A million measured orientations are not an answer. The orientation distribution function, or ODF, is the answer: a single function that says how much of the material sits at each orientation, in place of the list of individual measurements it was built from.
Being a density is the whole point, and it is where most misunderstandings start. An ODF does not assign a volume fraction to one exact orientation - the fraction of any material sitting at exactly one orientation is zero, in the same way that no one in a room is exactly 180 centimetres tall. It assigns a fraction to a region of orientations, and the function tells you how that fraction concentrates. The value is reported in multiples of a random distribution: 1 everywhere means no texture at all, and a peak of 20 means twenty times as much material near that orientation as random chance would put there.
Below, the same data as points and as a density. The black dots are individual measured orientations; the colours are the ODF estimated from them, both shown in the same pole figure.
mtexdata forsterite silent
ori = ebsd('Forsterite').orientations;
odf = calcDensity(ori,'halfwidth',10*degree);
h = Miller(1,0,0,ori.CS);
plotPDF(odf,h,'contourf','figSize','small')
hold on
plotPDF(ori(1:200:end),h,'MarkerSize',3,'MarkerFaceColor','k','MarkerEdgeColor','k')
hold off
The step from dots to colours involved a choice: how wide a region each measurement should be spread over, the halfwidth above. Too small and the density is a bumpy record of exactly which orientations happened to be measured; too large and real peaks are flattened away. There is no value that is correct independently of the data, which is why Density Estimation is a page of its own rather than a default nobody mentions.
A function on rotations, not on a sphere
An ODF lives on the set of all rotations, which is three-dimensional and curved and has no faithful picture on a flat page. Everything you will ever see of an ODF is therefore a projection or a slice, and each kind throws away something different: a pole figure integrates along a line, a section shows one plane and hides the rest. Reading these plots is a skill, and it is worth knowing which information each one has discarded.
Crystal symmetry applies here exactly as it does to a single orientation, and specimen symmetry may apply too, so an ODF is periodic in ways that are not obvious from any single view.
Where to start
Theory defines the function and the normalisation above. Calculations builds ODFs directly - uniform, unimodal, fibre - which is the fastest way to learn to read the plots, since you know the answer in advance.
The viewing pages come next. Plot is the overview; Pole Figures and Inverse Pole Figures are the two projections; Euler Angle Sections and Sigma Sections are the two common ways to slice. Sigma sections are usually the more honest of the two, because the distortion of Euler space near the poles makes Euler sections misleading about how concentrated a texture really is.
Components and Properties extract numbers - texture index, entropy, volume fractions, modes - which is what most papers actually report.
How an ODF is represented internally matters once you compute with it. Series Expansion stores coefficients, Radial ODF stores a sum of bumps, Fibre ODF stores density along a curve, and Bingham ODF fits a specific statistical model. They are interchangeable in use and very different in cost and in what they can represent sharply.
Density Estimation and Random Sampling are the two directions between a list of orientations and a density. Sample Symmetry asks whether the specimen symmetry you assumed is really there, Shapes concerns the shape of the peaks themselves, and Import and Export handle files.
Next
ODFs are estimated from maps in EBSD and from diffraction data in Pole Figures. The general machinery behind them is Orientation Functions. Using an ODF to predict a material property is Tensors and Elasticity.