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  1. We propose a set of techniques to efficiently importance sample the derivatives of a wide range of Bidirectional Reflectance Distribution Function (BRDF) models. In differentiable rendering, BRDFs are replaced by their differential BRDF counterparts, which are real-valued and can have negative values. This leads to a new source of variance arising from their change in sign. Real-valued functions cannot be perfectly importance sampled by a positive-valued PDF, and the direct application of BRDF sampling leads to high variance. Previous attempts at antithetic sampling only addressed the derivative with the roughness parameter of isotropic microfacet BRDFs. Our work generalizes BRDF derivative sampling to anisotropic microfacet models, mixture BRDFs, Oren-Nayar, Hanrahan-Krueger, among other analytic BRDFs.

    Our method first decomposes the real-valued differential BRDF into a sum of single-signed functions, eliminating variance from a change in sign. Next, we importance sample each of the resulting single-signed functions separately. The first decomposition, positivization, partitions the real-valued function based on its sign, and is effective at variance reduction when applicable. However, it requires analytic knowledge of the roots of the differential BRDF, and for it to be analytically integrable too. Our key insight is that the single-signed functions can have overlapping support, which significantly broadens the ways we can decompose a real-valued function. Our product and mixture decompositions exploit this property, and they allow us to support several BRDF derivatives that positivization could not handle. For a wide variety of BRDF derivatives, our method significantly reduces the variance (up to 58× in some cases) at equal computation cost and enables better recovery of spatially varying textures through gradient-descent-based inverse rendering.

     
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    Free, publicly-accessible full text available June 30, 2025
  2. Computations in physical simulation, computer graphics, and probabilistic inference often require the differentiation of discontinuous processes due to contact, occlusion, and changes at a point in time. Popular differentiable programming languages, such as PyTorch and JAX, ignore discontinuities during differentiation. This is incorrect forparametric discontinuities—conditionals containing at least one real-valued parameter and at least one variable of integration. We introduce Potto, the first differentiable first-order programming language to soundly differentiate parametric discontinuities. We present a denotational semantics for programs and program derivatives and show the two accord. We describe the implementation of Potto, which enables separate compilation of programs. Our prototype implementation overcomes previous compile-time bottlenecks achieving an 88.1x and 441.2x speed up in compile time and a 2.5x and 7.9x speed up in runtime, respectively, on two increasingly large image stylization benchmarks. We showcase Potto by implementing a prototype differentiable renderer with separately compiled shaders.

     
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    Free, publicly-accessible full text available April 29, 2025
  3. Physics-based differentiable rendering is becoming increasingly crucial for tasks in inverse rendering and machine learning pipelines. To address discontinuities caused by geometric boundaries and occlusion, two classes of methods have been proposed: 1) the edge-sampling methods that directly sample light paths at the scene discontinuity boundaries, which require nontrivial data structures and precomputation to select the edges, and 2) the reparameterization methods that avoid discontinuity sampling but are currently limited to hemispherical integrals and unidirectional path tracing.

    We introduce a new mathematical formulation that enjoys the benefits of both classes of methods. Unlike previous reparameterization work that focused on hemispherical integral, we derive the reparameterization in the path space. As a result, to estimate derivatives using our formulation, we can apply advanced Monte Carlo rendering methods, such as bidirectional path tracing, while avoiding explicit sampling of discontinuity boundaries. We show differentiable rendering and inverse rendering results to demonstrate the effectiveness of our method.

     
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  4. We introduce SLANG.D, an extension to the Slang shading language that incorporates first-class automatic differentiation support. The new shading language allows us to transform a Direct3D-based path tracer to be fully differentiable with minor modifications to existing code. SLANG.D enables a shared ecosystem between machine learning frameworks and pre-existing graphics hardware API-based rendering systems, promoting the interchange of components and ideas across these two domains.

    Our contributions include a differentiable type system designed to ensure type safety and semantic clarity in codebases that blend differentiable and non-differentiable code, language primitives that automatically generate both forward and reverse gradient propagation methods, and a compiler architecture that generates efficient derivative propagation shader code for graphics pipelines. Our compiler supports differentiating code that involves arbitrary control-flow, dynamic dispatch, generics and higher-order differentiation, while providing developers flexible control of checkpointing and gradient aggregation strategies for best performance. Our system allows us to differentiate an existing real-time path tracer, Falcor, with minimal change to its shader code. We show that the compiler-generated derivative kernels perform as efficiently as handwritten ones. In several benchmarks, the SLANG.D code achieves significant speedup when compared to prior automatic differentiation systems.

     
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  5. Emerging research in computer graphics, inverse problems, and machine learning requires us to differentiate and optimize parametric discontinuities. These discontinuities appear in object boundaries, occlusion, contact, and sudden change over time. In many domains, such as rendering and physics simulation, we differentiate the parameters of models that are expressed as integrals over discontinuous functions. Ignoring the discontinuities during differentiation often has a significant impact on the optimization process. Previous approaches either apply specialized hand-derived solutions, smooth out the discontinuities, or rely on incorrect automatic differentiation. We propose a systematic approach to differentiating integrals with discontinuous integrands, by developing a new differentiable programming language. We introduce integration as a language primitive and account for the Dirac delta contribution from differentiating parametric discontinuities in the integrand. We formally define the language semantics and prove the correctness and closure under the differentiation, allowing the generation of gradients and higher-order derivatives. We also build a system, Teg, implementing these semantics. Our approach is widely applicable to a variety of tasks, including image stylization, fitting shader parameters, trajectory optimization, and optimizing physical designs. 
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