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Initial commit: FishServer monorepo (FishAction, FishMeasure, fish_api)

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FishMeasure/pointcloud_classifier/Pointnet_Pointnet2_pytorch/visualizer/eulerangles.py Executable file
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# emacs: -*- mode: python-mode; py-indent-offset: 4; indent-tabs-mode: nil -*-
# vi: set ft=python sts=4 ts=4 sw=4 et:
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#
# See COPYING file distributed along with the NiBabel package for the
# copyright and license terms.
#
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''' Module implementing Euler angle rotations and their conversions
See:
* http://en.wikipedia.org/wiki/Rotation_matrix
* http://en.wikipedia.org/wiki/Euler_angles
* http://mathworld.wolfram.com/EulerAngles.html
See also: *Representing Attitude with Euler Angles and Quaternions: A
Reference* (2006) by James Diebel. A cached PDF link last found here:
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.110.5134
Euler's rotation theorem tells us that any rotation in 3D can be
described by 3 angles. Let's call the 3 angles the *Euler angle vector*
and call the angles in the vector :math:`alpha`, :math:`beta` and
:math:`gamma`. The vector is [ :math:`alpha`,
:math:`beta`. :math:`gamma` ] and, in this description, the order of the
parameters specifies the order in which the rotations occur (so the
rotation corresponding to :math:`alpha` is applied first).
In order to specify the meaning of an *Euler angle vector* we need to
specify the axes around which each of the rotations corresponding to
:math:`alpha`, :math:`beta` and :math:`gamma` will occur.
There are therefore three axes for the rotations :math:`alpha`,
:math:`beta` and :math:`gamma`; let's call them :math:`i` :math:`j`,
:math:`k`.
Let us express the rotation :math:`alpha` around axis `i` as a 3 by 3
rotation matrix `A`. Similarly :math:`beta` around `j` becomes 3 x 3
matrix `B` and :math:`gamma` around `k` becomes matrix `G`. Then the
whole rotation expressed by the Euler angle vector [ :math:`alpha`,
:math:`beta`. :math:`gamma` ], `R` is given by::
R = np.dot(G, np.dot(B, A))
See http://mathworld.wolfram.com/EulerAngles.html
The order :math:`G B A` expresses the fact that the rotations are
performed in the order of the vector (:math:`alpha` around axis `i` =
`A` first).
To convert a given Euler angle vector to a meaningful rotation, and a
rotation matrix, we need to define:
* the axes `i`, `j`, `k`
* whether a rotation matrix should be applied on the left of a vector to
be transformed (vectors are column vectors) or on the right (vectors
are row vectors).
* whether the rotations move the axes as they are applied (intrinsic
rotations) - compared the situation where the axes stay fixed and the
vectors move within the axis frame (extrinsic)
* the handedness of the coordinate system
See: http://en.wikipedia.org/wiki/Rotation_matrix#Ambiguities
We are using the following conventions:
* axes `i`, `j`, `k` are the `z`, `y`, and `x` axes respectively. Thus
an Euler angle vector [ :math:`alpha`, :math:`beta`. :math:`gamma` ]
in our convention implies a :math:`alpha` radian rotation around the
`z` axis, followed by a :math:`beta` rotation around the `y` axis,
followed by a :math:`gamma` rotation around the `x` axis.
* the rotation matrix applies on the left, to column vectors on the
right, so if `R` is the rotation matrix, and `v` is a 3 x N matrix
with N column vectors, the transformed vector set `vdash` is given by
``vdash = np.dot(R, v)``.
* extrinsic rotations - the axes are fixed, and do not move with the
rotations.
* a right-handed coordinate system
The convention of rotation around ``z``, followed by rotation around
``y``, followed by rotation around ``x``, is known (confusingly) as
"xyz", pitch-roll-yaw, Cardan angles, or Tait-Bryan angles.
'''
import math
import sys
if sys.version_info >= (3,0):
from functools import reduce
import numpy as np
_FLOAT_EPS_4 = np.finfo(float).eps * 4.0
def euler2mat(z=0, y=0, x=0):
''' Return matrix for rotations around z, y and x axes
Uses the z, then y, then x convention above
Parameters
----------
z : scalar
Rotation angle in radians around z-axis (performed first)
y : scalar
Rotation angle in radians around y-axis
x : scalar
Rotation angle in radians around x-axis (performed last)
Returns
-------
M : array shape (3,3)
Rotation matrix giving same rotation as for given angles
Examples
--------
>>> zrot = 1.3 # radians
>>> yrot = -0.1
>>> xrot = 0.2
>>> M = euler2mat(zrot, yrot, xrot)
>>> M.shape == (3, 3)
True
The output rotation matrix is equal to the composition of the
individual rotations
>>> M1 = euler2mat(zrot)
>>> M2 = euler2mat(0, yrot)
>>> M3 = euler2mat(0, 0, xrot)
>>> composed_M = np.dot(M3, np.dot(M2, M1))
>>> np.allclose(M, composed_M)
True
You can specify rotations by named arguments
>>> np.all(M3 == euler2mat(x=xrot))
True
When applying M to a vector, the vector should column vector to the
right of M. If the right hand side is a 2D array rather than a
vector, then each column of the 2D array represents a vector.
>>> vec = np.array([1, 0, 0]).reshape((3,1))
>>> v2 = np.dot(M, vec)
>>> vecs = np.array([[1, 0, 0],[0, 1, 0]]).T # giving 3x2 array
>>> vecs2 = np.dot(M, vecs)
Rotations are counter-clockwise.
>>> zred = np.dot(euler2mat(z=np.pi/2), np.eye(3))
>>> np.allclose(zred, [[0, -1, 0],[1, 0, 0], [0, 0, 1]])
True
>>> yred = np.dot(euler2mat(y=np.pi/2), np.eye(3))
>>> np.allclose(yred, [[0, 0, 1],[0, 1, 0], [-1, 0, 0]])
True
>>> xred = np.dot(euler2mat(x=np.pi/2), np.eye(3))
>>> np.allclose(xred, [[1, 0, 0],[0, 0, -1], [0, 1, 0]])
True
Notes
-----
The direction of rotation is given by the right-hand rule (orient
the thumb of the right hand along the axis around which the rotation
occurs, with the end of the thumb at the positive end of the axis;
curl your fingers; the direction your fingers curl is the direction
of rotation). Therefore, the rotations are counterclockwise if
looking along the axis of rotation from positive to negative.
'''
Ms = []
if z:
cosz = math.cos(z)
sinz = math.sin(z)
Ms.append(np.array(
[[cosz, -sinz, 0],
[sinz, cosz, 0],
[0, 0, 1]]))
if y:
cosy = math.cos(y)
siny = math.sin(y)
Ms.append(np.array(
[[cosy, 0, siny],
[0, 1, 0],
[-siny, 0, cosy]]))
if x:
cosx = math.cos(x)
sinx = math.sin(x)
Ms.append(np.array(
[[1, 0, 0],
[0, cosx, -sinx],
[0, sinx, cosx]]))
if Ms:
return reduce(np.dot, Ms[::-1])
return np.eye(3)
def mat2euler(M, cy_thresh=None):
''' Discover Euler angle vector from 3x3 matrix
Uses the conventions above.
Parameters
----------
M : array-like, shape (3,3)
cy_thresh : None or scalar, optional
threshold below which to give up on straightforward arctan for
estimating x rotation. If None (default), estimate from
precision of input.
Returns
-------
z : scalar
y : scalar
x : scalar
Rotations in radians around z, y, x axes, respectively
Notes
-----
If there was no numerical error, the routine could be derived using
Sympy expression for z then y then x rotation matrix, which is::
[ cos(y)*cos(z), -cos(y)*sin(z), sin(y)],
[cos(x)*sin(z) + cos(z)*sin(x)*sin(y), cos(x)*cos(z) - sin(x)*sin(y)*sin(z), -cos(y)*sin(x)],
[sin(x)*sin(z) - cos(x)*cos(z)*sin(y), cos(z)*sin(x) + cos(x)*sin(y)*sin(z), cos(x)*cos(y)]
with the obvious derivations for z, y, and x
z = atan2(-r12, r11)
y = asin(r13)
x = atan2(-r23, r33)
Problems arise when cos(y) is close to zero, because both of::
z = atan2(cos(y)*sin(z), cos(y)*cos(z))
x = atan2(cos(y)*sin(x), cos(x)*cos(y))
will be close to atan2(0, 0), and highly unstable.
The ``cy`` fix for numerical instability below is from: *Graphics
Gems IV*, Paul Heckbert (editor), Academic Press, 1994, ISBN:
0123361559. Specifically it comes from EulerAngles.c by Ken
Shoemake, and deals with the case where cos(y) is close to zero:
See: http://www.graphicsgems.org/
The code appears to be licensed (from the website) as "can be used
without restrictions".
'''
M = np.asarray(M)
if cy_thresh is None:
try:
cy_thresh = np.finfo(M.dtype).eps * 4
except ValueError:
cy_thresh = _FLOAT_EPS_4
r11, r12, r13, r21, r22, r23, r31, r32, r33 = M.flat
# cy: sqrt((cos(y)*cos(z))**2 + (cos(x)*cos(y))**2)
cy = math.sqrt(r33*r33 + r23*r23)
if cy > cy_thresh: # cos(y) not close to zero, standard form
z = math.atan2(-r12, r11) # atan2(cos(y)*sin(z), cos(y)*cos(z))
y = math.atan2(r13, cy) # atan2(sin(y), cy)
x = math.atan2(-r23, r33) # atan2(cos(y)*sin(x), cos(x)*cos(y))
else: # cos(y) (close to) zero, so x -> 0.0 (see above)
# so r21 -> sin(z), r22 -> cos(z) and
z = math.atan2(r21, r22)
y = math.atan2(r13, cy) # atan2(sin(y), cy)
x = 0.0
return z, y, x
def euler2quat(z=0, y=0, x=0):
''' Return quaternion corresponding to these Euler angles
Uses the z, then y, then x convention above
Parameters
----------
z : scalar
Rotation angle in radians around z-axis (performed first)
y : scalar
Rotation angle in radians around y-axis
x : scalar
Rotation angle in radians around x-axis (performed last)
Returns
-------
quat : array shape (4,)
Quaternion in w, x, y z (real, then vector) format
Notes
-----
We can derive this formula in Sympy using:
1. Formula giving quaternion corresponding to rotation of theta radians
about arbitrary axis:
http://mathworld.wolfram.com/EulerParameters.html
2. Generated formulae from 1.) for quaternions corresponding to
theta radians rotations about ``x, y, z`` axes
3. Apply quaternion multiplication formula -
http://en.wikipedia.org/wiki/Quaternions#Hamilton_product - to
formulae from 2.) to give formula for combined rotations.
'''
z = z/2.0
y = y/2.0
x = x/2.0
cz = math.cos(z)
sz = math.sin(z)
cy = math.cos(y)
sy = math.sin(y)
cx = math.cos(x)
sx = math.sin(x)
return np.array([
cx*cy*cz - sx*sy*sz,
cx*sy*sz + cy*cz*sx,
cx*cz*sy - sx*cy*sz,
cx*cy*sz + sx*cz*sy])
def quat2euler(q):
''' Return Euler angles corresponding to quaternion `q`
Parameters
----------
q : 4 element sequence
w, x, y, z of quaternion
Returns
-------
z : scalar
Rotation angle in radians around z-axis (performed first)
y : scalar
Rotation angle in radians around y-axis
x : scalar
Rotation angle in radians around x-axis (performed last)
Notes
-----
It's possible to reduce the amount of calculation a little, by
combining parts of the ``quat2mat`` and ``mat2euler`` functions, but
the reduction in computation is small, and the code repetition is
large.
'''
# delayed import to avoid cyclic dependencies
import nibabel.quaternions as nq
return mat2euler(nq.quat2mat(q))
def euler2angle_axis(z=0, y=0, x=0):
''' Return angle, axis corresponding to these Euler angles
Uses the z, then y, then x convention above
Parameters
----------
z : scalar
Rotation angle in radians around z-axis (performed first)
y : scalar
Rotation angle in radians around y-axis
x : scalar
Rotation angle in radians around x-axis (performed last)
Returns
-------
theta : scalar
angle of rotation
vector : array shape (3,)
axis around which rotation occurs
Examples
--------
>>> theta, vec = euler2angle_axis(0, 1.5, 0)
>>> print(theta)
1.5
>>> np.allclose(vec, [0, 1, 0])
True
'''
# delayed import to avoid cyclic dependencies
import nibabel.quaternions as nq
return nq.quat2angle_axis(euler2quat(z, y, x))
def angle_axis2euler(theta, vector, is_normalized=False):
''' Convert angle, axis pair to Euler angles
Parameters
----------
theta : scalar
angle of rotation
vector : 3 element sequence
vector specifying axis for rotation.
is_normalized : bool, optional
True if vector is already normalized (has norm of 1). Default
False
Returns
-------
z : scalar
y : scalar
x : scalar
Rotations in radians around z, y, x axes, respectively
Examples
--------
>>> z, y, x = angle_axis2euler(0, [1, 0, 0])
>>> np.allclose((z, y, x), 0)
True
Notes
-----
It's possible to reduce the amount of calculation a little, by
combining parts of the ``angle_axis2mat`` and ``mat2euler``
functions, but the reduction in computation is small, and the code
repetition is large.
'''
# delayed import to avoid cyclic dependencies
import nibabel.quaternions as nq
M = nq.angle_axis2mat(theta, vector, is_normalized)
return mat2euler(M)