Deformation & Rotations

DVC measures how a 3-D volume deforms between two loading states. This page explains how the package converts a DVC displacement field into a local rotation field and uses it to warp DCT volumes and update crystal orientations.

The displacement field

DVC produces a displacement field \(\mathbf{U}(\mathbf{X})\) defined on a regular grid in the reference (undeformed) configuration. A material point at position \(\mathbf{X}\) in the reference moves to:

\[ \mathbf{x} = \mathbf{X} + \mathbf{U}(\mathbf{X}) \]

in the deformed configuration. The field \(\mathbf{U}\) is typically stored as a 4-D array of shape \((N_x, N_y, N_z, 3)\).

Deformation gradient

The local deformation is fully described by the deformation gradient tensor:

\[ \mathbf{F} = \mathbf{I} + \nabla \mathbf{U}, \qquad F_{ij} = \delta_{ij} + \frac{\partial U_i}{\partial X_j} \]

\(\mathbf{F}\) is a \(3 \times 3\) tensor field computed numerically using central finite differences (numpy.gradient) with the DVC grid spacing as the step size.

Spatial derivatives

The gradient is taken with respect to the reference coordinates \(\mathbf{X}\), consistent with the Lagrangian description used by DVC.

Polar decomposition

Any invertible tensor \(\mathbf{F}\) (with \(\det \mathbf{F} > 0\)) admits a unique right polar decomposition:

\[ \mathbf{F} = \mathbf{R}\,\mathbf{U}_s \]

where \(\mathbf{R} \in \mathrm{SO}(3)\) is a proper rotation matrix and \(\mathbf{U}_s\) is a symmetric positive-definite stretch tensor.

SVD route

The rotation \(\mathbf{R}\) is extracted efficiently via the singular value decomposition:

\[ \mathbf{F} = \mathbf{U}\,\mathbf{S}\,\mathbf{V}^\top \implies \mathbf{R} = \mathbf{U}\,\mathbf{V}^\top \]

This is evaluated in batch over all voxels using numpy.linalg.svd, which vectorises over arbitrary leading dimensions:

U, S, Vt = np.linalg.svd(F)          # F shape: (..., 3, 3)
R = U @ Vt                            # R shape: (..., 3, 3)

A sign correction ensures \(\det \mathbf{R} = +1\) (proper rotation) even near singular points where numerical noise can introduce a reflection.

Warping DCT volumes

To map DCT data from the reference into the deformed configuration, a pull-back (inverse) warp is used. For each output voxel position \(\mathbf{x}\), the source position in the reference is approximated as:

\[ \mathbf{X}(\mathbf{x}) \approx \mathbf{x} - \mathbf{U}(\mathbf{x}) \]

This is exact for small displacements and a good approximation for typical DVC fields. The warped volume is then:

\[ f_\text{deformed}(\mathbf{x}) = f_\text{ref}\!\left(\mathbf{x} - \mathbf{U}(\mathbf{x})\right) \]

implemented via scipy.ndimage.map_coordinates with order-0 (nearest-neighbour) for integer arrays and order-1 (trilinear) for float arrays.

Coordinate mapping

The DVC field covers only a ROI within the PCT frame. Voxels outside the ROI receive zero displacement (identity warp), localised using dvc.origin and dvc.spacing:

\[ \mathbf{p}_\text{DVC} = \frac{\mathbf{p}_\text{PCT} - \mathbf{o}}{\Delta} \]

where \(\mathbf{o}\) is the ROI origin and \(\Delta\) is the DVC grid spacing, both in PCT voxel units.

Rotating grain orientations

The rotation field \(\mathbf{R}(\mathbf{X})\) describes how each material element has rotated from the reference to the deformed state. The crystal orientation at a voxel transforms as:

\[ \mathbf{g}_\text{new} = \mathbf{R}\,\mathbf{g}_\text{old} \]

where \(\mathbf{g}\) is the orientation matrix mapping crystal axes to sample axes (see Crystal Orientations & IPF).

Interpolation to the DCT grid

\(\mathbf{R}\) is defined on the (coarser) DVC grid and must be interpolated to the (finer) DCT grid. The nine matrix components are interpolated independently with trilinear interpolation, after which each matrix is re-projected to \(\mathrm{SO}(3)\) via a second SVD step to restore orthogonality lost during interpolation:

\[ \tilde{\mathbf{R}} = \text{interp}(\mathbf{R}_\text{DVC}) \xrightarrow{\text{SVD}} \hat{\mathbf{R}} \in \mathrm{SO}(3) \]

Voxels outside the DVC ROI are assigned the identity rotation (\(\mathbf{R} = \mathbf{I}\)), meaning no lattice rotation is applied there.

References

Sutton M A et al. (2009). Image Correlation for Shape, Motion and Deformation Measurements. Springer.
Bonet J, Wood R D (2008). Nonlinear Continuum Mechanics for Finite Element Analysis, 2nd ed. Cambridge University Press. (§3 — kinematics)
Guizar-Sicairos M et al. (2011). Phase tomography from x-ray coherent diffractive imaging. Physical Review Letters 107, 022607.