crosscorr#
HR-EBSD: deviatoric strain and lattice rotation by pattern cross-correlation.
Where refine_hr() matches a pattern against a
dynamical master, HR-EBSD is purely differential: it cross-correlates regions of a
test pattern against a reference pattern of the same phase and fits the
displacement gradient tensor that maps one onto the other. Nothing about the master
pattern enters, so the model-fidelity ceiling that caps absolute pattern-center
accuracy near 1e-3 does not apply here.
The shift equation#
A lattice displacement gradient beta = F - I sends a crystal direction r to
(I + beta) r. In gnomonic screen coordinates (x, y, z) with z = pcz, a
feature therefore moves by, to first order,
qx = b11 x + b12 y + b13 z - (x/z)(b31 x + b32 y + b33 z)qy = b21 x + b22 y + b23 z - (y/z)(b31 x + b32 y + b33 z)
Two equations per region of interest, linear in beta. Adding c I to beta
leaves both exactly unchanged – the hydrostatic component is invisible to a
projection – so only the 8 trace-free components are measurable and hrebsd()
solves under trace(beta) = 0. Recovering the hydrostatic part needs an elastic
boundary condition (usually sigma_33 = 0 at the free surface) and the stiffness
tensor; that is left to the caller.
Reference remapping#
The first-order shift equation degrades once the lattice rotation is large, and a
pattern-center error coupled to a lattice rotation w leaks in as a phantom strain
of order (dpc / pcz) * w. So hrebsd() defaults to two passes: measure
F, re-project the reference through the gnomonic geometry into the deformed
configuration (remap_reference()), then re-correlate and compose. The residual
rotation entering the second pass is only as large as the first pass’s error, which
relaxes the pattern-center requirement by one to two orders of magnitude. The remap
is exact rather than first-order, and it absorbs any pattern-center difference
between the reference and test points as well.
How much of this is verified#
The geometry: the design matrix reproduces the exact projective displacement of a
region to ~1e-4 px, and the solver inverts it to ~1e-3 of beta.
The locator, on real 2048px direct-detection patterns (64 regions on a uniform grid,
5% independent noise added, true shifts from 0.04 to 13 px, known exactly by Fourier
shift): median error 0.002-0.003 px, worst 0.006-0.009 px at 128-256 px regions.
Plain correlation with one refinement iteration measured best; whitening was neither
needed nor helpful. Feeding that precision through the shift equation puts beta
around 1e-5 with ~64 regions, and the error falls as 1 / sqrt(M).
After Wilkinson, Meaden & Dingley (2006); remapping after Britton & Wilkinson (2012).
Functions
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Deviatoric strain and lattice rotation relative to a reference pattern. |
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Re-project a whole reference pattern into the deformed configuration. |
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Region-of-interest centers on a |
Classes
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What |