def decode(
self, symbol_stream: np.ndarray,
channel_state: ChannelStateInformation, stream_noises: np.ndarray
) -> Tuple[np.ndarray, ChannelStateInformation, np.ndarray]:
for time_idx, (symbols, csi, noise) in enumerate(
zip(symbol_stream.T, channel_state.samples(),
stream_noises.T)):
noise_variance = np.mean(noise)
# Combine the responses of all superimposed transmit antennas for equalization
transform = np.sum(csi.linear[:, :, 0, :], axis=2, keepdims=False)
# Compute the pseudo-inverse from the singular-value-decomposition of the linear channel transform
# noinspection PyTupleAssignmentBalance
u, s, vh = svd(transform.todense(),
full_matrices=False,
check_finite=False)
u *= s / (s**2 + noise_variance)
equalizer = (u @ vh).T.conj()
symbol_stream[:, time_idx] = equalizer @ symbols
channel_state.state[:, :,
time_idx, :] = np.tensordot(equalizer,
csi.linear[:, :,
0, :],
axes=(1, 0))
stream_noises[:, time_idx] = noise * (s**2 + noise_variance)
return symbol_stream, channel_state, stream_noises
def extract_noise_components(realigned_file,
mask_file,
num_components=5,
extra_regressors=None):
"""Derive components most reflective of physiological noise
Parameters
----------
realigned_file: a 4D Nifti file containing realigned volumes
mask_file: a 3D Nifti file containing white matter + ventricular masks
num_components: number of components to use for noise decomposition
extra_regressors: additional regressors to add
Returns
-------
components_file: a text file containing the noise components
"""
from scipy.linalg.decomp_svd import svd
import numpy as np
import nibabel as nb
from nipype.utils import NUMPY_MMAP
from nipype.utils.filemanip import filename_to_list
import os
imgseries = nb.load(realigned_file, mmap=NUMPY_MMAP)
components = None
for filename in filename_to_list(mask_file):
mask = nb.load(filename, mmap=NUMPY_MMAP).get_data()
if len(np.nonzero(mask > 0)[0]) == 0:
continue
voxel_timecourses = imgseries.get_data()[mask > 0]
voxel_timecourses[np.isnan(np.sum(voxel_timecourses, axis=1)), :] = 0
# remove mean and normalize by variance
# voxel_timecourses.shape == [nvoxels, time]
X = voxel_timecourses.T
stdX = np.std(X, axis=0)
stdX[stdX == 0] = 1.
stdX[np.isnan(stdX)] = 1.
stdX[np.isinf(stdX)] = 1.
X = (X - np.mean(X, axis=0)) / stdX
u, _, _ = svd(X, full_matrices=False)
if components is None:
components = u[:, :num_components]
else:
components = np.hstack((components, u[:, :num_components]))
if extra_regressors:
regressors = np.genfromtxt(extra_regressors)
components = np.hstack((components, regressors))
components_file = os.path.join(os.getcwd(), 'noise_components.txt')
np.savetxt(components_file, components, fmt=b"%.10f")
return components_file
def extract_noise_components(realigned_file,
mask_file,
num_components=5,
extra_regressors=None):
"""Derive components most reflective of physiological noise
Parameters
----------
realigned_file: a 4D Nifti file containing realigned volumes
mask_file: a 3D Nifti file containing white matter + ventricular masks
num_components: number of components to use for noise decomposition
extra_regressors: additional regressors to add
Returns
-------
components_file: a text file containing the noise components
"""
from scipy.linalg.decomp_svd import svd
import numpy as np
import nibabel as nb
from nipype.utils import NUMPY_MMAP
import os
imgseries = nb.load(realigned_file, mmap=NUMPY_MMAP)
components = None
for filename in filename_to_list(mask_file):
mask = nb.load(filename, mmap=NUMPY_MMAP).get_data()
if len(np.nonzero(mask > 0)[0]) == 0:
continue
voxel_timecourses = imgseries.get_data()[mask > 0]
voxel_timecourses[np.isnan(np.sum(voxel_timecourses, axis=1)), :] = 0
# remove mean and normalize by variance
# voxel_timecourses.shape == [nvoxels, time]
X = voxel_timecourses.T
stdX = np.std(X, axis=0)
stdX[stdX == 0] = 1.
stdX[np.isnan(stdX)] = 1.
stdX[np.isinf(stdX)] = 1.
X = (X - np.mean(X, axis=0)) / stdX
u, _, _ = svd(X, full_matrices=False)
if components is None:
components = u[:, :num_components]
else:
components = np.hstack((components, u[:, :num_components]))
if extra_regressors:
regressors = np.genfromtxt(extra_regressors)
components = np.hstack((components, regressors))
components_file = os.path.join(os.getcwd(), 'noise_components.txt')
np.savetxt(components_file, components, fmt=b"%.10f")
return components_file
def orthogonal_procrustes(A, ref_matrix, reflection=False):
# Adaptation of scipy.linalg.orthogonal_procrustes -> https://github.com/scipy/scipy/blob/v0.16.0/scipy/linalg/_procrustes.py#L14
# Info here: http://compgroups.net/comp.soft-sys.matlab/procrustes-analysis-without-reflection/896635
# goal is to find unitary matrix R with det(R) > 0 such that ||A*R - ref_matrix||^2 is minimized
from scipy.linalg.decomp_svd import svd # Singular Value Decomposition, factors matrices
from scipy.linalg import det
import numpy as np
A = np.asarray_chkfinite(A)
ref_matrix = np.asarray_chkfinite(ref_matrix)
if A.ndim != 2:
raise ValueError('expected ndim to be 2, but observed %s' % A.ndim)
if A.shape != ref_matrix.shape:
raise ValueError('the shapes of A and ref_matrix differ (%s vs %s)' % (A.shape, ref_matrix.shape))
u, w, vt = svd(ref_matrix.T.dot(A).T)
# Goal: minimize ||A*R - ref||^2, switch to trace
# trace((A*R-ref).T*(A*R-ref)), now we distribute
# trace(R'*A'*A*R) + trace(ref.T*ref) - trace((A*R).T*ref) - trace(ref.T*(A*R)), trace doesn't care about order, so re-order
# trace(R*R.T*A.T*A) + trace(ref.T*ref) - trace(R.T*A.T*ref) - trace(ref.T*A*R), simplify
# trace(A.T*A) + trace(ref.T*ref) - 2*trace(ref.T*A*R)
# Thus, to minimize we want to maximize trace(ref.T * A * R)
# u*w*v.T = (ref.T*A).T
# ref.T * A = w * u.T * v
# trace(ref.T * A * R) = trace (w * u.T * v * R)
# differences minimized when trace(ref.T * A * R) is maximized, thus when trace(u.T * v * R) is maximized
# This occurs when u.T * v * R = I (as u, v and R are all unitary matrices so max is 1)
# R is a rotation matrix so R.T = R^-1
# u.T * v * I = R^-1 = R.T
# R = u * v.T
# Thus, R = u.dot(vt)
R = u.dot(vt) # Get the rotation matrix, including reflections
if not reflection and det(R) < 0: # If we don't want reflection
# To remove reflection, we change the sign of the rightmost column of u (or v) and the scalar associated
# with that column
u[:,-1] *= -1
w[-1] *= -1
R = u.dot(vt)
scale = w.sum() # Get the scaled difference
return R,scale
def decode(
self, symbol_stream: np.ndarray,
channel_state: ChannelStateInformation, stream_noises: np.ndarray
) -> Tuple[np.ndarray, ChannelStateInformation, np.ndarray]:
equalized_symbols = np.empty(
(channel_state.num_receive_streams, channel_state.num_samples),
dtype=complex)
equalized_noises = np.empty(
(channel_state.num_receive_streams, channel_state.num_samples),
dtype=float)
equalized_channel_state = ChannelStateInformation(
channel_state.state_format)
# Equalize in space in a first step
for idx, (symbols, stream_state, noise) in enumerate(
zip(symbol_stream, channel_state.received_streams(),
stream_noises)):
noise_variance = np.mean(noise)
# Combine the responses of all superimposed transmit antennas for equalization
linear_state = stream_state.linear
transform = np.sum(linear_state[0, ::], axis=0, keepdims=False)
# Compute the pseudo-inverse from the singular-value-decomposition of the linear channel transform
# noinspection PyTupleAssignmentBalance
u, s, vh = svd(transform.todense(),
full_matrices=False,
check_finite=False)
u *= s / (s**2 + noise_variance)
equalizer = (u @ vh).T.conj()
equalized_symbols[idx, :] = equalizer @ symbols
equalized_csi_slice = tensordot(equalizer,
linear_state,
axes=(1, 2)).transpose(
(1, 2, 0, 3))
equalized_channel_state.append_linear(equalized_csi_slice, 0)
equalized_noises[idx, :] = noise[:stream_state.num_samples] * (
s**2 + noise_variance)
return equalized_symbols, channel_state, equalized_noises
def orthogonal_procrustes(A, ref_matrix, reflection=False):
'''
Using the orthogonal procrustes method, we find the unitary matrix R with
det(R) > 0 such that ||A*R - ref_matrix||^2 is minimized. This varies
from that within scipy by the addition of the reflection term, allowing
and disallowing inversion. NOTE - This means that the rotation matrix is
used for right side multiplication!
**Parameters**
A: *list,* :class:`squid.structures.atom.Atom`
A list of atoms for which R will minimize the frobenius
norm ||A*R - ref_matrix||^2.
ref_matrix: *list,* :class:`squid.structures.atom.Atom`
A list of atoms for which *A* is being rotated towards.
reflection: *bool, optional*
Whether inversion is allowed (True) or not (False).
**Returns**
R: *list, list, float*
Right multiplication rotation matrix to best overlay A onto the
reference matrix.
scale: *float*
Scalar between the matrices.
**Derivation**
Goal: minimize ||A\*R - ref||^2, switch to trace
trace((A\*R-ref).T\*(A\*R-ref)), now we distribute
trace(R'\*A'\*A\*R) + trace(ref.T\*ref) - trace((A\*R).T\*ref) -
trace(ref.T\*(A\*R)), trace doesn't care about order, so re-order
trace(R\*R.T\*A.T\*A) + trace(ref.T\*ref) - trace(R.T\*A.T\*ref) -
trace(ref.T\*A\*R), simplify
trace(A.T\*A) + trace(ref.T\*ref) - 2\*trace(ref.T\*A\*R)
Thus, to minimize we want to maximize trace(ref.T \* A \* R)
u\*w\*v.T = (ref.T\*A).T
ref.T \* A = w \* u.T \* v
trace(ref.T \* A \* R) = trace (w \* u.T \* v \* R)
differences minimized when trace(ref.T \* A \* R) is maximized, thus
when trace(u.T \* v \* R) is maximized
This occurs when u.T \* v \* R = I (as u, v and R are all unitary
matrices so max is 1)
R is a rotation matrix so R.T = R^-1
u.T \* v \* I = R^-1 = R.T
R = u \* v.T
Thus, R = u.dot(vt)
**References**
* https://github.com/scipy/scipy/blob/v0.16.0/scipy/linalg/
_procrustes.py#L14
* http://compgroups.net/comp.soft-sys.matlab/procrustes-analysis
-without-reflection/896635
'''
assert hasattr(A, "__len__") and hasattr(ref_matrix, "__len__"),\
"Error - A and ref_matrix must be lists of atomic coordinates!"
cast.assert_vec(A[0], length=3, numeric=True)
cast.assert_vec(ref_matrix[0], length=3, numeric=True)
A = np.asarray_chkfinite(A)
ref_matrix = np.asarray_chkfinite(ref_matrix)
if A.ndim != 2:
raise ValueError('expected ndim to be 2, but observed %s' % A.ndim)
if A.shape != ref_matrix.shape:
raise ValueError('the shapes of A and ref_matrix differ (%s vs %s)' %
(A.shape, ref_matrix.shape))
u, w, vt = svd(A.T.dot(ref_matrix))
R = u.dot(vt) # Get the rotation matrix, including reflections
if not reflection and scipy.linalg.det(R) < 0:
# To remove reflection, we change the sign of the rightmost column of
# u (or v) and the scalar associated
# with that column
u[:, -1] *= -1
w[-1] *= -1
R = u.dot(vt)
scale = w.sum() # Get the scaled difference
return R, scale
def pinv2(a,
rank=None,
cond=None,
rcond=None,
return_rank=False,
check_finite=True):
"""
Compute the (Moore-Penrose) pseudo-inverse of a matrix.
Calculate a generalized inverse of a matrix using its
singular-value decomposition and including all 'large' singular
values.
Parameters
----------
a : (M, N) array_like
Matrix to be pseudo-inverted.
cond, rcond : float or None
Cutoff for 'small' singular values; singular values smaller than this
value are considered as zero. If both are omitted, the default value
``max(M,N)*largest_singular_value*eps`` is used where ``eps`` is the
machine precision value of the datatype of ``a``.
.. versionchanged:: 1.3.0
Previously the default cutoff value was just ``eps*f`` where ``f``
was ``1e3`` for single precision and ``1e6`` for double precision.
return_rank : bool, optional
If True, return the effective rank of the matrix.
check_finite : bool, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
B : (N, M) ndarray
The pseudo-inverse of matrix `a`.
rank : int
The effective rank of the matrix. Returned if `return_rank` is True.
Raises
------
LinAlgError
If SVD computation does not converge.
Examples
--------
>>> from scipy import linalg
>>> a = np.random.randn(9, 6)
>>> B = linalg.pinv2(a)
>>> np.allclose(a, np.dot(a, np.dot(B, a)))
True
>>> np.allclose(B, np.dot(B, np.dot(a, B)))
True
"""
a = _asarray_validated(a, check_finite=check_finite)
u, s, vh = decomp_svd.svd(a, full_matrices=False, check_finite=False)
if rank is None:
if rcond is not None:
cond = np.max(s) * rcond
if cond in [None, -1]:
t = u.dtype.char.lower()
cond = np.max(s) * max(a.shape) * np.finfo(t).eps
rank = np.sum(s > cond)
elif rank == float('inf'):
assert (cond is not None)
rank = len(s)
s[s < cond] = cond
print(rank, flush='True')
u = u[:, :rank]
u /= s[:rank]
B = np.transpose(np.conjugate(np.dot(u, vh[:rank])))
if return_rank:
return B, rank
else:
return B