Supplemental information
Reducing weighting matrix
The size of weighting matrix is important in the determination of reconstruction time
and computing memory. When the number of voxel and pixel nodes are l×m×n (Nvox) and
o×p (Npix), the size of weighting matrix is 3(Npix×projection number)×Nvox. The weighting
matrix is really sparse, and a lot of zero components exist in the matrix. To remove the zero
components for reducing the size of weighting matrix, the matrix is divided into many slices
along z direction and the velocity in each slice is independently reconstructed. The vertical
regions (r ranges) affected by the velocity information in each slice of the voxel are
determined by the reconstructed voxel size, pixel size, and Dso. Then, the distance Dso can be
used to make a single line in the projected plane correspond to each slice of the voxel. In this
study, the weighting matrix can be reduced to 3(o×projection number)×(l×m). Using this
reducing matrix, the reconstruction time is decreased, and the computing memory is
effectively used. For example, if the camera has 1024 × 1024 pixels resolution and the size of
interrogation window is 32 × 32 pixels with 50 % overlapping in the PIV analysis, the pixel
nodes are 64 × 64. In addition, the voxel nodes and the number of projections are determined
as 64 × 64 × 64 and 7, respectively. Then, the size of the reduced matrix (1344 × 262144,
about 3.5 × 108) is much smaller than that of the original matrix (86016 × 262144, about 2.3
× 1010).
The LSMR method and SMART algorithm
The 3D velocity information is reconstructed using the least squares minimum residue
(LSMR) method and simultaneous multiplicative algebraic reconstruction technique
(SMART) algorithm. The LSMR method utilizes a conjugate-gradient type algorithm to solve
sparse linear equations and sparse least-square problems based on the Golub–Kahan process.
It computes a solution x for the problem of Ax = b. Here, A is a rectangular matrix of
dimension m × n, and B is a vector of length m. If the system is inconsistent, it solves the
least-square problem min ∥b - Ax∥2. In our system, the A and b correspond to the weighting
matrix W and the measured velocity vector P, respectively. In the Golub-Kahan process, the
solution is determined when backward error estimate (the Frobenius norm of residual vector,
∥rk∥ = ∥b - Axk∥) is smaller than the quantity depending on ATOL and BTOL which are
involved in the practical stopping criteria of the computing process. The stopping rules of the
computing for obtaining the solution are as follows:
S1: Stop if ∥rk∥ ≤ BTOL∥b∥ + ATOL∥A∥∥xk∥
S2: Stop if ∥ATrk∥ ≤ ATOL∥A∥∥rk∥
If Ax = b ise consistent, S1 is applied. Otherwise, the rule S2 is applied.
The accuracy of the LSMR method depends on the dimensionless quantities ATOL
and BTOL. In order to reduce the effect of those quantities, the SMART algorithm is applied
after finding the LSMR computation. Conventional tomographic PIV techniques commonly
use the SMART algorithm. The solution is determined based on the product of the ratio of the
recorded velocity to the projected velocity at each pixel node for each iteration k:
V jk 1

Pi
k
 V j  
k

i   n WinVn

Ni




Wij




1 / Ni
where Ni is the total number of pixel nodes that observe a given voxel j. The relaxation
parameter μ is typically chosen in the range of 0–2.
Synthetic particle images and RMS error estimation
Three different flow images were simulated. The first image was in response to a
circular pipe flow with only one velocity component. The two others were axisymmetric and
asymmetric flows similar with those used by Dubsky et al. The 3D displacement of the
axisymmetric flow between two consecutive images is expressed by the following:
x  c1 ( y  x 2 y  y 3 )
y  c2 ( x  xy 2  x 3 )
z  c3 (1  x 2  y 2 )
and the 3D displacement of the asymmetric flow is similarly expressed by:
x  c1 (1  x 2  y 2 )
y  c2 (1  x 2  y 2 )
z  c3 (2 x 3  x 5  xy 2  y 2 x 3  x)
where c1, c2, and c3 are 5, 5, and 15 pixels in both cases. The reconstruction accuracy was
estimated by the root mean square (RMS) error, obtained by the following formula
RMS 
 v
3n
 vexact 
2
recon
3n
where vrecon and vexact are the reconstructed and exact velocity at each voxel node. The digit n
denotes the total number of voxel nodes.
Scan angle and iteration number effect on the reconstruction accuracy
The effects of the scan angle and iteration number of SMART method were
investigated by simulation study. The variation in RMS error in relation to the scan angle for
the asymmetric flow case is shown in Fig. S1(a). When the scan angle was larger than 30°,
the results were acceptable qualitatively and quantitatively. Therefore, only the results for
scan angles larger than 30° are depicted in Fig. S1(a). No peculiar trend in RMS error is
shown. The minimum RMS errors for 11 and for 16 projections were obtained at the scan
angles of 60° and 45°, respectively. The corresponding optimal angle increments are 6° and 3°
for 11 and 16 projections. The effect of iteration number of SMART method on the RMS
error method for asymmetric flow is illustrated in Fig. S1(b). The RMS error monotonically
decreased as the iteration number increased. The RMS error remained constant when the
iteration number exceeded 50.
Image pre-processing
Figure S2(a) shows a typical X-ray image captured without any objects. Given the
honeycomb structure of the fiber optics plate (FOP) attached in front of the CCD camera, the
corresponding unwanted image pattern was observed. In addition, X-ray illumination was
uneven because of the anode heel effect. Therefore, flat field correction (FFC) pre-processing
was adopted to improve the quality of the captured X-ray images in this study. The FFC is
expressed as follows:
FFC image 
object image  offset image
 scale factor
gain image  offset image
where the gain image is the image captured without test samples under the same experimental
conditions, and the offset image is that taken without X-ray illumination. Compared with the
raw image [Fig. S2(b)], the contrast of the tracer particles was much higher [Fig. S2(c)]. This
implies that FFC pre-processing significantly improves the image contrast. The projected
velocity field was obtained after FFC image correction. Thereafter, the 3D velocity fields
were reconstructed from the projected velocity field information.
List of Figures
Figure S1. (a) Variations in RMS error with respect to scan angles at 11 projections (red
circle) and 16 projections (black triangle) for the asymmetric flow case (iteration k = 50). (b)
Effect of iteration number on 11 projections with a 60° scan angle (red circle) and on 16
projections with a 45° scan angle (black triangle) for asymmetric flow.
(a)
(b)
Figure S2. Typical X-ray images (a) without test samples, (b) raw image, and (c) preprocessed by FFC.
(a)
(b)
(c)