A Simple Approach to Differentiable Rendering of SDFs

TL;DR

This paper tackles the non-differentiability of physically based rendering caused by visibility boundaries by introducing a boundary-relaxation strategy that widens sampling from the exact silhouette boundary $\mathcal{B}$ to a thin nearby region $\mathcal{A}$ around the surface represented by Signed Distance Fields (SDF). It derives a relaxed-boundary integral that reuses forward samples and scales with $\varepsilon$, yielding a simple, low-variance differentiable renderer suitable for end-to-end inverse rendering tasks. The approach achieves competitive or superior results compared to prior methods in both forward derivative accuracy and end-to-end reconstruction quality, while maintaining architectural simplicity and broad applicability across shapes and scales. The work highlights a practical bias-variance tradeoff controlled by $\varepsilon$, demonstrating robust performance across scenes and suggesting avenues for extending the framework to other representations and improved bias handling in future work.

Abstract

We present a simple algorithm for differentiable rendering of surfaces represented by Signed Distance Fields (SDF), which makes it easy to integrate rendering into gradient-based optimization pipelines. To tackle visibility-related derivatives that make rendering non-differentiable, existing physically based differentiable rendering methods often rely on elaborate guiding data structures or reparameterization with a global impact on variance. In this article, we investigate an alternative that embraces nonzero bias in exchange for low variance and architectural simplicity. Our method expands the lower-dimensional boundary integral into a thin band that is easy to sample when the underlying surface is represented by an SDF. We demonstrate the performance and robustness of our formulation in end-to-end inverse rendering tasks, where it obtains results that are competitive with or superior to existing work.

Figures (10)

• Figure 1: Normal velocity. Each direction $\omega$ in the visibility boundary $\mathcal{B}$ corresponds to a distant silhouette point $\mathbf{x}^*$, where the associated ray grazes an occluder. The dot products of the velocity and the normal give the normal velocities in each space, which are inversely proportional to $||\mathbf{x}-\mathbf{x}^*||$.
• Figure 2: Relaxation. We color image plane samples whose corresponding $\mathbf{y}$s satisfy the relaxation conditions blue/green to visualize the sampling of the silhouette after relaxation. (a) Relaxing the directional derivative condition leads to rays that intersect with the surface almost tangentially. (b) Relaxing the SDF condition leads to rays that graze the surface with no intersection. Here we set $\varepsilon_1 = 0.2$ and $\varepsilon_2 = 0.002$. We see that relaxing the SDF condition leads to more uniform samples around the silhouette.
• Figure 3: Relaxed silhouette point. Extending along the normal of a silhouette point $\mathbf{x}^*$ gives a set of relaxed silhouette points $\mathbf{y}^*$ (red curve) that can be approximated to first order by a line segment. A relaxed silhouette point $\mathbf{y}^*$ can be seen as a silhouette point of the $\lambda$-level set, where $\lambda = \mathop{\mathrm{SDF}}\nolimits(\mathbf{y}^*)$.
• Figure 4: Forward derivative. We use forward-mode differentiation to compute the derivative of the rendered image with respect to a translation along the $x$ axis. We test different shapes under different sizes of square area light and compare with SDF convolution bangaru2020warpedsampling, SDF Reparameterization Vicini2022sdf, and mesh Projective Sampling Zhang2023Projective. For all tests we use a direct integrator with $512\times512$ resolution and $1024$ spp.
• Figure 5: Higher-order light transport. We place a metallic bunny in front of a mirror to validate our differentiable renderer on higher-order light transport. In the first row, we show our rendering under different maximum depths. In the second and third row, we compute the forward derivatives with respect to a translation in the $y$ axis.