X-ray rheography uncovers planar granular flows despite non-planar walls

Extremely useful techniques exist to observe the interior of deforming opaque materials, but these methods either require that the sample is replaced with a model material or that the motion is stopped intermittently. For example, X-ray computed tomography cannot measure the continuous flow of materials due to the significant scanning time required for density reconstruction. Here we resolve this technological gap with an alternative X-ray method that does not require such tomographs. Instead our approach uses correlation analysis of successive high-speed radiographs from just three directions to directly reconstruct three-dimensional velocities. When demonstrated on a steady granular system, we discover a compressible flow field that has planar streamlines despite curved confining boundaries, in surprising contrast to Newtonian fluids. More generally, our new X-ray technique can be applied using synchronous source/detector pairs to investigate transient phenomena in various soft matter such as biological tissues, geomaterials and foams.

ρ b the bulk density. We consider an idealised system where the sample is composed of either solid glass particles, each of the same uniform material density, or interstitial air. Since the X-ray attenuation coefficient in air is several orders of magnitude lower than in glass, these contributions are negligible compared to the solid phase. If we also assume a parallel X-ray beam with no scattering then the expression (1) can be simplified to where µ is the constant attenuation coefficient and D(x) represents the integrated thickness of solid material that a ray at in-plane position x travels through.
Using expression (2), initial radiographs are generated by assuming that 4000 spherical par- The units in (3) are assumed to be pixels per time step, with the normalised coordinates y, z ∈ [0, 1] and u 0 a constant magnitude. Such a flow field bears superficial resemblance to steady uniform chute flow 2 , where motion is primarily in the downslope direction and is independent of the xcoordinate. The domain is taken to be periodic in the x direction.
Deconvolution process errors. The first step in the velocity reconstruction process is obtaining the velocity probability density functions (PDFs) in each interrogation window. This requires splitting the radiographs into discrete windows and computing the auto-correlation (A) and crosscorrelation (C) functions. Note that, due to the one-dimensional displacement field, the definitions are altered slightly and become for successive images I 1 and I 2 . Specifically, a two-dimensional patch is still used but we are only seeking one-dimensional displacements, meaning the computed A and C are already onedimensional functions. These correlation functions are then averaged over many time steps and are used to directly compute the one-dimensional PDFs by solving the deconvolution inverse problem, given by equations (4)-(6) in the main text. Finally, these deconvolutions are averaged over the x-direction to give a single PDF for each distinct z position (for side-on view radiographs) or y position (for top-down view radiographs).
To calculate the errors introduced during this deconvolution process, the analytical veloc- magnitude, u 0 in equation (3), is displayed on Supplementary Figure 2b. Here we see that larger displacements lead to less accurate deconvolutions, because it becomes more difficult to correlate individual particles between images. Nevertheless, in all cases the errors remain less than 10% and therefore within acceptable bounds.
Discretisation process errors. The next stage in the reconstruction process is the discretisation of the velocity PDFs into 'candidate' vectors by splitting into equally-spaced percentiles. Intuitively, the errors between these candidate vectors and the exact underlying velocity field must tend to zero as the discretisation gets successively finer. This is because the PDFs can be thought of as the large Matching process errors. The final stage involves taking the two sets of candidate arrays, from the side view and top view, and using them to reconstruct the full internal field in a single (y, z) slice. We refer to this as a Sudoku-style problem due to the parallels with such puzzles. The approach is based around minimising the 'matching error' between the two sets of observations, given by equations (8) and (9)  fine enough. The relative errors for other flow regimes and parameters may differ, but these results should at least provide useful guidelines.
The next question to be addressed is whether the accumulation of the different errors has a detrimental effect on the final reconstruction. To this effect, the whole process has been followed from artificial radiograph generation to matching of candidate arrays, and Supplementary Figure   6 shows the final results. It can be seen that the comparison to the analytical velocity is generally very good, with mean errors being calculated as less than 15%. There does, however, appear to be some smearing of velocity gradients in the reconstructed flow field, which may be a result of the enforced regularisation in the deconvolution process, as well as the path-averaging approach.