (vector) part.
Note - rotation matrices here apply to column vectors, that is, they are applied on the left of the vector. For example:
>>> import numpy as np
>>> from nibabel.quaternions import quat2mat
>>> q = [0, 1, 0, 0] # 180 degree rotation around axis 0
>>> M = quat2mat(q) # from this module
>>> vec = np.array([1, 2, 3]).reshape((3,1)) # column vector
>>> tvec = np.dot(M, vec)
|
Rotation matrix of angle theta around vector |
|
Quaternion for rotation of angle theta around vector |
|
Conjugate of quaternion |
|
Return identity quaternion |
|
Compute unit quaternion from last 3 values |
|
Return multiplicative inverse of quaternion q |
|
Return True is this is very nearly a unit quaternion |
|
Calculate quaternion corresponding to given rotation matrix |
|
Multiply two quaternions |
|
Returns True if q1 and q2 give near equivalent transforms |
|
Return norm of quaternion |
|
Convert quaternion to rotation of angle around axis |
|
Calculate rotation matrix corresponding to quaternion |
|
Apply transformation in quaternion q to vector v |
angle_axis2mat¶
- nibabel.quaternions.angle_axis2mat(theta, vector, is_normalized=False)¶
Rotation matrix of angle theta around vector
- Parameters:
- thetascalar
angle of rotation
- vector3 element sequence
vector specifying axis for rotation.
- is_normalizedbool, optional
True if vector is already normalized (has norm of 1). Default False
- Returns:
- matarray shape (3,3)
rotation matrix for specified rotation
Notes
From: https://en.wikipedia.org/wiki/Rotation_matrix#Axis_and_angle
angle_axis2quat¶
- nibabel.quaternions.angle_axis2quat(theta, vector, is_normalized=False)¶
Quaternion for rotation of angle theta around vector
- Parameters:
- thetascalar
angle of rotation
- vector3 element sequence
vector specifying axis for rotation.
- is_normalizedbool, optional
True if vector is already normalized (has norm of 1). Default False
- Returns:
- quat4 element sequence of symbols
quaternion giving specified rotation
Notes
Formula from http://mathworld.wolfram.com/EulerParameters.html
Examples
>>> q = angle_axis2quat(np.pi, [1, 0, 0]) >>> np.allclose(q, [0, 1, 0, 0]) True
conjugate¶
- nibabel.quaternions.conjugate(q)¶
Conjugate of quaternion
- Parameters:
- q4 element sequence
w, i, j, k of quaternion
- Returns:
- conjqarray shape (4,)
w, i, j, k of conjugate of q
eye¶
- nibabel.quaternions.eye()¶
Return identity quaternion
fillpositive¶
- nibabel.quaternions.fillpositive(xyz, w2_thresh=None)¶
Compute unit quaternion from last 3 values
- Parameters:
- xyziterable
iterable containing 3 values, corresponding to quaternion x, y, z
- w2_threshNone or float, optional
threshold to determine if w squared is non-zero. If None (default) then w2_thresh set equal to 3 *
np.finfo(xyz.dtype).eps
, if possible, otherwise 3 *np.finfo(np.float64).eps
- Returns:
- wxyzarray shape (4,)
Full 4 values of quaternion
Notes
If w, x, y, z are the values in the full quaternion, assumes w is positive.
Gives error if w*w is estimated to be negative
w = 0 corresponds to a 180 degree rotation
The unit quaternion specifies that np.dot(wxyz, wxyz) == 1.
If w is positive (assumed here), w is given by:
w = np.sqrt(1.0-(x*x+y*y+z*z))
w2 = 1.0-(x*x+y*y+z*z) can be near zero, which will lead to numerical instability in sqrt. Here we use the system maximum float type to reduce numerical instability
Examples
>>> import numpy as np >>> wxyz = fillpositive([0,0,0]) >>> np.all(wxyz == [1, 0, 0, 0]) True >>> wxyz = fillpositive([1,0,0]) # Corner case; w is 0 >>> np.all(wxyz == [0, 1, 0, 0]) True >>> np.dot(wxyz, wxyz) 1.0
inverse¶
- nibabel.quaternions.inverse(q)¶
Return multiplicative inverse of quaternion q
- Parameters:
- q4 element sequence
w, i, j, k of quaternion
- Returns:
- invqarray shape (4,)
w, i, j, k of quaternion inverse
isunit¶
- nibabel.quaternions.isunit(q)¶
Return True is this is very nearly a unit quaternion
mat2quat¶
- nibabel.quaternions.mat2quat(M)¶
Calculate quaternion corresponding to given rotation matrix
- Parameters:
- Marray-like
3x3 rotation matrix
- Returns:
- q(4,) array
closest quaternion to input matrix, having positive q[0]
Notes
Method claimed to be robust to numerical errors in M
Constructs quaternion by calculating maximum eigenvector for matrix K (constructed from input M). Although this is not tested, a maximum eigenvalue of 1 corresponds to a valid rotation.
A quaternion q*-1 corresponds to the same rotation as q; thus the sign of the reconstructed quaternion is arbitrary, and we return quaternions with positive w (q[0]).
References
Bar-Itzhack, Itzhack Y. (2000), “New method for extracting the quaternion from a rotation matrix”, AIAA Journal of Guidance, Control and Dynamics 23(6):1085-1087 (Engineering Note), ISSN 0731-5090
Examples
>>> import numpy as np >>> q = mat2quat(np.eye(3)) # Identity rotation >>> np.allclose(q, [1, 0, 0, 0]) True >>> q = mat2quat(np.diag([1, -1, -1])) >>> np.allclose(q, [0, 1, 0, 0]) # 180 degree rotn around axis 0 True
mult¶
- nibabel.quaternions.mult(q1, q2)¶
Multiply two quaternions
- Parameters:
- q14 element sequence
- q24 element sequence
- Returns:
- q12shape (4,) array
Notes
See : https://en.wikipedia.org/wiki/Quaternions#Hamilton_product
nearly_equivalent¶
- nibabel.quaternions.nearly_equivalent(q1, q2, rtol=1e-05, atol=1e-08)¶
Returns True if q1 and q2 give near equivalent transforms
q1 may be nearly numerically equal to q2, or nearly equal to q2 * -1 (because a quaternion multiplied by -1 gives the same transform).
- Parameters:
- q14 element sequence
w, x, y, z of first quaternion
- q24 element sequence
w, x, y, z of second quaternion
- Returns:
- equivbool
True if q1 and q2 are nearly equivalent, False otherwise
Examples
>>> q1 = [1, 0, 0, 0] >>> nearly_equivalent(q1, [0, 1, 0, 0]) False >>> nearly_equivalent(q1, [1, 0, 0, 0]) True >>> nearly_equivalent(q1, [-1, 0, 0, 0]) True
norm¶
- nibabel.quaternions.norm(q)¶
Return norm of quaternion
- Parameters:
- q4 element sequence
w, i, j, k of quaternion
- Returns:
- nscalar
quaternion norm
quat2angle_axis¶
- nibabel.quaternions.quat2angle_axis(quat, identity_thresh=None)¶
Convert quaternion to rotation of angle around axis
- Parameters:
- quat4 element sequence
w, x, y, z forming quaternion
- identity_threshNone or scalar, optional
threshold below which the norm of the vector part of the quaternion (x, y, z) is deemed to be 0, leading to the identity rotation. None (the default) leads to a threshold estimated based on the precision of the input.
- Returns:
- thetascalar
angle of rotation
- vectorarray shape (3,)
axis around which rotation occurs
Notes
A quaternion for which x, y, z are all equal to 0, is an identity rotation. In this case we return a 0 angle and an arbitrary vector, here [1, 0, 0]
Examples
>>> theta, vec = quat2angle_axis([0, 1, 0, 0]) >>> np.allclose(theta, np.pi) True >>> vec array([1., 0., 0.])
If this is an identity rotation, we return a zero angle and an arbitrary vector
>>> quat2angle_axis([1, 0, 0, 0]) (0.0, array([1., 0., 0.]))
quat2mat¶
- nibabel.quaternions.quat2mat(q)¶
Calculate rotation matrix corresponding to quaternion
- Parameters:
- q4 element array-like
- Returns:
- M(3,3) array
Rotation matrix corresponding to input quaternion q
Notes
Rotation matrix applies to column vectors, and is applied to the left of coordinate vectors. The algorithm here allows non-unit quaternions.
References
Algorithm from https://en.wikipedia.org/wiki/Rotation_matrix#Quaternion
Examples
>>> import numpy as np >>> M = quat2mat([1, 0, 0, 0]) # Identity quaternion >>> np.allclose(M, np.eye(3)) True >>> M = quat2mat([0, 1, 0, 0]) # 180 degree rotn around axis 0 >>> np.allclose(M, np.diag([1, -1, -1])) True
rotate_vector¶
- nibabel.quaternions.rotate_vector(v, q)¶
Apply transformation in quaternion q to vector v
- Parameters:
- v3 element sequence
3 dimensional vector
- q4 element sequence
w, i, j, k of quaternion
- Returns:
- vdasharray shape (3,)
v rotated by quaternion q
Notes
See: https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation#Describing_rotations_with_quaternions