Triple product wavelet integrals for all-frequency relighting

This paper presents a comprehensive study of triple integrals, extending the conventional understanding of volumetric integration to encompass a broader class of functions and domains. We introduce a novel framework for defining and evaluating these integrals, particularly focusing on cases where the integrand or the region of integration exhibits discontinuities, singularities. Utilizing concepts from measure theory and functional analysis, we develop techniques for establishing the existence and uniqueness of total triple integrals under relaxed regularity conditions. A significant contribution is the establishment of a generalized total triple integrals. We further explore applications of these integrals in physics, specifically in modelling heterogeneous materials and non-equilibrium systems, where conventional integration methods prove inadequate. Triple integrals play a crucial role in various digital applications, particularly in fields like computer graphics, image processing, machine learning, and signal processing. These integrals enable the computation of volumetric properties, aiding in tasks such as 3D rendering, object recognition, and spatial data analysis. In computer graphics, triple integrals help simulate realistic lighting, shading, and textures. In image processing, they assist in reconstructing 3D models from medical scans and volumetric data. Machine learning utilizes triple integrals in probability density estimation and feature extraction from multidimensional datasets. Other applications include 3D scanning, AR/VR development, and robotics, where they contribute to path planning and spatial mapping. Overall, triple integrals are indispensable in advancing digital technologies that rely on 3D data representation and analysis.

The bispectrum is the Fourier transform of the triple correlation, sometimes also referred to as triple product integral. We are concerned here with the bispectrum of the autotriple correlation. Bispectrum analysis can be used to solve phase problems in signal processing, since the knowledge of the bispectrum of a signal usually allows one to reconstruct both amplitude and phase of the Fourier transform signal. We present mathematical proof based on the theory of analytic functions and discuss the restrictions involved. A recursive algorithm is outlined for the reconstruction from sampled data. In addition, possibilities for noise reduction by averaging redundant information will be described. Examples are included for 1-D signals.

In this paper we present the first practical method for importance sampling functions represented as spherical harmonics (SH). Given a spherical probability density function (PDF) represented as a vector of SH coefficients, our method warps an input point set to match the target PDF using hierarchical sample warping. Our approach is efficient and produces high quality sample distributions. As a by‐product of the sampling procedure we produce a multi‐resolution representation of the density function as either a spherical mip‐map or Haar wavelet. By exploiting this implicit conversion we can extend the method to distribute samples according to the product of an SH function with a spherical mip‐map or Haar wavelet. This generalization has immediate applicability in rendering, e.g., importance sampling the product of a BRDF and an environment map where the lighting is stored as a single high‐resolution wavelet and the BRDF is represented in spherical harmonics. Since spherical harmonics can be efficiently rotated, this product can be computed on‐the‐fly even if the BRDF is stored in local‐space. Our sampling approach generates over 6 million samples per second while significantly reducing precomputation time and storage requirements compared to previous techniques.

Fourier method for large-scale surface modeling and registration

The highest fidelity representations of realistic real-world materials currently used comprise Bidirectional Texture Functions (BTF). The BTF is a six dimensional function depending on view and illumination directions as well as on planar texture coordinates. The huge size of such measurements, typically in the form of thousands of images covering all possible combinations of illumination and viewing angles, has prohibited their practical exploitation and obviously some compression and modelling method of these enormous BTF data spaces is inevitable. The proposed approach combines BTF spatial clustering with cluster index modelling by means of an efficient Markov random field model. This method allows to generate seamless cluster index of arbitrary size to cover large virtual 3D objects surfaces. The method represents original BTF data using a set of local spatially dependent Bidirectional Reflectance Distribution Function (BRDF) values which are combined according to synthesised cluster index and illumination / viewing directions. BTF data compression using this method is about 1:100 and their synthesis is very fast.KeywordsContextual NeighbourTexture SynthesisViewing AngleIllumination DirectionCluster RepresentativeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In ordinary CT scan, so called full field-of-view (FFOV) scan, in which the x-ray beam span covers the whole section of the body, a large number of projections are necessary to reconstruct high resolution images. However, excessive x-ray dose is a great concern in FFOV scan. Region-of-interest (ROI) scan is a method to visualize the ROI in high resolution while reducing the x-ray dose. But, ROI scan suffers from bright-band artifacts which may hamper CT-number accuracy. In this study, we propose an image reconstruction method to eliminate the band artifacts in the ROI scan. In addition to the ROI scan with high sampling rate in the view direction, we get FFOV projection data with much lower sampling rate. Then, we reconstruct images in the compressed sensing (CS) framework with dual resolutions, that is, high resolution in the ROI and low resolution outside the ROI. For the dual-resolution image reconstruction, we implemented the dual-CS reconstruction algorithm in which data fidelity and total variation (TV) terms were enforced twice in the framework of adaptive steepest descent projection onto convex sets (ASD-POCS). The proposed method has remarkably reduced the bright-band artifacts at around the ROI boundary, and it has also effectively suppressed the streak artifacts over the entire image. We expect the proposed method can be greatly used for dual-resolution imaging with reducing the radiation dose, artifacts and scan time.

Real‐time global illumination is a highly desirable yet challenging task in computer graphics. Existing works well solving this problem are mostly based on some kind of precomputed data (caches), while the final results depend significantly on the quality of the caches. In this paper, we propose a learning‐based pipeline that can reproduce a wide range of complex light transport phenomena, including high‐frequency glossy interreflection, at any viewpoint in real time (> 90 frames per‐second), using information from imperfect caches stored at the barycentre of every triangle in a 3D scene. These caches are generated at a precomputation stage by a physically‐based offline renderer at a low sampling rate (e.g., 32 samples per‐pixel) and a low image resolution (e.g., 64×16). At runtime, a deep radiance reconstruction method based on a dedicated neural network is then involved to reconstruct a high‐quality radiance map of full global illumination at any viewpoint from these imperfect caches, without introducing noise and aliasing artifacts. To further improve the reconstruction accuracy, a new feature fusion strategy is designed in the network to better exploit useful contents from cheap G‐buffers generated at runtime. The proposed framework ensures high‐quality rendering of images for moderate‐sized scenes with full global illumination effects, at the cost of reasonable precomputation time. We demonstrate the effectiveness and efficiency of the proposed pipeline by comparing it with alternative strategies, including real‐time path tracing and precomputed radiance transfer.

Back‐face culling is a preprocessing technique used in computer graphics to speed up the rendering of polyhedra. In this paper we show how this technique can be modified to reduce unnecessary checking of boundary elements in collison detection for a physical‐based simulation and animation systems. At each time step, we determine a priori which faces cannot be part of the contact between two polyhedra and thus can be culled. In the computer graphics technique, the normal vector of a polygon is compared with the view direction. Here, the normal is compared to one or possibly several relative‐velocity vectors, and the face is culled when its motion is in the opposite direction of the normal vector.We also give an algorithm that takes linear time in terms of the number of faces, and on the average eliminates half of the polygons. Owing to its low computational overhead, when it is used as a front end to a collision detection system, a noticeable improvement in performance can be achieved.

<p>Spherical arrays facilitate processing and analysis of sound fields with the potential for high resolution in three dimensions in the spherical harmonic domain. Using the captured sound field, robust source localisation systems are required for speech acquisition, speaker tracking and environment mapping. Source localisation becomes a challenging problem in reverberant environments and under noisy conditions, leading to potentially poor performance in cocktail party scenarios. This paper evaluates the performance of a low-complexity localisation approach using spherical harmonics in reverberant environments for multiple speakers. Eigen-beams are used to estimate pseudo-intensity vectors pointing in the direction of sound intensity. This paper proposes a clustering approach in which the intensity vectors of active sound sources and strong reflections are extracted, yielding an estimate of the source direction in azimuth and inclination as an approach to source localisation.</p>

Precomputed radiance transfer (PRT) methods established for handling global illumination (GI) of objects from area lights in real time and many techniques proposed for rotating the light using linear algebra rotation matrices. Rotating area lights efficiently are crucial part for computer graphics since it is one of the main components of real‐time rendering. Matrices commonly used for handling such rotations are not quite efficient and require high memory consumption; as a result, the need for proposing new more efficient rotation algorithms has been established. In this work, we use the conformal geometric algebra (CGA) as the mathematical background for “GI in real‐time” under distant Image‐based lighting (IBL) illumination, for diffuse surfaces with self‐shadowing by efficiently rotating the environment light using CGA entities. Our work is based on spherical harmonics (SH), which are used for approximating natural, area‐light illumination as irradiance maps. Our main novelty is that we extend the PRT algorithm by representing SH for the first time with CGA.The main intuition is that SH of band index 1 are represented using CGA entities and SH with band index larger than 1 are represented in terms of CGA‐SH of band 1. Specifically, we propose a new method for representing SH with CGA entities and rotating SH by rotating CGA entities. In this way, we can visualize the SH rotations, rotate them faster than rotation matrices, and we provide a unique visual representation and intuition regarding their rotation, in stark contrast to usual rotation matrices, and we achieve consistently better visual results from Ivanic rotation matrices during light rotation. Via our CGA expressed SH, we provide a significant boost on the PRT algorithm since we represent SH rotations by CGA rotors (4 numbers) as opposed to 9 × 9 sparse matrices that are usually used. With our algorithm, we pave the way for including scaling (dilation) and translation of light coefficients using CGA motors.

Front tracking with moving-least-squares surfaces

Computing the light field due to an area light source remains an interesting problem in computer graphics. The paper presents a series approximation of the light field due to an unoccluded area source, by expanding the light field in spherical harmonics. The source can be nonuniform and need not be a planar polygon. The resulting formulas give expressions whose cost and accuracy can be chosen between the exact and expensive Lambertian solution for a diffuse polygon, and the fast but inexact method of replacing the area source by a point source of equal power. The formulas break the computation of the light vector into two phases: the first phase represents the light source's shape and brightness with numerical coefficients, and the second uses these coefficients to compute the light field at arbitrary locations. The author examines the accuracy of the formulas for spherical and rectangular Lambertian sources, and applies them to obtain light gradients. The author also shows how to use the formulas to estimate light from uniform polygonal sources, sources with polynomially varying radiosity, and luminous textures.

We present a new, real-time method for rendering diffuse and glossy objects in low-frequency lighting environments that captures soft shadows, interreflections, and caustics. As a preprocess, a novel global transport simulator creates functions over the object's surface representing transfer of arbitrary, low-frequency incident lighting into transferred radiance which includes global effects like shadows and interreflections from the object onto itself. At run-time, these transfer functions are applied to actual incident lighting. Dynamic, local lighting is handled by sampling it close to the object every frame; the object can also be rigidly rotated with respect to the lighting and vice versa. Lighting and transfer functions are represented using low-order spherical harmonics. This avoids aliasing and evaluates efficiently on graphics hardware by reducing the shading integral to a dot product of 9 to 25 element vectors for diffuse receivers. Glossy objects are handled using matrices rather than vectors. We further introduce functions for radiance transfer from a dynamic lighting environment through a preprocessed object to neighboring points in space. These allow soft shadows and caustics from rigidly moving objects to be cast onto arbitrary, dynamic receivers. We demonstrate real-time global lighting effects with this approach.

Radio-frequency simultaneous localization and mapping (RF-SLAM) methods jointly infer the position of mobile transmitters and receivers in wireless networks, together with a geometric map of the propagation environment. An inferred map of specular surfaces can be used to exploit non-line-of-sight components of the multipath channel to increase robustness, bypass obstructions, and improve overall communication and positioning performance. While performance bounds for user location are well established, the literature lacks performance bounds for map information. This paper derives the mapping error bound (MEB), i.e., the posterior Cramér–Rao lower bound on the position and orientation of specular surfaces, for RF-SLAM. In particular, we consider a very general scenario with single- and double-bounce reflections, as well as distributed anchors. We demonstrate numerically that a state-of-the-art RF-SLAM algorithm asymptotically approaches this MEB. The bounds assess not only the localization (position and orientation) but also the mapping performance of RF-SLAM algorithms in terms of global features.

An angular approximation of the transport equation based on a collocation technique results as an intermediary step in the search for a set of modified discrete ordinates (DO) equations, which eliminates ray effects. The collocation equations are similar to the DO ones with the only difference being that the scattering term is evaluated with a full Galerkin matrix instead of with the DO quadrature formula. The Galerkin quadrature offers the advantage of a better treatment of scattering anisotropy and a correct evaluation of the singular scattering associated to multigroup transport correction. However, the construction of the Galerkin matrix requires the existence of two equivalent bases in a final-dimensional representation space: an interpolatory basis to retain the collocative nature of the DO approximation and a spherical harmonic basis to represent scattering terms accurately. Up to now, the relationship between these two bases was heuristic, stemming from trial and errors. In this work we analyze the symmetries of the angular direction set and also use the factorized form of the spherical harmonics to derive a set of necessary conditions for the construction of the spherical harmonic basis. These conditions give an analytical explanation to previous heuristic techniques and fully extend them to three-dimensional geometries. We have adopted an assembling method for which extensive numerical tests show that the necessary conditions permit the construction of the Galerkin quadrature from level-symmetric, triangular, and product direction sets up to a high number of polar cosines. Our results can also be generalized to calculate Galerkin matrices for nonregular quadrature formulas. However, these necessary conditions are not sufficient, and we give numerical proof of this fact using different assembling techniques. Our assembling technique allows for the construction of Galerkin matrices from triangular direction sets (for which the DO quadrature is notoriously poor), which have positive weights for up to 44 polar cosines. In three dimensions this quadrature has 2024 angular directions and is able to exactly integrate scattering of anisotropy order 24.