Regular Grid Sampling

• Regular grid sampling is the method of arranging data points in uniform, periodic lattices (e.g., rectilinear or hexagonal) to enable precise interpolation and signal reconstruction.
• It is widely applied in signal processing, spatial statistics, computational physics, and machine learning, leveraging its analytical tractability and computational efficiency.
• Hybrid strategies that blend regular grids with clustered samples optimize parameter estimation and prediction, balancing spatial coverage with local detail.

Regular grid sampling refers to the acquisition or arrangement of data points at organized, equally spaced locations within a domain, forming structured patterns such as rectilinear, hexagonal, or other periodic lattices. This approach underlies numerous methodologies in signal processing, spatial statistics, computational physics, astronomy, materials science, and machine learning. Regular grid sampling is valued for its analytical tractability, computational efficiency, and the strong theoretical guarantees it provides, especially in the context of interpolation, reconstruction, and asymptotic statistical analysis.

1. Mathematical Foundations and Classical Theory

At its core, regular grid sampling organizes points in a domain (typically $\mathbb{R}^d$) according to simple generating rules:

• Rectilinear (Cartesian) grids: Points are located at coordinates $x_i = x_0 + i h$, $y_j = y_0 + j k$, etc., where $h, k$ are uniform spacings.
• Hexagonal grids: Each point has six equidistant neighbors, providing denser packing and optimal isotropic coverage for circularly band-limited signals.
• Generalized regular grids: In higher dimensions or nonstandard coordinate systems, lattice points are generated by integer linear combinations of basis vectors with fixed spacings, possibly nonorthogonal.

The sampling theorem (in 1D and regular extension to higher dimensions) ties the density of the grid to perfect reconstruction for bandlimited signals. In more complex contexts, unions of lattices (e.g., Manhattan grids) or perturbations of a perfect grid (via random displacement) are used to investigate robustness and expressiveness (Prelee et al., 2015).

2. Statistical Inference and Asymptotic Behavior

Regular grid sampling plays a key role in spatial statistics and the theory of Gaussian processes. For covariance parameter estimation, the spatial sampling design—especially its deviation from a perfect grid quantified by a regularity parameter $\varepsilon$—directly influences asymptotic properties of estimators (Bachoc, 2013). Sample locations of the form $v_i + \varepsilon X_i$ (where $v_i$ forms a grid and $X_i$ is random) create a controllable tradeoff between regularity and clustering.

• Maximum Likelihood (ML) estimation: The asymptotic covariance of the parameter estimator depends deterministically on $\varepsilon$. For example, in the Matérn covariance case,

$$\Sigma_{\mathrm{ML}} = \frac{1}{2} M\left(\frac{f_\theta^2}{f^2}\right),$$

where $f$ is the Fourier transform of the grid-based kernel sequence.

• Estimator performance: Larger $\varepsilon$ (greater irregularity and clustering) decreases the asymptotic variance for ML, since closely spaced samples provide more information about small-scale covariance structure. However, cross-validation (CV) estimators may perform worse under irregular designs due to heteroscedasticity of local prediction errors.
• Prediction: For Kriging, integrated prediction error is minimized with a regular, space-filling design ($\varepsilon = 0$); clustered designs increase prediction error despite aiding parameter estimation.

A practical outcome is a hybrid sampling strategy—combining an underlying regular grid for prediction with targeted clusters to improve covariance identification (Bachoc, 2013).

3. Efficiency, Symmetry, and Grid Design in Computational Physics

In computational materials science, especially density functional theory (DFT), regular grids—such as Monkhorst-Pack (MP) grids in reciprocal space—are used for Brillouin zone integrations. However, generalized regular (GR) grids, which relax the requirement of integer subdivisions along reciprocal lattice vectors, offer substantial efficiency improvements (Morgan et al., 2018, Morgan et al., 2019).

• Grid efficiency is measured by the number of irreducible $k$-points needed for a target convergence threshold (e.g., 1 meV/atom). GR grids reduce this number by 60% over MP grids and 20% over simultaneously commensurate (SC) grids for metallic systems (Morgan et al., 2018).
• Grid construction: Algorithms for generating GR grids exploit symmetry-preserving supercells (parameterized via Hermite Normal Form matrices) and Minkowski reduction for vector uniformity, dramatically narrowing the search space for efficient grid candidates (Morgan et al., 2019).
• Statistical integration: Leading-order error terms in numerical integration are canceled for appropriately chosen “special points” in regular grids, and further symmetry reduction in GR grids enhances this property.

Applications include high-throughput DFT, where computational cost is tightly linked to sampling grid selection (Morgan et al., 2018, Morgan et al., 2019).

4. Reconstruction, Interpolation, and Missing Data

Regular grids facilitate efficient algorithms for interpolation, error correction, and signal reconstruction:

• FFT-based interpolation from missing or nonuniform samples: For band-limited signals sampled on a regular grid but missing data at some points, direct algorithms based on erasure polynomials and FFTs recover missing values with $O(N\log N)$ complexity, outperforming pseudo-inverse and burst error recovery techniques in speed and numerical stability (Selva, 2014).
• Frequency-selective and mesh-to-grid resampling: In image processing, reconstructing data from nonregular positions onto a regular grid is achieved via frequency-selective reconstruction or mesh-to-grid modeling. These use sparsity and optical transfer function-inspired weighting to iteratively select basis functions, yielding high PSNR and efficient suppression of artifacts (Heimann et al., 2022, Seiler et al., 2022).
• Compressive sampling on grids: In spherical acoustic field measurement, random selection of grid points from an equiangular grid (rather than arbitrary position sampling) can still guarantee recovery of band-limited fields, leveraging the unitary structure of the discrete Fourier transform (DFT) matrix associated with regular grids. This approach reduces measurement number—sometimes to one-third of the classical Nyquist requirement—while substantially improving denoising capacity (Valdez et al., 2022).

5. Advanced Grid Types: Multidimensional and Hexagonal Grids

Regular grid sampling extends beyond rectilinear layouts:

• Manhattan sampling: This is the union of several rectangular lattices, each with dense sampling in certain dimensions and coarse in others, enabling perfect reconstruction for signals band-limited to unions of their Nyquist regions. A bi-step lattice representation provides a compact algebraic formalism for efficient algorithm design (Prelee et al., 2015).
• Hexagonal grids and interlaced scanning: Hexagonal grids enable optimal sampling for circularly band-limited signals, providing better isotropy and packing. Recent work details interlaced hexagonal scanning with progressive sampling phases, ensuring full field-of-view coverage early in acquisition and facilitating real-time visualization and efficient dose usage in imaging (Hinkle et al., 2022).

Tables can summarize grid types and properties:

Grid Type	Key Feature	Domain of Application
Rectilinear	Orthogonal, uniform spacing	Image/Signal Processing
Generalized Regular	Arbitrary lattice generation	DFT, Materials Science
Manhattan (bi-step)	Unions of dense/coarse lattices	Multidimensional Imaging
Hexagonal	6-fold symmetry, isotropy	Drift-sensitive Imaging

6. Practical Design Trade-Offs and Applications

Regular grid sampling reveals fundamental trade-offs:

• Parameter estimation vs. prediction: For latent parameter inference (e.g., covariance function in spatial statistics), clustering (irregularity) is often beneficial; for optimal prediction, uniform (regular) coverage is essential (Bachoc, 2013).
• Regular vs. random grid layouts: In astronomical surveys, regular sparse grids introduce periodic features (aliasing) in the window function, which can interfere with the statistical reliability at specific scales (e.g., BAO). Random or quasi-random sampling mitigates this but does not improve overall statistical power—both yield similar global performance if coverage and density are matched (Paykari et al., 2013).
• Computational and physical efficiency: For resource-intensive simulations/experiments, efficient and symmetric grid construction (e.g., GR grids) directly reduces cost, especially in large-scale computations or high-throughput settings (Morgan et al., 2018).

The interplay of these trade-offs guides practical sampling strategies in fields ranging from geostatistics to cosmological survey design and large-scale condensed matter modeling.

7. Extensions and Directions for Future Research

Recent advancements suggest several ongoing and future directions:

• Adaptive and hybrid designs: Incorporating local adaptivity into regular grids to balance estimator efficiency and prediction accuracy, potentially with two-tier structures (regular backbone plus targeted clusters) (Bachoc, 2013).
• Grid design automation and integration: Automated generation of optimal grids (e.g., for Brillouin zone integration) tailored to symmetry and application constraints (Morgan et al., 2019).
• Multiresolution and progressive grids: Interlacing strategies enable multi-scale analysis, dose fractionation, and dynamic adjustment of resolution during data acquisition (Hinkle et al., 2022).
• Learning-based representations: Conversion of irregular geometric data (e.g., skeleton graphs) into learned grid structures enables direct application of regular convolutional architectures to otherwise irregular data, as in skeleton-to-grid action recognition (Cai et al., 2023) and hybrid grid-point representations in 3D neural modeling (Li et al., 4 Jan 2024).

Continued development of grid sampling methodologies promises further theoretical and practical advances for data acquisition, modeling, and analysis across scientific disciplines.