Planar-Adapted Gaussian Representation

• Planar-adapted Gaussian representation is a modeling approach where 3D Gaussians are aligned to local planar surfaces by collapsing one covariance axis to produce thin, disk-like primitives.
• It enhances reconstruction quality by aligning the covariance structure with planar geometry, thus improving depth, normal estimation, and multi-view consistency across varied applications.
• Empirical studies report improved PSNR, SSIM, and reduced depth errors, with faster training times and enhanced performance in tasks such as indoor scanning and RF radiance modeling.

Searching arXiv for the cited papers and closely related planar Gaussian methods. Planar-adapted Gaussian representation denotes a class of Gaussian-based models in which Gaussian primitives are constrained, parameterized, or regularized by planes or locally planar surface structure rather than treated as unconstrained volumetric ellipsoids. In the cited literature, the formulation appears in planar reconstruction from RGB images, high-fidelity surface reconstruction, dynamic scene modeling, indoor reconstruction, rendering of planar transmission and reflection, and radio-frequency radiance modeling for Spatial-CSI. Across these settings, the central operation is to make the Gaussian primitive surface-aligned: by collapsing one covariance axis, embedding 2D Gaussians in a 3D plane, constraining centers to fitted planes, or attaching explicit planar interaction models to otherwise standard Gaussians (Zanjani et al., 2024, Chen et al., 2024, Cai et al., 2024, Zhang et al., 23 Aug 2025).

1. Primitive definition and geometric parameterization

A common starting point is the standard 3D Gaussian Splatting parameterization, in which each Gaussian carries a centroid $\boldsymbol\mu_i\in\mathbb R^3$, a covariance $\Sigma_i\in\mathbb R^{3\times3}$, an opacity or density term, and a color or radiance term. In several planar-adapted variants, the covariance is factorized as $\Sigma_i=R_i\,S_i\,S_i^T\,R_i^T$, with $S_i=\mathrm{diag}(s_{i1},s_{i2},s_{i3})$, and one principal axis is driven to become very small so that the primitive behaves like a thin disk rather than a volumetric blob (Chen et al., 2024, Jin et al., 27 Oct 2025, Gan et al., 20 Oct 2025).

In PGSR, each Gaussian is explicitly “flattened” along its smallest principal axis. The Gaussian center is $\boldsymbol\mu_i\in\mathbb R^3$, the normal $\mathbf n_i$ is the unit-length eigenvector of $\Sigma_i$ associated with the smallest eigenvalue, and the in-plane covariance is written as

$$\Sigma_i^\parallel = R_i\,\mathrm{diag}(0,s_2,s_3)\,R_i^T.$$

The residual thickness along the normal direction is penalized through

$L_s=\sum_i \min(s_1,s_2,s_3)\longrightarrow 0.$

This turns the Gaussian into a thin, oriented planar disk (Chen et al., 2024).

RF-PGS adopts an analogous construction in radio-frequency modeling. A single Planar Gaussian is parameterized by center position $\mathbf x_g\in\mathbb R^3$, rotation quaternion $\Sigma_i\in\mathbb R^{3\times3}$0 or rotation matrix $\Sigma_i\in\mathbb R^{3\times3}$1, scale vector $\Sigma_i\in\mathbb R^{3\times3}$2, and base density $\Sigma_i\in\mathbb R^{3\times3}$3, with covariance

$\Sigma_i\in\mathbb R^{3\times3}$4

Planarity is enforced by shrinking one scale dimension through

$\Sigma_i\in\mathbb R^{3\times3}$5

The surface normal $\Sigma_i\in\mathbb R^{3\times3}$6 is the column of $\Sigma_i\in\mathbb R^{3\times3}$7 corresponding to the minimum $\Sigma_i\in\mathbb R^{3\times3}$8 (Zhang et al., 23 Aug 2025).

Other planar-adapted parameterizations are more structured. DynaSurfGS augments each Gaussian with a local plane normal $\Sigma_i\in\mathbb R^{3\times3}$9 and offset $\Sigma_i=R_i\,S_i\,S_i^T\,R_i^T$0, with $\Sigma_i=R_i\,S_i\,S_i^T\,R_i^T$1, and rewrites the covariance as

$\Sigma_i=R_i\,S_i\,S_i^T\,R_i^T$2

so that the third axis is aligned with the plane normal and $\Sigma_i=R_i\,S_i\,S_i^T\,R_i^T$3 drives the Gaussian into a pancake-shaped primitive (Cai et al., 2024). In contrast, 3D Gaussian Flats represents planar regions by explicit 3D planes $\Sigma_i=R_i\,S_i\,S_i^T\,R_i^T$4 with origin $\Sigma_i=R_i\,S_i\,S_i^T\,R_i^T$5 and unit normal $\Sigma_i=R_i\,S_i\,S_i^T\,R_i^T$6, then places 2D Gaussians with in-plane parameters $\Sigma_i=R_i\,S_i\,S_i^T\,R_i^T$7 and $\Sigma_i=R_i\,S_i\,S_i^T\,R_i^T$8 on those planes and embeds them in 3D via

$\Sigma_i=R_i\,S_i\,S_i^T\,R_i^T$9

This yields Gaussians whose mass lies exactly on the plane (Taktasheva et al., 19 Sep 2025).

2. Rendering, depth estimation, and surface queries

Planar adaptation is not only a geometric regularizer; it directly changes how depth, normals, and surface interactions are computed. In PGSR, each planar Gaussian is projected to image space through the standard anisotropic 2D Gaussian kernel

$S_i=\mathrm{diag}(s_{i1},s_{i2},s_{i3})$0

converted into opacity $S_i=\mathrm{diag}(s_{i1},s_{i2},s_{i3})$1, and composited front-to-back with

$S_i=\mathrm{diag}(s_{i1},s_{i2},s_{i3})$2

From these weights, PGSR renders a blended normal map

$S_i=\mathrm{diag}(s_{i1},s_{i2},s_{i3})$3

and a blended signed distance map

$S_i=\mathrm{diag}(s_{i1},s_{i2},s_{i3})$4

Its “unbiased depth” is then

$S_i=\mathrm{diag}(s_{i1},s_{i2},s_{i3})$5

The cited formulation states that dividing by $S_i=\mathrm{diag}(s_{i1},s_{i2},s_{i3})$6 cancels the ray-accumulation weight and yields the exact intersection depth on the blended plane (Chen et al., 2024).

The same geometric logic is extended to dynamic scenes in DynaSurfGS. Normals are rendered by $S_i=\mathrm{diag}(s_{i1},s_{i2},s_{i3})$7-blending the per-splat normals, a distance map is accumulated as

$S_i=\mathrm{diag}(s_{i1},s_{i2},s_{i3})$8

and the depth is recovered as

$S_i=\mathrm{diag}(s_{i1},s_{i2},s_{i3})$9

A local plane normal is reconstructed from depth at each pixel and four neighbors, and normal consistency is enforced between the depth-derived normal and the Gaussian-rendered normal (Cai et al., 2024).

In RF-PGS, the analogous surface query is a ray–surface intersection in radio space. For a receiver at $\boldsymbol\mu_i\in\mathbb R^3$0 and query direction $\boldsymbol\mu_i\in\mathbb R^3$1, the intersection depth of a Planar Gaussian is computed from

$\boldsymbol\mu_i\in\mathbb R^3$2

and per-Gaussian depths are alpha-blended to find the first surface intersection. The RF radiance stage then decomposes each multipath component path loss into free-space path loss and surface interaction gain, with the latter encoded by spherical-harmonic coefficients stored on the intersected Planar Gaussians. The cited text states that accurate geometry ensures that each SH-modeled leaf precisely corresponds to a physical surface patch (Zhang et al., 23 Aug 2025).

3. Optimization objectives and structural supervision

Planar-adapted Gaussian models are typically trained by combining the baseline photometric objective with planar, geometric, and multi-view constraints. In PGSR, the full objective is

$\boldsymbol\mu_i\in\mathbb R^3$3

where $\boldsymbol\mu_i\in\mathbb R^3$4 is the flattening term, $\boldsymbol\mu_i\in\mathbb R^3$5 penalizes the discrepancy between a local normal reconstructed from neighboring depths and the rendered normal, $\boldsymbol\mu_i\in\mathbb R^3$6 is a multi-view photometric term based on normalized cross-correlation under a plane-induced homography, and $\boldsymbol\mu_i\in\mathbb R^3$7 enforces forward–backward homography cycle consistency. The paper also introduces a camera exposure compensation model,

$\boldsymbol\mu_i\in\mathbb R^3$8

inside the photometric loss (Chen et al., 2024).

RF-PGS adds RF-specific regularization. Beyond the minimum-scale loss $\boldsymbol\mu_i\in\mathbb R^3$9, it uses curvature-aware refinement based on two alpha-blended maps, plane depth $\mathbf n_i$0 and perpendicular distance $\mathbf n_i$1. A wedge point is identified when $\mathbf n_i$2 is large and $\mathbf n_i$3 is small, and wedge Gaussians are enlarged to capture diffraction in a single primitive. Surface uniformity is imposed by estimating a local plane normal from neighboring intersection points,

$\mathbf n_i$4

and penalizing the angular discrepancy between $\mathbf n_i$5 and $\mathbf n_i$6. The total geometry loss is

$\mathbf n_i$7

supplemented by a multi-view consistency term based on NCC between neighboring receiver poses (Zhang et al., 23 Aug 2025).

A separate line of work uses planar semantics rather than only covariance collapse. Planar Gaussian Splatting lifts 2D segmentation embeddings and 2D normals into Gaussians, storing a plane descriptor $\mathbf n_i$8 and a surface normal $\mathbf n_i$9 per Gaussian. It then constructs a tree-structured Gaussian mixture and performs bottom-up probabilistic merges using a criterion that combines photometric likelihood, descriptor similarity, and normal alignment. The merge probability is

$\Sigma_i$0

and planar instances are recovered by agglomerative clustering on the final hierarchy. The abstract states that this method achieves state-of-the-art performance in 3D planar reconstruction without requiring either 3D plane labels or depth supervision (Zanjani et al., 2024).

More recent structured methods incorporate external priors. PlanarGS builds Language-Prompted Planar Priors through GroundedSAM proposals, cross-view fusion, and geometric inspection using depth-derived normals and plane-distance maps. It adds planar prior supervision,

$\Sigma_i$1

and geometric prior supervision,

$\Sigma_i$2

where $\Sigma_i$3 aligns rendered depth with plane-consistent depth, and the depth and normal priors are obtained from DUSt3R. GSPlane imposes an even stronger structural constraint by representing planar Gaussian centers as convex combinations of three basis points on a fitted plane,

$\Sigma_i$4

and introduces a Dynamic Gaussian Re-classifier that reverts persistently high-gradient planar Gaussians to unconstrained means in $\Sigma_i$5 (Jin et al., 27 Oct 2025, Gan et al., 20 Oct 2025).

4. Major variants in the literature

The literature does not reduce planar adaptation to a single primitive design. It includes covariance-collapse models, explicit plane-coordinate models, hierarchical plane parsing, and planar transport or reflection models.

Method	Planar mechanism	Task domain
PGS	Gaussians augmented with plane descriptors and normals; hierarchical Gaussian mixtures	3D planar reconstruction
PGSR	Flattened 3D Gaussians with unbiased depth and normal rendering	Surface reconstruction
DynaSurfGS	Planar-based Gaussians plus 4D neural-voxel deformation	Dynamic reconstruction
RF-PGS	Planar Gaussians with RF-specific optimizations and structured RF radiance	Spatial-CSI / RF modeling
3D Gaussian Flats	Hybrid constrained planar 2D Gaussians and freeform 3D Gaussians	Photometric scene reconstruction
GSPlane	Plane-constrained Gaussian coordinates and planar-guided remeshing	Structured planar reconstruction
PlanarGS	3DGS with language-prompted planar priors and geometric supervision	Indoor reconstruction
TR-Gaussians	Learnable planes and mirrored Gaussians for reflection/transmission	Planar reflection and transmission rendering

TR-Gaussians illustrates a distinct branch of the family. Rather than flattening every Gaussian onto a plane, it keeps ordinary 3D Gaussians for the real scene, introduces an explicit plane $\Sigma_i$6, and generates mirrored Gaussians across that plane to model reflections. Reflection and transmission are blended through a Fresnel-based, view-dependent weighting,

$\Sigma_i$7

with the final composition

$\Sigma_i$8

Its optimization is staged, and includes depth variance, gradient conflict, and reflection mask losses (Liu et al., 17 Nov 2025).

This diversity counters a recurrent simplification: planar adaptation is not exhausted by shrinking one covariance axis. The cited methods show at least four distinct mechanisms—flattened covariances, 2D Gaussians embedded in 3D planes, plane-constrained Gaussian coordinates, and explicit planar interaction models. This suggests that “planar-adapted Gaussian representation” is best understood as a design principle rather than a single architecture.

5. Reported empirical behavior

The reported empirical behavior consistently links planar adaptation to better geometric fidelity, improved multi-view consistency, or more physically structured interaction models. In RF-PGS, the comparison against RF-3DGS is especially explicit. The cited results report mean depth error $\Sigma_i$9, a PSNR improvement from $$\Sigma_i^\parallel = R_i\,\mathrm{diag}(0,s_2,s_3)\,R_i^T.$$0 over RF-3DGS, geometry-stage training in $$\Sigma_i^\parallel = R_i\,\mathrm{diag}(0,s_2,s_3)\,R_i^T.$$1, full RF training $$\Sigma_i^\parallel = R_i\,\mathrm{diag}(0,s_2,s_3)\,R_i^T.$$2, inference $$\Sigma_i^\parallel = R_i\,\mathrm{diag}(0,s_2,s_3)\,R_i^T.$$3, and primitive budgets of $$\Sigma_i^\parallel = R_i\,\mathrm{diag}(0,s_2,s_3)\,R_i^T.$$4–$$\Sigma_i^\parallel = R_i\,\mathrm{diag}(0,s_2,s_3)\,R_i^T.$$5 Gaussians. On a 2.4 GHz simulation with 50 training samples, RF-PGS reports $$\Sigma_i^\parallel = R_i\,\mathrm{diag}(0,s_2,s_3)\,R_i^T.$$6, $$\Sigma_i^\parallel = R_i\,\mathrm{diag}(0,s_2,s_3)\,R_i^T.$$7, and $$\Sigma_i^\parallel = R_i\,\mathrm{diag}(0,s_2,s_3)\,R_i^T.$$8, versus RF-3DGS with $$\Sigma_i^\parallel = R_i\,\mathrm{diag}(0,s_2,s_3)\,R_i^T.$$9, $L_s=\sum_i \min(s_1,s_2,s_3)\longrightarrow 0.$0, and $L_s=\sum_i \min(s_1,s_2,s_3)\longrightarrow 0.$1. The same paper states that RF-PGS reconstructs Tx-side path-loss spectra up to $L_s=\sum_i \min(s_1,s_2,s_3)\longrightarrow 0.$2 fidelity using only Rx-side supervision, achieves $L_s=\sum_i \min(s_1,s_2,s_3)\longrightarrow 0.$3 of the theoretical MIMO capacity with only 10 training samples in the 2.4 GHz setup, and after 30 s fine-tuning on 100 real 60 GHz NIST samples recovers about $L_s=\sum_i \min(s_1,s_2,s_3)\longrightarrow 0.$4 PSNR and $L_s=\sum_i \min(s_1,s_2,s_3)\longrightarrow 0.$5 SSIM from the direct simulation-to-field gap (Zhang et al., 23 Aug 2025).

In scene reconstruction, 3D Gaussian Flats reports on ScanNet++ an $L_s=\sum_i \min(s_1,s_2,s_3)\longrightarrow 0.$6, $L_s=\sum_i \min(s_1,s_2,s_3)\longrightarrow 0.$7, $L_s=\sum_i \min(s_1,s_2,s_3)\longrightarrow 0.$8, $L_s=\sum_i \min(s_1,s_2,s_3)\longrightarrow 0.$9, $\mathbf x_g\in\mathbb R^3$0, and $\mathbf x_g\in\mathbb R^3$1 Gaussians, of which $\mathbf x_g\in\mathbb R^3$2 are planar. On ScanNetv2 it reports $\mathbf x_g\in\mathbb R^3$3, $\mathbf x_g\in\mathbb R^3$4, and $\mathbf x_g\in\mathbb R^3$5. For planar-surface mesh extraction on the DSLR subset, it reports $\mathbf x_g\in\mathbb R^3$6, accuracy $\mathbf x_g\in\mathbb R^3$7, and completeness $\mathbf x_g\in\mathbb R^3$8. Its ablations also attribute large degradations to removing the mask loss, alternating plane optimization, snapping of Gaussians to planes, or relocation densification (Taktasheva et al., 19 Sep 2025).

GSPlane emphasizes mesh structure and compactness. On ScanNetV2, the cited table reports a 2DGS baseline with $\mathbf x_g\in\mathbb R^3$9-score $\Sigma_i\in\mathbb R^{3\times3}$00 and $\Sigma_i\in\mathbb R^{3\times3}$01 vertices, versus 2DGS + GSPlane with $\Sigma_i\in\mathbb R^{3\times3}$02-score $\Sigma_i\in\mathbb R^{3\times3}$03 and $\Sigma_i\in\mathbb R^{3\times3}$04 vertices. On Tanks & Temples, the cited table reports 3DGS with $\Sigma_i\in\mathbb R^{3\times3}$05-score $\Sigma_i\in\mathbb R^{3\times3}$06, planar vertices $\Sigma_i\in\mathbb R^{3\times3}$07, total vertices $\Sigma_i\in\mathbb R^{3\times3}$08, versus 3DGS + GSPlane with $\Sigma_i\in\mathbb R^{3\times3}$09-score $\Sigma_i\in\mathbb R^{3\times3}$10, planar vertices $\Sigma_i\in\mathbb R^{3\times3}$11, and total vertices $\Sigma_i\in\mathbb R^{3\times3}$12. On the top 20 ScanNet planes, the reported planar-wise Chamfer distances are about $\Sigma_i\in\mathbb R^{3\times3}$13 for PlanarRecon, about $\Sigma_i\in\mathbb R^{3\times3}$14 for AirPlanes, and about $\Sigma_i\in\mathbb R^{3\times3}$15 for 3DGS + GSPlane (Gan et al., 20 Oct 2025).

PlanarGS reports that on Replica, ScanNet++, and MuSHRoom it cuts Chamfer Distance roughly in half versus vanilla 3DGS, raises $\Sigma_i\in\mathbb R^{3\times3}$16-score at the $\Sigma_i\in\mathbb R^{3\times3}$17 threshold from about $\Sigma_i\in\mathbb R^{3\times3}$18 to about $\Sigma_i\in\mathbb R^{3\times3}$19, improves normal consistency from about $\Sigma_i\in\mathbb R^{3\times3}$20 to about $\Sigma_i\in\mathbb R^{3\times3}$21, and maintains or slightly improves novel-view PSNR/SSIM over 3DGS. PGSR, for its part, states a training time of about 30k iterations, about 1 hr on $\Sigma_i\in\mathbb R^{3\times3}$22, followed by TSDF fusion of rendered depths for mesh extraction, and characterizes itself as $\Sigma_i\in\mathbb R^{3\times3}$23 faster than NeRF in training while retaining real-time rendering in milliseconds (Jin et al., 27 Oct 2025, Chen et al., 2024).

6. Scope, misconceptions, and adjacent meanings

The dominant contemporary use of planar-adapted Gaussian representation concerns surface-aligned primitives for rendering, reconstruction, and RF propagation. Two adjacent literatures, however, use “planar” structure differently. In the adaptive fast Gauss transform, the central representation is a plane-wave spectral approximation of the Gaussian kernel rather than a surface-aligned Gaussian primitive. The Gaussian

$\Sigma_i\in\mathbb R^{3\times3}$24

is approximated by a truncated plane-wave expansion with fixed expansion length $\Sigma_i\in\mathbb R^{3\times3}$25, diagonal translations, and adaptive tree structures. The paper states that this avoids classical Hermite expansions and yields $\Sigma_i\in\mathbb R^{3\times3}$26 complexity per box (Greengard et al., 2023).

A different meaning appears in geometric modeling of plane curves. There, each segment of a polygonal approximation carries tangent direction $\Sigma_i\in\mathbb R^{3\times3}$27, normal direction $\Sigma_i\in\mathbb R^{3\times3}$28, segment length $\Sigma_i\in\mathbb R^{3\times3}$29, and user-defined normal uncertainty $\Sigma_i\in\mathbb R^{3\times3}$30, producing a Gaussian component with mean at the segment midpoint,

$\Sigma_i\in\mathbb R^{3\times3}$31

and covariance

$\Sigma_i\in\mathbb R^{3\times3}$32

The mixture weights are chosen proportional to segment length. This is a planar-adapted Gaussian representation of uncertainty on a plane curve, not a splatting model (Darijani et al., 24 May 2026).

Within the 3DGS-derived literature, a common misconception is that planar adaptation is universally beneficial. The cited papers qualify that claim. PlanarGS explicitly notes that curved or highly organic geometry sees no direct benefit, and that outdoor scenes lacking large, consistent planes or scenes where language prompts cannot easily enumerate all plane types remain challenging (Jin et al., 27 Oct 2025). 3D Gaussian Flats is motivated specifically by flat, texture-less surfaces, where purely photometric reconstruction is ill-conditioned (Taktasheva et al., 19 Sep 2025). RF-PGS, meanwhile, enlarges wedge Gaussians in high-curvature regions to capture diffraction, indicating that strict local planarity alone is not sufficient for every propagation phenomenon (Zhang et al., 23 Aug 2025).

Taken together, the literature presents planar-adapted Gaussian representation as a structured response to a specific failure mode of unconstrained Gaussian models: ambiguous geometry on flat regions, weak surface alignment, or physically uninterpretable interaction points. The cited methods differ in implementation, but they share a common objective: to bind Gaussian support more tightly to planar or locally planar geometry so that rendering, reconstruction, or propagation queries become more accurate, more structured, and more computationally useful.