Direct Conditional Parameterization for N-Dimensional Splatting

N-dimensional splatting extends 3D Gaussian Splatting with view direction and time to model view-dependent and dynamic effects. Each primitive is lifted into a joint distribution over position and conditioning variables, then conditionally sliced at each rendering query. N-DGS and UBS share this step across Gaussian and Beta kernels: recovering the conditional mean, opacity, and covariance requires per-primitive matrix inversions and regression products in every forward pass. We introduce direct conditional parameterization, replacing this covariance-derived readout with explicit per-effect parameters: a Cholesky precision factor for opacity and a factorized query-to-position regression operator with learnable per-dimension coupling. Unrestricted conditional coordinates recover a valid joint covariance; our factorization adds spatial-scale and coupling priors. Applied to both kernels, the parameterization yields dGS (Gaussian) and dBS (Beta). Across five static and dynamic benchmarks, they match or exceed their covariance-derived counterparts on quality, with gains up to +1.26 dB in the main MCMC evaluation. Production slicing is 6.9-7.7 times faster than on-the-fly covariance-derived conditioning; complete-render profiling gives 1.25-1.91 times speedups on four trained scenes. Direct conditioning provides a structured, kernel-agnostic alternative for N-dimensional splatting. Project page: https://gaozhongpai.github.io/dgs/.