Differentiable volume rendering enables gradient-based optimization of volumetric scenes, but unbiased estimators suffer from high gradient variance. We observe that the extinction gradients split into two components on structurally different integration domains: a scattering term evaluated at a single path vertex, and a transmittance term integrated along the ray segment. Because the domains are mismatched, existing estimators sample the two components at different locations, leaving the negative correlation between their opposite-signed contributions unexploited. We expose this overlooked correlation and exploit it through a principle we call sample matching : evaluate both components at shared sample locations. To enable this, we derive the first reformulation of the differential path integral that couples the two contributions within a single integrand, yielding an unbiased Monte Carlo estimator that ties them together by construction. For efficiency, the estimator reuses partially sampled light paths and amortizes in-scattering cost by evaluating gradients at multiple probe points per segment. On voxel-grid reconstruction, our estimator reduces gradient variance by up to 80% over differential ratio tracking (DRT), yielding faster convergence and higher reconstruction quality.

2D Gaussian Splatting has recently emerged as a significant method in 3D reconstruction, enabling novel view synthesis and geometry reconstruction simultaneously. While the well-known Gaussian kernel is broadly used, its lack of anisotropy and deformation ability leads to dim and vague edges at object silhouettes, limiting the reconstruction quality of current Gaussian splatting methods. To enhance the representation power, we draw inspiration from quantum physics and propose to use the Gaussian-Hermite kernel as the new primitive in Gaussian splatting. The new kernel takes a unified mathematical form and extends the Gaussian function, which serves as the zero-rank special case in the updated general formulation. Our experiments demonstrate that the proposed Gaussian-Hermite kernel achieves improved performance over traditional Gaussian Splatting kernels on both geometry reconstruction and novel-view synthesis tasks. Specifically, on the DTU dataset, our method yields more accurate geometry reconstruction, while on datasets such as MipNeRF360 and our customized Detail dataset, it achieves better results in novel-view synthesis. These results highlight the potential of the Gaussian-Hermite kernel for high-quality 3D reconstruction and rendering.

In this paper, we propose the reduced model for the full dynamics of a bicycle and analyze its nonlinear behavior under a proportional control law for steering. Based on the Gibbs-Appell equations for the Whipple bicycle, we obtain a second-order nonlinear ordinary differential equation (ODE) that governs the bicycle’s controlled motion. Two types of equilibrium points for the governing equation are found, which correspond to the bicycle’s uniform straight forward and circular motions, respectively. By applying the Hurwitz criterion to the linearized equation, we find that the steer coefficient must be negative, consistent with the human’s intuition of turning toward a fall. Under this condition, a critical angular velocity of the rear wheel exists, above which the uniform straight forward motion is stable, and slightly below which a pair of symmetrical stable uniform circular motions will occur. These theoretical findings are verified by both numerical simulations and experiments performed on a powered autonomous bicycle.