LegendreImage: Generalized Fourier Expansion Splatting for Image Representation

Existing 2D Gaussian image representations mainly scale their representations through the number and spatial allocation of primitives, while the function space represented by each individual primitive remains comparatively limited. We revisit primitive representation and investigate whether richer local function spaces can provide an additional dimension of representation design. We introduce Generalized Fourier Expansion Splatting (GFES), which views the signal carried by each primitive as a function over an intrinsic domain and parameterizes it through a truncated generalized Fourier expansion. This formulation separates the representation of the primitive-carried signal from its spatial localization: a spatial weighting function determines where the primitive contributes, while the function expansion determines how its signal varies over the corresponding domain. The basis is chosen according to the domain, measure, and function space of the represented signal. Following this principle, we instantiate GFES as LegendreImage by combining compact raised-cosine localization with a two-dimensional Legendre polynomial expansion to represent spatially varying color within individual primitives. Adaptive primitive allocation yields LegendreImage+. On DIV2K, LegendreImage+ surpasses GaussianImage++ from the 2.4M-parameter regime onward and reaches 51.83 dB at a 4M trainable-parameter budget, outperforming GaussianImage++ by 3.77 dB. These results show that the per-primitive function space is an important dimension of representation design, complementary to scaling through the number and spatial allocation of primitives.