#!/usr/bin/env python
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"""
Module for functions related to finding axis of rotation.
"""
from __future__ import (absolute_import, division, print_function,
unicode_literals)
import numpy as np
from scipy import ndimage
from tomopy.util.misc import fft2, write_tiff
from scipy.optimize import minimize
from skimage.feature import register_translation
from tomopy.misc.corr import circ_mask
from tomopy.misc.morph import downsample
from tomopy.recon.algorithm import recon
import tomopy.util.dtype as dtype
import os.path
import logging
logger = logging.getLogger(__name__)
__author__ = "Doga Gursoy, Luis Barroso-Luque, Nghia Vo"
__copyright__ = "Copyright (c) 2015, UChicago Argonne, LLC."
__docformat__ = 'restructuredtext en'
__all__ = ['find_center',
'find_center_vo',
'find_center_pc',
'write_center',
'mask_empty_slice',
]
[docs]def find_center(
tomo, theta, ind=None, init=None,
tol=0.5, mask=True, ratio=1., sinogram_order=False):
"""
Find rotation axis location.
The function exploits systematic artifacts in reconstructed images
due to shifts in the rotation center. It uses image entropy
as the error metric and ''Nelder-Mead'' routine (of the scipy
optimization module) as the optimizer :cite:`Donath:06`.
Parameters
----------
tomo : ndarray
3D tomographic data.
theta : array
Projection angles in radian.
ind : int, optional
Index of the slice to be used for reconstruction.
init : float
Initial guess for the center.
tol : scalar
Desired sub-pixel accuracy.
mask : bool, optional
If ``True``, apply a circular mask to the reconstructed image to
limit the analysis into a circular region.
ratio : float, optional
The ratio of the radius of the circular mask to the edge of the
reconstructed image.
sinogram_order: bool, optional
Determins whether data is a stack of sinograms (True, y-axis first axis)
or a stack of radiographs (False, theta first axis).
Returns
-------
float
Rotation axis location.
"""
tomo = dtype.as_float32(tomo)
theta = dtype.as_float32(theta)
if sinogram_order:
dy, dt, dx = tomo.shape
else:
dt, dy, dx = tomo.shape
if ind is None:
ind = dy // 2
if init is None:
init = dx // 2
# extract slice we are using to find center
if sinogram_order:
tomo_ind = tomo[ind:ind + 1]
else:
tomo_ind = tomo[:, ind:ind + 1, :]
hmin, hmax = _adjust_hist_limits(
tomo_ind, theta, mask, sinogram_order)
# Magic is ready to happen...
res = minimize(
_find_center_cost, init,
args=(tomo_ind, theta, hmin, hmax, mask, ratio, sinogram_order),
method='Nelder-Mead',
tol=tol)
return res.x
def _adjust_hist_limits(tomo_ind, theta, mask, sinogram_order):
# Make an initial reconstruction to adjust histogram limits.
rec = recon(tomo_ind,
theta,
sinogram_order=sinogram_order,
algorithm='gridrec')
# Apply circular mask.
if mask is True:
rec = circ_mask(rec, axis=0)
# Adjust histogram boundaries according to reconstruction.
return _adjust_hist_min(rec.min()), _adjust_hist_max(rec.max())
def _adjust_hist_min(val):
if val < 0:
val = 2 * val
elif val >= 0:
val = 0.5 * val
return val
def _adjust_hist_max(val):
if val < 0:
val = 0.5 * val
elif val >= 0:
val = 2 * val
return val
def _find_center_cost(
center, tomo_ind, theta, hmin, hmax, mask, ratio,
sinogram_order=False):
"""
Cost function used for the ``find_center`` routine.
"""
logger.info('Trying rotation center: %s', center)
center = np.array(center, dtype='float32')
rec = recon(
tomo_ind, theta, center,
sinogram_order=sinogram_order,
algorithm='gridrec')
if mask is True:
rec = circ_mask(rec, axis=0)
hist, e = np.histogram(rec, bins=64, range=[hmin, hmax])
hist = hist.astype('float32') / rec.size + 1e-12
val = -np.dot(hist, np.log2(hist))
logger.info("Function value = %f" % val)
return val
[docs]def find_center_vo(tomo, ind=None, smin=-50, smax=50, srad=6, step=0.25,
ratio=0.5, drop=20):
"""
Find rotation axis location using Nghia Vo's method. :cite:`Vo:14`.
Parameters
----------
tomo : ndarray
3D tomographic data.
ind : int, optional
Index of the slice to be used for reconstruction.
smin, smax : int, optional
Coarse search radius. Reference to the horizontal center of the sinogram.
srad : float, optional
Fine search radius.
step : float, optional
Step of fine searching.
ratio : float, optional
The ratio between the FOV of the camera and the size of object.
It's used to generate the mask.
drop : int, optional
Drop lines around vertical center of the mask.
Returns
-------
float
Rotation axis location.
"""
tomo = dtype.as_float32(tomo)
(depth, height, width) = tomo.shape
if ind is None:
ind = height // 2
if height > 10:
# Averaging sinograms to improve SNR
_tomo = np.mean(tomo[:, ind - 5:ind + 5, :], axis=1)
else:
_tomo = tomo[:, ind, :]
else:
_tomo = tomo[:, ind, :]
# Denoising
# There's a critical reason to use different window sizes
# between coarse and fine search.
_tomo_cs = ndimage.filters.gaussian_filter(_tomo, (3, 1))
_tomo_fs = ndimage.filters.gaussian_filter(_tomo, (2, 2))
# Coarse and fine searches for finding the rotation center.
if _tomo.shape[0] * _tomo.shape[1] > 4e6: # If data is large (>2kx2k)
_tomo_coarse = downsample(
np.expand_dims(_tomo_cs, 1), level=2)[:, 0, :]
init_cen = _search_coarse(
_tomo_coarse, smin / 4.0, smax / 4.0, ratio, drop)
fine_cen = _search_fine(_tomo_fs, srad, step,
init_cen * 4, ratio, drop)
else:
init_cen = _search_coarse(_tomo_cs, smin, smax, ratio, drop)
fine_cen = _search_fine(_tomo_fs, srad, step, init_cen, ratio, drop)
logger.debug('Rotation center search finished: %i', fine_cen)
return fine_cen
def _search_coarse(sino, smin, smax, ratio, drop):
"""
Coarse search for finding the rotation center.
"""
(nrow, ncol) = sino.shape
cen_fliplr = (ncol - 1.0) / 2.0
smin = np.int16(np.clip(smin + cen_fliplr, 0, ncol - 1) - cen_fliplr)
smax = np.int16(np.clip(smax + cen_fliplr, 0, ncol - 1) - cen_fliplr)
# Flip left-right the [0:Pi ] sinogram to make a full [0;2Pi] sinogram
flip_sino = np.fliplr(sino)
# Below image is used for compensating the shift of the [Pi;2Pi] sinogram
# It helps to avoid local minima.
comp_sino = np.flipud(sino)
list_shift = np.arange(smin, smax + 1)
list_metric = np.zeros(len(list_shift), dtype='float32')
mask = _create_mask(2 * nrow, ncol, 0.5 * ratio * ncol, drop)
sino_sino = np.vstack((sino, flip_sino))
abs_fft2_sino = np.empty_like(sino_sino)
for i in list_shift:
_sino = sino_sino[nrow:]
_sino[...] = np.roll(flip_sino, i, axis=1)
if i >= 0:
_sino[:, 0:i] = comp_sino[:, 0:i]
else:
_sino[:, i:] = comp_sino[:, i:]
fft2sino = np.fft.fftshift(fft2(sino_sino))
np.abs(fft2sino, out=abs_fft2_sino)
abs_fft2_sino *= mask
list_metric[i - smin] = abs_fft2_sino.mean()
minpos = np.argmin(list_metric)
if minpos == 0:
logger.debug('WARNING!!!Global minimum is out of searching range')
logger.debug('Please extend smin: %i', smin)
if minpos == len(list_metric) - 1:
logger.debug('WARNING!!!Global minimum is out of searching range')
logger.debug('Please extend smax: %i', smax)
init_cen = cen_fliplr + list_shift[minpos] / 2.0
return init_cen
def _search_fine(sino, srad, step, init_cen, ratio, drop):
"""
Fine search for finding the rotation center.
"""
(nrow, ncol) = sino.shape
cen_fliplr = (ncol - 1.0) / 2.0
srad = np.clip(np.abs(srad), 1.0, ncol / 4.0)
step = np.clip(np.abs(step), 0.1, srad)
init_cen = np.clip(init_cen, srad, ncol - srad - 1)
list_cor = init_cen + np.arange(-srad, srad + step, step)
flip_sino = np.fliplr(sino)
comp_sino = np.flipud(sino) # Used to avoid local minima
list_metric = np.zeros(len(list_cor), dtype=np.float32)
mask = _create_mask(2 * nrow, ncol, 0.5 * ratio * ncol, drop)
for i, cor in enumerate(list_cor):
shift = 2.0 * (cor - cen_fliplr)
sino_shift = ndimage.interpolation.shift(
flip_sino, (0, shift), order=3, prefilter=True)
if shift >= 0:
shift_int = np.int16(np.ceil(shift))
sino_shift[:, :shift_int] = comp_sino[:, :shift_int]
else:
shift_int = np.int16(np.floor(shift))
sino_shift[:, shift_int:] = comp_sino[:, shift_int:]
sinojoin = np.vstack((sino, sino_shift))
list_metric[i] = np.mean(np.abs(
np.fft.fftshift(fft2(sinojoin))) * mask)
cor = list_cor[np.argmin(list_metric)]
return cor
def _create_mask(nrow, ncol, radius, drop):
"""
Make a binary mask to select coefficients outside the double-wedge region.
Eq.(3) in https://doi.org/10.1364/OE.22.019078
Parameters
----------
nrow : int
Image height.
ncol : int
Image width.
radius: int
Radius of an object, in pixel unit.
drop : int
Drop lines around vertical center of the mask.
Returns
-------
2D binary mask.
"""
du = 1.0 / ncol
dv = (nrow - 1.0) / (nrow * 2.0 * np.pi)
cen_row = np.int16(np.ceil(nrow / 2.0) - 1)
cen_col = np.int16(np.ceil(ncol / 2.0) - 1)
drop = min(drop, np.int16(np.ceil(0.1 * nrow)))
mask = np.zeros((nrow, ncol), dtype='float32')
for i in range(nrow):
pos = np.int16(np.round(((i - cen_row) * dv / radius) / du))
(pos1, pos2) = np.clip(np.sort(
(-pos + cen_col, pos + cen_col)), 0, ncol - 1)
mask[i, pos1:pos2 + 1] = 1.0
mask[cen_row - drop:cen_row + drop + 1, :] = 0.0
mask[:, cen_col - 1:cen_col + 2] = 0.0
return mask
[docs]def find_center_pc(proj1, proj2, tol=0.5, rotc_guess=None):
"""
Find rotation axis location by finding the offset between the first
projection and a mirrored projection 180 degrees apart using
phase correlation in Fourier space.
The ``register_translation`` function uses cross-correlation in Fourier
space, optionally employing an upsampled matrix-multiplication DFT to
achieve arbitrary subpixel precision. :cite:`Guizar:08`.
Parameters
----------
proj1 : ndarray
2D projection data.
proj2 : ndarray
2D projection data.
tol : scalar, optional
Subpixel accuracy
rotc_guess : float, optional
Initual guess value for the rotation center
Returns
-------
float
Rotation axis location.
"""
imgshift = 0.0 if rotc_guess is None else rotc_guess - \
(proj1.shape[1] - 1.0) / 2.0
proj1 = ndimage.shift(proj1, [0, -imgshift], mode='constant', cval=0)
proj2 = ndimage.shift(proj2, [0, -imgshift], mode='constant', cval=0)
# create reflection of second projection
proj2 = np.fliplr(proj2)
# Determine shift between images using scikit-image pcm
shift = register_translation(proj1, proj2, upsample_factor=1.0 / tol)
# Compute center of rotation as the center of first image and the
# registered translation with the second image
center = (proj1.shape[1] + shift[0][1] - 1.0) / 2.0
return center + imgshift
[docs]def write_center(
tomo, theta, dpath='tmp/center', cen_range=None, ind=None,
mask=False, ratio=1., sinogram_order=False, algorithm='gridrec', filter_name='parzen'):
"""
Save images reconstructed with a range of rotation centers.
Helps finding the rotation center manually by visual inspection of
images reconstructed with a set of different centers.The output
images are put into a specified folder and are named by the
center position corresponding to the image.
Parameters
----------
tomo : ndarray
3D tomographic data.
theta : array
Projection angles in radian.
dpath : str, optional
Folder name to save output images.
cen_range : list, optional
[start, end, step] Range of center values.
ind : int, optional
Index of the slice to be used for reconstruction.
mask : bool, optional
If ``True``, apply a circular mask to the reconstructed image to
limit the analysis into a circular region.
ratio : float, optional
The ratio of the radius of the circular mask to the edge of the
reconstructed image.
sinogram_order: bool, optional
Determins whether data is a stack of sinograms (True, y-axis first axis)
or a stack of radiographs (False, theta first axis).
algorithm : {str, function}
One of the following string values.
'art'
Algebraic reconstruction technique :cite:`Kak:98`.
'bart'
Block algebraic reconstruction technique.
'fbp'
Filtered back-projection algorithm.
'gridrec'
Fourier grid reconstruction algorithm :cite:`Dowd:99`,
:cite:`Rivers:06`.
'mlem'
Maximum-likelihood expectation maximization algorithm
:cite:`Dempster:77`.
'osem'
Ordered-subset expectation maximization algorithm
:cite:`Hudson:94`.
'ospml_hybrid'
Ordered-subset penalized maximum likelihood algorithm with
weighted linear and quadratic penalties.
'ospml_quad'
Ordered-subset penalized maximum likelihood algorithm with
quadratic penalties.
'pml_hybrid'
Penalized maximum likelihood algorithm with weighted linear
and quadratic penalties :cite:`Chang:04`.
'pml_quad'
Penalized maximum likelihood algorithm with quadratic penalty.
'sirt'
Simultaneous algebraic reconstruction technique.
'tv'
Total Variation reconstruction technique
:cite:`Chambolle:11`.
'grad'
Gradient descent method with a constant step size
filter_name : str, optional
Name of the filter for analytic reconstruction.
'none'
No filter.
'shepp'
Shepp-Logan filter (default).
'cosine'
Cosine filter.
'hann'
Cosine filter.
'hamming'
Hamming filter.
'ramlak'
Ram-Lak filter.
'parzen'
Parzen filter.
'butterworth'
Butterworth filter.
'custom'
A numpy array of size `next_power_of_2(num_detector_columns)/2`
specifying a custom filter in Fourier domain. The first element
of the filter should be the zero-frequency component.
'custom2d'
A numpy array of size `num_projections*next_power_of_2(num_detector_columns)/2`
specifying a custom angle-dependent filter in Fourier domain. The first element
of each filter should be the zero-frequency component.
"""
tomo = dtype.as_float32(tomo)
theta = dtype.as_float32(theta)
if sinogram_order:
dy, dt, dx = tomo.shape
else:
dt, dy, dx = tomo.shape
if ind is None:
ind = dy // 2
if cen_range is None:
center = np.arange(dx / 2 - 5, dx / 2 + 5, 0.5)
else:
center = np.arange(*cen_range)
stack = dtype.empty_shared_array((len(center), dt, dx))
for m in range(center.size):
if sinogram_order:
stack[m] = tomo[ind]
else:
stack[m] = tomo[:, ind, :]
# Reconstruct the same slice with a range of centers.
rec = recon(stack,
theta,
center=center,
sinogram_order=True,
algorithm=algorithm,
filter_name=filter_name,
nchunk=1)
# Apply circular mask.
if mask is True:
rec = circ_mask(rec, axis=0)
# Save images to a temporary folder.
dpath = os.path.abspath(dpath)
if not os.path.exists(dpath):
os.makedirs(dpath)
for m in range(len(center)):
write_tiff(data=rec[m], fname=dpath, digit='{0:.2f}'.format(center[m]))
[docs]def mask_empty_slice(tomo, threshold=0.25):
"""
Generate a mask to indicate whether current slice contains sample
At APS 1ID, some of the projection images contains large empty area above
the sample, resulting in empty layers.
Parameters
----------
tomo : ndarray
3D tomographic data.
threshold : float, optional
determine whether a layer is considered to be empty
Returns
-------
nparray:
a mask indicate the emptyness of each layer
"""
projs_sum = np.sum(tomo, axis=0)
projs_sum /= projs_sum.max()
projs_sumsum = np.sum(projs_sum, axis=0)
projs_sumsum /= projs_sumsum.max()
stds = np.array([np.std(projs_sum[i, :])
for i in range(projs_sum.shape[0])])
std_ref = np.std(projs_sumsum) * threshold
return np.array([std < std_ref for std in stds])