calcDensity

estimate a smooth density from orientations, directions, angles or real-valued samples.

Syntax

odf = calcDensity(ori)

odf = calcDensity(ori, halfwidth=5*degree)

odf = calcDensity(ori, kernel=SO3AbelPoissonKernel(halfwidth=10*degree))

odf = calcDensity(grains.meanOrientation, weights=grains.area)

odf = calcDensity(ebsd)

odf = calcDensity(ori, harmonic=True, bandwidth=16)

sF = calcDensity(v, halfwidth=10*degree, weights=w)

sF = calcDensity(v, halfwidth=10*degree, antipodal=True)

f = calcDensity(rho, periodic=True, sigma=5*degree)

f = calcDensity(x, range=[0, 1])

f = calcDensity(X, range=[[0, 0], [1, 1]], rng=rng)

Description

The estimate replaces the sample cloud with a smooth function. Narrow kernels retain finer detail; wider kernels suppress more sampling noise. Use odf(oriQuery), sF(vQuery) or f(xQuery) to evaluate it. For a joint density in two dimensions, use f(xQuery, yQuery); the coordinate arrays broadcast together.

The input determines the domain and the meaning of the density:

For orientations and directions, an explicit halfwidth uses a de la Vallee Poussin kernel; halfwidth is the angle at which the kernel falls to half its peak. Without a halfwidth or kernel, calcKernel selects the width by unbiased cross validation with a noise floor. Angular halfwidths are in radians; write 5*degree for five degrees.

For an EBSD map, select one phase before calling. Its grain ids are used automatically when available, so neighbouring pixels within a grain are grouped during width selection. Map orientations without grain groups use a halfwidth of 10 degrees and issue a warning. Automatic selection also falls back to 10 degrees when it cannot be performed, for example with fewer than two valid points. NaN orientations and directions are omitted; if none remain, their density is uniform. These empty-sample rules do not apply to numeric angles or real-valued data.

Numeric arrays are interpreted as real-valued data by default. Use periodic=True for angles; supplying sigma, kernel or antipodal also selects the circle estimator, even when antipodal=False. Circular data use a Gaussian kernel with Silverman’s width rule unless sigma or kernel is supplied. Real-valued data use a diffusion estimate in one dimension and a fitted Gaussian mixture in several dimensions.

Additional options depend on the input. Omitted weights give every sample equal weight; supplied weights are normalised internally.

Multidimensional real data currently ignore weights and bandwidth. For orientations, bandwidth is used when the estimate is built with harmonic=True; Fourier=True converts the result to harmonics afterwards. Use harmonic=True to request a specific degree. MATLAB’s ‘order’ is bandwidth; for a Bingham model use calcBinghamODF.

Input

Keywords

Output

Example

Estimate an orientation density from a reproducible synthetic sample. Compare two angular halfwidths along one Euler-angle section through the density.

import numpy as np
import matplotlib.pyplot as plt
from mtex import crystalFrame, orientation, calcDensity, degree

rng = np.random.default_rng(7)
cs = crystalFrame('m-3m')
angles = rng.normal([25, 35, 45], [8, 8, 8], size=(80, 3)) * degree
ori = orientation.byEuler(angles, cs)
narrow = calcDensity(ori, halfwidth=7.5*degree)
broad = calcDensity(ori, halfwidth=20*degree)
phi1 = np.linspace(0, 70, 141)
probe = orientation.byEuler(phi1*degree, 35*degree, 45*degree, cs)
plt.plot(phi1, narrow(probe), label='7.5 degrees')
plt.plot(phi1, broad(probe), label='20 degrees')
plt.xlabel('phi1 (degrees), Phi=35 degrees, phi2=45 degrees')
plt.ylabel('Orientation density (mrd)')
plt.legend()
print('Mean orientation density:', round(float(narrow.mean()), 3))
plt.show()

The broader kernel smooths more strongly. Both ODFs have mean one over orientation space; this one-dimensional section is not itself a probability density normalized to area one. Numeric real-valued inputs to calcDensity instead return a probability density in data units.

Mean orientation density: 1.0

See also

calcKernel, calcBinghamODF, unimodalODF, interpolate, calcDensityReal, DensityEstimation

Used on

EBSD2ODF, EBSDAdvancedMaps, EBSDOrientationAnalysis, CPOSeismicProperties, DensityEstimation, OptimalKernel, CSLBoundaries, SubGrainBoundaries, TiltAndTwistBoundaries, EllipseBasedParameters, GrainOrientationParameters, Grain_dispersion_axes

MATLAB’s calcDensity

@article{bachmann2010mtex,
  author  = {F. Bachmann and R. Hielscher and H. Schaeben},
  title   = {Texture Analysis with MTEX - Free and Open Source Software Toolbox},
  journal = {Solid State Phenomena},
  volume  = {160},
  pages   = {63-68},
  year    = {2010},
  doi     = {10.4028/www.scientific.net/SSP.160.63},
  url     = {https://doi.org/10.4028/www.scientific.net/SSP.160.63}
}