Refining boundary sample space for differentiable rendering

The objective of physics-based differentiable rendering (PBDR) is to propagate gradients between scene parameters and the intensities of image pixels in a manner that is physically correct. The gradients obtained can be applied in optimization algorithms for the reconstruction of 3D geometry or materials, or they can be further propagated into neural network to learn neural representations of the scene. However, applying automatic differentiation techniques directly to the primary rendering process will result in biased gradients, as the rendering integral contains moving high-dimensional discontinuities. Based on how these discontinuities are managed—either implicitly or explicitly—existing PBDR methods can be categorized into two groups: reparameterization methods and boundary sampling methods. Boundary sampling methods need to construct paths that have one segment tangent to the geometry being differentiated in order to estimate a boundary integral to address the discontinuities explicitly. Such paths are usually constructed by sampling the tangent segment first and then extending it to complete the paths for subsequent processing. Fortunately, the number of dimensions in the space composed of such tangent segments is only three. In scenes comprised solely of triangle meshes, the first dimension is used to parameterize all the edges on the mesh, which determines a point on the tangent segment. The remaining two dimensions are used to parameterize the direction of the tangent segment. However, state-of-the-art boundary sampling methods parameterize the first dimension uniformly, which is inefficient because only a small portion of the edges contributes to the boundary integral, resulting in wasted parameter space. In this paper, we parameterize the first dimension by considering both edge length and contributions, thereby allocating more parameter space to important edges. Experiments demonstrate that our methods achieve lower variance gradients in the forward differentiable rendering process and improved geometry reconstruction quality in the inverse rendering results.