Image Registration

Registration is the process of finding a geometric transformation that brings two volumes into the same frame of reference. Here we deal exclusively with rigid translation — the same physical sample is imaged by two different techniques, so only a shift (and possibly a flip) is needed to align them.

Why the modalities need aligning

DCT and PCT are acquired on the same sample but in separate experiments, often on different beamlines. The sample may be:

Mounted differently between experiments (translation, rotation).
Scanned at a different voxel size — DCT voxel sizes are typically 5–20 µm, while PCT can reach 0.5–2 µm.
Stored with a different spatial origin in the reconstruction software.

The registration pipeline corrects for all of these.

Upscaling

Before any alignment can be attempted, both volumes must share the same voxel size. A zoom factor is computed from the ratio of voxel sizes:

\[ z = \frac{\Delta x_\text{DCT}}{\Delta x_\text{PCT}} \]

and applied independently to each spatial axis using scipy.ndimage.zoom (order-0 nearest-neighbour for integer arrays such as GIDvol; order-1 bilinear for float arrays).

Phase cross-correlation

Once the volumes share a voxel size and have been padded to the same shape, the translation is estimated using phase cross-correlation in the Fourier domain.

Given a fixed image \(f\) and a moving image \(g\), define the cross-power spectrum:

\[ C(\mathbf{u}) = \frac{F(\mathbf{u})\, \overline{G(\mathbf{u})}} {\lvert F(\mathbf{u})\, \overline{G(\mathbf{u})} \rvert} \]

where \(F\) and \(G\) are the Fourier transforms of \(f\) and \(g\), and \(\mathbf{u}\) is the frequency vector. The inverse Fourier transform of \(C\) is an impulse at the true translation vector:

\[ \mathcal{F}^{-1}\{C\}(\mathbf{x}) = \delta(\mathbf{x} - \mathbf{t}) \]

The peak location gives \(\mathbf{t}\) to sub-voxel accuracy (controlled by upsample_factor).

Why normalise?

Dividing by the magnitude of the cross-power spectrum makes the estimator insensitive to the absolute intensity scales of the two modalities — essential here since DCT produces a binary mask while PCT produces a grey-level absorption contrast image.

Why use masks

Correlating the raw DCT grain-ID volume against the PCT grey-level volume would fail: the two signals carry completely different physical information. Instead, both volumes are reduced to binary masks that capture the sample geometry — the one quantity both modalities share.

Volume	Mask derivation
DCT	`Mask` dataset loaded directly from the HDF5 file
PCT	Otsu threshold on the grey-level volume; optional binary closing to fill internal voids

The shift found from the mask correlation is then applied to all DCT volumes (grain IDs, IPF colours, etc.).

Sub-pixel accuracy

Phase cross-correlation can localise the peak to sub-voxel precision by upsampling the cross-power spectrum in a small neighbourhood around the integer-pixel peak (Guizar-Sicairos et al., 2008). The upsample_factor parameter controls the precision: a value of 10 gives 0.1-voxel accuracy.

References

Guizar-Sicairos M, Thurman S T, Fienup J R (2008). Efficient subpixel image registration algorithms. Optics Letters 33(2), 156–158.
Lewis J P (1995). Fast normalized cross-correlation. Vision Interface 10.