Shift–Average Upsampling Procedure

• Shift–average upsampling procedure is a mathematically grounded method that reconstructs undersampled signals using phase projection and local averaging.
• It is implemented in pseudo-periodic signal processing, shift-invariant subspaces, and neural networks, achieving improved effective resolution and exact shift equivariance.
• The approach is applied in biomedical imaging and deep learning, yielding substantial improvements in temporal resolution and reduced reconstruction errors.

The shift–average upsampling procedure encompasses a family of mathematically grounded techniques for reconstructing or upsampling signals—typically in the contexts of pseudo-periodic time series, shift-invariant function spaces, and shift-equivariant deep networks—by combining phase-based alignment, local averaging, and explicit inversion of structured downsampling operations. These procedures enable substantial improvement of effective sampling rate, exact recovery guarantees under structural assumptions, or perfect shift equivariance in neural architectures, and have direct implications in biomedical signal analysis and deep learning.

1. Mathematical Foundations and Core Algorithms

The shift–average upsampling procedure arises in several forms. One prominent instantiation is hypersampling by analytic phase projection, especially suited for pseudo-periodic signals that are undersampled in the original acquisition but have access to a reference signal that is sufficiently oversampled. Let $x(t)$ denote the undersampled signal of interest (e.g., MRI time series) and $r(t)$ a simultaneously acquired, well-sampled reference (e.g., pulse oximetry or EEG) (Voss, 2018). The procedure:

• Constructs a monocomponent reference $r_M(t)$ via filtering.
• Computes the analytic extension $r_a(t) = r_M(t) + iH[r_M(t)] = p(t)e^{i\varphi(t)}$, where $H[\cdot]$ is the Hilbert transform.
• Extracts the continuous unwrapped phase $\varphi_u(t)$.
• Realigns samples by interpolating $\varphi_u(t_k)$ onto the acquisition times of $x(t)$, then mapping to a cycle coordinate $\tau_k = (\varphi_k \bmod 2\pi)/(2\pi) \in [0,1)$.

Averaging proceeds by partitioning the unit interval $[0,1)$ into $M$ bins $B_m$, collecting the values $x(t_k)$ according to their $\tau_k$, and forming bin averages $$y_m = \frac{1}{|B_m|} \sum_{k: \tau_k \in B_m} x(t_k)$$. Optionally, the resulting sequence can be smoothed via filtering or interpolation, yielding an estimate of a high-resolution pseudo-periodic cycle.

Alternative formulations arise in shift-invariant subspaces of mixed Lebesgue spaces through "random average sampling," where a locally averaged signal is sampled at randomly drawn points within an observation window. Under suitable generator conditions, explicit reconstruction is achieved by inversion of a sampling matrix built from the convolution of the generator with the averaging kernel (Garg et al., 2021).

In convolutional neural networks, especially symmetric encoder–decoder architectures, shift–average upsampling is realized as adaptive polyphase upsampling (APS-U), which inverts adaptive polyphase downsampling (APS-D) by combining zero insertion and a signal-dependent shift, leading to perfect shift equivariance (Chaman et al., 2021).

2. Workflow and Implementation Details

The core steps of shift–average upsampling in its canonical signal-processing form follow the algorithmic pipeline (Voss, 2018):

1. Acquisition: Undersampled $x(t_k)$ and highly sampled $r(t)$.
2. Reference preprocessing: Filtering $r(t)$ to $r_M(t)$ for monocomponentness.
3. Analytic extension: Compute $r_a(t)$, obtain $\varphi(t) = \arg(r_a(t))$, and unwrap to $\varphi_u(t)$.
4. Phase projection: Interpolate $\varphi_u$ to $t_k$, compute $\tau_k$.
5. Binning: Choose $M$, construct bins $B_m$, aggregate $x(t_k)$ by phase.
6. Averaging and smoothing: Form $y_m$, optionally smooth to produce continuous $y(\tau)$.

A similar structure is seen in the random average setting, but here reconstruction leverages a possibly random sample set and explicit matrix inversion. The algorithmic procedure consists of sampling, measurement acquisition, sampling matrix assembly, computation of a pseudoinverse, construction of reconstruction kernels, and application of the reconstruction formula (Garg et al., 2021).

In the neural network context, APS-U is implemented by zero-inserting between samples, followed by a grid realignment using the index selected by APS-D, ensuring channelwise and batchwise invariance under shifts. The selection index from APS-D must be retained and passed through the upsampling operation.

3. Theoretical Guarantees, Resolution, and Tradeoffs

In phase-projection hypersampling, the effective resolution is tied to the number $N$ of low-rate observations and the average cycle length $\bar{T}$ of $r(t)$, providing an effective temporal resolution $\Delta t_\mathrm{eff} = \bar{T} / N$—often three orders of magnitude finer than the original sampling (Voss, 2018). The reference $r(t)$ must be sampled at least twice the maximal instantaneous frequency of the process, with higher oversampling rates improving phase estimation.

The bin count $M$ determines a tradeoff between nominal resolution (large $M$) and noise (fewer samples per bin). Typically, one chooses $M \lesssim N$, ensuring at least $1$–$5$ samples per bin on average.

For shift-invariant subspaces, invertibility and recovery guarantees are provided by explicit probabilistic sampling inequalities: given sufficient sample size $nm$, the sampling operator is bounded and invertible with exponentially high probability, leading to exact recovery in the finite-dimensional space (Garg et al., 2021).

APS-U in neural networks is mathematically proven to restore perfect shift equivariance lost through standard downsampling/upsampling layers, provided the adaptive selection index is consistently applied throughout the architecture (Chaman et al., 2021).

4. Comparison with Classical Methods

Analytic phase projection (APP), the core of hypersampling, is differentiated from retrospective gating by its ability to track local phase nonlinearity throughout the cycle. While retrospective gating relies on discrete template markers and assumes linear phase evolution between templates, APP projects directly to the instantaneous analytic phase of the reference, yielding higher effective upsampling factors and more accurate reconstruction under non-linear or variable reference dynamics. In empirical and simulated settings, APP displays reduced reconstruction errors and avoids phase misassignments that can bias the estimated cycle shape, particularly when gaps or non-uniformities are present in the sample times (Voss, 2018).

In shift-invariant subspace sampling, the explicit inversion of the sampling matrix and the form of the reconstruction formula guarantee (up to numerical accuracy) perfect recovery, with observed errors matching analytic expectations to machine precision (Garg et al., 2021).

Within deep learning, standard upsampling methods (nearest-neighbor, bilinear, transposed convolution) fail to preserve shift equivariance due to the fixed alignment of the sampling grid; APS-U overcomes this by explicitly realigning the upsampled feature map using the adaptive index from APS-D, yielding provable shift-equivariant mappings (Chaman et al., 2021).

5. Practical Applications

Applications of shift–average upsampling span neuroimaging, biomedical signal reconstruction, and shift-equivariant deep learning:

• MRI pulse-wave imaging: Hypersampling with analytic phase projection reconstructs high-resolution cerebral arterial and venous waveforms by linking slowly acquired MRI data with a high-rate reference (e.g., pulse oximetry), achieving effective temporal resolutions on the millisecond scale and matching known physiological delays (Voss, 2018).
• EEG source localization: Potential for cycle alignment and hypersampling in multimodal EEG–MRI or EEG–optical setups, exploiting EEG rhythms as phase references for slow optical or MRI signals.
• Hybrid optical–MRI systems: Use of fast optical reference to phase-align and upsample slower MRI or similar acquisitions in multimodal neuroimaging.
• Shift-invariant approximation: In shift-invariant subspaces, the upsampling method achieves exact signal reconstruction from random local averages given sufficient sampling, with exponential decay in reconstruction error (Garg et al., 2021).
• Deep learning architectures: Integration of APS-D and APS-U in convolutional neural networks, particularly symmetric encoder–decoder architectures (e.g., U-Net), confers exact shift equivariance in feature maps and reconstruction outputs, crucial for robustness to input translations in image reconstruction tasks (Chaman et al., 2021).

6. Limitations and Requirements

Shift–average upsampling techniques rely on several important conditions:

• Reference requirements: For analytic phase projection, the reference $r(t)$ must be monocomponent or rendered so by filtering, with high sampling relative to pseudo-periodic content. The accuracy of the Hilbert-transform phase and the absence of multicomponent artifacts are critical.
• Sample coverage: Sufficient spanning of cycles ($N$) is required to achieve desired temporal or phase resolution. Too few cycles or poor cycle coverage limits the effective upsampling benefit.
• Matrix invertibility/stability: In shift-invariant subspace sampling, guarantees of recovery depend on sample number scaling appropriately with subspace dimension and conditions on generator functions.
• Noise–resolution balance: Increasing nominal resolution by bin subdivision or small kernel support can elevate variance when average sample count per bin drops.
• Pathologies: Extreme phase resets, missing cycles, or reference artifacts can degrade reconstruction accuracy, requiring preemptive outlier rejection or adaptive binning strategies.

7. Numerical, Empirical, and Theoretical Results

Empirical evaluations in both hypersampling and random average sampling demonstrate near-perfect recovery under model assumptions. In MRI pulse-wave applications, effective temporal resolutions of $\sim$1.9 ms have been reported, corresponding to a three orders of magnitude increase over original MRI sampling intervals and enabling physiologically meaningful waveform reconstructions (Voss, 2018).

Numerical simulations in shift-invariant spline-based subspaces report $L^\infty$ reconstruction errors of $10^{-15}$–$10^{-13}$ for modest sample sizes, consistent with the theoretical performance bounds (Garg et al., 2021).

In APS-U–augmented neural networks, shift equivariance is realized exactly and empirically confirmed to outperform standard data augmentation or anti-aliasing strategies, without introducing additional trainable parameters or loss in reconstructed image quality (Chaman et al., 2021).

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References

• (Voss, 2018) Hypersampling of pseudo-periodic signals by analytic phase projection
• (Garg et al., 2021) Random average sampling and reconstruction in shift-invariant subspaces of mixed Lebesgue spaces
• (Chaman et al., 2021) Truly shift-equivariant convolutional neural networks with adaptive polyphase upsampling