KC-3DGS: Kurtosis-Constrained Gaussian Splatting for High-Fidelity View Synthesis

3D Gaussian Splatting (3DGS) enables real-time novel view synthesis by representing scenes as collections of anisotropic Gaussians optimized via differentiable rasterization. However, standard pixel-space losses (L1, SSIM) constrain only aggregate reconstruction error, permitting the optimization to redistribute error across frequency scales. This leads to oversmoothing and structural artifacts, particularly in sparse-view settings where supervision is limited. We propose KC-3DGS, which augments 3DGS training with wavelet-domain supervision based on natural image statistics. Our method combines three components: (1) a multi-scale wavelet coefficient alignment loss that explicitly penalizes missing high-frequency detail, (2) a supervised kurtosis concentration loss that encourages rendered images to match the heavy-tailed frequency statistics of ground-truth images, and (3) a cross-band covariance penalty that promotes frequency specialization. We provide theoretical analysis showing that pixel-space losses admit a family of indistinguishable perturbations under wavelet redistribution, and that our joint objective excludes degenerate solutions. Experiments across MipNeRF360, Tanks&Temples, MVImgNet, DeepBlending, and WRIVA-ULTRRA demonstrate consistent improvements in perceptual quality. On the challenging WRIVA-ULTRRA outdoor dataset, KC-3DGS achieves a 9.48% improvement in DreamSim while also improving PSNR, SSIM, and LPIPS. In sparse-view settings with only 12 training images, our method improves PSNR by up to 0.5 dB on MipNeRF360 while maintaining perceptual quality. The approach integrates seamlessly into existing 3DGS pipelines as a plug-and-play regularization strategy.