def misorientationStats(misorient, *wts):
'''
MisorientationStats - Misorientation correlation statistics.
USAGE:
stats = MisorientationStats(misorient, locations)
stats = MisorientationStats(misorient, locations, wts)
INPUT:
misorient is 4 x n,
a list of misorientation quaternions,
assumed to have been derived from properly clustered
orientation data
wts is 1 x n, (optional)
a list of weights; if not specified, uniform weights are used
OUTPUT:
stats is a structure with five components:
W is a 3 x 3 matrix (A in Barton paper)
Winv is a 3 x 3 matrix (A^-1 in Barton paper)
wi is 3 x n, the unnormalized axial vectors
REFERENCE:
"A Methodology for Determining Average Lattice Orientation and
Its Application to the Characterization of Grain Substructure",
Nathan R. Barton and Paul R. Dawson,
Metallurgical and Materials Transactions A,
Volume 32A, August 2001, pp. 1967--1975
'''
misorient = utl.mat2d_row_order(misorient)
d, n = misorient.shape
if len(wts) == 0:
wts = np.tile(1.0 / n, (3, n))
else:
wts = np.tile(wts, (3, 1))
wts1 = np.tile(wts[0, :], (4, 1))
misOriCen = sph.SphereAverage(misorient, **{'wts': wts1})
misorient = misorient - np.tile(misOriCen, (1, n))
ang = utl.mat2d_row_order(2 * np.arccos(misorient[0, :]))
wsc = np.zeros(ang.shape)
limit = (ang < np.finfo(float).eps)
nlimit = (ang > np.finfo(float).eps)
angn = ang[nlimit]
wsc[nlimit] = angn / np.sin(angn / 2)
wsc[limit] = 2
wi = misorient[1:4, :] * np.tile(wsc.T, (3, 1))
angax = np.zeros((4, n))
angax[0, :] = np.linalg.norm(wi, axis=0)
angax[1:4, :] = normalize(wi, axis=0)
Winv = np.sum(utl.RankOneMatrix(wi * wts, wi), axis=2)
#We needed to scale this up if it was close to being ill-conditioned
if (np.abs(np.linalg.det(Winv)) < 1e-6):
if (np.abs(np.linalg.det(Winv)) < 1e-16):
W = np.zeros((3, 3))
else:
Wtemp = np.multiply(1e9, Winv)
W = np.multiply(1e9, np.linalg.inv(Wtemp))
else:
W = np.linalg.inv(Winv)
stat = {'W': W, 'Winv': Winv, 'wi': wi, 'angaxis': angax}
return stat
def bartonStats(misorient, locations, *wts):
'''
MisorientationStats - Misorientation correlation statistics.
USAGE:
stats = MisorientationStats(misorient, locations)
stats = MisorientationStats(misorient, locations, wts)
INPUT:
misorient is 4 x n,
a list of misorientation quaternions,
assumed to have been derived from properly clustered
orientation data
locations is d x n, (d <= 3)
a list of spatial locations corresponding to the
misorientations
wts is 1 x n, (optional)
a list of weights; if not specified, uniform weights are used
OUTPUT:
stats is a structure with five components:
W is a 3 x 3 matrix (A in Barton paper)
X is a d x d matrix (M in Barton paper)
WX is a 3 x d matrix (cross-correlation
of normalized variables; X in
Barton paper)
wi is 3 x n, the unnormalized axial vectors
xi is d x n, the unnormalized spatial directions
from the centroid
REFERENCE:
"A Methodology for Determining Average Lattice Orientation and
Its Application to the Characterization of Grain Substructure",
Nathan R. Barton and Paul R. Dawson,
Metallurgical and Materials Transactions A,
Volume 32A, August 2001, pp. 1967--1975
'''
locations = utl.mat2d_row_order(locations)
misorient = utl.mat2d_row_order(misorient)
d, n = misorient.shape
if len(wts) == 0:
wts = np.tile(1.0 / n, (3, n))
else:
wts = np.tile(wts, (3, 1))
wts1 = np.tile(wts[0, :], (4, 1))
misOriCen = sph.SphereAverage(misorient, **{'wts': wts1})
misorient = misorient - np.tile(misOriCen, (1, n))
ang = utl.mat2d_row_order(2 * np.arccos(misorient[0, :]))
wsc = np.zeros(ang.shape)
limit = (ang < np.finfo(float).eps)
nlimit = (ang > np.finfo(float).eps)
angn = ang[nlimit]
wsc[nlimit] = angn / np.sin(angn / 2)
wsc[limit] = 2
wi = misorient[1:4, :] * np.tile(wsc.T, (3, 1))
wi = wi * np.tile(ang.T, (3, 1))
cen = utl.mat2d_row_order(np.sum(locations * wts, axis=1))
xi = locations - np.tile(cen, (1, n))
Winv = np.sum(utl.RankOneMatrix(wi * wts, wi), axis=2)
Xinv = np.sum(utl.RankOneMatrix(xi * wts, xi), axis=2)
#We needed to scale this up if it was close to being ill-conditioned
if (np.abs(np.linalg.det(Winv)) < 1e-6):
if (np.abs(np.linalg.det(Winv)) < 1e-16):
W = np.zeros((3, 3))
else:
Wtemp = np.multiply(1e9, Winv)
W = np.multiply(1e9, np.linalg.inv(Wtemp))
else:
W = np.linalg.inv(Winv)
Whalf = sci.linalg.sqrtm(W)
if (np.abs(np.linalg.det(Xinv)) < 1e-6):
Xtemp = np.multiply(1e9, Xinv)
X = np.multiply(1e9, np.linalg.inv(Xtemp))
else:
X = np.linalg.inv(Xinv)
Xhalf = sci.linalg.sqrtm(X)
wibar = np.dot(Whalf, wi)
xibar = np.dot(Xhalf, xi)
WX = np.sum(utl.RankOneMatrix(wibar * wts, xibar), axis=2)
stat = {
'W': W,
'Winv': Winv,
'Xinv': Xinv,
'X': X,
'WX': WX,
'wi': wi,
'xi': xi
}
return stat