Spherical Harmonics (SH) are fundamental mathematical tools for compactly representing low-frequency spherical functions in computer graphics. Evaluating products of SH-represented functions is a core operation in many rendering algorithms and often becomes a major computational bottleneck. Existing SH product methods face two key challenges: insufficient computational efficiency, and the inflexible requirement that all input functions share the same SH order. To overcome these limitations, we introduce the Generalized Spherical Harmonics Product , a novel formulation that supports inputs with different SH orders and allows flexible truncation of the output. We further propose an efficient and practical computation approach based on Spherical Grids (SphGrid). We establish explicit bidirectional conversions between SphGrid representations and SH coefficients, thus enabling SphGrid to serve as an intermediate representation. Under this framework, SH product computation is reduced to simple point-wise multiplications on the grid. We then derive sufficient conditions on the grid resolution to guarantee exact computation. Using lower resolutions yields an approximate variant that provides a favorable trade-off between accuracy and efficiency. Our approach is not only more general but also demonstrates superior performance even for the specialized case of identical input orders. In particular, our accurate method achieves a 3.5–6.0× speedup over the state-of-the-art method in a practical setting for graphics, while our approximate variant yields lower errors and a 2.5–6.5× speedup over the state-of-the-art approximate method. We rigorously analyze the correctness and efficiency of our method and demonstrate its effectiveness in real-time rendering applications.

Rendering realistic appearances of homogeneous translucent materials, such as milk and marble, poses challenges due to the complexity of subsurface scattering. In this paper, we present a neural method for real-time rendering of homogeneous translucent objects. Based on the observation that light propagation inside a highly scattered media is like a diffusion process (Stam 1995), we propose a neural data structure named diffusion block to mimic the behavior of the diffusion process. The diffusion block is built upon a recent network structure named DiffusionNet (Sharp et al. 2022) with a few modifications to adapt to our problem of translucent rendering. Our network is lightweight and efficient, leading to a real-time rendering method. Furthermore, our method supports dynamic material properties and diverse lighting conditions. Comparisons with state-of-the-art real-time translucent rendering methods demonstrate the superiority of our method in rendering quality.

Spherical Harmonics (SH) have been proven as a powerful tool for rendering, especially in real-time applications such as Precomputed Radiance Transfer (PRT). Spherical harmonics are orthonormal basis functions and are efficient in computing dot products. However, computations of triple product and multiple product operations are often the bottlenecks that prevent moderately high-frequency use of spherical harmonics. Specifically state-of-the-art methods for accurate SH triple products of order n have a time complexity of O ( n 5 ), which is a heavy burden for most real-time applications. Even worse, a brute-force way to compute k -multiple products would take O ( n 2 k ) time. In this paper, we propose a fast and accurate method for spherical harmonics triple products with the time complexity of only O ( n 3 ), and further extend it for computing k -multiple products with the time complexity of O ( kn 3 + k 2 n 2 log ( kn )). Our key insight is to conduct the triple and multiple products in the Fourier space, in which the multiplications can be performed much more efficiently. To our knowledge, our method is theoretically the fastest for accurate spherical harmonics triple and multiple products. And in practice, we demonstrate the efficiency of our method in rendering applications including mid-frequency relighting and shadow fields.

Fingerprint positioning is considered to be one of the most promising indoor positioning technique. The most significant technological bottleneck of fingerprint positioning technology is the high labor cost of the radio map establishing process. Many radio map fast establishing algorithms were proposed to solve this problem. There is no evaluation norm for these algorithms but positioning accuracy, which is supposed to be unreliable. This paper presents a radio map establishing algorithm evaluation criterion. To avoid the influence of other factors, the proposed evaluation criterion focuses on the quality of the radio map only. The Euclidean distance between radio maps was calculated to indicate the radio map quality. Simulation results show that the evaluation criterion can effectively measure the quality of radio maps. Some typical radio map establishing algorithms are also compared in this paper based on the proposed evaluation criterion.

Fingerprint localization algorithm is widely used in indoor scenario positioning. Received Signal Strength (RSS) is commonly used as the signal feature of fingerprint algorithm. But RSS could be changed due to the temperature or environment reason, which makes it an unreliable signal feature. This paper proposes a novel positioning algorithm based on the formal K Nearest Neighbor (KNN) algorithm. The proposed algorithm enhances the robustness of positioning system take example by the Receiver Autonomous Integrity Monitoring (RAIM) receive technique of Global Navigation Satellite System (GNSS) technologies. Simulation shows the novel algorithm is helpful to avoid the influence of RSS dramatic changes.