Hypersampling by Analytic Phase Projection

• The paper introduces a novel APP method to reconstruct high-temporal-resolution estimates from undersampled, pseudo-periodic data.
• It employs the Hilbert transform to create a continuous phase and aggregates measurements across cycles to boost effective sampling rates and reduce noise.
• Empirical validations in MRI and physiological recordings show up to a 10^3-fold resolution increase while preserving waveform fidelity.

Hypersampling by analytic phase projection (APP) is a methodology for reconstructing high-temporal-resolution estimates of pseudo-periodic signals from undersampled experimental data. APP achieves substantial increases in effective sampling rate by utilizing a well-resolved reference signal, projecting the target time series onto the phase of the reference, and aggregating measurements across multiple cycles. The method generalizes the classical retrospective gating approach, offering robust performance even in the presence of waveform variability and allowing for quantitative recovery of fast transient dynamics, such as those found in physiological recordings or time-resolved MRI studies (Voss, 2018).

1. Pseudo-Periodic Signal Framework and Phase Construction

A pseudo-periodic signal $r(t)$ exhibits approximate repetition with variations in period, amplitude, or waveform. Such signals are prevalent in physiological domains (e.g., cardiac and EEG signals). For phase-based analysis, APP requires a continuous phase construct, which is achieved by band-limiting $r(t)$ to obtain a monocomponent $r_M(t)$—a signal whose instantaneous frequency remains non-negative and whose phase is strictly increasing except at isolated resets.

The analytic signal is constructed via the Hilbert transform:

$$H\{r_M\}(t) = \frac{1}{\pi} \mathrm{P.V.} \int_{-\infty}^\infty \frac{r_M(\tau)}{t - \tau}\,d\tau \ ,$$

yielding the analytic form

$$a_r(t) = r_M(t) + j H\{r_M\}(t) = p(t) e^{j\varphi_r(t)} \ ,$$

where $p(t)$ is the instantaneous amplitude and $\varphi_r(t)$ is the analytic phase.

Key assumptions:

• $r_M(t)$ must be sampled at a rate satisfying the Nyquist criterion for its bandwidth.
• The band-limited reference must describe the same pseudo-periodic process as the target $x(t)$ across the observational period (Voss, 2018).

2. Analytic Phase Projection (APP) Algorithm

Given coarse or undersampled time series $x(t_i)$ and a well-sampled reference $r(t)$0, APP reconstructs a high-resolution template cycle by mapping each $r(t)$1 to the analytic phase $r(t)$2 and accumulating samples in phase bins:

1. Compute the analytic phase $r(t)$3 from $r(t)$4 using the Hilbert transform and extract $r(t)$5 at the $r(t)$6 sampling times by interpolation.
2. Create a uniform phase grid $r(t)$7 over $r(t)$8 and assign each $r(t)$9 to the nearest bin.
3. For each phase bin $r_M(t)$0:

$r_M(t)$1

where $r_M(t)$2 is the number of assignments to bin $r_M(t)$3.

1. (Optionally) Apply light smoothing in phase to $r_M(t)$4 to counter measurement scatter.

This process effectively projects all measurements onto a phase-canonicalized cycle, leveraging cycles-of-interest even with temporally sparse data (Voss, 2018).

3. Effective Sampling Rate and Temporal Super-Resolution

APP enhances the effective sampling interval over $r_M(t)$5 pseudo-cycles of average period $r_M(t)$6 by a factor proportional to the number of cycles and samples:

• The phase-collapsed estimate amasses $r_M(t)$7 total points, providing an effective time-step:

$r_M(t)$8

• In practice, with thousands of cycles (e.g., in long-duration fMRI with cardiac reference), APP can yield sub-millisecond temporal resolution from data originally sampled at seconds per frame:
  • For fMRI at TR = 2 s with $r_M(t)$91,000 cardiac cycles across 441 volumes: $$H\{r_M\}(t) = \frac{1}{\pi} \mathrm{P.V.} \int_{-\infty}^\infty \frac{r_M(\tau)}{t - \tau}\,d\tau \ ,$$0s$$H\{r_M\}(t) = \frac{1}{\pi} \mathrm{P.V.} \int_{-\infty}^\infty \frac{r_M(\tau)}{t - \tau}\,d\tau \ ,$$1 ms, corresponding to a $$H\{r_M\}(t) = \frac{1}{\pi} \mathrm{P.V.} \int_{-\infty}^\infty \frac{r_M(\tau)}{t - \tau}\,d\tau \ ,$$2-fold increase (Voss, 2018).

The relationship between desired phase-bin size $$H\{r_M\}(t) = \frac{1}{\pi} \mathrm{P.V.} \int_{-\infty}^\infty \frac{r_M(\tau)}{t - \tau}\,d\tau \ ,$$3 and temporal resolution is $$H\{r_M\}(t) = \frac{1}{\pi} \mathrm{P.V.} \int_{-\infty}^\infty \frac{r_M(\tau)}{t - \tau}\,d\tau \ ,$$4, where $$H\{r_M\}(t) = \frac{1}{\pi} \mathrm{P.V.} \int_{-\infty}^\infty \frac{r_M(\tau)}{t - \tau}\,d\tau \ ,$$5 is number of phase bins.

4. Comparison to Retrospective Gating and Validation

Retrospective gating segments cycles using discrete triggers (e.g., R-peaks), stacking cycles of fixed length and assuming linear phase evolution between triggers. In contrast, APP uses a continuous, possibly nonlinear phase, providing resilience to cycle-to-cycle variability and local time warping. Advantages validated by numerical comparison include:

• Temporal Resolution: Set by the total number of measurements, not the original recording interval.
• Noise Reduction: Averaging data from multiple cycles in phase reduces random noise by a factor of $$H\{r_M\}(t) = \frac{1}{\pi} \mathrm{P.V.} \int_{-\infty}^\infty \frac{r_M(\tau)}{t - \tau}\,d\tau \ ,$$6.
• Waveform Fidelity: APP accurately reconstructs waveforms in the presence of nonlinear phase or cycle distortion; retrospective gating is less effective under such variability.
• Empirical Results: MRI data processed with APP exhibited classic arterial pulse features with millisecond-scale resolution and physiologically consistent inter-voxel delays (Voss, 2018).

5. Extensions and Application Domains

The APP framework is modular and extensible to a variety of pseudo-periodic reference signals and measurement modalities:

• EEG-MRI Data Fusion: EEG rhythms (e.g., alpha, beta, delta) serve as reference for phase projection, enabling the localization of deep-brain sources in MRI if the associated frequencies are captured in the imaging.
• Optical-MRI Hybrids: Fast optical recordings provide the reference phase $$H\{r_M\}(t) = \frac{1}{\pi} \mathrm{P.V.} \int_{-\infty}^\infty \frac{r_M(\tau)}{t - \tau}\,d\tau \ ,$$7 to hypersample much slower volumetric MRI, revealing hemodynamic or metabolic transients.
• Parameter Recommendations:
  • Bin widths $$H\{r_M\}(t) = \frac{1}{\pi} \mathrm{P.V.} \int_{-\infty}^\infty \frac{r_M(\tau)}{t - \tau}\,d\tau \ ,$$8 in the range $$H\{r_M\}(t) = \frac{1}{\pi} \mathrm{P.V.} \int_{-\infty}^\infty \frac{r_M(\tau)}{t - \tau}\,d\tau \ ,$$9–$$a_r(t) = r_M(t) + j H\{r_M\}(t) = p(t) e^{j\varphi_r(t)} \ ,$$0 radians are typical for millisecond resolution.
  • Each phase bin should contain $$a_r(t) = r_M(t) + j H\{r_M\}(t) = p(t) e^{j\varphi_r(t)} \ ,$$15–10 samples to ensure robustness against noise.
  • Reference must be sampled per its bandwidth (e.g., up to $$a_r(t) = r_M(t) + j H\{r_M\}(t) = p(t) e^{j\varphi_r(t)} \ ,$$250 Hz for cardiac or EEG).

APP has demonstrated practical performance gains in standard fMRI workflows, opening avenues for leveraging pseudo-periodic structure in other slow imaging or sensing modalities (Voss, 2018).

6. Analytic Phase Projection and Exponential Signal Models

A parallel approach known as analytic phase projection for hypersampling (sometimes as APM) targets signals comprised of sums of exponentials:

$$a_r(t) = r_M(t) + j H\{r_M\}(t) = p(t) e^{j\varphi_r(t)} \ ,$$3

Here, deliberate sub-Nyquist sampling with stride $$a_r(t) = r_M(t) + j H\{r_M\}(t) = p(t) e^{j\varphi_r(t)} \ ,$$4 produces aliased spectra; ambiguity is resolved by a secondary, interleaved grid with relatively prime stride $$a_r(t) = r_M(t) + j H\{r_M\}(t) = p(t) e^{j\varphi_r(t)} \ ,$$5. The core steps:

• Form data matrices at the two strides, solve for eigenvalues (as in Prony’s, matrix pencil, or MUSIC methods), and use the Chinese Remainder Theorem (CRT) to uniquely unwrap frequencies.
• This collateral exploitation of aliasing reconditions ill-posed parametric estimation, notably when frequency clusters are otherwise inseparable.
• The method is a generic algorithmic wrapper for classical parametric solvers and is not itself a spectral estimator (Cuyt et al., 2017).

Results confirm that APM applied to tightly clustered or colliding exponentials achieves superior resolution compared to single-rate estimators for the same sample count, with robust handling of noise and model-order selection.

7. Practical Recommendations and Limitations

Parameter	Typical Range / Advice	Purpose
Bin width $$a_r(t) = r_M(t) + j H\{r_M\}(t) = p(t) e^{j\varphi_r(t)} \ ,$$6	$$a_r(t) = r_M(t) + j H\{r_M\}(t) = p(t) e^{j\varphi_r(t)} \ ,$$7–$$a_r(t) = r_M(t) + j H\{r_M\}(t) = p(t) e^{j\varphi_r(t)} \ ,$$8 rad	Millisecond timing features
Reference $$a_r(t) = r_M(t) + j H\{r_M\}(t) = p(t) e^{j\varphi_r(t)} \ ,$$9	Satisfy Nyquist for $p(t)$0	No loss of high-frequency info
Cycles $p(t)$1	As large as acquisition allows	Improves resolution, SNR

• Choose $p(t)$2 and $p(t)$3 (APM) such that clusters are well-separated but not so large as to re-introduce conditioning problems (Cuyt et al., 2017).
• Over-estimate model order and use rank diagnostics to guard against omission of weak modes.
• APP requires the availability of a well-sampled, monocomponent reference sharing the same cycle type as the target series.
• The method presumes that the underlying pseudo-periodic process does not change type (e.g., transition between cardiac and respiratory rhythm); abrupt reference cycle loss or change degrades performance.
• Large stride factors ($p(t)$4) may amplify numerical instability unless handled with stabilization techniques (SVD, regularization).

APP is not strictly limited to biomedical imaging contexts and has plausible extensions in any domain where pseudo-periodic, undersampled data are acquired in synchrony with a well-resolved reference cycle (Voss, 2018, Cuyt et al., 2017).