Monte Carlo integration is widely used in computer graphics, especially in rendering, but we identify two key limitations. First, for each sample, we can often obtain rich auxiliary information, such as sample position, or geometric information. However, a classical Monte Carlo estimator cannot effectively use this information and only averages the function values. Second, the Monte Carlo formulation makes it difficult to adapt the sampling distribution toward truly informative regions, which can be critical for reconstructing the signal. To address these limitations, we argue that a sampler and integrator beyond the standard Monte Carlo methods is needed. We therefore propose an end-to-end sampling-integration approach that jointly learns both a sampler and an integrator using neural networks, enabling samples to be drawn and used in a more coupled and principled manner. By training on a dataset of integrands, the estimator can further use learned priors over integrand structure and specialize to a family of problems. We evaluate our method on diverse applications in lighting, transmittance, generalized winding number, and walk-on-spheres, spanning both linear and nonlinear cases, and a broad range of low- and high-dimensional settings. Even though the networks add computational overheads, in the equal-sample setting, our method achieves substantial improvements, providing a powerful alternative to traditional quadrature rules and sampling methods.

Integral linear operators play a key role in many graphics problems, but solutions obtained via Monte Carlo methods often suffer from high variance. A common strategy to improve the efficiency of integration across various inputs is to precompute the kernel function. Traditional methods typically rely on basis expansions for both the input and output functions. However, using fixed output bases can restrict the precision of output reconstruction and limit the compactness of the kernel representation. In this work, we introduce a new method that approximates both the kernel and the input function using Gaussian mixtures. This formulation allows the integral operator to be evaluated analytically, leading to improved flexibility in kernel storage and output representation. Moreover, our method naturally supports the sequential application of multiple operators and enables closed-form operator composition, which is particularly beneficial in tasks involving chains of operators. We demonstrate the versatility and effectiveness of our approach across a variety of graphics problems, including environment map relighting, boundary value problems, and fluorescence rendering.

Variance reduction techniques are widely used for reducing the noise of Monte Carlo integration. However, these techniques are typically designed with the assumption that the integrand is scalar-valued. Recognizing that rendering and inverse rendering broadly involve vector-valued integrands, we identify the limitations of classical variance reduction methods in this context. To address this, we introduce ratio control variates, an estimator that leverages a ratio-based approach instead of the conventional difference-based control variates. Our analysis and experiments demonstrate that ratio control variables can significantly reduce the mean squared error of vector-valued integration compared to existing methods and are broadly applicable to various rendering and inverse rendering tasks.

We propose a real-time path guiding method, Voxel Path Guiding (VXPG), that significantly improves fitting efficiency under limited sampling budget. Our key idea is to use a spatial irradiance voxel data structure across all shading points to guide the location of path vertices. For each frame, we first populate the voxel data structure with irradiance and geometry information. To sample from the data structure for a shading point, we need to select a voxel with high contribution to that point. To importance sample the voxels while taking visibility into consideration, we adapt techniques from offline many-lights rendering by clustering pairs of shading points and voxels. Finally, we unbiasedly sample within the selected voxel while taking the geometry inside into consideration. Our experiments show that VXPG achieves significantly lower perceptual error compared to other real-time path guiding and virtual point light methods under equal-time comparison. Furthermore, our method does not rely on temporal information, but can be used together with other temporal reuse sampling techniques such as ReSTIR to further improve sampling efficiency.

Tactic analysis is a major issue in badminton as the effective usage of tactics is the key to win. The tactic in badminton is defined as a sequence of consecutive strokes. Most existing methods use statistical models to find sequential patterns of strokes and apply 2D visualizations such as glyphs and statistical charts to explore and analyze the discovered patterns. However, in badminton, spatial information like the shuttle trajectory, which is inherently 3D, is the core of a tactic. The lack of sufficient spatial awareness in 2D visualizations largely limited the tactic analysis of badminton. In this work, we collaborate with domain experts to study the tactic analysis of badminton in a 3D environment and propose an immersive visual analytics system, TIVEE, to assist users in exploring and explaining badminton tactics from multi-levels. Users can first explore various tactics from the third-person perspective using an unfolded visual presentation of stroke sequences. By selecting a tactic of interest, users can turn to the first-person perspective to perceive the detailed kinematic characteristics and explain its effects on the game result. The effectiveness and usefulness of TIVEE are demonstrated by case studies and an expert interview.