Velocity Spectrum Imaging: Principles & Applications

• Velocity Spectrum Imaging is a set of techniques that reconstruct full local velocity distributions instead of a single mean value, using Fourier encoding and model-based inversion.
• It integrates methods from MRI, ultrasound, and charged-particle imaging to resolve velocity-dependent information from indirect or sparse measurements.
• Applications range from resolving intravoxel flow patterns in medical imaging to probing astrophysical and radar velocity spectra for enhanced data interpretation.

Searching arXiv for papers directly relevant to “Velocity Spectrum Imaging” and closely related velocity-imaging uses of the term. Velocity Spectrum Imaging (VSI) denotes a family of imaging approaches in which the target of reconstruction is not merely a single mean velocity per spatial location, but a distribution of velocities, a velocity-resolved spectrum, or a velocity-dependent momentum-space observable associated with each measurement location or acquisition condition. In the strictest sense used in magnetic resonance imaging, VSI reconstructs an intra-voxel convective velocity distribution by treating velocity encoding as a Fourier-encoding dimension and inverting measurements acquired over multiple velocity-encoding moments (Hernandez-Garcia et al., 27 Aug 2025). More broadly, the term also encompasses methods that recover velocity-resolved observables from projected data, sparse temporal sampling, or spectroscopic measurements, including velocity-map imaging, spectral Doppler, and X-ray or reflection-matrix approaches that infer a velocity fluctuation spectrum or velocity-dependent momentum distribution rather than a single bulk value (Weger et al., 2013, Cohen et al., 2017, Zhuravleva et al., 2012). Across these domains, the common theme is that velocity is treated as a structured latent variable with internal distributional content rather than as a scalar summary.

1. Conceptual scope and definitions

Velocity Spectrum Imaging is most explicitly formulated in MRI as a method to “measure the velocity distribution of water inside each voxel of an MR image” (Hernandez-Garcia et al., 27 Aug 2025). In that usage, the objective is to estimate a voxelwise velocity density $p(v)$, motivated by the coexistence of multiple convective populations within one voxel, such as capillaries, CSF spaces, and perivascular spaces (Hernandez-Garcia et al., 27 Aug 2025). Conventional convective flow MRI generally reports an average voxel velocity; VSI instead aims to recover “what fractions of spins in this voxel are moving at which velocities” (Hernandez-Garcia et al., 27 Aug 2025).

A broader but technically coherent use of the term appears in methods that resolve velocity-dependent or momentum-dependent distributions from indirect observables. In velocity-map imaging, the relevant object is often a photoelectron or ion momentum distribution reconstructed from 2D projections, sometimes augmented by tomography or spectral conditioning (Weger et al., 2013, Wang et al., 2022). In spectral Doppler ultrasound, the objective is the blood velocity distribution over slow time, estimated as a Doppler power spectrum (Cohen et al., 2017). In X-ray cluster spectroscopy, the target is the 3D velocity fluctuation power spectrum of intracluster gas, inferred from line centroid shifts and line broadening (Zhuravleva et al., 2012, Eckert et al., 24 Oct 2025). In reflection-matrix ultrasound, a local focusing response is scanned over candidate wave speeds and converted into a speed-of-sound tomogram (Bureau et al., 2024). These are not identical inverse problems, but they share a common structure: velocity enters as a distributed degree of freedom encoded into measured data and recovered through model-based inversion.

This suggests a useful distinction between two regimes. In “strict VSI,” the unknown is an explicit per-voxel velocity distribution, as in MRI with velocity encoding preparation pulses (Hernandez-Garcia et al., 27 Aug 2025). In a broader “velocity-resolved imaging” sense, the unknown may be a velocity power spectrum, a blood-flow spectrogram, or a photon-energy-resolved momentum distribution, provided that the method reconstructs velocity-dependent structure rather than a single mean (Cohen et al., 2017, Wang et al., 2022, Eckert et al., 24 Oct 2025).

2. Fourier-encoded MRI formulation

The most direct formalization of VSI is the MRI method “Velocity Spectrum Imaging using velocity encoding preparation pulses” (Hernandez-Garcia et al., 27 Aug 2025). The method modifies velocity-selective RF pulse trains so that stationary spins are rephased while spins moving with constant velocity accumulate phase proportional to velocity. The accumulated phase is written as

$\phi = \gamma \int G(t)\, v(t)\, t\, dt \tag{1}$

and, under constant velocity during the pulse, becomes

$\phi = m_g v .$

With a $-90_y$ tip-up pulse, the longitudinal magnetization from a velocity component $v$ is

$$M_z(v) = M_{xy}(v)\cos(\phi) = M_{xy}(v)\cos(m_g v) \tag{2}$$

and after integrating over all velocity populations in the voxel,

$$M_z(m_g) = \int M_{xy}(v)\cos(m_g v)\, dv . \tag{3}$$

Using a $-90_x$ tip-up pulse provides the sine-encoded complement, so that combining cosine and sine encodings yields

$$M_z(m_g) = \int \left\{ M_{xy}(v)\cos(m_g v) + i\, M_{xy}(v)\sin(m_g v)\right\} dv \tag{4}$$

which is equivalent to a Fourier relationship between the acquired signal as a function of gradient first moment and the underlying velocity distribution (Hernandez-Garcia et al., 27 Aug 2025).

The paper explicitly interprets the first gradient moment as a “velocity k-space” coordinate. Sampling over multiple values of $m_g$ or $\phi = \gamma \int G(t)\, v(t)\, t\, dt \tag{1}$0 is therefore analogous to conventional spatial phase encoding, except that the encoded variable is velocity rather than position (Hernandez-Garcia et al., 27 Aug 2025). The corresponding velocity field-of-view and resolution are

$\phi = \gamma \int G(t)\, v(t)\, t\, dt \tag{1}$1

and

$\phi = \gamma \int G(t)\, v(t)\, t\, dt \tag{1}$2

Aliasing occurs when the true velocity distribution exceeds the sampled velocity bandwidth, exactly as in undersampled Fourier imaging (Hernandez-Garcia et al., 27 Aug 2025).

In the implementation reported in (Hernandez-Garcia et al., 27 Aug 2025), image reconstruction is first performed from spiral k-space using Total Variation regularized model-based conjugate-gradient SENSE, after which the per-voxel series across encoding moments is rephased, detrended with a third-order polynomial, windowed with a Hanning function, Fourier transformed, and normalized to estimate the spin-density fraction at each velocity. The result is a voxelwise 1D velocity spectrum along each encoded axis; in human studies, spectra were acquired independently along the right-left, anterior-posterior, and superior-inferior laboratory axes (Hernandez-Garcia et al., 27 Aug 2025).

The phantom experiments show the intended behavior. In multi-tube laminar flow, VSI spectra approximated the expected laminar distributions, and the highest-velocity tube exhibited wrap-around into negative velocity bins when the true velocity exceeded the sampled range (Hernandez-Garcia et al., 27 Aug 2025). In a loop phantom, opposite sides of the loop appeared in positive and negative velocity bins, demonstrating directional discrimination (Hernandez-Garcia et al., 27 Aug 2025). Human brain measurements produced voxelwise spectra with the largest spin fractions in the lowest velocity bins, including near zero, and revealed anatomical differences among white matter, gray matter, CSF, the sagittal sinus, and the cerebral aqueduct (Hernandez-Garcia et al., 27 Aug 2025).

The method is explicitly presented as a proof of concept rather than a fully mature quantitative tool. The paper states that diffusion is also encoded by the gradients and is not clearly separable in the current implementation; cardiac pulsatility and bulk motion violate ideal constant-velocity assumptions; scan times remain long; and the current implementation is not yet sensitive enough to resolve much slower perivascular flows associated with glymphatic hypotheses (Hernandez-Garcia et al., 27 Aug 2025).

3. Velocity-resolved momentum imaging and spectroscopy

A second major lineage of VSI arises in charged-particle imaging, especially velocity-map imaging (VMI). Here the measurement is a 2D projection of a 3D momentum distribution, and velocity-resolved information is recovered either by tomographic reconstruction, event-driven timing, or additional spectral conditioning.

The attoclock implementation of VMI in (Weger et al., 2013) is a canonical example of reconstructing a 3D velocity distribution from multiple 2D projections. The paper records many VMI images while rotating the polarization ellipse and reconstructs the 3D photoelectron momentum distribution by filtered back-projection: $\phi = \gamma \int G(t)\, v(t)\, t\, dt \tag{1}$3 with

$\phi = \gamma \int G(t)\, v(t)\, t\, dt \tag{1}$4

A Hann filter is applied to suppress artifacts, and the reconstructed distribution is then projected into the polarization plane for attoclock analysis (Weger et al., 2013). This is not VSI in the MRI sense of estimating an intra-voxel velocity density, but it is a velocity-spectrum imaging method in the sense that it reconstructs a full 3D momentum distribution from projection data.

The paper emphasizes that standard VMIS records only 2D projections and often not in the polarization plane of interest; tomography removes this limitation and makes VMI suitable for high-count-rate attoclock measurements (Weger et al., 2013). The practical advantages over COLTRIMS include higher tolerable ionization yield per pulse, higher momentum resolution, and shorter acquisition times, at the cost of losing event-by-event coincidence capability (Weger et al., 2013).

Time-stamped event-driven VMI extends this logic by adding a temporal dimension. “Single-shot MHz velocity-map-imaging using two Timepix3 cameras” (Bromberger et al., 2021) records per-hit tuples $\phi = \gamma \int G(t)\, v(t)\, t\, dt \tag{1}$5 from phosphor flashes generated by MCP detectors, enabling retrospective slicing and 3D ion velocity reconstruction. The transverse ion velocities are reconstructed as

$\phi = \gamma \int G(t)\, v(t)\, t\, dt \tag{1}$6

with $\phi = \gamma \int G(t)\, v(t)\, t\, dt \tag{1}$7, while the longitudinal component is recovered from arrival-time deviations in the extraction field $\phi = \gamma \int G(t)\, v(t)\, t\, dt \tag{1}$8 (Bromberger et al., 2021). This event-stream architecture provides full $\phi = \gamma \int G(t)\, v(t)\, t\, dt \tag{1}$9 datasets for every hit and allows arbitrary slices through reconstructed velocity distributions from a single acquisition (Bromberger et al., 2021).

Another extension resolves a spectral coordinate jointly with momentum. “Photon Energy-Resolved Velocity Map Imaging from Spectral Domain Ghost Imaging” (Wang et al., 2022) combines spectral-domain ghost imaging with Abel-type inversion in a single regression. The measured VMI image is modeled as

$\phi = m_g v .$0

where $\phi = m_g v .$1 is the shot-resolved FEL spectrum and $\phi = m_g v .$2 is the photon-energy-dependent 3D momentum distribution. Expanding $\phi = m_g v .$3 in pBasex basis functions and fitting all shots simultaneously yields a photon-energy-resolved kinetic-energy and binding-energy reconstruction (Wang et al., 2022). The method resolves the $\phi = m_g v .$4 spin-orbit splitting in argon despite a $\phi = m_g v .$5 eV average FEL bandwidth, recovering a binding-energy linewidth of $\phi = m_g v .$6 eV FWHM, far narrower than the bandwidth-limited conventional VMI result (Wang et al., 2022). This is a particularly clear instance of multidimensional velocity-spectrum imaging: the recovered observable is simultaneously momentum-resolved and photon-energy-resolved.

Instrumentation also matters. A 2025 paper on a UV VMI spectrometer redesign shows that scattered UV photons can generate severe electrode photoelectron background, and that suppressing this requires thin electrodes, blocked transport pathways, and optical baffles rather than relying primarily on window quality (Ladda et al., 20 Mar 2025). The result is background suppression exceeding $\phi = m_g v .$7 while preserving practical resolution, with measured photoelectron energy resolution

$\phi = m_g v .$8

at $\phi = m_g v .$9 eV (Ladda et al., 20 Mar 2025). This is not a new VSI inversion method, but it is directly relevant because velocity-spectrum imaging depends critically on preserving contrast and interpretability in weak momentum distributions.

4. Spectral Doppler and sparse slow-time sampling

In ultrasound, VSI often refers to spectral Doppler, where the desired quantity is the blood velocity distribution over time at a fixed spatial gate. “Sparse Doppler Sensing Based on Nested Arrays” (Cohen et al., 2017) recasts spectral Doppler as a second-order sparse sensing problem in slow time.

For a blood-filled resolution cell, the slow-time signal is modeled as

$-90_y$0

where $-90_y$1 are Doppler frequencies linked to axial velocities through

$-90_y$2

The goal is not reconstruction of the slow-time waveform itself but recovery of the power spectrum, i.e. the variances $-90_y$3 of the Doppler components (Cohen et al., 2017).

The key observation is that the covariance

$-90_y$4

can be vectorized as

$-90_y$5

so the available information is governed by the set of slow-time lags rather than by the directly sampled pulse indices (Cohen et al., 2017). A nested-array transmission scheme is introduced in which only $-90_y$6 Doppler emissions are sent, but their pairwise difference set spans the full lag range $-90_y$7 required for a length-$-90_y$8 Doppler aperture (Cohen et al., 2017).

For a perfect-square observation window $-90_y$9, the minimum number of emissions required for perfect noise-free recovery is

$v$0

For $v$1, this gives $v$2, i.e. about $v$3 of the full uniform Doppler transmissions (Cohen et al., 2017). Two recovery methods are proposed. NEST uses the filled covariance lag sequence and FFT-based reconstruction; NESPRIT reconstructs continuous Doppler frequencies via ESPRIT to avoid off-grid leakage (Cohen et al., 2017). The significance for duplex ultrasound is immediate: the saved slow-time emissions can be reassigned to B-mode while preserving spectral Doppler capability.

This suggests a broader VSI principle: velocity distributions need not be measured by dense direct sampling if the second-order structure preserves the full correlation aperture. That idea is domain-specific in its implementation but general in its logic.

5. Statistical velocity spectra in astrophysics and radar

A further branch of VSI reconstructs not local velocity distributions in pixels or voxels, but the spatial power spectrum of a velocity field from indirect observables.

In galaxy clusters, the line-of-sight bulk velocity and line width measured from X-ray emission lines can be related to the 3D velocity field. A foundational analysis shows that the projected velocity power spectrum and line-width profile are linked to the underlying 3D spectrum through emissivity-weighted projection kernels (Zhuravleva et al., 2012). The expected line-of-sight dispersion is

$v$4

while the projected centroid-map power spectrum is

$v$5

The paper argues that $v$6 acts as a practical proxy for the 3D velocity structure function evaluated near the effective line-of-sight depth $v$7 (Zhuravleva et al., 2012). This is a form of velocity-spectrum imaging in which the target is the 3D fluctuation spectrum rather than a direct velocity map.

A recent XRISM analysis of the Coma cluster turns this into a practical inverse pipeline (Eckert et al., 24 Oct 2025). The 3D spectrum is parameterized as

$v$8

with free parameters $v$9. Gaussian random velocity fields are generated in Fourier space,

$$M_z(v) = M_{xy}(v)\cos(\phi) = M_{xy}(v)\cos(m_g v) \tag{2}$$0

projected to emissivity-weighted bulk velocities and dispersions, convolved with the XRISM PSF, region-averaged, and compared to observed data via simulation-based inference (Eckert et al., 24 Oct 2025). Applied to Coma, the method infers a large injection scale

$$M_z(v) = M_{xy}(v)\cos(\phi) = M_{xy}(v)\cos(m_g v) \tag{2}$$1

and a Mach number within $$M_z(v) = M_{xy}(v)\cos(\phi) = M_{xy}(v)\cos(m_g v) \tag{2}$$2

$$M_z(v) = M_{xy}(v)\cos(\phi) = M_{xy}(v)\cos(m_g v) \tag{2}$$3

showing that the measured large bulk shifts and modest line widths imply substantial power on cluster-scale motions (Eckert et al., 24 Oct 2025). This is a clear example of statistical velocity-spectrum imaging from sparse projected spectroscopy.

A conceptually related but methodologically different case appears in coherent agile radar (Thornton, 18 Jun 2026). There the concern is not direct reconstruction of a velocity spectrum but the Fisher-information limit for radial velocity under pulse-to-pulse carrier and bandwidth agility. In the resolved-bin slow-time model, the effective velocity information is

$$M_z(v) = M_{xy}(v)\cos(\phi) = M_{xy}(v)\cos(m_g v) \tag{2}$$4

where $$M_z(v) = M_{xy}(v)\cos(\phi) = M_{xy}(v)\cos(m_g v) \tag{2}$$5, $$M_z(v) = M_{xy}(v)\cos(\phi) = M_{xy}(v)\cos(m_g v) \tag{2}$$6 is the carrier sequence, and the projection removes nuisance directions associated with range and phase (Thornton, 18 Jun 2026). The result shows that randomized or orthogonalized carrier hops are nearly harmless, whereas ramp-correlated hops can severely degrade velocity information (Thornton, 18 Jun 2026). While not an imaging algorithm, it is directly relevant to any velocity-spectrum imaging system built on coherent slow-time processing.

6. Related inverse problems that are not strict VSI

Several adjacent methods recover velocity information from imaging data without estimating a true per-location velocity spectrum.

A variational optical-flow velocimetry system computes dense per-pixel 2D velocity fields in real time on a GPU and explicitly shows that those fields support spectral analysis in a DNS turbulent benchmark (Pimienta et al., 30 Sep 2025). The optical-flow constraint is

$$M_z(v) = M_{xy}(v)\cos(\phi) = M_{xy}(v)\cos(m_g v) \tag{2}$$7

and the method yields one velocity vector per pixel at rates up to thousands of Hz depending on image size (Pimienta et al., 30 Sep 2025). The paper directly compares wavenumber spectra against DNS and finds that OFV tracks the DNS spectrum at large and intermediate scales and outperforms CC-PIV, with divergence beginning near $$M_z(v) = M_{xy}(v)\cos(\phi) = M_{xy}(v)\cos(m_g v) \tag{2}$$8 and effective cutoff near $$M_z(v) = M_{xy}(v)\cos(\phi) = M_{xy}(v)\cos(m_g v) \tag{2}$$9 (Pimienta et al., 30 Sep 2025). This is not VSI in the strict spectral-inversion sense, but it provides an acquisition front-end for spatial and temporal velocity spectrum estimation.

Ultrafast coded vector Doppler imaging similarly reconstructs a single in-plane vector velocity per pixel, not a velocity distribution, but then analyzes the temporal waveform to derive pulsatility and resistive index (Yan et al., 24 Apr 2025). The vector projection model is

$$M_z(m_g) = \int M_{xy}(v)\cos(m_g v)\, dv . \tag{3}$$0

solved by weighted least squares,

$$M_z(m_g) = \int M_{xy}(v)\cos(m_g v)\, dv . \tag{3}$$1

with iterative residual-based weight updates (Yan et al., 24 Apr 2025). The method supports spectrum-like temporal analysis over a cardiac cycle but does not reconstruct a full velocity distribution per pixel (Yan et al., 24 Apr 2025).

A joint inversion framework for velocity-encoded MRI reconstructs magnitude, phase, and segmentation from undersampled k-space and derives one effective velocity component per voxel from phase differences (Corona et al., 2019). Its forward model is

$$M_z(m_g) = \int M_{xy}(v)\cos(m_g v)\, dv . \tag{3}$$2

and velocity is recovered through

$$M_z(m_g) = \int M_{xy}(v)\cos(m_g v)\, dv . \tag{3}$$3

The paper is explicit that this is not VSI in the strict sense because it reconstructs a single effective velocity component per voxel, not an intravoxel velocity distribution (Corona et al., 2019).

These distinctions matter because the phrase “velocity spectrum imaging” can easily become ambiguous. Some methods estimate velocity fields that are later analyzed spectrally; some estimate statistical velocity power spectra; some estimate true local velocity distributions. The literature now contains all three.

7. Limitations, ambiguities, and methodological trade-offs

A recurring issue across the literature is that velocity-distribution inference is more ambitious than mean-velocity estimation and therefore more sensitive to model assumptions, sampling design, and nuisance effects.

In MRI VSI, the main limitations are diffusion contamination, pulsatility, long scan time, and finite velocity bandwidth. The paper explicitly notes that the current implementation is not yet able to resolve much slower perivascular velocities and that the practical in-vivo results should be treated as preliminary (Hernandez-Garcia et al., 27 Aug 2025). The method currently reconstructs 1D spectra along separate laboratory axes rather than a fully joint $$M_z(m_g) = \int M_{xy}(v)\cos(m_g v)\, dv . \tag{3}$$4 distribution (Hernandez-Garcia et al., 27 Aug 2025).

In charged-particle VMI, 3D momentum reconstruction usually requires cylindrical symmetry or tomography, and event-wise correlations are lost unless coincidence or covariance schemes are added (Weger et al., 2013, Rading et al., 2019). High-intensity XUV experiments benefit from covariance analysis, but covariance is statistical and demands careful synchronization and intensity-fluctuation correction (Rading et al., 2019).

In spectral Doppler, sparse slow-time methods preserve the full lag aperture in theory, but practical recovery still depends on stationarity, covariance estimation quality, and noise; off-grid leakage motivates higher-complexity continuous-frequency methods such as NESPRIT (Cohen et al., 2017).

In cluster X-ray work, the main challenge is that only projected emissivity-weighted moments of the line-of-sight velocity are observable. Recovering a 3D velocity fluctuation spectrum therefore depends strongly on assumptions of isotropy, Gaussianity, and parametric spectral form, with cosmic variance often dominating measurement noise (Zhuravleva et al., 2012, Eckert et al., 24 Oct 2025).

In reflection-matrix ultrasound, the local optimized speed first yields inverse depth-averaged slowness, not local speed directly; conversion to $$M_z(m_g) = \int M_{xy}(v)\cos(m_g v)\, dv . \tag{3}$$5 requires differentiation and currently neglects refraction in first approximation (Bureau et al., 2024). The authors explicitly identify this inversion stage as a current limitation (Bureau et al., 2024).

A broader conceptual ambiguity also persists: the same phrase can denote a true local velocity-density inversion, a statistical power-spectrum recovery, or a velocity-resolved spectrogram. A plausible implication is that the field benefits from keeping these meanings distinct. The MRI usage in (Hernandez-Garcia et al., 27 Aug 2025) is the narrowest and most literal. The broader literature shows that velocity-resolved imaging problems naturally organize around similar encoding and inverse principles even when the recovered object differs.

8. Significance and outlook

Velocity Spectrum Imaging is significant because it generalizes the measurement target from “how fast, on average, is material moving here?” to “what is the local or statistical distribution of velocities, and how is that distribution encoded in the data?” The value of that shift is clearest where multiple sub-resolution populations coexist, where projection obscures the latent velocity structure, or where weak anisotropies and correlations are otherwise washed out.

In MRI, VSI introduces a Fourier-encoded velocity dimension analogous to spatial k-space and demonstrates that noninvasive voxelwise velocity spectra are experimentally accessible, at least in proof-of-principle form (Hernandez-Garcia et al., 27 Aug 2025). In charged-particle imaging, tomographic VMI, event-driven readout, and spectral-domain ghost imaging show how momentum distributions can be reconstructed jointly with angular, temporal, or photon-energy resolution (Weger et al., 2013, Bromberger et al., 2021, Wang et al., 2022). In ultrasound, nested-array spectral Doppler shows that blood velocity spectra can be recovered from dramatically fewer Doppler emissions by exploiting the difference-set structure of covariance lags (Cohen et al., 2017). In astrophysical spectroscopy, sparse projected velocity measurements can already constrain 3D fluctuation spectra when embedded in realistic forward models (Eckert et al., 24 Oct 2025).

The current frontier is likely to be shaped by three converging trends. First, richer acquisition schemes are turning velocity into an explicit encoding dimension rather than a nuisance parameter. Second, forward models increasingly include the full measurement chain—projection, PSF, covariance structure, and noise—rather than relying on approximate summary formulas. Third, simulation-based and high-dimensional inverse methods are making it practical to infer velocity-resolved latent structure from data that would previously have supported only mean-value estimates.

Taken together, these developments suggest that VSI is best understood not as a single technique but as a unifying inverse-problem paradigm: velocity is encoded indirectly, often through phase, projection, or covariance; the observable is higher-dimensional than a scalar mean; and the reconstruction seeks a distribution, spectrum, or velocity-resolved field whose structure is physically more informative than average flow alone (Hernandez-Garcia et al., 27 Aug 2025, Cohen et al., 2017, Wang et al., 2022, Eckert et al., 24 Oct 2025).