TayloRF: Approximate Volume Integral Using Explicit Polynomials Representation

Neural Radiance Fields (NeRFs) have gained popularity by demonstrating impressive capabilities in synthesizing realistic novel views from multiple view images. It approximates the continuous integration of rays as a finite Riemann sum of the estimated colors and densities of the sampled points. Although this allows for efficient rendering, approximating divergent integrals under varying directions and interval lengths with piecewise constant features does not account for high-order variations within integration intervals, leading to ambiguous representations and limited reconstruction quality. In this paper, we propose to model the distribution within each interval using explicit Taylor-like explicit polynomials parameterized by spatial gradients. These polynomials encode the length and direction information of integrals, helping to disambiguate interval distributions and mitigate integral approximation errors in volume rendering. Our framework includes a learnable gradient estimator, regularized via an interval boundary loss to ensure numerical consistency and smoothness, alongside an adaptive interval scaling module that enhances stability and handles varying interval sizes. Our proposed method allows an easy integration with existing NeRF-based rendering frameworks. Experimental results on both synthetic and real-world scenes demonstrate that our method significantly boosts the rendering quality of various NeRF models, achieving state-of-the-art performance.