In this paper, we propose a ratio estimator of the direct-illumination equation that allows us to combine analytic illumination techniques with stochastic raytraced shadows while maintaining correctness. Our main contribution is to show that the shadowed illumination can be split into the product of the unshadowed illumination and the illumination-weighted shadow. These terms can be computed separately — possibly using different techniques — without affecting the exactness of the final result given by their product. This formulation broadens the utility of analytic illumination techniques to raytracing applications, where they were hitherto avoided because they did not incorporate shadows. We use such methods to obtain sharp and noise-free shading in the unshadowed-illumination image and we compute the weighted-shadow image with stochastic raytracing. The advantage of restricting stochastic evaluation to the weighted-shadow image is that the final result exhibits noise only in the shadows. Furthermore, we denoise shadows separately from illumination so that even aggressive denoising only overblurs shadows, while high-frequency shading details (textures, normal maps, etc.) are preserved.
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