import numpy.lib.stride_tricks
import numpy as np
from scipy.fft import rfftn, irfftn
from numpy import ma
from tqdm import tqdm
from openpiv.pyprocess import get_field_shape, find_first_peak
"""This module contains a pure python implementation of the basic
cross-correlation algorithm for PIV image processing."""
__licence_ = """
Copyright (C) 2011 www.openpiv.net
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
"""
[docs]def get_coordinates(image_size, search_area_size, window_size, overlap):
"""Compute the x, y coordinates of the centers of the interrogation windows.
Parameters
----------
image_size: two elements tuple
a three dimensional tuple for the pixel size of the image
window_size: tuple
the size of the interrogation window.
search_area_size: tuple
the size of the search area window.
overlap: tuple
the number of pixel by which two adjacent interrogation
windows overlap.
Returns
-------
x : 23 np.ndarray
a three dimensional array containing the x coordinates of the
interrogation window centers, in pixels.
y : 23 np.ndarray
a three dimensional array containing the y coordinates of the
interrogation window centers, in pixels.
z : 23 np.ndarray
a three dimensional array containing the y coordinates of the
interrogation window centers, in pixels.
"""
# get shape of the resulting flow field
field_shape = get_field_shape(image_size, search_area_size, overlap)
# compute grid coordinates of the search area centers
x = (
np.arange(field_shape[1]) * (window_size[1] - overlap[1])
+ (search_area_size[1] - 1) / 2.0
)
y = (
np.arange(field_shape[0]) * (window_size[0] - overlap[0])
+ (search_area_size[0] - 1) / 2.0
)
z = (
np.arange(field_shape[2]) * (window_size[2] - overlap[2])
+ (search_area_size[2] - 1) / 2.0
)
# moving coordinates further to the center, so that the points at the extreme left/right or top/bottom
# have the same distance to the window edges. For simplicity only integer movements are allowed.
x += (
image_size[1]
- 1
- (
(field_shape[1] - 1) * (window_size[1] - overlap[1])
+ (search_area_size[1] - 1)
)
) // 2
y += (
image_size[0]
- 1
- (
(field_shape[0] - 1) * (window_size[0] - overlap[0])
+ (search_area_size[0] - 1)
)
) // 2
z += (
image_size[2]
- 1
- (
(field_shape[2] - 1) * (window_size[2] - overlap[2])
+ (search_area_size[2] - 1)
)
) // 2
return np.meshgrid(x, y, z)
[docs]def find_second_peak_3D(corr, i=None, j=None, z=None, width=2):
"""
Find the value of the second largest peak.
The second largest peak is the height of the peak in
the region outside a 3x3 submatrix around the first
correlation peak.
Parameters
----------
corr: np.ndarray
the correlation map.
i,j,z : ints
row, column and layer location of the first peak.
width : int
the half size of the region around the first correlation
peak to ignore for finding the second peak.
Returns
-------
i : int
the row index of the second correlation peak.
j : int
the column index of the second correlation peak.
z : int
the 3rd index of the second correlation peak.
corr_max2 : int
the value of the second correlation peak.
"""
if i is None or j is None or z is None:
(i, j, z), tmp = find_first_peak(corr) # TODO: why tmp?
# create a masked view of the corr
tmp = corr.view(ma.MaskedArray)
# set width x width square submatrix around the first correlation peak as masked.
# Before check if we are not too close to the boundaries, otherwise we
# have negative indices
iini = max(0, i - width)
ifin = min(i + width + 1, corr.shape[0])
jini = max(0, j - width)
jfin = min(j + width + 1, corr.shape[1])
zini = max(0, z - width)
zfin = min(z + width + 1, corr.shape[1])
tmp[iini:ifin, jini:jfin, zini:zfin] = ma.masked
(i, j, z), corr_max2 = find_first_peak(tmp)
return (i, j, z), corr_max2
[docs]def find_subpixel_peak_position(corr, subpixel_method="gaussian"):
"""
Find subpixel approximation of the correlation peak.
This function returns a subpixels approximation of the correlation
peak by using one of the several methods available. If requested,
the function also returns the signal to noise ratio level evaluated
from the correlation map.
Parameters
----------
corr : np.ndarray
the correlation map.
subpixel_method : string
one of the following methods to estimate subpixel location of the peak:
'centroid' [replaces default if correlation map is negative],
'gaussian' [default if correlation map is positive],
'parabolic'.
Returns
-------
subp_peak_position : two elements tuple
the fractional row and column indices for the sub-pixel
approximation of the correlation peak.
"""
# initialization
# default_peak_position = (np.floor(corr.shape[0] / 2.), np.floor(corr.shape[1] / 2.))
default_peak_position = (0, 0, 0)
# the peak locations
(peak1_i, peak1_j, peak1_z), dummy = find_first_peak(corr)
try:
# the peak and its neighbours: left, right, down, up
c = corr[peak1_i, peak1_j, peak1_z]
cl = corr[peak1_i - 1, peak1_j, peak1_z]
cr = corr[peak1_i + 1, peak1_j, peak1_z]
cd = corr[peak1_i, peak1_j - 1, peak1_z]
cu = corr[peak1_i, peak1_j + 1, peak1_z]
cf = corr[peak1_i, peak1_j, peak1_z - 1]
cb = corr[peak1_i, peak1_j, peak1_z + 1]
# gaussian fit
if (
np.any(np.array([c, cl, cr, cd, cu, cf, cb]) < 0)
and subpixel_method == "gaussian"
):
subpixel_method = "centroid"
try:
if subpixel_method == "centroid":
subp_peak_position = (
((peak1_i - 1) * cl + peak1_i * c + (peak1_i + 1) * cr)
/ (cl + c + cr),
((peak1_j - 1) * cd + peak1_j * c + (peak1_j + 1) * cu)
/ (cd + c + cu),
((peak1_z - 1) * cf + peak1_z * c + (peak1_z + 1) * cb)
/ (cf + c + cb),
)
elif subpixel_method == "gaussian":
with numpy.errstate(divide="ignore"):
subp_peak_position = (
peak1_i
+ (
(np.log(cl) - np.log(cr))
/ (2 * np.log(cl) - 4 * np.log(c) + 2 * np.log(cr))
),
peak1_j
+ (
(np.log(cd) - np.log(cu))
/ (2 * np.log(cd) - 4 * np.log(c) + 2 * np.log(cu))
),
peak1_z
+ (
(np.log(cf) - np.log(cb))
/ (2 * np.log(cf) - 4 * np.log(c) + 2 * np.log(cb))
),
)
elif subpixel_method == "parabolic":
subp_peak_position = (
peak1_i + (cl - cr) / (2 * cl - 4 * c + 2 * cr),
peak1_j + (cd - cu) / (2 * cd - 4 * c + 2 * cu),
peak1_z + (cf - cb) / (2 * cf - 4 * c + 2 * cb),
)
except:
subp_peak_position = default_peak_position
except IndexError:
subp_peak_position = default_peak_position # TODO: is this a good idea??
return np.array(subp_peak_position) - np.array(default_peak_position)
[docs]def sig2noise_ratio(corr, sig2noise_method="peak2peak", width=2):
"""
Computes the signal to noise ratio from the correlation map.
The signal to noise ratio is computed from the correlation map with
one of two available method. It is a measure of the quality of the
matching between to interogation windows.
Parameters
----------
corr : 2d np.ndarray
the correlation map.
sig2noise_method: string
the method for evaluating the signal to noise ratio value from
the correlation map. Can be `peak2peak`, `peak2mean` or None
if no evaluation should be made.
width : int, optional
the half size of the region around the first
correlation peak to ignore for finding the second
peak. [default: 2]. Only used if ``sig2noise_method==peak2peak``.
Returns
-------
sig2noise : float
the signal to noise ratio from the correlation map.
"""
# compute first peak position
(peak1_i, peak1_j, peak1_z), corr_max1 = find_first_peak(corr)
# now compute signal to noise ratio
if sig2noise_method == "peak2peak":
# find second peak height
(peak1_i, peak1_j, peak1_z), corr_max2 = find_second_peak_3D(
corr, peak1_i, peak1_j, peak1_z, width=width
)
# if it's an empty interrogation window
# if the image is lacking particles, totally black it will correlate to very low value, but not zero
# if the first peak is on the borders, the correlation map is also wrong
if corr_max1 < 1e-3 or any(
[
x == 0 or x == corr.shape[i]
for i, x in enumerate([peak1_i, peak1_j, peak1_z])
]
):
return 0.0
elif sig2noise_method == "peak2mean":
# find mean of the correlation map
corr_max2 = corr.mean()
else:
raise ValueError("wrong sig2noise_method")
# avoid dividing by zero
try:
sig2noise = corr_max1 / corr_max2
except ValueError:
sig2noise = np.inf
return sig2noise
[docs]def correlate_windows(
window_a, window_b, correlation_method="fft", nfftx=None, nffty=None, nfftz=None
):
"""Compute correlation function between two interrogation windows.
The correlation function can be computed by using the correlation
theorem to speed up the computation.
Parameters
----------
window_a : 2d np.ndarray
a two dimensions array for the first interrogation window,
window_b : 2d np.ndarray
a two dimensions array for the second interrogation window.
correlation_method : string
one method is currently implemented: 'fft'.
nfftx : int
the size of the 2D FFT in x-direction,
[default: 2 x windows_a.shape[0] is recommended].
nffty : int
the size of the 2D FFT in y-direction,
[default: 2 x windows_a.shape[1] is recommended].
nfftz : int
the size of the 2D FFT in z-direction,
[default: 2 x windows_a.shape[2] is recommended].
Returns
-------
corr : 3d np.ndarray
a three dimensional array of the correlation function.
Note that due to the wish to use 2^N windows for faster FFT
we use a slightly different convention for the size of the
correlation map. The theory says it is M+N-1, and the
'direct' method gets this size out
the FFT-based method returns M+N size out, where M is the window_size
and N is the search_area_size
It leads to inconsistency of the output
"""
if correlation_method == "fft":
window_b = np.conj(window_b[::-1, ::-1, ::-1])
if nfftx is None:
nfftx = nextpower2(window_b.shape[0] + window_a.shape[0])
if nffty is None:
nffty = nextpower2(window_b.shape[1] + window_a.shape[1])
if nfftz is None:
nfftz = nextpower2(window_b.shape[2] + window_a.shape[2])
f2a = rfftn(normalize_intensity(window_a), s=(nfftx, nffty, nfftz))
f2b = rfftn(normalize_intensity(window_b), s=(nfftx, nffty, nfftz))
corr = irfftn(f2a * f2b).real
corr = corr[
: window_a.shape[0] + window_b.shape[0],
: window_b.shape[1] + window_a.shape[1],
: window_b.shape[2] + window_a.shape[2],
]
return corr
# elif correlation_method == 'direct':
# return convolve2d(normalize_intensity(window_a),
# normalize_intensity(window_b[::-1, ::-1, ::-1]), 'full')
else:
raise ValueError("method is not implemented")
[docs]def normalize_intensity(window):
"""Normalize interrogation window by removing the mean value.
Parameters
----------
window : 2d np.ndarray
the interrogation window array
Returns
-------
window : 2d np.ndarray
the interrogation window array, with mean value equal to zero.
"""
return window - window.mean()
[docs]def nextpower2(i):
""" Find 2^n that is equal to or greater than. """
n = 1
while n < i:
n *= 2
return n
[docs]def extended_search_area_piv3D(
frame_a,
frame_b,
window_size,
overlap=(0, 0, 0),
dt=(1.0, 1.0, 1.0),
search_area_size=None,
correlation_method="fft",
subpixel_method="gaussian",
sig2noise_method=None,
width=2,
nfftx=None,
nffty=None,
nfftz=None,
):
"""Standard PIV cross-correlation algorithm, with an option for
extended area search that increased dynamic range. The search region
in the second frame is larger than the interrogation window size in the
first frame.
This is a pure python implementation of the standard PIV cross-correlation
algorithm. It is a zero order displacement predictor, and no iterative
process is performed.
Parameters
----------
frame_a : 3d np.ndarray
an two dimensions array of integers containing grey levels of
the first frame.
frame_b : 3d np.ndarray
an two dimensions array of integers containing grey levels of
the second frame.
window_size : tuple
the size of the (square) interrogation window, [default: 32 pix].
overlap : tuple
the number of pixels by which two adjacent windows overlap
[default: 16 pix].
dt : tuple
the time delay separating the two frames [default: 1.0].
correlation_method : string
only one method is currently implemented: 'fft'
subpixel_method : string
one of the following methods to estimate subpixel location of the peak:
'centroid' [replaces default if correlation map is negative],
'gaussian' [default if correlation map is positive],
'parabolic'.
sig2noise_method : string
defines the method of signal-to-noise-ratio measure,
('peak2peak' or 'peak2mean'. If None, no measure is performed.)
nfftx : int
the size of the 3D FFT in x-direction,
[default: 2 x windows_a.shape[0] is recommended]
nffty : int
the size of the 3D FFT in y-direction,
[default: 2 x windows_a.shape[1] is recommended]
nfftz : int
the size of the 3D FFT in z-direction,
[default: 2 x windows_a.shape[2] is recommended]
width : int
the half size of the region around the first
correlation peak to ignore for finding the second
peak. [default: 2]. Only used if ``sig2noise_method==peak2peak``.
search_area_size : tuple
the size of the interrogation window in the second frame,
default is the same interrogation window size and it is a
fallback to the simplest FFT based PIV
Returns
-------
u : 3d np.ndarray
a three dimensional array containing the u velocity component,
in pixels/seconds.
v : 3d np.ndarray
a three dimensional array containing the v velocity component,
in pixels/seconds.
w : 3d np.ndarray
a three dimensional array containing the w velocity component,
in pixels/seconds.
sig2noise : 3d np.ndarray, (optional: only if sig2noise_method is not None)
a three dimensional array the signal to noise ratio for each
window pair.
"""
# checking if the input is correct
window_size, overlap, search_area_size = check_input(
window_size, overlap, search_area_size, frame_a, frame_b
)
# get field shape
field_shape = get_field_shape(frame_a.shape, search_area_size, overlap)
u = np.zeros(field_shape)
v = np.zeros(field_shape)
w = np.zeros(field_shape)
# if we want sig2noise information, allocate memory
if sig2noise_method is not None:
sig2noise = np.zeros(field_shape)
# shift for x and y coordinates of the search area windows so that the centers of search area windows have
# the same distances to the image edge at all sides. For simplicity only shifts by integers are allowed
x_centering = (
frame_a.shape[1]
- 1
- (
(field_shape[1] - 1) * (window_size[1] - overlap[1])
+ (search_area_size[1] - 1)
)
) // 2
y_centering = (
frame_a.shape[0]
- 1
- (
(field_shape[0] - 1) * (window_size[0] - overlap[0])
+ (search_area_size[0] - 1)
)
) // 2
z_centering = (
frame_a.shape[2]
- 1
- (
(field_shape[2] - 1) * (window_size[2] - overlap[2])
+ (search_area_size[2] - 1)
)
) // 2
# loop over the interrogation windows
# i, j are the row, column indices of the center of each interrogation
# window
for k in tqdm(range(field_shape[0])):
for m in range(field_shape[1]):
for l in range(field_shape[2]):
# centers of search area. (window_size - overlap) defines the distance between each center
# and (search_area_size - 1)/2.0 moves the center points away from the left or top image edge
y = k * (window_size[0] - overlap[0]) + (search_area_size[0] - 1) / 2.0
x = m * (window_size[1] - overlap[1]) + (search_area_size[1] - 1) / 2.0
z = l * (window_size[2] - overlap[2]) + (search_area_size[2] - 1) / 2.0
# moving the coordinates a bit to the center, to guarantee that the distance of a extreme
# point at the image edges is symmetric all all edges
x += x_centering
y += y_centering
z += z_centering
# left, right, top, bottom, front, back indices of the search area edges
# note that x - (search_area_size +/- 1)/2 always returns an integer due to the definition of x and y
# see also "get_coordinates()"
il = int(y - (search_area_size[0] - 1) / 2.0)
ir = int(y + (search_area_size[0] + 1) / 2.0)
it = int(x - (search_area_size[1] - 1) / 2.0)
ib = int(x + (search_area_size[1] + 1) / 2.0)
ifr = int(z - (search_area_size[2] - 1) / 2.0)
iba = int(z + (search_area_size[2] + 1) / 2.0)
# picking the search area from frame b
window_b = frame_b[il:ir, it:ib, ifr:iba]
# left, right, top, bottom, front, back indices of the interrogation window
# Sometimes the interrogation window cannot be placed in the middle of the search area, e.g.
# in the case of window_size=3 search_area_size=4. In this case the interrogation window
# is shifted 0.5 pixels to the left/top, which is achieved by rounding the indices
# down during the int() conversion
il = int(y - (window_size[0] - 1) / 2)
ir = int(y + (window_size[0] + 1) / 2)
it = int(x - (window_size[1] - 1) / 2)
ib = int(x + (window_size[1] + 1) / 2)
ifr = int(z - (window_size[2] - 1) / 2)
iba = int(z + (window_size[2] + 1) / 2)
# picking the interrogation window from frame a
window_a = frame_a[il:ir, it:ib, ifr:iba]
if np.any(window_a):
corr = correlate_windows(
window_a,
window_b,
correlation_method=correlation_method,
nfftx=nfftx,
nffty=nffty,
nfftz=nfftz,
)
# get subpixel approximation for peak position row and column index
row, col, z = find_subpixel_peak_position(
corr, subpixel_method=subpixel_method
)
row -= (search_area_size[0] + window_size[0] - 1) // 2
col -= (search_area_size[1] + window_size[1] - 1) // 2
z -= (search_area_size[2] + window_size[2] - 1) // 2
# get displacements, apply coordinate system definition
u[k, m, l], v[k, m, l], w[k, m, l] = col, -row, -z
# get signal to noise ratio
if sig2noise_method is not None:
sig2noise[k, m, l] = sig2noise_ratio(
corr, sig2noise_method=sig2noise_method, width=width
)
# return output if user wanted sig2noise information
if sig2noise_method is not None:
return u / dt[0], v / dt[1], w / dt[2], sig2noise
else:
return u / dt[0], v / dt[1], w / dt[2]