# -*- coding: utf-8 -*-
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
from __future__ import unicode_literals
import numpy as np
import abel
import scipy
from scipy.special import eval_legendre
from scipy import ndimage
###############################################################################
# linbasex - inversion procedure based on 1-dimensional projections of
# velocity-map images
#
# As described in:
# Gerber, Thomas, Yuzhu Liu, Gregor Knopp, Patrick Hemberger, Andras Bodi,
# Peter Radi, and Yaroslav Sych,
# “Charged Particle Velocity Map Image Reconstruction with One-Dimensional
# Projections of Spherical Functions.”
# Review of Scientific Instruments 84, no. 3 (March 1, 2013):
# 033101–033101 – 10.
# doi:10.1063/1.4793404.
#
# 2016-04- Thomas Gerber and Daniel Hickstein - theory and code updates
# 2016-04- Stephen Gibson core code extracted from the supplied jupyter
# notebook (see #167: https://github.com/PyAbel/PyAbel/issues/167)
#
###############################################################################
_linbasex_parameter_docstring = \
r"""Inverse Abel transform using 1d projections of images.
`Gerber, Thomas, Yuzhu Liu, Gregor Knopp, Patrick Hemberger, Andras Bodi,
Peter Radi, and Yaroslav Sych.
Charged Particle Velocity Map Image Reconstruction with One-Dimensional Projections of Spherical Functions.` Review of Scientific Instruments 84, no. 3 (March 1, 2013): 033101–033101 – 10.
<http://dx.doi.org/10.1063/1.4793404>`_
``linbasex``models the image using a sum of Legendre polynomials at each
radial pixel, As such, it should only be applied to situations that can
be adequately represented by Legendre polynomials, i.e., images that
feature spherical-like structures. The reconstructed 3D object is
obtained by adding all the contributions, from which slices are derived.
Parameters
----------
IM : numpy 2D array
image data must be square shape of odd size
proj_angles : list
projection angles, in radians (default :math:`[0, \pi/2]`)
e.g. :math:`[0, \pi/2]` or :math:`[0, 0.955, \pi/2]` or :math:`[0, \pi/4, \pi/2, 3\pi/4]`
legendre_orders : list
orders of Legendre polynomials to be used as the expansion
even polynomials [0, 2, ...] gerade
odd polynomials [1, 3, ...] ungerade
all orders [0, 1, 2, ...].
In a single photon experiment there are only anisotropies up to
second order. The interaction of 4 photons (four wave mixing) yields
anisotropies up to order 8.
radial_step : int
number of pixels per Newton sphere (default 1)
smoothing: float
convolve Beta array with a Gaussian function of 1/e 1/2 width `smoothing`.
rcond : float
(default 0.0005) scipy.linalg.lstsq fit conditioning value.
set rcond to zero to switch conditioning off.
Note: In the presence of noise the equation system may be ill posed.
Increasing rcond smoothes the result, lowering it beyond a minimum
renders the solution unstable. Tweak rcond to get a "reasonable"
solution with acceptable resolution.
clip : int
clip first vectors (smallest Newton spheres) to avoid singularities
(default 0)
norm_range : tuple
(low, high)
normalization of Newton spheres, maximum in range Beta[0, low:high].
Note: Beta[0, i] the total number of counts integrated over sphere i,
becomes 1.
threshold : float
threshold for normalization of higher order Newton spheres (default 0.2)
Set all Beta[j], j>=1 to zero if the associated Beta[0] is smaller
than threshold.
return_Beta : bool
return the Beta array of Newton spheres, as the tuple: radial-grid, Beta
for the case :attr:`legendre_orders=[0, 2]`
Beta[0] vs radius -> speed distribution
Beta[2] vs radius -> anisotropy of each Newton sphere
see 'Returns'.
direction : str
"inverse" - only option for this method.
Abel transform direction.
verbose : bool
print information about processing (normally used for debugging)
Returns
-------
inv_IM : numpy 2D array
inverse Abel transformed image
radial-grid, Beta, projections : tuple
(if :attr:`return_Beta=True`)
contributions of each spherical harmonic :math:`Y_{i0}` to the 3D
distribution contain all the information one can get from an experiment.
For the case :attr:`legendre_orders=[0, 2]`:
Beta[0] vs radius -> speed distribution
Beta[1] vs radius -> anisotropy of each Newton sphere.
projections : are the radial projection profiles at angles `proj_angles`
"""
def _linbasex_transform_with_basis(IM, Basis, proj_angles=[0, np.pi/2],
legendre_orders=[0, 2], radial_step=1,
rcond=0.0005, smoothing=0.5, threshold=0.2,
clip=0, norm_range=(0, -1)):
"""linbasex inverse Abel transform evaluated with supplied basis set Basis.
"""
IM = np.atleast_2d(IM)
rows, cols = IM.shape
# Number of used polynoms
pol = len(legendre_orders)
# How many projections
proj = len(proj_angles)
QLz = np.zeros((proj, cols)) # array for projections.
# Rotate and project VMI-image for each angle (as many as projections)
# if proj_angles == [0, np.pi/2]:
# # If coordinates of the detector coincide with the projection
# # directions unnecessary rotations are avoided
# # i.e. proj_angles=[0, np.pi/2] degrees
# QLz[0] = np.sum(IM, axis=1)
# QLz[1] = np.sum(IM, axis=0)
# else:
for i in range(proj):
Rot_IM = scipy.ndimage.interpolation.rotate(IM,
proj_angles[i]*180/np.pi, axes=(1, 0), reshape=False)
QLz[i, :] = np.sum(Rot_IM, axis=1)
# arrange all projections for input into "lstsq"
bb = np.concatenate(QLz, axis=0)
Beta = _beta_solve(Basis, bb, pol, rcond=rcond)
inv_IM, Beta_convol = _Slices(Beta, legendre_orders, smoothing=smoothing)
# normalize
Beta = _single_Beta_norm(Beta_convol, threshold=threshold,
norm_range=norm_range)
radial = np.linspace(clip, cols//2, len(Beta[0]))
# Fix Me! Issue #202 the correct scaling factor for inv_IM intensity?
return inv_IM, radial, Beta, QLz
linbasex_transform_full.__doc__ = _linbasex_parameter_docstring
def _beta_solve(Basis, bb, pol, rcond=0.0005):
# set rcond to zero to switch conditioning off
# array for solutions. len(Basis[0,:])//pol is an integer.
Beta = np.zeros((pol, len(Basis[0])//pol))
# solve equation
Sol = np.linalg.lstsq(Basis, bb, rcond)
# arrange solutions into subarrays for each β.
Beta = Sol[0].reshape((pol, len(Sol[0])//pol))
return Beta
def _SL(i, x, y, Beta_convol, index, legendre_orders):
"""Calculates interpolated β(r), where r= radius"""
r = np.sqrt(x**2 + y**2 + 0.1) # + 0.1 to avoid divison by zero.
# normalize: divison by circumference.
BB = np.interp(r, index, Beta_convol[i, :], left=0)/(2*np.pi*r)
return BB*eval_legendre(legendre_orders[i], x/r)
def _Slices(Beta, legendre_orders, smoothing=0.5):
"""Convolve Beta with a Gaussian function of 1/e width smoothing.
"""
pol = len(legendre_orders)
NP = len(Beta[0]) # number of points in 3_d plot.
index = range(NP)
Beta_convol = np.zeros((pol, NP))
Slice_3D = np.zeros((pol, 2*NP, 2*NP))
# Convolve Beta's with smoothing function
if smoothing > 0:
# smoothing function
Basis_s = np.fromfunction(lambda i: np.exp(-(i - (NP)/2)**2 /
(2*smoothing**2))/(smoothing*2.5), (NP,))
for i in range(pol):
Beta_convol[i] = np.convolve(Basis_s, Beta[i], mode='same')
else:
Beta_convol = Beta
# Calculate ordered slices:
for i in range(pol):
Slice_3D[i] = np.fromfunction(lambda k, l: _SL(i, (k-NP), (l-NP),
Beta_convol, index, legendre_orders), (2*NP, 2*NP))
# Sum ordered slices up
Slice = np.sum(Slice_3D, axis=0)
return Slice, Beta_convol
[docs]def int_beta(Beta, radial_step=1, threshold=0.1, regions=None):
"""Integrate beta over a range of Newton spheres.
Parameters
----------
Beta : numpy array
Newton spheres
radial_step : int
number of pixels per Newton sphere (default 1)
threshold : float
threshold for normalisation of higher orders, 0.0 ... 1.0.
regions : list of tuple radial ranges
[(min0, max0), (min1, max1), ...]
Returns
-------
Beta_in : numpy array
integrated normalized Beta array [Newton sphere, region]
"""
pol = Beta.shape[0]
# Define new array for normalized beta's, independent of Beat_norm
Beta_n = np.zeros(Beta.shape)
# Normalized to Newton sphere with maximal counts.
max_counts = max(Beta[0, :])
Beta_n[0] = Beta[0]/max_counts
for i in range(1, pol):
Beta_n[i] = np.where(Beta[0]/max_counts > threshold, Beta[i]/Beta[0],
0)
Beta_int = np.zeros((pol, len(regions))) # arrays for results
for j, reg in enumerate(regions):
for i in range(pol):
Beta_int[i, j] = np.sum(Beta_n[i, range(*reg)])/(reg[1]-reg[0])
return Beta_int
def _single_Beta_norm(Beta, threshold=0.2, norm_range=(0, -1)):
"""Normalize Newton spheres.
Parameters
----------
Beta : numpy array
Newton spheres
threshold : float
choose only Beta's for which Beta0 is greater than the maximal Beta0
times threshold in the chosen range
Set all βi, i>=1 to zero if the associated β0 is smaller than threshold
norm_range : tuple (int, int)
(low, high)
normalize to Newton sphere with maximum counts in chosen range.
Beta[0, low:high]
Return
------
Beta : numpy array
normalized Beta array
"""
pol = Beta.shape[0]
Beta_norm = np.zeros_like(Beta)
# Normalized to Newton sphere with maximum counts in chosen range.
max_counts = Beta[0, norm_range[0]:norm_range[1]].max()
Beta_norm[0] = Beta[0]/max_counts
for i in range(1, pol):
Beta_norm[i] = np.where(Beta[0]/max_counts > threshold,
Beta[i]/Beta[0], 0)
return Beta_norm
def _bas(ord, angle, COS, TRI):
"""Define Basis vectors for a given polynomial order "order" and a
given projection angle "angle".
"""
basis_vec = scipy.special.eval_legendre(ord, angle) *\
scipy.special.eval_legendre(ord, COS) * TRI
return basis_vec
def _bs_linbasex(cols, proj_angles=[0, np.pi/2], legendre_orders=[0, 2],
radial_step=1, clip=0, **kwargs):
pol = len(legendre_orders)
proj = len(proj_angles)
# Calculation of Base vectors
# Define triangular matrix containing columns :math:`x/y` (representing :math:`\cos(\theta))`.
n = cols//2 + cols % 2
Index = np.indices((n, n))
Index[:, 0, 0] = 1
cos = Index[0]*np.tri(n, n, k=0)[::-1, ::-1]/np.diag(Index[0])
# Concatenate to "bi"-triangular matrix
COS = np.concatenate((-cos[::-1, :], cos[1:, :]), axis=0)
TRI = np.concatenate((np.tri(n, n, k=0,)[:-1, ::-1],
np.tri(n, n, k=0,)[::-1, ::-1]), axis=0)
# radial_step: use only each radial_step other vector.
# Keep the base vector with full span
COS = COS[:, ::-radial_step]
TRI = TRI[:, ::-radial_step]
COS = COS[:, ::-1] # rearrange base vectors again in ascending order
TRI = TRI[:, ::-1]
if clip > 0:
# clip first vectors (smallest Newton spheres) to avoid singularities
COS = COS[:, clip:]
# It is difficult to trace the effect on the SVD solver used below.
TRI = TRI[:, clip:] # usually no clipping works fine.
# Calculate base vectors for each projection and each order.
B = np.zeros((pol, proj, len(COS[:, 0]), len(COS[0, :])))
Norm = np.sum(_bas(0, 1, COS, TRI), axis=0) # calculate normalization
cos_an = np.cos(proj_angles) # angles in radians
for p in range(pol):
for u in range(proj):
B[p, u] = _bas(legendre_orders[p], cos_an[u], COS, TRI)/Norm
# concatenate vectors to one matrix of bases
Bpol = np.concatenate((B), axis=2)
Basis = np.concatenate((Bpol), axis=0)
return Basis
linbasex_transform.__doc__ += _linbasex_parameter_docstring