CoherentRaster: Raster-Based Coherence

• CoherentRaster is a framework of raster-based formulations that treats coherence as a fundamental attribute across applications such as InSAR, THz imaging, radar fusion, and light field rendering.
• It integrates spectral, phase, and cross-view coherence through techniques like frequency-domain analysis, autoencoding, and computational sorting to enhance estimation and efficiency.
• Practical implementations demonstrate that preserving coherence in raster outputs substantially improves prediction accuracy, processing speed, and resource utilization.

CoherentRaster denotes a family of raster- or grid-based computational formulations in which coherence is treated as a first-class object of estimation, preservation, or acceleration. In the cited materials, the designation appears explicitly as the name of a light-field rendering framework for 3D Gaussian Splatting (Sim et al., 6 May 2026), and it also appears as an implementation-oriented label for frequency-domain raster analysis of stationary random fields (Kleiber, 2015), interferometric SAR coherence mapping and regression (Mukherjee et al., 2020, Sica et al., 5 Jun 2026), coherent terahertz raster imaging (Ravaro et al., 2013), cooperative multistatic radar backprojection on spatial rasters (Tagliaferri et al., 2023), and rasterization-seeded coherent ray processing on GPUs (Reis et al., 2023). Taken together, these sources suggest a unifying principle: raster outputs become substantially more informative or efficient when the relevant notion of coherence—spectral, interferometric, phase, multiview, or ray-space—is modeled directly rather than treated as an incidental by-product.

1. Scope and terminological usage

Among the cited works, the formal system name "CoherentRaster" is used by "CoherentRaster: Efficient 3D Gaussian Splatting for Light Field Displays" (Sim et al., 6 May 2026). In the other sources, the term is not the paper title but an implementation-oriented designation for raster workflows that compute or exploit coherence on regular grids. This produces a polysemous but technically consistent usage: the raster is the sampled support, and coherence is the structure being estimated, enforced, or reused.

Domain	Coherence object	Representative source
Multivariate spatial statistics	Frequency-specific linear dependence	(Kleiber, 2015)
InSAR	Magnitude of complex correlation or learned proxy	(Mukherjee et al., 2020, Sica et al., 5 Jun 2026)
THz coherent imaging	Per-pixel complex field amplitude and phase	(Ravaro et al., 2013)
Cooperative radar imaging	Phase-synchronized multistatic image fusion	(Tagliaferri et al., 2023)
GPU ray tracing	Ray-space coherence from rasterization seeds	(Reis et al., 2023)
Light field rendering	Cross-view and memory-access coherence	(Sim et al., 6 May 2026)

A central consequence of this breadth is that coherence is not a single scalar concept. In William Kleiber’s spatial-statistical setting it is a function of spatial frequency; in InSAR it is the magnitude of a normalized complex correlation coefficient or its learned surrogate; in coherent sensing it is phase alignment across measurements; and in graphics it refers to structural regularity that improves culling, sorting, and memory efficiency (Kleiber, 2015, Mukherjee et al., 2020, Tagliaferri et al., 2023, Sim et al., 6 May 2026).

2. Spectral coherence for gridded random fields

In "Coherence for Random Fields," Kleiber develops coherence, phase, and gain for multidimensional stationary processes by working with the spectral density matrix rather than only with cross-covariance functions (Kleiber, 2015). For a mean-zero, weakly stationary $p$-variate random field $Z(s)$ on $s \in \mathbb{R}^d$, the covariance structure is represented by a matrix-valued covariance function $C(h)$ and, via the multivariate Bochner–Cramér representation, by a matrix of marginal and cross-spectral densities $f(\omega) = (f_{ij}(\omega))$. In this framework, coherence between two processes is defined by

$$\gamma(\omega) = \frac{f_{12}(\omega)}{\sqrt{f_{11}(\omega)f_{22}(\omega)}},$$

with squared coherence $|\gamma(\omega)|^2 \in [0,1]$. For raster data, $\omega$ is a two-dimensional spatial frequency, so coherence becomes a scale- and direction-specific measure of linear dependence between layers.

The same paper defines gain and phase through

$$A(\omega) = \frac{f_{12}(\omega)}{f_{11}(\omega)}, \qquad G(\omega)=|A(\omega)|, \qquad \phi(\omega)=\arg A(\omega).$$

This yields a direct interpretation of spatial shifts and frequency-selective amplification. If one field is approximately a shifted, rescaled version of another, then the phase varies linearly with frequency and the gain is constant. The prediction result is equally important: the optimal linear predictor of one process from another has spectral density

$f_{1|2}(\omega)=f_{11}(\omega)|\gamma(\omega)|^2,$

so squared coherence is literally the frequency-wise variance explained by optimal linear dependence.

A major contribution of the paper is diagnostic rather than merely definitional. Several common multivariate constructions impose unrealistic coherence behavior. Separable models force constant squared coherence; covariance convolution and process convolution can force $Z(s)$0; and kernel-convolution models with a common latent process can make coherence constant across all frequencies. This is not a minor technicality: by the prediction theorem, such models cannot represent real raster data whose dependence changes with spatial scale. The linear model of coregionalization is somewhat more flexible, but standard parameterizations still constrain gain and coherence strongly.

The multivariate Matérn class receives a particularly explicit coherence interpretation. In the common-range case, if $Z(s)$1, coherence is constant across frequency; if $Z(s)$2, coherence is larger at low frequencies and decays as $Z(s)$3. In the common-smoothness case, the cross-range parameter $Z(s)$4 can make coherence stronger either at low or at high frequencies. The supplied implementation guide emphasizes that this distinction matters for gridded practice: $Z(s)$5 governs low-frequency sharing, whereas $Z(s)$6 can induce high-frequency coherence, which parsimonious Matérn parameterizations cannot express.

Estimation proceeds from smoothed multivariate periodogram matrices on a regular grid. The raw multivariate periodogram is asymptotically unbiased but not consistent, so local smoothing is required: $Z(s)$7 with coherence estimator

$Z(s)$8

The raster-oriented workflow in the supplied material uses mean removal, optional detrending, tapering, 2D FFTs, cross-periodograms, and small frequency-domain smoothers such as the 3×3 low-pass filter used in the paper’s examples. The sea level pressure and geopotential-height case studies show why this matters empirically: coherence was substantial at low frequencies, phase shifts were present at specific bands, and some high-frequency shared structure was not well captured by standard multivariate models.

3. InSAR coherence rasters: classification and regression

In interferometric SAR, coherence is the similarity of two complex SAR signals over the same area and is typically defined as the magnitude of the normalized complex correlation coefficient. The supplied InSAR materials distinguish two CoherentRaster paradigms: classification from complex interferograms and regression from detected backscatter without precise coregistration (Mukherjee et al., 2020, Sica et al., 5 Jun 2026).

"CNN-based InSAR Coherence Classification" frames coherence mapping as demarcation of coherent versus incoherent regions in an interferogram (Mukherjee et al., 2020). The input is a complex interferogram represented by real and imaginary channels. The proposed workflow first trains a denoising autoencoder, then computes raw coherence between the noisy interferogram and its filtered version using 7×7 patches, thresholds the result with Otsu’s method, and refines the binary labels with an MRF energy

$Z(s)$9

using graph cuts with $s \in \mathbb{R}^d$0. The classifier itself is a shallow three-layer SeparableConv2D network with filter counts 16-16-1, ReLU activations in the hidden layers, sigmoid output, binary cross-entropy loss, Adam optimization, and Xavier initialization. Training used 135 interferograms of size 1000×1000, with 500 patches per interferogram for both the denoiser and classifier. On 100 simulated interferograms, the reported averaged scores were accuracy 0.8425, precision 0.8399, and recall 0.9107 for the proposed method, exceeding Boxcar, NLInSAR, and NLSAR in the paper’s comparison. The reported inference time was about 0.67 seconds per 1000×1000 interferogram on an NVIDIA GTX 1070 GPU.

"Beyond Backscatter: InSAR coherence from detected SAR images" moves the problem into a distinct regime by regressing coherence directly from detected magnitudes, such as Sentinel-1 GRD or analysis-ready data, without accurate coregistration (Sica et al., 5 Jun 2026). The model is a Residual U-Net with 4 encoder levels, a bridge, and 4 decoder levels; it takes two $s \in \mathbb{R}^d$1 channels, clips them in dB to $s \in \mathbb{R}^d$2, normalizes them to $s \in \mathbb{R}^d$3, and is trained with MSE loss and Adam. Training used 128×128 patches, batch size 64, 20 epochs, and a step-based learning-rate schedule beginning at $s \in \mathbb{R}^d$4 and dropping by a factor of 0.1 every 10 epochs. The targets were high-resolution coherence maps computed from precisely coregistered Sentinel-1 SLC pairs using Φ-Net, with training on 12-day VV pairs and testing across 0, 6, 12, and 36 days, cross-polarization VH, large perpendicular baselines, and even ALOS L-band HH.

The key quantitative result is that learned regression from detected imagery substantially outperformed an established intensity-based baseline. On a 12-day Sentinel-1 experiment over San Francisco Bay in slant-range SLC magnitudes, overall RMSE was 0.108 for the proposed model versus 0.275 for the baseline; the single-input ablation rose to 0.261, confirming the importance of paired observations. Generalization was reported across Atacama desert, Rondonia, Schwyz Alps, Oberpfaffenhofen, and Rosamond Dry Lake, with RMSE values 0.055, 0.105, 0.139, 0.126, and 0.215, respectively. The GRD transfer results were qualitatively consistent with SLC-derived targets, though stripe-like artifacts could appear because of GRD resampling.

These two lines of work formalize different roles for a coherence raster. In the classification setting, the raster is a binary or probabilistic reliability mask for filtering and unwrapping. In the regression setting, it is a continuous proxy for SLC-derived coherence available from globally accessible detected products. A common misconception is that such learned products replace interferometry itself. The regression paper states the opposite: the method predicts coherence but does not reconstruct interferometric phase and therefore cannot replace full InSAR processing when phase is needed (Sica et al., 5 Jun 2026).

4. Coherent raster sensing in terahertz and multistatic radar imaging

In coherent sensing, a CoherentRaster is a spatial image whose pixels retain complex-field information, not merely intensity. The supplied terahertz and automotive radar sources both operate in this regime, although with very different hardware and synchronization strategies [(Ravaro et al., 2013); (Tagliaferri et al., 2023)].

"Continuous-wave coherent imaging with terahertz quantum cascade lasers using electro-optic harmonic sampling" demonstrates raster-scan coherent imaging with a 2.5 THz quantum cascade laser phase-locked to a near-infrared femtosecond-laser comb (Ravaro et al., 2013). The system uses two electro-optic modules with 2 mm-thick ZnTe crystals, balanced detection, and a PLL of approximately 2 MHz bandwidth. The beat note between the THz field and harmonics of the comb is mixed to 70 kHz and demodulated by a lock-in amplifier, producing per-pixel complex field values

$s \in \mathbb{R}^d$5

Raster scanning uses a motorized XY stage with continuous line scans along $s \in \mathbb{R}^d$6 at 2.2 mm/s, line step $s \in \mathbb{R}^d$7, matched 100 $s \in \mathbb{R}^d$8m sampling along the line, and lock-in time constant $s \in \mathbb{R}^d$9 ms. The reported detection noise floor is 3 pW/Hz, long-term phase stability is less than 3 degrees per hour, and measured spatial resolution on a 10-cent Euro coin is about 160 $C(h)$0m, consistent with diffraction-limited performance at $C(h)$1.

The automotive multistatic case is conceptually similar in that the raster is again formed by coherent backprojection, but the central challenge shifts to phase synchronization across networked sensors. "Cooperative Coherent Multistatic Imaging and Phase Synchronization in Networked Sensing" defines a cooperative imaging model for FMCW vehicular radar and forms a global raster by summing monostatic and bistatic images after compensating timing, frequency, and phase offsets (Tagliaferri et al., 2023). For node pair $C(h)$2, the backprojected image is

$C(h)$3

and the coherent fusion is

$C(h)$4

after synchronization. The paper gives a three-step synchronization procedure: frequency-offset estimation, coarse synchronization through image-domain coregistration and timing-offset estimation, and fine phase synchronization using calibration points and an alternating optimization over sensor positions, clock phases, and target positions. Feasibility is assessed with a hybrid Cramér–Rao bound.

The detection-theoretic benefit is explicit. Coherent fusion improves output SNR approximately linearly in the number of synchronized images, whereas non-coherent power summation does not improve SNR in the same way. The experiment used $C(h)$5 GHz, bandwidth $C(h)$6 MHz, 8 channels, synthetic aperture length approximately 9 cm, and effective inter-vehicle separation approximately 5 m. After multistatic synchronization and coherent fusion, the system detected the legs of a pedestrian with about 40 cm separation standing roughly 30–40 cm from a parked car, a case in which single low-end monostatic or bistatic images could not separate the pedestrian from the strong nearby return.

The two sensing examples highlight a shared structural requirement. Coherent rasters are only meaningful when phase is stable enough that amplitude and phase can be interpreted per pixel or fused across pixels and sensors. In the THz system this is achieved by phase-locking a single source to a comb; in the automotive system it requires explicit estimation of $C(h)$7, $C(h)$8, $C(h)$9, and geometry-dependent propagation terms.

5. Rasterization as a source of computational coherence

In graphics systems, coherence is often a property of computation rather than of a measured physical field. The supplied ray-tracing and light-field rendering sources use rasterization to seed structured workloads whose subsequent processing becomes more efficient when coherence is preserved (Reis et al., 2023, Sim et al., 6 May 2026).

"Ray-Tracing With a Coherent Ray-Space Hierarchy" starts from primary visibility obtained by rasterization and uses it to generate coherent secondary rays on the GPU (Reis et al., 2023). The method hashes rays by quantized origin, direction, bounce index, and type, compresses identical keys into chunks, radix-sorts the chunks, and builds an $f(\omega) = (f_{ij}(\omega))$0-level ray-space hierarchy whose nodes bound bundles of rays by spheres and cones. Geometry is also partitioned into bounding spheres, so culling can occur bundle-versus-geometry before per-ray tests. The supplied implementation-oriented material emphasizes why this is a raster-seeded CoherentRaster construction: screen-space adjacency produces camera-space adjacency, which produces coherent origin bins in ray-space. On a GTX TITAN at 512×512 resolution and RSH depth 2, the method reduced intersection tests by 63.79% versus a prior RSH in OFFICE, 26.47% in CORNELL, and 35.86% in SPONZA; relative to brute force, the reductions were 98.14%, 91.17%, and 97.62%, respectively.

The explicitly named CoherentRaster framework appears in "CoherentRaster: Efficient 3D Gaussian Splatting for Light Field Displays" (Sim et al., 6 May 2026). Light field displays use an interlaced image in which each RGB subpixel is assigned to one of $f(\omega) = (f_{ij}(\omega))$1 viewpoints by a viewpoint index matrix $f(\omega) = (f_{ij}(\omega))$2. This mapping destroys the spatial coherence on which conventional rasterizers rely, because adjacent screen-space subpixels often belong to different views. The paper modifies 3D Gaussian Splatting in two complementary ways. First, it performs subpixel-level rasterization, rendering only the subpixels that actually contribute to the interlaced output. Second, it introduces Cross-view Coherent Attribute Reuse and View-coherent Remapping. The former clusters neighboring views and reuses slowly varying attributes—screen-space covariance, depth, and SH color—while still computing per-view 2D means. The latter precomputes a per-tile permutation $f(\omega) = (f_{ij}(\omega))$3 that sorts subpixels by viewpoint index so that warp lanes process subpixels with similar view assignments, restoring coalesced memory reads.

The resulting pipeline projects Gaussians, generates 64-bit keys packing tile, cluster, and depth, radix-sorts them, and blends front-to-back over per-tile, per-cluster lists. Hardware results are reported on an RTX 5090 with 32 GB memory. On Synthetic Blender at 2K resolution with 63 views, CoherentRaster achieved 88 FPS with cluster size $f(\omega) = (f_{ij}(\omega))$4, compared with 28 FPS for Subpixel-3DGS and 5.8 FPS for Full-frame 3DGS; at 4K with 71 views, it achieved 56 FPS compared with 19 FPS and 4.1 FPS. On Mip-NeRF 360, the method reached 30 FPS at 2K and 16 FPS at 4K with $f(\omega) = (f_{ij}(\omega))$5. The key-count reduction was also substantial: at 2K on Synthetic Blender, Gaussian–tile pairs fell from 76.0M without reuse to 15.6M with reuse; at 4K, from 134.3M to 27.5M. The ablation study showed 28 FPS without reuse or remap, 67 FPS with remap only, 34 FPS with reuse only, and 88 FPS with both.

This graphics usage reveals a different sense of coherence from the sensing literature, but the operational logic is parallel. In both cases, coherence is exploited to avoid redundant work. In sensing, coherent processing preserves linear or phase relationships to improve inference. In rendering, coherent processing preserves cross-view or ray-space regularity to improve memory locality, warp behavior, and culling efficiency.

6. Cross-domain principles, limitations, and recurring misconceptions

Across these domains, coherence is always a structured relation indexed by a raster, but the structure itself changes with the application. In spatial statistics it is a frequency-dependent function that may vary strongly with scale and direction, so constant-coherence models are diagnostically restrictive (Kleiber, 2015). In interferometric SAR it is either estimated from complex samples or learned from intensity cues, but the learned product does not recreate phase (Mukherjee et al., 2020, Sica et al., 5 Jun 2026). In THz and cooperative radar imaging it is inseparable from phase synchronization, and performance degrades when residual phase errors increase [(Ravaro et al., 2013); (Tagliaferri et al., 2023)]. In ray tracing and light field rendering it is a computational regularity that can be hashed, sorted, clustered, and remapped for better GPU execution (Reis et al., 2023, Sim et al., 6 May 2026).

Several misconceptions recur when these literatures are collapsed into a single label. One is that coherence is inherently scalar and global. The random-field work shows that coherence is a function over spatial frequency; the Matérn analysis further shows that smoothness and range parameters can govern low- versus high-frequency dependence in different ways (Kleiber, 2015). A second misconception is that stronger smoothing always yields better coherence estimates. Both the periodogram-based random-field pipeline and the InSAR workflows describe an explicit bias–variance trade-off: too little smoothing leaves estimates unstable, while too much hides structure (Kleiber, 2015, Mukherjee et al., 2020). A third is that coherent rasters are automatically physically interpretable. The THz and automotive examples make clear that phase drift, feedback, timing offsets, geometry errors, and phase wrapping directly affect interpretation [(Ravaro et al., 2013); (Tagliaferri et al., 2023)]. A fourth is that computational coherence can be assumed rather than engineered. The light-field and ray-space systems succeed precisely because coherence is recovered through sorting, clustering, and remapping after naïve raster layouts or secondary-ray generation would otherwise destroy it (Reis et al., 2023, Sim et al., 6 May 2026).

The limitations are correspondingly domain-specific. Stationarity is required for the formal definition of coherence in random fields. InSAR classification methods inherit bias from label generation and can oversmooth narrow coherent structures if MRF regularization is too strong. Detected-image coherence regression degrades under strong domain shift, difficult terrain, or when precise phase information is required. THz raster imaging remains sensitive to optical feedback and mechanical drift. Cooperative radar imaging requires sub-wavelength-equivalent calibration accuracy and nontrivial communication of geometry and timing metadata. Light-field CoherentRaster assumes small enough view clusters that covariance, depth, and SH color vary smoothly within a cluster; larger baselines or stronger view dependence can introduce artifacts.

What emerges from the collected literature is not a single canonical algorithm but a reproducible design pattern. CoherentRaster systems compute directly on the representation where coherence is most useful—frequency grids, interferometric pixels, backprojected spatial rasters, sorted ray bundles, or viewpoint-indexed subpixels—and then organize estimation or rendering so that the coherence structure remains explicit rather than being averaged away.