Temporal Signal Synchronization

Spatial registration answers “where is this anatomy?” Temporal synchronization answers “when did this event occur?” A dynamic MR sequence, PET time-activity curve, physiological trace, or gated reconstruction can be spatially aligned while its samples still refer to different moments.

RITK estimates one constant delay between two equally sampled finite signals. It does not resample unequal acquisition rates, model a delay that changes over time, or perform dynamic time warping. Those are different contracts.

Lag convention

Let (x_i) be the reference signal and (y_i) the moving signal. At integer lag (k), RITK evaluates only the index pairs that exist in both arrays:

[ r(k)= \frac{ \sum_i (x_i-\bar{x}k)(y{i+k}-\bar{y}_k) }{ \sqrt{ \sum_i(x_i-\bar{x}k)^2 \sum_i(y{i+k}-\bar{y}_k)^2 } }. ]

The overlap-specific means prevent a shorter edge overlap from being compared against full-signal means. Pearson normalization also makes the estimated lag invariant to a positive scale and offset applied to either signal.

A positive (k) means the moving signal is delayed. To align it, sample the moving signal at (i+k). The reported shift in seconds is

[ \Delta t = k,T_\mathrm{frame}. ]

This sign convention is visible in the worked example: the unaligned orange curve lags the reference, and the aligned curve is evaluated at reference_index + shift_frames.

From integer peak to fractional estimate

The integer lag with maximum normalized correlation is the discrete estimate. When correlations exist on both sides of an interior maximum, RITK fits a three-point parabola:

[ \delta = \frac{r(k-1)-r(k+1)} {2[r(k-1)-2r(k)+r(k+1)]}, \qquad \hat{k}=k+\delta. ]

The offset is accepted only for a finite concave peak and is bounded to the neighboring interval. A peak at the configured search boundary remains integer-valued. Parabolic refinement is an estimator; digitized correlation curves can bias it. Céspedes et al. compare parabolic and cosine fits and describe this limitation in Methods for Estimation of Subsample Time Delays of Digitized Echo Signals.

Correlation ties are deterministic: RITK chooses the smallest absolute lag, then the smaller signed lag. This prevents scheduling or iterator order from changing a result.

Validated configuration and typed failures

use leto::Array1;
use ritk_registration::{
    TemporalSync, TemporalSyncConfig, TemporalSyncStatus,
};

let reference = Array1::from_vec(
    [8],
    vec![0.0, 0.2, 0.9, 0.4, -0.3, -0.8, -0.2, 0.3],
)?;
let moving = Array1::from_vec(
    [8],
    vec![0.0, 0.0, 0.2, 0.9, 0.4, -0.3, -0.8, -0.2],
)?;

let config = TemporalSyncConfig::try_new(
    0.04, // seconds per sample
    3,    // search ±3 frames
    0.8,  // acceptance threshold
)?;
let result = TemporalSync::with_config(config)
    .synchronize(&reference, &moving)?;

assert!(result.shift_frames() > 0.0);
assert_eq!(result.status(), TemporalSyncStatus::Accepted);
Ok::<(), Box<dyn std::error::Error>>(())

TemporalSyncConfig::try_new rejects non-finite or non-positive frame spacing, a zero search range, and a minimum correlation outside [0, 1]. Synchronization then rejects:

• unequal signal lengths;
• fewer than three samples;
• the first NaN or infinite sample, with signal identity and index; and
• a zero-variance signal whose lag is not identifiable.

Two identical constant signals are not “perfectly synchronized.” They contain no timing feature, so every lag explains them equally well. RITK reports TemporalSyncError::UnidentifiableSignal instead of inventing a zero shift.

Acceptance is not numerical success

minimum_correlation classifies a measured estimate:

• TemporalSyncStatus::Accepted, or
• TemporalSyncStatus::BelowMinimumCorrelation.

A below-threshold result still exposes its shift and correlation. This lets a quality-control pipeline display or record the observation without confusing weak evidence with a numerical failure.

The correlation coefficient can be negative. The acceptance threshold is non-negative because a negatively correlated peak is not evidence that the signals share the same polarity and timing.

Residual metrics and units

After estimating (\hat{k}), RITK linearly interpolates (y) at (i+\hat{k}). Only coordinates inside the moving signal contribute:

[ \mathrm{RMS} = \sqrt{\frac{1}{N_\mathrm{overlap}} \sum_{i\in\mathrm{overlap}} [x_i-\tilde{y}(i+\hat{k})]^2}. ]

residual_rms and residual_max_abs are in signal amplitude units, not seconds. shift_seconds is the timing quantity. overlap_samples states the exact RMS denominator, so a large shift cannot make the error look smaller by dividing by samples that were never compared.

Search allocation and diagnostic profiles

synchronize scans lags once and retains only the current best peak and its neighbors. Search scratch is constant-size; it does not allocate full lag and correlation arrays. Call correlation_profile only when a graph or diagnostic record needs every lag:

let profile = synchronizer.correlation_profile(&reference, &moving)?;
for sample in &profile {
    if let Some(correlation) = sample.correlation() {
        println!("lag={} r={correlation}", sample.lag_frames());
    }
}
Ok::<(), ritk_registration::TemporalSyncError>(())

A profile sample is None when that particular overlap is locally constant. The normal peak selector ignores it. Full-signal zero variance remains a typed input error.

The direct method performs (O(NS)) work for (N) samples and search radius (S). It is appropriate when the physically plausible delay window is small. The committed temporal_sync benchmark covers 4,096 samples and a ±64-frame search without changing the workload between implementations.

Method reference

Xiao, Ding, and Hu describe lagged cross-correlation and maximum-lag selection for multimodal physiological signals in Section 2.3, Equation 1, and Algorithm 1. RITK pins its overlap, normalization, tie, sign, interpolation, and failure conventions above because a method name alone does not define those choices.