Spectral Optimization of kD-Sample Points for Integrands in Realtime-Path Tracing

几年来，蒙特卡罗光线跟踪方法，如路径跟踪，是离线渲染的标准，也是实时渲染的未来。然而，在这种情况下，每张图像的有限计算时间使得鲁棒和高效的渲染算法成为必要，这些算法能够产生具有低方差和高时间稳定性的图像，每个像素只有很少甚至单个计算光传输路径。高效蒙特卡罗方法的最重要组成部分——仅次于重要性采样、去噪和离群值去除，近年来取得了很大进展——是生成用于数值积分的样本点集，这能够减少渲染图像中的感知积分误差。为此，提议的项目应超越已知的分层或低差异采样技术（例如，在准蒙特卡罗方法中使用），并探索更适合实时光线追踪的新方法。在这种情况下，与离线方法相比，不可能在每个单个像素内争取蒙特卡罗积分的收敛。相反，在整个积分维度和局部像素邻域内良好的样本点分布，从而减少图像空间中低频分量的误差，是至关重要的。如果随后应用后滤波去噪，这也允许最好的方差减少和低偏差。良好样本分布的必要性以及相对较低的维数激发了分布的预计算，相比之下，由于路径长度可变和较高，这对离线渲染没有吸引力。在图像空间中，采样点的分布与误差频谱之间的相互作用是复杂的，有待于我们的研究。为此，我们将分析和模拟实际的被积函数。对于中等维度，这可以通过使用仪器光线跟踪实现的被积体密集采样的傅立叶变换来实现。结果光谱的可视化将提供对现实积分结构的见解，并有助于开发通用模型，这些模型以后不再需要对特定的3D场景进行采样。对给定样本点分布的图像空间中的误差频谱的预测使我们能够优化样本点的频谱，从而减轻误差的频率成分，这些频率成分被认为是分散的，或者很难或不可能通过后滤波去除。根据这些光谱优化采样点的方法也是本项目的主题。目标是所有这些昂贵的预计算只对每一类积分执行一次，并且在运行时可以简单地从平铺纹理中读取预计算的样本点。

Since several years, Monte Carlo-raytracing methods, such as path tracing, are standard in offline rendering, and the future of real-time rendering. Limited computation time per image in this scenario, however, make robust and efficient rendering algorithms necessary which are able to produce images with low variance and high temporal stability with only few, or even a single, computed light transport path per pixel. The most important ingredient of efficient Monte Carlo-methods -- next to importance sampling, denoising and outlier removal, where much progress has been made in recent years -- is the generation of sets of sample points for the numeric integration, which are able to reduce the perceived integration error in rendered images.For this, the proposed project shall exceed the known techniques for stratification or low-discrepancy sampling (e.g. used in quasi-Monte Carlo methods) and explore new approaches which are better suited for real-time raytracing. In this setting, in contrast to offline methods, it is not possible to strive for convergence of the Monte Carlo-integration within every single pixel. Instead, a good distribution of sample points across the integration dimensions and in a local pixel neighborhood, and by this the reduction of low-frequency components of the error in image space, is of utmost importance. This also allows for a best possible variance reduction and low bias if post-filtering for denoising is applied subsequently. The necessity of good sample distributions and at the same time the relatively low dimensionality motivates a precomputation of distributions, which, in contrast, would not be attractive for offline rendering due to variable and high path lengths.The interplay between a distribution of sample points and the frequency spectrum of the error in image space is complex and subject to our studies. For this, we will analyze and model realistic integrands. For moderate dimensionality, this can be achieved using Fourier transformations of dense sampling of the integrand performed with instrumented raytracing implementations. Visualizations of the resulting spectra will provide insights in the structure of realistic integrands and help to develop generic models, which later will not require sampling of specific 3D scenes anymore.A prediction of the spectrum of the error in image space for a given distribution of sample points enables us to optimize the spectrum of sample points such that frequency components of the error, which are perceived as distracting, or are difficult or impossible to remove with post-filtering, are mitigated. Optimization approaches for sample points according to these spectra are also subject of this project. The goal is that all these costly precomputations are only performed once for each class of integrands, and at runtime precomputed sample points can simply be read from tileable textures.