KC-3DGS: Kurtosis-Constrained 3DGS

• KC-3DGS is a training objective for 3D Gaussian Splatting that uses multi-scale wavelet alignment and supervised kurtosis concentration to preserve high-frequency detail.
• It incorporates a cross-band covariance penalty to control error distribution across frequency subbands, mitigating oversmoothing and structural artifacts.
• KC-3DGS serves as a plug-and-play regularizer for existing 3DGS pipelines, improving reconstruction quality without altering the inference process.

KC-3DGS, short for Kurtosis-Constrained Gaussian Splatting, is a training objective for 3D Gaussian Splatting (3DGS) designed to improve high-fidelity novel view synthesis by augmenting conventional pixel-space supervision with wavelet-domain constraints derived from natural image statistics. It targets a specific limitation of standard 3DGS optimization: losses such as $L1$ and SSIM constrain aggregate reconstruction error but do not determine how that error is distributed across frequency subbands, which permits oversmoothing and structural artifacts, especially under sparse-view supervision. KC-3DGS addresses this by combining multi-scale wavelet coefficient alignment, supervised kurtosis concentration, and a cross-band covariance penalty within a plug-and-play regularization framework that leaves inference unchanged and integrates into existing 3DGS pipelines (Banerjee et al., 2 Jun 2026).

1. Problem setting and representation

3D Gaussian Splatting represents a scene as a set of $N$ anisotropic 3D Gaussians,

$G=\{G_i\}_{i=1..N}.$

Each Gaussian is parameterized by a mean $\mu_i \in \mathbb{R}^3$, a covariance

$$\Sigma_i = R_i S_i S_i^T R_i^T \in \mathbb{R}^{3 \times 3},$$

an opacity $\alpha_i \in [0,1]$, and view-dependent color encoded by spherical harmonics coefficients $c_i \in \mathbb{R}^{3 \times d_{SH}}$ (Banerjee et al., 2 Jun 2026).

Given a pinhole camera with intrinsics $K$ and extrinsics $[R|t]$, the 3D mean is projected to image coordinates by

$$\mu_{\text{cam}} = R\mu_i + t,\qquad u_i = \Pi(\mu_{\text{cam}}) = (K[I|0])[\mu_{\text{cam}};1].$$

Its screen-space footprint has covariance

$N$0

where $N$1 is the Jacobian of the perspective projection $N$2 at $N$3. The per-pixel contribution of Gaussian $N$4 at image coordinate $N$5 is a 2D elliptical Gaussian

$N$6

View-dependent RGB is evaluated through spherical harmonics as

$N$7

with $N$8 the SH coefficients and $N$9 the SH basis (Banerjee et al., 2 Jun 2026).

Front-to-back alpha compositing, with Gaussians sorted by depth, yields the rendered color $G=\{G_i\}_{i=1..N}.$0 and alpha $G=\{G_i\}_{i=1..N}.$1:

$G=\{G_i\}_{i=1..N}.$2

$G=\{G_i\}_{i=1..N}.$3

Standard training minimizes a pixel-space distortion loss

$G=\{G_i\}_{i=1..N}.$4

optionally together with adaptive density control (ADC) to split or prune Gaussians (Banerjee et al., 2 Jun 2026).

The motivation for KC-3DGS is that these pixel-space losses constrain only aggregate per-pixel error. Because the 2D discrete wavelet transform is orthonormal and energy preserving, the same error energy can be redistributed across frequency subbands without changing the value of $G=\{G_i\}_{i=1..N}.$5, while SSIM is described as largely insensitive to precise spectral allocation of error. This permits optimization to suppress or smear high-frequency content and compensate with low-frequency deviations, producing oversmoothing and structural artifacts. The effect is reported as particularly acute in sparse-view setups, where weak gradients and limited multi-view constraints can cause ADC to spawn floaters and prune fine-detail Gaussians prematurely (Banerjee et al., 2 Jun 2026).

2. Wavelet-domain formulation and loss components

KC-3DGS augments $G=\{G_i\}_{i=1..N}.$6 with wavelet-domain supervision based on natural image statistics. The method applies a $G=\{G_i\}_{i=1..N}.$7-level 2D discrete wavelet transform $G=\{G_i\}_{i=1..N}.$8 per RGB channel. For an image $G=\{G_i\}_{i=1..N}.$9,

$\mu_i \in \mathbb{R}^3$0

where $\mu_i \in \mathbb{R}^3$1 is the coarsest lowpass approximation band and $\mu_i \in \mathbb{R}^3$2 are detail subbands at scale $\mu_i \in \mathbb{R}^3$3, with $\mu_i \in \mathbb{R}^3$4 the finest scale. KC-3DGS uses a 3-level orthonormal 2D DWT with Daubechies-3 filters, applied independently to RGB channels, with symmetric extension at boundaries (Banerjee et al., 2 Jun 2026).

The first component is multi-scale wavelet coefficient alignment. Predicted and ground-truth detail coefficients are denoted $\mu_i \in \mathbb{R}^3$5 and $\mu_i \in \mathbb{R}^3$6. KC-3DGS explicitly penalizes per-band reconstruction errors:

$\mu_i \in \mathbb{R}^3$7

The reported implementation uses $\mu_i \in \mathbb{R}^3$8 and $\mu_i \in \mathbb{R}^3$9, summarized for RGB as

$$\Sigma_i = R_i S_i S_i^T R_i^T \in \mathbb{R}^{3 \times 3},$$0

The stated rationale is to counteract the $$\Sigma_i = R_i S_i S_i^T R_i^T \in \mathbb{R}^{3 \times 3},$$1 energy falloff of natural images by giving greater weight to finer scales (Banerjee et al., 2 Jun 2026).

The second component is supervised kurtosis concentration. For a set of coefficients $$\Sigma_i = R_i S_i S_i^T R_i^T \in \mathbb{R}^{3 \times 3},$$2, excess kurtosis is defined as

$$\Sigma_i = R_i S_i S_i^T R_i^T \in \mathbb{R}^{3 \times 3},$$3

with

$$\Sigma_i = R_i S_i S_i^T R_i^T \in \mathbb{R}^{3 \times 3},$$4

KC-3DGS computes per-subband excess kurtosis and a kurtosis spread

$$\Sigma_i = R_i S_i S_i^T R_i^T \in \mathbb{R}^{3 \times 3},$$5

The supervised loss matches both bandwise kurtosis and global spread:

$$\Sigma_i = R_i S_i S_i^T R_i^T \in \mathbb{R}^{3 \times 3},$$6

The reported configuration uses $$\Sigma_i = R_i S_i S_i^T R_i^T \in \mathbb{R}^{3 \times 3},$$7 and $$\Sigma_i = R_i S_i S_i^T R_i^T \in \mathbb{R}^{3 \times 3},$$8. In practice, coefficients are standardized per band before computing kurtosis so that the statistic is scale-free (Banerjee et al., 2 Jun 2026).

The third component is the cross-band covariance penalty, introduced as a frequency-specialization term. A $$\Sigma_i = R_i S_i S_i^T R_i^T \in \mathbb{R}^{3 \times 3},$$9 matrix of flattened, centered band responses is formed from predicted coefficients, with $\alpha_i \in [0,1]$0. Writing

$\alpha_i \in [0,1]$1

the empirical covariance is

$\alpha_i \in [0,1]$2

An unsupervised version would penalize

$\alpha_i \in [0,1]$3

but KC-3DGS uses a supervised variant that compares the off-diagonal covariance magnitude against the corresponding ground-truth covariance $\alpha_i \in [0,1]$4:

$\alpha_i \in [0,1]$5

This design is reported as more robust to genuine cross-band correlations, such as those produced by corners, in the ground truth (Banerjee et al., 2 Jun 2026).

The overall objective is

$\alpha_i \in [0,1]$6

Typical values reported for WRIVA-ULTRRA with FasterGS are $\alpha_i \in [0,1]$7, $\alpha_i \in [0,1]$8, and $\alpha_i \in [0,1]$9; similar weights are stated to work well for Splatfacto (Banerjee et al., 2 Jun 2026).

3. Theoretical analysis

The theoretical analysis begins from the observation that pixel-space losses admit what the authors term wavelet redistribution. Let $c_i \in \mathbb{R}^{3 \times d_{SH}}$0 and let $c_i \in \mathbb{R}^{3 \times d_{SH}}$1 be an orthonormal $c_i \in \mathbb{R}^{3 \times d_{SH}}$2-level discrete wavelet transform. By Parseval identity,

$c_i \in \mathbb{R}^{3 \times d_{SH}}$3

If the detail subbands are indexed by $c_i \in \mathbb{R}^{3 \times d_{SH}}$4, then the per-subband energies

$c_i \in \mathbb{R}^{3 \times d_{SH}}$5

can vary while preserving the same total pixel loss as long as

$c_i \in \mathbb{R}^{3 \times d_{SH}}$6

This is stated as a $c_i \in \mathbb{R}^{3 \times d_{SH}}$7-dimensional family of spectrally different perturbations that produce the same $c_i \in \mathbb{R}^{3 \times d_{SH}}$8 loss; the argument is extended to $c_i \in \mathbb{R}^{3 \times d_{SH}}$9 and SSIM through norm equivalence and SSIM’s insensitivity to spectral allocation, respectively. The practical implication drawn in the paper is that pixel-space supervision alone cannot determine where reconstruction error should reside in frequency space, leaving high-frequency content underconstrained (Banerjee et al., 2 Jun 2026).

A further result states that subband supervision strictly refines pixel supervision. For orthonormal $K$0,

$K$1

with equality if and only if $K$2 has support in a single subband. The paper therefore states that $K$3, but not vice versa. This formalizes the claim that explicit subband alignment is stronger than aggregate pixel-level alignment (Banerjee et al., 2 Jun 2026).

The weighting of fine scales is justified by the assumption that natural images exhibit $K$4 spectra with $K$5. Under that model, per-coefficient energy decays across scales, so an unweighted sum over wavelet subbands is dominated by coarse-scale terms. Weighting $K$6 is described as an approximate equalization across scales that combats the under-penalization of fine-scale errors (Banerjee et al., 2 Jun 2026).

The analysis also identifies a degeneracy in unsupervised kurtosis concentration. Minimizing only the kurtosis spread,

$K$7

is said to be solved by any image with Gaussianized detail subbands, including a constant image, because such subbands have $K$8. Since natural images exhibit positive excess kurtosis in wavelet bands, the paper concludes that an unsupervised kurtosis-only objective would drive oversmoothing rather than prevent it (Banerjee et al., 2 Jun 2026).

To exclude that failure mode, the paper proves a joint identifiability statement for an objective of the form

$K$9

with $[R|t]$0 and ground-truth band kurtoses $[R|t]$1. Under those conditions, any global minimizer must satisfy positive kurtosis in all bands. The supervised KC-3DGS formulation is described as stronger still, because it anchors each band’s kurtosis directly to the ground truth. A related lemma states that, under unit marginal variances,

$[R|t]$2

which provides a principled decorrelation interpretation for the cross-band covariance penalty (Banerjee et al., 2 Jun 2026).

This suggests that KC-3DGS is not merely an empirical regularizer but an attempt to make the spectral allocation of reconstruction error identifiable under 3DGS training.

4. Integration into 3DGS pipelines

KC-3DGS is implemented as an additive loss module rather than a modification of the scene representation, rendering kernels, or adaptive density control. The reported training loop retains the baseline 3DGS optimizer and update schedule. For each training step, a batch of training views or patches is sampled; the scene is rasterized with front-to-back compositing and SH colors; ADC is run according to the baseline schedule; pixel loss is computed; wavelet coefficients are extracted for rendered and ground-truth images; the three wavelet-domain losses are evaluated; and the total loss is backpropagated to update Gaussian positions, rotations, scales, opacities, and SH coefficients (Banerjee et al., 2 Jun 2026).

The implementation uses differentiable convolutions in PyTorch to realize the DWT, so gradients flow through the wavelet transform and the differentiable rasterizer back to Gaussian parameters. Boundary handling follows symmetric extension as in standard DWT toolkits. The method is described as plug-and-play and reproducible within Nerfstudio Splatfacto, FasterGS, and OctreeGS without modifying rasterizers (Banerjee et al., 2 Jun 2026).

Several stabilization choices are reported. Wavelet depth is fixed at $[R|t]$3 with db3 filters. Kurtosis is computed after per-band standardization, with an added $[R|t]$4 to the variance for numerical stability. Minibatch statistics are used, and $[R|t]$5 norms are preferred. The same optimizer and schedules as the baseline 3DGS implementation are used, for example Adam with the default learning-rate schedule, and the same ADC cadence is preserved. KC losses are active from the start of training; no warm-up is reported as necessary (Banerjee et al., 2 Jun 2026).

The computational profile is asymmetric between training and inference. Inference is unchanged because the wavelet machinery is used only during optimization, so real-time rendering is preserved. Training adds DWT computation and per-band statistics. On MipNeRF360 with FasterGS, average per-scene training time is reported to increase from approximately 6 minutes to approximately 16 minutes, with DWT becoming the bottleneck relative to the optimized rasterizer. On Tanks&Temples with Splatfacto, training is reported as comparable, from approximately 47 minutes baseline to approximately 44 minutes with KC-3DGS. Memory overhead is described as modest, consisting primarily of storing several wavelet pyramids per GPU batch (Banerjee et al., 2 Jun 2026).

5. Experimental evaluation

The reported evaluation covers MipNeRF360, Tanks&Temples, MVImgNet, DeepBlending, and WRIVA-ULTRRA, using PSNR, SSIM, LPIPS (VGG), and DreamSim as metrics. Sparse-view setups use 12 training images for MipNeRF360, Tanks&Temples, and MVImgNet with MASt3R poses from the SPARS3R release. WRIVA-ULTRRA uses official sparse splits with 10, 15, and 50 views. Splatfacto is used for sparse-view ablations, while FasterGS is used for full-scene evaluations (Banerjee et al., 2 Jun 2026).

The strongest quantitative gains are reported on WRIVA-ULTRRA under FasterGS. DreamSim improves from $[R|t]$6 to $[R|t]$7, described as a $[R|t]$8 relative reduction, while PSNR increases from $[R|t]$9 to $$\mu_{\text{cam}} = R\mu_i + t,\qquad u_i = \Pi(\mu_{\text{cam}}) = (K[I|0])[\mu_{\text{cam}};1].$$0, SSIM from $$\mu_{\text{cam}} = R\mu_i + t,\qquad u_i = \Pi(\mu_{\text{cam}}) = (K[I|0])[\mu_{\text{cam}};1].$$1 to $$\mu_{\text{cam}} = R\mu_i + t,\qquad u_i = \Pi(\mu_{\text{cam}}) = (K[I|0])[\mu_{\text{cam}};1].$$2, and LPIPS decreases from $$\mu_{\text{cam}} = R\mu_i + t,\qquad u_i = \Pi(\mu_{\text{cam}}) = (K[I|0])[\mu_{\text{cam}};1].$$3 to $$\mu_{\text{cam}} = R\mu_i + t,\qquad u_i = \Pi(\mu_{\text{cam}}) = (K[I|0])[\mu_{\text{cam}};1].$$4 (Banerjee et al., 2 Jun 2026).

Sparse-view gains are also reported on multiple datasets. On MipNeRF360 with 12 training images and Splatfacto, PSNR improves from $$\mu_{\text{cam}} = R\mu_i + t,\qquad u_i = \Pi(\mu_{\text{cam}}) = (K[I|0])[\mu_{\text{cam}};1].$$5 to $$\mu_{\text{cam}} = R\mu_i + t,\qquad u_i = \Pi(\mu_{\text{cam}}) = (K[I|0])[\mu_{\text{cam}};1].$$6 for a gain of $$\mu_{\text{cam}} = R\mu_i + t,\qquad u_i = \Pi(\mu_{\text{cam}}) = (K[I|0])[\mu_{\text{cam}};1].$$7 dB, SSIM from $$\mu_{\text{cam}} = R\mu_i + t,\qquad u_i = \Pi(\mu_{\text{cam}}) = (K[I|0])[\mu_{\text{cam}};1].$$8 to $$\mu_{\text{cam}} = R\mu_i + t,\qquad u_i = \Pi(\mu_{\text{cam}}) = (K[I|0])[\mu_{\text{cam}};1].$$9, and DreamSim from $N$00 to $N$01 for a $N$02 improvement. On MVImgNet sparse-view, PSNR improves from $N$03 to $N$04, LPIPS from $N$05 to $N$06, and DreamSim from $N$07 to $N$08 for a $N$09 improvement. Tanks&Temples sparse-view is reported as showing similar overall performance, with DreamSim from $N$10 to $N$11 for a $N$12 change. On DeepBlending full-scene with FasterGS, DreamSim improves from $N$13 to $N$14 for a $N$15 improvement. On MipNeRF360 full-scene with FasterGS, DreamSim improves from $N$16 to $N$17 for an $N$18 improvement, while a slight drop in PSNR and SSIM is reported as a distortion–perception trade-off under dense supervision (Banerjee et al., 2 Jun 2026).

The method is further evaluated for generalization to other 3DGS variants on WRIVA-ULTRRA with OctreeGS. At 50 views, DreamSim improves from $N$19 to $N$20 for a $N$21 change; at 15 views, from $N$22 to $N$23 for $N$24; and at 10 views, from $N$25 to $N$26 for $N$27. The paper characterizes these results as evidence that gains hold across Splatfacto, FasterGS, and OctreeGS (Banerjee et al., 2 Jun 2026).

Qualitatively, KC-3DGS is reported to reduce streaks and floaters, sharpen window grids and rooftops, preserve foliage and ground-plane textures such as asphalt markings, and reduce color bleeding across WRIVA-ULTRRA and Tanks&Temples. Wavelet-band analyses are described as showing recovery of high-frequency structure in the $N$28, $N$29, and $N$30 subbands and concentration of errors on structural regions such as edges and corners, which pixel losses are said to underweight (Banerjee et al., 2 Jun 2026).

6. Ablations, interpretation, and limitations

The reported ablations distinguish the contributions of the individual terms. In sparse-view settings, adding only wavelet alignment, denoted “L1+Wavelet,” can improve PSNR and SSIM but degrade LPIPS and DreamSim. The full combination,

$N$31

is reported to provide the best perceptual trade-off. Removing scale weighting or using overly heavy scale weights degrades perceptual metrics. Excessively large $N$32 or $N$33 harms both distortion and perception metrics; examples given for WRIVA are $N$34 and $N$35 (Banerjee et al., 2 Jun 2026).

The interpretive account offered in the paper is explicitly statistical. Natural images are described as heavy-tailed in wavelet bands, with positive excess kurtosis, because sharp edges and high-frequency textures produce rare large coefficients. Pixel-space losses do not enforce that structure, so optimization may collapse the tails and blur the image. Matching per-band kurtosis and its spread is therefore presented as a mechanism for recovering sparse, heavy-tailed coefficient distributions. Likewise, the covariance penalty is motivated by the claim that redundant responses across multiple bands cause gradient interference, unstable refinement, and texture smearing; penalizing off-diagonal covariance encourages complementary band specialization while the supervised formulation preserves genuine ground-truth correlations such as those at corners (Banerjee et al., 2 Jun 2026).

The paper also identifies several limitations. Under dense coverage and strong pixel supervision, KC-3DGS can slightly reduce PSNR and SSIM while improving perceptual metrics, which it frames as a perception–distortion trade-off. Potential artifacts include amplification of sensor noise in poorly constrained regions and subtle ringing if wavelet choices and weights are aggressive. Performance is said to depend on the wavelet family, transform depth $N$36, and the loss weights. Although db3 with $N$37 is reported as robust, other configurations may be preferable for particular data. Highly specular or transparent surfaces and extreme sparsity remain difficult. Proposed future directions include adaptive scheduling of the $N$38 coefficients, learned or steerable wavelet-like transforms, scene-adaptive frequency priors, cross-scene priors, and integration with anti-aliasing approaches such as Mip-Splatting and geometry-regularized variants (Banerjee et al., 2 Jun 2026).

A plausible implication is that KC-3DGS is best understood as a frequency-aware regularization layer for 3DGS training rather than as an alternative scene representation. Its contribution lies in constraining how reconstruction error is distributed across scales and orientations, with the strongest reported gains appearing in sparse-view and real-world outdoor settings where conventional pixel losses leave high-frequency structure particularly underdetermined (Banerjee et al., 2 Jun 2026).