14th International Symposium on Particle Image Velocimetry – ISPIV 2021
August 1–5, 2021

Development and Uncertainty Characterization of
Rotating 3D Velocimetry using a Single Plenoptic

Camera
M. Moaven, A. Gururaj, Z. P. Tan‡, S. Morris, B. S. Thurow 1∗, V. Raghav 2∗

Auburn University, Department of Aerospace Engineering, Auburn, AL, USA
1∗ thurow@auburn.edu
2∗ raghav@auburn.edu

Abstract
Rotating 3D velocimetry (R3DV) is a single-camera PIV technique designed to track the evolution of flow
over a rotor in the rotating reference frame. A high-speed (stationary) plenoptic camera capable of 3D
imaging captures the motion of particles within the volume of interest through a revolving mirror from the
central hub of a hydrodynamic rotor facility, a by-product being an undesired image rotation. R3DV employs
a calibration method adapted for rotation such that during MART reconstruction, voxels are mapped to pixel
coordinates based on the mirror’s instantaneous azimuthal position. Interpolation of calibration polynomial
coefficients using a fitted Fourier series is performed to bypass the need to physically calibrate volumes
corresponding to each fine azimuth angle. Reprojection error associated with calibration is calculated on
average to be less than 0.6 of a pixel. Experimental uncertainty of cross-correlated 3D/3C vector fields is
quantified by comparing vectors obtained from imaging quiescent flow via a rotating mirror to an idealized
model based purely on rotational kinematics. The uncertainty shows no dependency on azimuth angle while
amounting to approximately less than 0.21 voxels per timestep in the in-plane directions and correspondingly
1.7 voxels in the radial direction, both comparable to previously established uncertainty estimations for
single-camera plenoptic PIV.

1 Motivation
There exist countless unsteady fluid dynamics problems involving rotation - an action tending to give rise
to highly 3D flow features; however, diagnostics of such flows have traditionally been constrained to phase-
averaged PIV and time-resolved experiments within a stationary frame of reference (Mulleners et al. (2012);
Percin and Oudheusden (2015)). Recording the three-dimensional evolution of rotating flows through time
is paramount to the success of aerodynamicists in understanding and predicting flows such as that around a
helicopter rotor.

Rotating 3D velocimetry (R3DV) is a technique capable of continuously tracking and reconstructing
time-resolved 3D flow over a rotor within the rotating frame of reference. Images are captured via a 45◦
inclined mirror that is rotating in conjunction with the wing, thereby transporting the rotating field of view
containing the flow as it evolves above the wing through a 90◦ turn and towards a stationary camera (see
Fig. 1). Herein lies the cornerstone of R3DV - a high-speed plenoptic camera capable of 3D imaging. The
feature that sets a plenoptic camera apart from a conventional camera is an array of microlenses placed
between the camera sensor and main lens. This simple modification enables incoming light rays to be
encoded based on their angle of incidence with the main lens. As a result, plenoptic images can be decoded
to resolve volumetric information (Fahringer et al. (2015)) and whilst 3D flow-field measurements have
traditionally relied on multi-camera configurations, the plenoptic camera can be the sole imaging device
in a 3D-PIV experiment. This provides a crucial advantage in an optically restricted environment such as
that of the current R3DV implementation. For a more comprehensive discussion of the R3DV technique
as well as details of the experimental configuration, the authors would like to point the reader to ”Rotating
three-dimensional velocimetry” by Gururaj et al. (2021).

‡Currently at: National Yang Ming Chiao Tung University



Figure 1: R3DV setup schematic including isosurfaces of vorticity (not to scale)

The purpose of this paper is to address unique challenges that arise from imaging with the configuration
used in R3DV, namely the development of a rotational calibration method and quantification of uncertainty
with a potential dependency on rotation. It can be deduced based on intuition that imaging with a stationary
camera via a rotating mirror results in a continuously rotating image. Fig. 2 illustrates this effect by showing
a calibration plate mounted at the position of the wing starting from a position (Fig. 2(a)) and undergoing
an approximately 40◦ azimuthal rotation (ψ) (Fig. 2(b)), resulting in the camera viewing the plate as having
been rotated by the same angle ψ. Ideally, the rotation should be rectified prior to cross-correlation such that
imaged flow particles are not seen as experiencing a false rotation. A simple 2D rotation matrix on plenoptic
images would be inappropriate in addressing this as a raw image contains thousands of microlens sub-images
meaning any such operation would impose an artificial rotation on the microlens array. As such, an existing
volumetric calibration method was adapted to a rotating field of view in order to seamlessly incorporate
rotation into camera calibration such that Fig. 2(b) is transformed to Fig. 2(c) to match the orientation of
Fig. 2(a). Section 3 discusses the details of this method as well as results from an error analysis including
a comparison to plenoptic calibration in a stationary frame benchtop setup. In PIV experiments designed
to examine the growth and formation of a leading-edge vortex (LEV) around a 45◦ pitch wing, R3DV was
shown to give qualitatively agreeable results with existing literature. Of course, the viability of a novel PIV
implementation cannot be assumed based solely on qualitative observation. As such, images of quiescent
flow were recorded via the rotating mirror without the presence of a wing - the impetus being to compare
resulting vector fields of the perceived rotating flow to a model based on rotational kinematics in order to
quantify the uncertainty of the method. With the imaged volume rotating through a wide range of angles, a
potential dependency of uncertainty on azimuth angle must simultaneously be investigated. Results of the
uncertainty analysis and a comparison to single camera plenoptic-PIV in the stationary frame are presented
in Section 4.

2 Rotating 3D Velocimetry

2.1 Technique
A typical tomographic or plenoptic-PIV experiment begins with recording images of a volume of illuminated
particles seeding the flow of interest (Fig. 3). A fundamental step specific to processing of plenoptic images
involves determining the projected microlens positions on the image sensor. Subsequent steps for both tomo-
PIV and plenoptic-PIV are based on the same principles of calibrating the volume with respect to the camera,



(a) (b)

(c)

Figure 2: Rotated field of view (b) resulting from mirror rotation of angle ψ rectified using rotational cali-
bration to generate image (c) with feature point coordinates consistent with that of image (a)

using said calibration to reconstruct the volume of interest, and finally cross-correlating consecutive images
to yield vector fields of particle displacements. As alluded to in the previous section, the unique nature
of R3DV’s rotating field of view is addressed within the volumetric calibration step. Here, a rotational
calibration technique establishes a relationship between the image and the volume while taking into account
its orientation. Applying this calibration during volumetric reconstruction generates an intensity field that
is dependent on the azimuth angle of the mirror at that instant meaning any rotation of the field of view as
a consequence of looking through the rotating mirror is eliminated. As a visual aid for the aforementioned
quiescent flow experiments, this results in the camera adopting the perspective of the mirror such that particle
images theoretically undergo only translation.

2.2 Setup
The configuration shown in Fig. 1 is an inverted schematic of the arrangement used in this work, with
the mirror and wing assembly contained in a 1.2 m × 1.2 m × 1.2 m acrylic water tank and an encoder
incorporated to track the mirror’s azimuth angle. The plenoptic camera assembly is composed of a 4096 ×
2304 pixel resolution Phantom VEO 4k 990L camera and a Modular Plenoptic Adaptor (Tan et al. (2019))
that houses a 471 × 362 microlens array, an optical relay system, as well as a 200 mm main imaging lens.
The calibration plate was held in place by a mounting wing consisting of evenly spaced holes to position
the plate at 10.16 mm increments in the depth direction. During uncertainty measurement acquisition,
water in the tank was seeded with fluorescent PMMA rhodamine particles at a density of approximately
0.04 particles per microlens (ppm). Volume illumination was provided by a LaVision LED-Flashlight 300
positioned normal to one of the tank’s sides and a 52 mm Tiffen Orange 21 lens filter was used to enhance
the signal-to-noise ratio of particle images.

2.3 Data Acquisition and Processing Parameters
The calibration plate was imaged at 9 consecutive positions on the mounting wing, covering a depth of
81.28 mm. The wingless experiments of section 4 designed for uncertainty analysis were performed for two
volumes corresponding to two different locations on the wing used in the flow visualization experiments



Image acquisition of laser
illuminated particle volume

Calibrate microlenses: find
all microlens center pro-
jections on image sensor

Volumetric calibration: establish
image to object space relationship

Volumetric reconstruction: gener-
ate 3D volume of particle positions

Cross-correlation of particle
motion: generate 3D/3C vector field

Figure 3: General roadmap of tomographic and conventional plenoptic-PIV (left, green) versus additional
step exclusive to plenoptic-PIV (right, red) (detailed tomographic-PIV workflow in Elsinga et al. (2006))

of Gururaj et al. (2021), across the same range of azimuth angles. Volume 1 denotes the volume placing
the focal plane near the tip of the wing while volume 2 was centered slightly inboard. Experiments for
each volume were repeated with three different rotation rates (approximately 1, 3, and 5 RPM) at an image
acquisition rate of 200 fps. The theoretical field of view at the focal plane was calculated to be 68 × 38.2
mm and 60.7 × 34 mm for volume 1 and volume 2, respectively. Seven iterations of a plenoptic-adapted
MART algorithm were used to reconstruct volumes at a grid resolution of 8 vox/mm. The cross-correlation
algorithm used here is based on a multi-pass 3D (VODIM) scheme. Cubic window sizes of [64, 48, 40, 40]
voxels in each dimension with overlaps of 75% were applied to generate vector fields of size 52 × 28 × 76
for volume 1 and 44 × 23 × 76 for volume 2, with vector spacings of 1.25 mm.

3 Rotational Volumetric Calibration
This section details the calibration technique developed for application in a rotating volume – the necessity
of which arises when a field of view is relayed via a rotating mirror to a stationary camera thus causing the
image to exhibit in-plane rotation (Fig. 2).

In a conventional (stationary) field of view, our group employs a technique known as direct light-field
calibration (DLFC) to relate a plenoptic image to object space (Hall et al. (2018)). Here, images of a
calibration pattern at known depths relative to the focal plane are captured, covering a total depth that
typically encloses the volume of interest. These images are then decoded into perspective views with each
perspective representing a discrete location (u,v) on the aperture plane. A generic third-order polynomial is
used as a mapping function to pair every location on the microlens plane (s, t) to its corresponding location
in object space (x,y,z) via each perspective:

s = P(x,y,z,u,v) = as0 +as1x+as2y+as3z+as4u+as5v+as6x
2 +as7xy+ · · · +as55v

3

t = P(x,y,z,u,v) = at0 +at1x+at2y+at3z+at4u+at5v+at6x
2 +at7xy+ · · · +at55v

3 (1)

Polynomial variables are stored in a matrix A where each row corresponds to a calibration feature point at
a unique depth and perspective view. Subsequently, the coefficients of Eq. 1 are calculated by method of
least-squares, as = A/s and at = A/t, where s and t represent vectors of s and t positions. The mapping
function is later employed during reconstruction of 3D volumes from plenoptic images.

For application to a rotating field of view, a method was devised whereby the coordinate system of the
calibration plate was rotated with its azimuth angle (ψ). As the azimuthal rotation is proportional to the



orientation of the field of view, assigned object-space coordinates of the calibration plate’s feature points
remain constant throughout its angular sweep. For example, Fig. 2(b) is imaged at an azimuth angle of
ψ≈ 40◦ after Fig. 2(a), therefore, the coordinate set (x,y) is rotated by ψ to become (x′,y′). This procedure
results in the calibration polynomial being a function of azimuth angle:

(s, t) = P(x′,y′,z,u,v,ψ) (2)

Calibration of the dot card in 2(b) with rotational DLFC results in 2(c) whereby the dot card appears at the
same position as 2(a) despite the azimuthal rotation increasing by ψ≈ 40◦.

A fixed volume would only require one calibration file whereas a continuously rotating volume with a
fine enough angular resolution would require hundreds. The impracticality of physically calibrating vol-
umes at every unique azimuth angle is avoided by exploiting the relationship between the mapping function
coefficients and the rotation of the field of view. It was found by imaging a calibration plate through a full
revolution that, conveniently, this is a periodic relation meaning that as the mirror rotates and thus the field
of view rotates, the coefficients of the mapping function vary sinusoidally, as seen in Fig. 4 which shows
three examples of coefficients plotted at every 10◦ for 360◦. Therefore, in the rotational implementation
of DLFC, calibration images are acquired for volumes at relatively broad increments of azimuth angle and
mapping functions are generated for any desired in-between angles using interpolation.

Figure 4: s-Coefficient values through a full revolution of calibration plate rotation (Gururaj et al. (2021))

A Fourier series was chosen as the most suitable representation of a periodic best fit curve and thus is
used here to aid interpolation. Each mapping function coefficient was initially represented by a four-term
Fourier series through its rotation:

(as,at) = f (ψ) = b0 +
n

∑
i=1

bicos(iwψ)+ cisin(iwψ) (3)

where w is a measure of the periodicity of a mapping function coefficient, n refers to the selected number
of harmonics, and b and c are cosine and sine coefficients, respectively. If the four-term series was deemed
to be overfitting the coefficients, the number of terms in the Fourier series was dropped to three and subse-
quently two if overfitting persisted. Goodness of fit was determined using a combination of the coefficient
of determination (R2) and a comparison of global absolute maxima between the Fourier curve and known
coefficients. In either case, a user-defined threshold was set, the violation of which resulted in sequentially
dropping the number of harmonics. For the rotating experiments presented here, images of the calibration
plate were captured across a depth of 81.28 mm at approximately every 10◦ increment for a total sweep of
90◦ while coefficients were interpolated for each intermediate angle at which flow images were acquired
according to encoder outputs. For example, in the case of the lowest rotation rate wingless experiment,



982 mapping functions were generated with each corresponding to a different azimuth angle using just 10
physically calibrated volumes.

In summary, the rotational DLFC technique involves acquiring images of a calibration plate at discrete
yet widely spaced azimuth angles across multiple depths throughout an angle sweep that encompasses the
relevant range of rotation of the wing. A mapping function is generated at each of these angles by applying
a generic third-order polynomial to the image and object space coordinates of known calibration points via
discretized perspective views. Coefficients for any angle within the swept range of the calibration plate are
interpolated for using a fitted Fourier series. This mapping function is then employed during reconstruction
meaning the calibrated volume at each azimuth angle is mapped to a common set of in-plane coordinates
(x′,y′) thus eliminating the effects of the rotating field of view when tracing particles into the volume.

A reprojection error analysis was performed for the physically calibrated volumes in R3DV. As a mea-
sure of comparison, the same camera configuration was used to calibrate a stationary volume on a benchtop
setup. In the latter, a 300 mm Thorlabs motorized translation stage was used to provide positioning of
the calibration plate with a precision of 0.5 µm. Reprojection error was calculated by applying the corre-
sponding mapping function to known calibration plate feature points in object space (x,y,z) to solve for
(srepro j, trepro j), the reprojected image space position. This was compared to (s, t), considered the true image
space location, for each feature point. Resulting pixel values were then scaled with respect to the plenoptic
camera’s perspective view resolution. In the rotational case, no trend in the reprojection error was observed
with respect to varying azimuth angles. Averaged across all 10 azimuth angles, the errors in rotational
DLFC applied with the R3DV setup as well as the benchtop conditions are shown in Table 1. For R3DV, an
increase in reprojection error of ∼0.15 occurs in the s-direction while the t-axis holds a ∼0.3 increase. Pos-
sible factors contributing to this rise include needing to image through a much thicker acrylic wall using the
R3DV setup relative to the benchtop as well as imperfections in R3DV’s 3D-printed calibration mounting
wing. The error in rotational calibration did not show a consistent dependency on ψ. Whilst the comparison
suggests the rotational calibration configuration has potential for improvements, subpixel accuracy is still
obtained and, as section 4 will show, the overall measurement uncertainty of the technique is on par with
stationary plenoptic-PIV. Therefore, the higher error of R3DV’s calibration is considered acceptable albeit a
task for future work to reduce.

Table 1: Reprojection errors of R3DV’s rotational DLFC versus conventional DLFC on benchtop setup

4 Uncertainty Analysis
In order to quantify uncertainty of the velocity vector fields, images of quiescent flow were acquired via the
mirror undergoing steady rotation. Cross-correlation was performed on these images to generate vectors of
velocities that represent the displacement in particle position due solely to imaging via the rotating mirror.
The initial vectors output in the camera’s rectangular coordinate system were transformed to a cylindrical
frame (Fig. 5) for convenience as follows:

ut = ucosθ+wsinθ (4)

uv = v (5)

ur = wcosθ−usinθ (6)

R′ =
R

cosθ
(7)

θ = tan−1(
x− x0

R
) (8)



where u, v, and w are velocities in the horizontal, vertical, and depth directions of the raw cross-correlated
vectors, respectively. Similarly, ut , uv, and ur are velocities in the tangential, vertical, and radial directions,
respectively, in relation to the cylindrical path of rotation.

Figure 5: Schematic of coordinate transformation from Cartesian (left) to cylindrical (right)

In quantifying uncertainty, vector fields were compared to a model described by rotational kinematics:

ut = ΩrotR = Ωrot(r−R0) (9)

uv = 0 (10)

ur = 0 (11)

where Ωrot is the rotation rate and r is measured relative to R0, the radial distance from the center of rotation
to the calibrated center plane. While the location of the in-plane center of rotation, x0, is not known from
physical parameters, it can be calculated using rotating images of the calibration board as follows. When
assigning object space positions to feature points during DLFC, (0, 0) is chosen arbitrarily as long as it
remains consistent to the same feature throughout every calibration image. The objective now becomes to
find the distance of the center of rotation to the calibrated (0, 0). This can be achieved by tracking the
position of a feature point through a series of images following the rotation of the calibration plate. A least-
squares algorithm is then used to fit a circle to the positions of the rotating dot in order to find its center of
rotation (Pratt (1987)). Finally, the known image magnification is applied to back out x0, the object-space
distance of the in-plane center of rotation to the calibrated center position. Ωrot and R0 are estimated through
a Levenberg-Marquardt least-squares minimization of velocity vectors after substituting Eq. (4) into Eq. (9).

Comparison of the model (Fig. 6(a)) to the time-average of the steady-state portion of wingless motion
(Fig. 6(b)) was determined to give the systematic error in assuming purely rotational motion, ∆u. Note the
velocity close to the edges of Fig. 6(b) drops towards 0 due to this portion of the volume being outside
the field of view across most azimuth angles. On average, an error less than 1% of the tangential velocity
occurred in the tangential direction, equivalent to less than 0.1 voxels per timestep. The model also assumed
zero velocity in the vertical and radial directions. Here, the comparison yielded ∆uv and ∆ur less than
0.15 and 0.6 voxels per frame, respectively. Voxel error values far below average particle displacements
in the direction of motion confirmed the PIV measurements obtained using this experimental setup could
be accurately modeled by pure rotation. It was noted that the in-plane systematic error increased further
from the center of the vector field. As an example, Fig. 7 plots the spatial variation of ∆ut normalized with
respect to ut for a portion of the center xy-plane of the RPM ≈ 1, volume 2 experimental configuration. No
consistent relationship was found with respect to error across the depth direction.

After establishing that the system could be described by the rotational model, quantification of measure-
ment uncertainty, δu′, was achieved by generating a model for the vector field at each azimuth angle, u′model ,
and comparing velocities from the cross-correlated vectors (Fig. 6(c)) to their respective ideal values. The
standard deviation of residual velocities from this comparison was taken as a value representative of mea-
surement uncertainty. An experimental setup consisting of different imaging conditions at each azimuth



(a) (b) (c)

Figure 6: Example of an instantaneous R3DV vector field’s velocity distribution (right) vs time-averaged
vector field (center) and a corresponding idealized model (left) calculated based on rotational kinematics

angle begs the question as to whether uncertainty possesses a dependency on azimuth angle. Fig. 8 plots δu′t
for a rotation of approximately 60◦ for volume 2 with the mirror rotating at approximately 5 RPM. Clearly,
the uncertainty fluctuates randomly around 0.15. This randomness with azimuth angle was seen in all six
test cases. Similarly, the δu′r and δu′v exhibit no relationship with azimuth angle. Accordingly, uncertainty
was averaged across all azimuth angles for each volume and each rotation rate to yield a single value for
the configurations shown in Table 2. Here, it is apparent the in-plane uncertainty is consistently higher for
the farther volume 1. It was hypothesized this was caused by a slight increase in particle density across the
two sets of experiments. Interestingly, volume 2 generally possesses a higher radial uncertainty. There also
exists a slight trend of increasing tangential uncertainty with higher rotation rates, although in proportion
to the mean tangential velocity it actually decreases. Due to the limited angular range of perspectives a
plenoptic camera can obtain, particle elongation is always expected to cause reduced depth precision rela-
tive to that of the in-plane directions in a single-camera plenoptic experiment and this is certainly observed
in Table 2. Overall, the authors assert the uncertainty for the current implementation of R3DV is less than
0.21 voxels per timestep for the in-plane components and better than 1.7 voxels per timestep in the radial
direction. These values are comparable to those obtained by Fahringer et al. who determined the uncertainty
of single-camera plenoptic-PIV in a stationary reference frame to be accurate to within 0.2 voxels and 1
voxel in the lateral and depth directions, respectively (Fahringer et al. (2015)).

5 Conclusions
The R3DV technique possesses unique challenges associated with imaging through a rotating mirror using
a stationary camera, with one such challenge being an undesired rotation of the images recorded by the
camera. The authors propose a rotational calibration method whereby the azimuth angle of the mirror is
incorporated in the calibration function to rectify the rotating field of view to a common axis. Despite
being higher than plenoptic calibration in a benchtop configuration, rotational DLFC’s reprojection errors
of less than 0.6 pixels were deemed acceptable for R3DV. A focus for future work will involve reducing
this error by improving the physical design of the calibration plate mounting system. Quantification of
uncertainty was achieved by comparing a model of rotation to vector fields obtained from quiescent flow
viewed through a rotating mirror. The error in assuming the motion in the setup adheres purely to laws
of rotational kinematics was first assessed and concluded to be insignificant. Comparison of instantaneous
vector fields to their respective ideal models yielded average uncertainties better than 0.21 and 1.7 voxels per



Figure 7: Spatial variation of systematic error in ut normalized with respect to the model ut across an xy-
plane of volume 2, RPM ≈ 1

Figure 8: Variation of uncertainty in tangential velocity across a 60◦ azimuthal rotation

timestep in the in-plane and out-of-plane directions, respectively, each with no reliance on azimuth angle. As
these values proved comparable to those found in previous work on plenoptic-PIV uncertainty estimation,
the current implementation of R3DV is seen to be a success. Optimization of cross-correlation parameters is
currently being explored with the aim of reducing uncertainty. Possibilities of future developments include
implementing particle tracking velocimetry via the plenoptic ray-bundling triangulation method developed
by Clifford et al. (2019) as well as investigating the fluid-structure interaction behavior of a rotor during the
LEV life cycle.



Table 2: Average measurement uncertainty across six wingless R3DV experiments

Acknowledgements
This research was sponsored by the Army Research Office and was accomplished under Grant Number
W911NF-19-1-0052 and W911NF-19-1-0124 (DURIP) monitored by Dr. Matthew Munson. The views and
conclusions contained in this document are those of the authors and should not be interpreted as representing
the official policies, either expressed or implied, of the Army Research Office or the U.S. Government. The
U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstand-
ing any copyright notation herein. The authors would also like to acknowledge the members of Advanced
Flow Diagnostics Laboratory, Applied Fluid Research Group, Dr. Chris Clifford, Dr. Eldon Triggs and Mr.
Andy Weldon in the Department of Aerospace Engineering at Auburn University.

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	Motivation
	Rotating 3D Velocimetry
	Technique
	Setup
	Data Acquisition and Processing Parameters

	Rotational Volumetric Calibration
	Uncertainty Analysis
	Conclusions