Rotating-Frame Acquisition: Methods & Applications

• Rotating-frame acquisition is a measurement strategy where sensing is performed in a rotating reference frame, integrating rotation directly into the forward model.
• It employs configurations such as fixed rotating platforms, synchronized sensor motion, and pulsed spin-lock drives to achieve coordinated data capture.
• Joint inversion techniques and precise synchronization enable accurate reconstruction of physical parameters in applications like fluid dynamics, photoacoustic imaging, and prethermal NMR.

Taken together, the cited literature uses rotating-frame acquisition for measurement configurations in which sensing or readout is carried out in a co-rotating mechanical frame or in an effective rotating spin frame, rather than being treated as a purely static laboratory-frame measurement. This suggests a family of acquisition strategies in which rotation is embedded directly into the forward model, the sensor geometry, the synchronization protocol, or the dynamical observable itself. In the fluid experiment of "Free surface of a liquid in a rotating frame with time-dependent velocity" (Monteiro et al., 2019), the rotating frame is implemented by mounting both the container and the smartphone on a rotating table; in "Photo-acoustic tomography in a rotating measurement setting" (Bal et al., 2016), both the optical source and the ultrasonic transducer array are mounted on a rigid rotating frame; and in "Prethermal rotating-frame solid echo in a dipolar nuclear-spin network" (Reynard-Feytis et al., 22 Jun 2026), cycle-resolved NMR readout is performed in the rotating frame established by a pulsed spin-lock Floquet drive.

1. Rotating-frame acquisition as a measurement architecture

In the mechanically rotating fluid experiment, an LG-G2 smartphone is rigidly fixed to the table beside a narrow prismatic tank, so that the camera and gyroscope measure the evolving free surface and the angular velocity in the same rotating frame. The apparatus is explicitly self-contained: the phone records video of the liquid-air interface through its rear camera, angular velocity $\Omega(t)$ via its built-in three-axis gyroscope logged with the “AndroSensor” app, and a proximity sensor signal used for synchronization (Monteiro et al., 2019).

In the rotating-measurement PAT formulation, the acquisition geometry is defined by a rigid frame rotating about the origin. For each angle $\theta \in [0,2\pi)$, the sensor positions are given by

$y^j(\theta)=R(\theta)\,y_0^j,$

with the boundary illumination rotated simultaneously according to

$$f_\theta(x)=f_0\bigl(R(-\theta)x\bigr), \qquad x\in \partial\Omega .$$

A consequence stated explicitly in the formulation is that one no longer has the option to reconstruct a map of sonic emission corresponding to a given optical illumination, so the optical and acoustic stages must be combined into a single inverse problem (Bal et al., 2016).

In the prethermal NMR setting, the rotating frame is not produced by mechanical motion but by a periodic train of spin-locking pulses. After optical hyperpolarization and a conventional $(\pi/2)_x$ tipping pulse, the system is driven by a Floquet sequence of $\vartheta_x$ pulses interleaved with short acquisition windows, and readout windows sample the transverse precession in the rotating frame, reconstructing the cycle-resolved quadratures

$$S_x(nT_{FC})=\langle I_x\rangle,\qquad S_y(nT_{FC})=\langle I_y\rangle .$$

Because $T_{FC}$ is much shorter than the typical dipolar timescale $(J^{-1}\simeq 1.5\,\mathrm{ms})$, the pulsed train realizes a Floquet-averaged Hamiltonian and quasi-continuous readout in the rotating frame without repeated re-initialization (Reynard-Feytis et al., 22 Jun 2026).

These realizations show that rotating-frame acquisition is not tied to a single discipline. The common feature is that rotation changes what is directly measured and therefore changes the appropriate forward model.

2. Continuum-mechanical formulation in a co-rotating frame

For the liquid free-surface problem, the theoretical model is written directly in the rotating frame of angular velocity $\Omega(t)$ about the vertical $\theta \in [0,2\pi)$0-axis. In steady rigid-body rotation, the velocity and centripetal acceleration are

$\theta \in [0,2\pi)$1

The stationary momentum balance in the non-inertial frame is

$\theta \in [0,2\pi)$2

Integration yields the pressure field

$\theta \in [0,2\pi)$3

where $\theta \in [0,2\pi)$4 is fixed by the total volume (Monteiro et al., 2019).

At the free surface, $\theta \in [0,2\pi)$5, so the interface becomes parabolic:

$\theta \in [0,2\pi)$6

Equivalently, relative to the mid-plane,

$\theta \in [0,2\pi)$7

The vertex height at $\theta \in [0,2\pi)$8 is

$\theta \in [0,2\pi)$9

The assumptions are explicit: inviscid, incompressible fluid in rigid-body rotation, transients neglected; a narrow rectangular prism so end effects are minimal; atmospheric pressure at the free surface; and conservation of volume (Monteiro et al., 2019).

The PAT rotating-frame formulation is likewise encoded at the level of the forward model. The diffusion problem for the fluence $y^j(\theta)=R(\theta)\,y_0^j,$0 is

$y^j(\theta)=R(\theta)\,y_0^j,$1

and the initial pressure field is

$y^j(\theta)=R(\theta)\,y_0^j,$2

The acoustic propagation is then governed by

$y^j(\theta)=R(\theta)\,y_0^j,$3

with initial data $y^j(\theta)=R(\theta)\,y_0^j,$4 and $y^j(\theta)=R(\theta)\,y_0^j,$5. Measurements are defined directly on the rotated detector positions through

$y^j(\theta)=R(\theta)\,y_0^j,$6

Here, rotating-frame acquisition is not merely a geometric convenience; it is part of the operator $y^j(\theta)=R(\theta)\,y_0^j,$7 itself (Bal et al., 2016).

In the NMR case, the corresponding rotating-frame structure appears in the effective Floquet Hamiltonian. To leading order in the Magnus expansion, the periodic $y^j(\theta)=R(\theta)\,y_0^j,$8-pulse train leaves an effective secular dipolar Hamiltonian aligned along $y^j(\theta)=R(\theta)\,y_0^j,$9,

$$f_\theta(x)=f_0\bigl(R(-\theta)x\bigr), \qquad x\in \partial\Omega .$$0

or equivalently

$$f_\theta(x)=f_0\bigl(R(-\theta)x\bigr), \qquad x\in \partial\Omega .$$1

where $$f_\theta(x)=f_0\bigl(R(-\theta)x\bigr), \qquad x\in \partial\Omega .$$2. Under $$f_\theta(x)=f_0\bigl(R(-\theta)x\bigr), \qquad x\in \partial\Omega .$$3, the total transverse magnetization $$f_\theta(x)=f_0\bigl(R(-\theta)x\bigr), \qquad x\in \partial\Omega .$$4 is approximately conserved, leading to a prethermal plateau (Reynard-Feytis et al., 22 Jun 2026).

3. Instrumentation, synchronization, and parameter extraction

The fluid experiment provides an explicit example of rotating-frame data acquisition in which synchronization is itself part of the measurement design. The acquisition protocol begins by starting video and sensor logging with a few seconds at zero rotation. A “lens-cover” event, achieved by covering camera and proximity sensor simultaneously, produces a dark frame and a proximity drop and therefore marks a common time fiducial $$f_\theta(x)=f_0\bigl(R(-\theta)x\bigr), \qquad x\in \partial\Omega .$$5 for the video and gyroscope streams. The gyroscope is factory-calibrated, with sampling rate $$f_\theta(x)=f_0\bigl(R(-\theta)x\bigr), \qquad x\in \partial\Omega .$$6 and resolution $$f_\theta(x)=f_0\bigl(R(-\theta)x\bigr), \qquad x\in \partial\Omega .$$7, and the camera orientation is fixed so that one camera axis is horizontal and aligned with $$f_\theta(x)=f_0\bigl(R(-\theta)x\bigr), \qquad x\in \partial\Omega .$$8 (Monteiro et al., 2019).

The same experiment also makes the extraction procedure explicit. Individual frames are extracted at selected times $$f_\theta(x)=f_0\bigl(R(-\theta)x\bigr), \qquad x\in \partial\Omega .$$9, with $(\pi/2)_x$0 frames covering the range of $(\pi/2)_x$1. In each frame, Tracker software is used to pick $(\pi/2)_x$2 points along the liquid-air interface, and these points are fitted to

$(\pi/2)_x$3

where $(\pi/2)_x$4 is the horizontal pixel coordinate and $(\pi/2)_x$5 is vertical. The concavity $(\pi/2)_x$6 is proportional to $(\pi/2)_x$7, and the vertex height in pixel units is

$(\pi/2)_x$8

A calibration ruler in the field of view provides the pixel-to-centimeter scale in horizontal and vertical directions, and each chosen frame at $(\pi/2)_x$9 is associated with $\vartheta_x$0 after bias subtraction and conversion to $\vartheta_x$1 (Monteiro et al., 2019).

The uncertainty budget is also resolved operationally. The sources listed are gyroscope resolution $\vartheta_x$2, frame extraction timing $\vartheta_x$3, pixel resolution $\vartheta_x$4, and parabolic-fit statistical error in $\vartheta_x$5 and $\vartheta_x$6. For $\vartheta_x$7 versus $\vartheta_x$8, the slope uncertainty comes from linear regression; for vertex height $\vartheta_x$9 versus $$S_x(nT_{FC})=\langle I_x\rangle,\qquad S_y(nT_{FC})=\langle I_y\rangle .$$0, the uncertainty combines fitting and pixel scale (Monteiro et al., 2019).

In the PAT setting, synchronization enters differently: the acquisition is indexed by angle, time, and sensor, and discretization is carried out as $$S_x(nT_{FC})=\langle I_x\rangle,\qquad S_y(nT_{FC})=\langle I_y\rangle .$$1, $$S_x(nT_{FC})=\langle I_x\rangle,\qquad S_y(nT_{FC})=\langle I_y\rangle .$$2, with rotated sensor positions generally off-grid. The implementation therefore uses bilinear interpolation in $$S_x(nT_{FC})=\langle I_x\rangle,\qquad S_y(nT_{FC})=\langle I_y\rangle .$$3D or trilinear interpolation in $$S_x(nT_{FC})=\langle I_x\rangle,\qquad S_y(nT_{FC})=\langle I_y\rangle .$$4D to evaluate $$S_x(nT_{FC})=\langle I_x\rangle,\qquad S_y(nT_{FC})=\langle I_y\rangle .$$5 at $$S_x(nT_{FC})=\langle I_x\rangle,\qquad S_y(nT_{FC})=\langle I_y\rangle .$$6, and in the adjoint pass the residuals are scattered back to the grid via the same interpolation stencil under the inverse rotation $$S_x(nT_{FC})=\langle I_x\rangle,\qquad S_y(nT_{FC})=\langle I_y\rangle .$$7. This makes interpolation part of the acquisition operator rather than a post hoc visualization step (Bal et al., 2016).

In the NMR experiment, acquisition is cycle resolved by construction. The pulsed spin-lock not only generates the rotating frame but also leaves short acquisition windows interleaved with the drive, allowing continuous interrogation of the same prethermal manifold. A practical consequence stated explicitly is that a single experimental shot can map $$S_x(nT_{FC})=\langle I_x\rangle,\qquad S_y(nT_{FC})=\langle I_y\rangle .$$8 up to many tens of milliseconds, avoiding the need for repeated tip-and-acquire cycles (Reynard-Feytis et al., 22 Jun 2026).

4. Inverse problems and reconstruction under rotation

In rotating-measurement PAT, the principal methodological consequence of the acquisition geometry is that the conventional two-step pipeline breaks down. The reconstruction target is the absorption coefficient $$S_x(nT_{FC})=\langle I_x\rangle,\qquad S_y(nT_{FC})=\langle I_y\rangle .$$9, recovered by minimizing

$T_{FC}$0

with either Tikhonov regularization,

$T_{FC}$1

or Total Variation,

$T_{FC}$2

The formal stationarity condition is

$T_{FC}$3

This is the analytic statement that acquisition and inversion must be handled jointly (Bal et al., 2016).

The adjoint is derived by introducing the residual

$T_{FC}$4

and the time-reversed adjoint field $T_{FC}$5 satisfying the backward wave equation with terminal data driven by the residual injected at the rotated sensors. The resulting adjoint operator is

$T_{FC}$6

with the volume-preserving property of the rotation used explicitly through the Jacobian equal to $T_{FC}$7. A steepest-descent update then reads

$T_{FC}$8

The forward operator, the adjoint, and the optimization are therefore all rotation dependent (Bal et al., 2016).

The numerical implementation described for a typical $T_{FC}$9D phantom on $(J^{-1}\simeq 1.5\,\mathrm{ms})$0 with a $(J^{-1}\simeq 1.5\,\mathrm{ms})$1 mesh reports resolution $(J^{-1}\simeq 1.5\,\mathrm{ms})$2, convergence of the residual $(J^{-1}\simeq 1.5\,\mathrm{ms})$3 to $(J^{-1}\simeq 1.5\,\mathrm{ms})$4 in $(J^{-1}\simeq 1.5\,\mathrm{ms})$5–$(J^{-1}\simeq 1.5\,\mathrm{ms})$6 iterations for TV regularization with $(J^{-1}\simeq 1.5\,\mathrm{ms})$7, stable recovery up to $(J^{-1}\simeq 1.5\,\mathrm{ms})$8 additive Gaussian noise, and per-iteration cost consisting of one forward diffusion solve, $(J^{-1}\simeq 1.5\,\mathrm{ms})$9 forward acoustic solves, and $\Omega(t)$0 adjoint acoustic solves, totaling $\Omega(t)$1 per iteration on a single GPU when $\Omega(t)$2 (Bal et al., 2016).

The fluid experiment uses a much simpler inference problem, but it still exemplifies model-based parameter estimation under rotation. The experimentally extracted parabola concavity and vertex height are each regressed against $\Omega(t)$3, producing a direct test of the rotating-frame model. The reported plot of $\Omega(t)$4 against $\Omega(t)$5 is linear, with measured slope $\Omega(t)$6 and theoretical $\Omega(t)$7, corresponding to agreement within $\Omega(t)$8. The plot of $\Omega(t)$9 versus $\theta \in [0,2\pi)$00 is also linear, with measured slope $\theta \in [0,2\pi)$01 and theoretical $\theta \in [0,2\pi)$02; the intercept at $\theta \in [0,2\pi)$03 gives $\theta \in [0,2\pi)$04, in agreement with direct measurement $\theta \in [0,2\pi)$05 (Monteiro et al., 2019).

5. Rotating-frame readout in prethermal NMR

The rotating-frame NMR experiment uses a pulsed spin-lock Floquet drive to stabilize a long-lived prethermal manifold of transverse $\theta \in [0,2\pi)$06 magnetization in diamond, and then probes that manifold by cycle-resolved inductive readout. If the prethermal $\theta \in [0,2\pi)$07-magnetization is tipped into the $\theta \in [0,2\pi)$08–$\theta \in [0,2\pi)$09 plane with a $\theta \in [0,2\pi)$10 pulse at $\theta \in [0,2\pi)$11 and the Floquet drive is continued, the rotating-frame free-induction signal is described by

$\theta \in [0,2\pi)$12

with $\theta \in [0,2\pi)$13 and fitted decay time

$\theta \in [0,2\pi)$14

By contrast,

$\theta \in [0,2\pi)$15

up to the prethermal heating time

$\theta \in [0,2\pi)$16

The rotating frame is therefore both the acquisition frame and the dynamical frame in which approximate conservation becomes visible (Reynard-Feytis et al., 22 Jun 2026).

Within this same driven manifold, a single echo pulse $\theta \in [0,2\pi)$17 applied after a delay $\theta \in [0,2\pi)$18 produces a rotating-frame solid echo. After evolution for $\theta \in [0,2\pi)$19, the pulse $\theta \in [0,2\pi)$20 is applied; after a further evolution of duration $\theta \in [0,2\pi)$21 under the same pulsed spin-lock, a revival is observed at

$\theta \in [0,2\pi)$22

The minimal two-spin picture attributes this to micromotion-induced transfer between operator subspaces. For a single pair with $\theta \in [0,2\pi)$23, one has stroboscopically

$\theta \in [0,2\pi)$24

The $\theta \in [0,2\pi)$25 pulse rotates these manifolds differently: the $\theta \in [0,2\pi)$26 subspace picks up a factor $\theta \in [0,2\pi)$27 and can be inverted, whereas the $\theta \in [0,2\pi)$28 subspace is either left unchanged or mixed into non-refocusable operators. Maximum contrast occurs near $\theta \in [0,2\pi)$29, with a small experimental offset $\theta \in [0,2\pi)$30 attributed to pulse-imperfection-induced phase shifts (Reynard-Feytis et al., 22 Jun 2026).

For an ensemble of dipolar-coupled spins, the echo amplitude is approximated as

$\theta \in [0,2\pi)$31

where $\theta \in [0,2\pi)$32 depends on details of micromotion mixing. The many-body echo envelope is fitted to

$\theta \in [0,2\pi)$33

Exact simulations of $\theta \in [0,2\pi)$34 random spin pairs with $\theta \in [0,2\pi)$35 uniformly in $\theta \in [0,2\pi)$36 under the same pulse train reproduce a strong, symmetric revival at $\theta \in [0,2\pi)$37 for ideal $\theta \in [0,2\pi)$38 pulses and a reduced echo amplitude for $\theta \in [0,2\pi)$39 errors (Reynard-Feytis et al., 22 Jun 2026).

The practical acquisition implications are explicit. NV-center optical pumping yields $\theta \in [0,2\pi)$40 polarizations $\theta \in [0,2\pi)$41, giving cycle-resolved inductive signals $\theta \in [0,2\pi)$42 in a single scan. The condition $\theta \in [0,2\pi)$43 ensures $\theta \in [0,2\pi)$44, prethermalization is robust, and the prethermal plateau persists for a duration stated to be four orders of magnitude longer than the conventional lab-frame $\theta \in [0,2\pi)$45 (Reynard-Feytis et al., 22 Jun 2026).

6. Comparative structure, limitations, and recurring misconceptions

The three cases describe distinct but structurally related forms of rotating-frame acquisition.

Domain	Rotating-frame mechanism	Acquired quantity
Fluid free surface	Smartphone fixed to rotating table	Video of interface and gyroscope $\theta \in [0,2\pi)$46
Photo-acoustic tomography	Optical source and US array on rigid rotating frame	$\theta \in [0,2\pi)$47
Prethermal NMR	Pulsed spin-lock Floquet drive	Cycle-resolved $\theta \in [0,2\pi)$48, $\theta \in [0,2\pi)$49

A first recurring misconception is that rotation can always be handled as a coordinate change applied after acquisition. The PAT formulation explicitly rules this out: because the light source rotates at the same time as the ultrasound measurements are acquired, one no longer has the option to reconstruct a map of sonic emission corresponding to a given optical illumination, and the two steps are combined into one (Bal et al., 2016).

A second misconception is that rotating-frame readout necessarily implies complete reversal of dephasing. The NMR result is more specific: analytical arguments and toy-model simulations attribute the revival to Floquet micromotion that transfers coherences between operator subspaces, so that only a subset of the many-body dephasing dynamics is inverted by the $\theta \in [0,2\pi)$50 pulse (Reynard-Feytis et al., 22 Jun 2026).

A third misconception is that co-rotating acquisition automatically yields a fully dynamical fluid model. In the liquid-surface experiment, the theoretical comparison is carried out under the assumptions of inviscid, incompressible fluid in rigid-body rotation with transients neglected. The experiment varies angular speed in small manual steps and compares selected frames with the steady parabolic prediction, so the success of the comparison validates that specific model and data-processing pipeline rather than an unrestricted time-dependent hydrodynamic description (Monteiro et al., 2019).

These limitations are integral to the topic. Rotating-frame acquisition is not a single universal algorithm; it is a measurement strategy whose meaning depends on whether rotation enters through sensor mounting, illumination geometry, effective Hamiltonian engineering, or all three simultaneously.