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On Floating-Point Normal Vectors
Abstract
In this paper we analyze normal vector representations. We derive the error of the most widely used representation, namely 3D floating-point normal vectors. Based on this analysis, we show that, in theory, the discretization error inherent to single precision floating-point normals can be achieved by 250.2 uniformly distributed normals, addressable by 51 bits. We review common sphere parameterizations and show that octahedron normal vectors perform best: they are fast and stable to compute, have a controllable error, and require only 1 bit more than the theoretical optimal discretization with the same error.
... To minimize the maximal error (angle between the original unit vector and the quantized unit vector) the chosen point set needs to have a distribution as uniform as possible onto the surface of the unit sphere. Among possible distributions, octahedral quantization (Meyer et al. 2010) is a good candidate for a fast compression and decompression of the unit vectors (Cigolle et al. 2014). More recently, Keinert et al. (2015) introduced a new inverse mapping for the spherical Fibonacci point set. ...
... This means that each 3D polyline from the dataset can be represented by its first point and a set of unit vectors. The (Meyer et al. 2010) projects a unit vector to an octahedron, then, to a unit square to encode the discretization of the resulting 2D coordinates stepsize needs to be constant on a per-streamline basis for this representation to work. ...
... As introduced in Section 2, when choosing a quantization method, we can prioritize the speed with the octahedral quantization (Meyer et al. 2010), or the precision with the spherical Fibonacci quantization (Keinert et al. 2015). (Meyer et al. 2010) It is a unit vector representation method that projects the vector [x, y, z] defined on the surface of the unit sphere to an octahedron by normalizing it using an L1-norm. ...
Diffusion MRI fiber tracking datasets can contain millions of 3D streamlines, and their representation can weight tens of gigabytes of memory. These sets of streamlines are called tractograms and are often used for clinical operations or research. Their size makes them difficult to store, visualize, process or exchange over the network. We propose a new compression algorithm well-suited for tractograms, by taking advantage of the way streamlines are obtained with usual tracking algorithms. Our approach is based on unit vector quantization methods combined with a spatial transformation which results in low compression and decompression times, as well as a high compression ratio. For instance, a 11.5GB tractogram can be compressed to a 1.02GB file and decompressed in 11.3 seconds. Moreover, our method allows for the compression and decompression of individual streamlines, reducing the need for a costly out-of-core algorithm with heavy datasets. Last, we open a way toward on-the-fly compression and decompression for handling larger datasets without needing a load of RAM (i.e. in-core handling), faster network exchanges and faster loading times for visualization or processing.
... A review of some techniques in this area was presented by Cigolle et al. [18]. It has also been shown by Meyer et al. [19] that 51-bits is sufficient to losslessly represent unit vectors formed of three 32-bit floating-point numbers. More recently, Smith et al. [20] considered applying lossy compression to this case. ...
... Higher dimensional terms, such as fluxes and solution gradients, are handled by applying compression in a row-wise fashion. For example, consider the inviscid flux tensorgiven by F = (19) where p is the pressure, E is the total energy, and γ is the ratio of specific heats. Here, each colour-coded row is treated as a three component vector and compressed independently of any other row. ...
... where the flux tensor, F, is defined in Eq. (19). The initial condition for the ICV was taken to be defined as ...
A novel inline data compression method is presented for single-precision vectors in three dimensions. The primary application of the method is for accelerating computational physics calculations where the throughput is bound by memory bandwidth. The scheme employs spherical polar coordinates, angle quantisation, and a bespoke floating-point representation of the magnitude to achieve a fixed compression ratio of 1.5. The anisotropy of this method is considered, along with companding and fractional splitting techniques to improve the efficiency of the representation. We evaluate the scheme numerically within the context of high-order computational fluid dynamics. For both the isentropic convecting vortex and the Taylor–Green vortex test cases, the results are found to be comparable to those without compression. Performance is evaluated for a vector addition kernel on an NVIDIA Titan V GPU; it is demonstrated that a speedup of 1.5 can be achieved.
... In our work, we improve over these local quantization approaches by expressing positions of mesh fragment vertices in the barycentric coordinate system relative to the containing tetrahedron. Hardware-friendly normal compression is achieved through an octahedral parametrization of normals [Meyer et al. 2010]. For network transmission, as for most compression schemes, we exploit high correlation between adjacent vertices by using predictive and entropy coding of prediction residuals. ...
... From the quantized coordinates remapped into [−1, 1] we compute nz = 1.0−|u|−|v| . Then if nz > 0 we are on the upper side of the octahedron and nxy = uv, otherwise we are on the lower part and we need to revert the nxy components according to these equations: nx = (1 − ny) · sign(u) and ny = (1 − nx) · sign(v), see [Meyer et al. 2010] for further details. Attribute decoding cost is thus negligible with respect to the other work performed by the shader (in particular, transformation, projection, and shading). ...
We present a software architecture for distributing and rendering gigantic 3D triangle meshes on common handheld devices. Our approach copes with strong bandwidth and hardware capabilities limitations in terms with a compression-domain adaptive multiresolution rendering approach. The method uses a regular conformal hierarchy of tetrahedra to spatially partition the input 3D model and to arrange mesh fragments at different resolution. We create compact GPU-friendly representations of these fragments by constructing cache-coherent strips that index locally quantized vertex data, exploiting the bounding tetrahedron for creating local barycentic parametrization of the geometry. For the first time, this approach supports local quantization in a fully adaptive seamless 3D mesh structure. For web distribution, further compression is obtained by exploiting local data coherence for entropy coding. At run-time, mobile viewer applications adaptively refine a local multiresolution model maintained in a GPU by asynchronously loading from a web server the required fragments. CPU and GPU cooperate for decompression, and a shaded rendering of colored meshes is performed at interactive speed directly from an intermediate compact representation using only 8bytes/vertex, therefore coping with both memory and bandwidth limitations. The quality and performance of the approach is demonstrated with the interactive exploration of gigatriangle-sized models on common mobile platforms.
... Because of numerical instabilities in the inverse mapping, doing computation with double precision, this approach is limited to about 8 millions of spherical Fibonacci points. If speed is the main concern, the octahedral quantization [Meyer et al. 2010] is a good tradeoff between error and performance. Using any quantization of unit vectors, we store, for each window, the number of compressed unit vectors and the quantizations associated to the mapped unit vectors. ...
... The compressed pattern is shown in Figure 5. Table 1: The grouping is done using 13 bits. Lin, oct and sf are respectively the method of Lindstrom [2014], the octahedral quantization [Meyer et al. 2010] and the Spherical Fibonacci point set quantization. The number after the name of the method corresponds to the number of bits used for quantization. ...
We propose a new efficient ray compression method to be used prior to transmission in a distributed Monte Carlo rendering engine. In particular, we introduce a new compression scheme for unorganized unit vector sets (ray directions) which, combined with state-of-the-art positional compression (ray origin), provides a significant gain in precision compared to classical ray quantization techniques. Given a ray set which directions lie in a subset of the Gauss sphere, our key idea consists in mapping it to the surface of the whole unit sphere, for which collaborative compression achieves a high signal-over-noise ratio. As a result, the rendering engine can distribute efficiently massive ray waves over large heterogeneous networks, with potentially low bandwidth. This opens a way toward Monte Carlo rendering in cloud computing, without the need to access specialized hardware such as rendering farms, to harvest the latent horsepower present in public institutions or companies.
... An alternative to the encoding of Aila and Karras is to use a more compact representation of ray direction such as the octahedron parametrization [Meyer et al. 2010]. The octahedron parametrization achieves high uniformity and high locality of its mapping to the 1D space using the Morton curve. ...
... The proposed Two Point ray reordering method performed the best for highly incoherent secondary rays. For shadow rays, other techniques such as the method proposed by Aila and Karras [2010] or the Octahedron method [Meyer et al. 2010] work better. Overall, the results indicate a great potential of ray reordering as a preprocessing step prior to the trace phase. ...
We study ray reordering as a tool for increasing the performance of existing GPU ray tracing implementations. We focus on ray reordering that is fully agnostic to the particular trace kernel. We summarize the existing methods for computing the ray sorting keys and discuss their properties. We propose a novel modification of a previously proposed method using the termination point estimation that is well-suited to tracing secondary rays. We evaluate the ray reordering techniques in the context of the wavefront path tracing using the RTX trace kernels. We show that ray reordering yields significantly higher trace speed on recent GPUs (1.3 − 2.0 ×), but to recover the reordering overhead in the hardware-accelerated trace phase is problematic.
... There are many such projections, including the classic graphics sphere maps and cube maps. As shown in Figure 3-10, an octahedral mapping [9] of the sphere produces relatively low distortion, has few boundary edges, and maps the entire sphere to a single square. These properties make it a popular modern choice. ...
The most fundamental step in any ray tracing program is the ray generation (or "raygen") program that produces the rays themselves. This chapter describes seven essential projections that should be in every toolkit and gives parameterized raygen shader code for each.
... The crucial difference between ambient occlusion and bent normals regarding the learning process is the dimension and kind of the learned output: While ambient occlusion is encoded in a single value in the range [0, 1], the representation of the bent normal, as a directional quantity, requires more than one value. As a representation of the bent normal, we use an octahedron normal vector (ONV, [Meyer et al. 2010]). Therefore, the ground truth data for each sample point consists of two values in the range [−1, 1]. ...
The computation of partial occlusion, as required for ambient occlusion or soft shadows, provides visually important cues but is notoriously expensive. In this paper we propose a novel solution to the ambient occlusion problem, combining signed distance scene representations and machine learning. We demonstrate how to learn and apply mappings which approximate a ray traced ground truth occlusion using only a few nearby samples of a signed distance representation. As representation for our trained mappings we use small feed-forward neural networks which are fast to evaluate, allowing for real-time occlusion queries. Our ambient occlusion approximation outperforms state-of-the-art methods in both quality and performance, yielding temporally stable and smooth results. Since our training data is different from typical machine learning approaches which mostly deal with 2D/3D image data and our techniques are also applicable to other occlusion problems (e.g. soft shadows), we give an in-depth overview of our framework. Furthermore, we discuss arising artifacts and possible extensions of our approach.
... oct [Meyer et al. 2010]: Map the sphere to an octahedron, project down into the z = 0 plane, and then reflect the −z-hemisphere over the appropriate diagonal as shown in Figure 3. The result fills the [−1, +1] 2 square. ...
... Corto is benchmarked in two variations that differ only in the normals prediction scheme. In Corto's default profile, which we call "Corto1", the normals are estimated from the quantized geometry and their differences with the quantized actual normals is encoded using the octahedron projection representation [24]. In Corto's second profile ("Corto2") the quantized normals in the octahedron projection representation are delta coded with respect to a neighboring quantized normal belonging to a quad incident to the normal's vertex. ...
This work provides a systematic understanding of the requirements of live 3D mesh coding, targeting (tele-)immersive media streaming applications. We thoroughly benchmark in rate-distortion and runtime performance terms, four static 3D mesh coding solutions that are openly available. Apart from mesh geometry and connectivity, our analysis includes experiments for compressing vertex normals and attributes, something scarcely found in literature. Additionally, we provide a theoretical model of the tele-immersion pipeline that calculates its expected frame-rate, as well as lower and upper bounds for its end-to-end latency. In order to obtain these measures, the theoretical model takes into account the compression performance of the codecs and some indicative network conditions. Based on the results we obtained through our codec benchmarking, we used our theoretical model to calculate and provide concrete measures for these tele-immersion pipeline’s metrics and discuss on the optimal codec choice depending on the network setup. This offers deep insight into the available solutions and paves the way for future research.
... We select between two different normal prediction methods, the Normals Quantized Coding (NQC) and the Normals Delta Coding (NDC). In the former, we store the differences between the normals estimated from the quantized geometry and the quantized actual normals, using an octahedron projection representation [69]. In the latter, the quantized normals in the octahedron projection representation are solely delta coded, with respect to a neighboring quantized normal belonging to a quad incident to the normal's vertex. ...
We introduce HUMAN4D, a large and multimodal 4D dataset that contains a variety of human activities simultaneously captured by a professional marker-based MoCap, a volumetric capture and an audio recording system. By capturing 2 female and 2 male professional actors performing various full-body movements and expressions, HUMAN4D provides a diverse set of motions and poses encountered as part of single- and multi-person daily, physical and social activities (jumping, dancing, etc.), along with multi-RGBD (mRGBD), volumetric and audio data. Despite the existence of multi-view color datasets captured with the use of hardware (HW) synchronization, to the best of our knowledge, HUMAN4D is the first and only public resource that provides volumetric depth maps with high synchronization precision due to the use of intra- and inter-sensor HW-SYNC. Moreover, a spatio-temporally aligned scanned and rigged 3D character complements HUMAN4D to enable joint research on time-varying and high-quality dynamic meshes. We provide evaluation baselines by benchmarking HUMAN4D with state-of-the-art human pose estimation and 3D compression methods. We apply OpenPose and AlphaPose reaching 70.02% and 82.95% mAPPCKh-0.5 on single- and 68.48% and 73.94% mAPPCKh-0.5 on two-person 2D pose estimation, respectively. In 3D pose, a recent multi-view approach named Learnable Triangulation, achieves 80.26% mAPPCK3D-10cm. For 3D compression, we benchmark Draco, Corto and CWIPC open-source 3D codecs, respecting online encoding and steady bit-rates between 7-155 and 2-90 Mbps for mesh- and point-based volumetric video, respectively. Qualitative and quantitative visual comparison between mesh-based volumetric data reconstructed in different qualities and captured RGB, showcases the available options with respect to 4D representations. HUMAN4D is introduced to enable joint research on spatio-temporally aligned pose, volumetric, mRGBD and audio data cues. The dataset and its code are available online.
... A review of some techniques in this area was presented by Cigolle et al. [18]. It has also been shown by Meyer et al. [19] that 51-bits is sufficient to losslessly represent unit vectors formed of three 32bit floating-point numbers. Some more recent work presented by Smith et al. [20] looked to apply lossy compression to this case with increased efficiency. ...
A novel inline data compression method is presented for single-precision vectors in three dimensions. The primary application of the method is for accelerating computational physics calculations where the throughput is bound by memory bandwidth. The scheme employs spherical polar coordinates, angle quantisation, and a bespoke floating-point representation of the magnitude to achieve a fixed compression ratio of 1.5. The anisotropy of this method is considered, along with companding and fractional splitting techniques to improve the efficiency of the representation. We evaluate the scheme numerically within the context of high-order computational fluid-dynamics. For both the isentropic convecting vortex and the Taylor--Green vortex test cases the results are found to be comparable to those without compression. Performance is evaluated for a vector addition kernel on an NVIDIA Titan V GPU; it is demonstrated that a speedup of 1.5 can be achieved.
We present a method for encoding unit vectors based on spherical coordinates that out-performs existing encoding methods both in terms of accuracy and encoding/decoding time. Given a tolerance ϵϵ, we solve a simple, discrete optimization problem to find a set of points on the unit sphere that can trivially be indexed such that the difference in angle between the encoded vector and the original are no more than ϵϵ apart. To encode a unit vector, we simply compute its spherical coordinates and round the result based on the prior optimization solution. We also present a moving frame method that further reduces the amount of data to be encoded when vectors have some coherence. Our method is extremely fast in terms of encoding and decoding both of which take constant time O(1). The accuracy of our encoding is also comparable or better than previous methods for encoding unit vectors.
We study the relationship between the noise in the vertex coordinates of a triangle mesh and normal noise. First, we compute in closed form the expectation for the angle $\theta$ between the new and the old normal when uniform noise is added to a single vertex of a triangle. Next, we propose and experimentally validate an approximation and lower and upper bounds for $\theta$ when uniform noise is added to all three vertices of the triangle. In all cases, for small amounts of spatial noise that do not severely distort the mesh, there is a linear correlation between $\theta$ and simple functions of the heights of the triangles and thus, $\theta$ can be computed efficiently. The addition of uniform spatial noise to a mesh can be seen as a dithered quantisation of its vertices. We use the obtained linear correlations between spatial and normal noise to compute the level of dithered quantisation of the mesh vertices when a tolerance for the average normal distortion is given.
The performance of shading and ray-tracing algorithms depends heavily on the quality of the surface normal information. As a result, in many visual applications normal information turns out to be more important than spatial information. This paper proposes a logistic model for the degradation of the normal information resulting from the quantisation of the vertex coordinates. The mesh is degraded by the randomization of each vertex coordinate after its t-th significant bit. The normal degradation is computed as a weighted average of the angle differences between the normals of the original triangles and the corresponding degraded triangles. The proposed model is validated experimentally. As an application, we use the proposed logistic model to estimate suitable levels of quantisation for 3D triangle meshes.
Ray-traced global illumination (GI) is becoming widespread in production rendering but incoherent secondary ray traversal limits practical rendering to scenes that fit in memory. Incoherent shading also leads to intractable performance with production-scale textures forcing renderers to resort to caching of irradiance, radiosity, and other values to amortize expensive shading. Unfortunately, such caching strategies complicate artist workflow, are difficult to parallelize effectively, and contend for precious memory. Worse, these caches involve approximations that compromise quality. In this paper, we introduce a novel path-tracing framework that avoids these tradeoffs. We sort large, potentially out-of-core ray batches to ensure coherence of ray traversal. We then defer shading of ray hits until we have sorted them, achieving perfectly coherent shading and avoiding the need for shading caches.
We propose a lossless, single-rate triangle mesh topology codec tailored for fast data-parallel GPU decompression. Our compression scheme coherently orders generalized triangle strips in memory. To unpack generalized triangle strips efficiently, we propose a novel parallel and scalable algorithm. We order vertices coherently to further improve our compression scheme. We use a variable bit-length code for additional compression benefits, for which we propose a scalable data-parallel decompression algorithm. For a set of standard benchmark models, we obtain (min: 3.7, med: 4.6, max: 7.6) bits per triangle. Our CUDA decompression requires only about 15% of the time it takes to render the model even with a simple shader. © 2012 Wiley Periodicals, Inc.
We discuss our experience in creating scalable systems for distributing and rendering gigantic 3D surfaces on web environments and common handheld devices. Our methods are based on compressed streamable coarse-grained multiresolution structures. By combining CPU and GPU compression technology with our multiresolution data representation, we are able to incrementally transfer, locally store and render with unprecedented performance extremely detailed 3D mesh models on WebGL-enabled browsers, as well as on hardware-constrained mobile devices.
We describe the design space for real-time photon density estimation, the key step of rendering global illumination (GI) via photon mapping. We then detail and analyze efficient GPU implementations of four best-of-breed algorithms. All produce reasonable results on NVIDIA GeForce 670 at 1920 × 1080 for complex scenes with multiple-bounce diffuse effects, caustics, and glossy reflection in real-time. Across the designs we conclude that tiled, deferred photon gathering in a compute shader gives the best combination of performance and quality.
We present HuMoRS, a networked 3D graphics system for interactively streaming and exploring massive 3D mesh models on mobile devices. The system integrates a networked architecture for adaptive on-device rendering of multiresolution surfaces with a simple and effective interactive camera controller customized for touch-enabled mobile devices. During interaction, knowledge of the currently rendered scene is exploited to automatically center a rotation pivot and to propose context-dependent precomputed viewpoints. Both the object of interest and the viewpoint database are resident on a web server and adaptive transmission is demonstrated over wireless and phone connections in a Cultural Heritage application for the exploration of sub-millimetric colored reconstructions of stone statues. We report also on a preliminary user-study comparing the performances of our camera navigation method with respect to the most popular Virtual TrackBall implementations, with and without pivoting.
Remote rendering methods enable devices with low computing power like smart phones or tablets to visualize massive data. By transmitting G-Buffers , Depth-Image-Based Rendering (DIBR) methods can be used to compensate the artefacts caused by the latency. However, the drawback is that a G-Buffer has at least twice as much data as an image.
We present a method for compressing G-Buffers which provides an efficient decoding suitable for web applications. Depending on the computing power of the device, software methods, which run on the CPU, may not be fast enough for an interactive experience. Therefore, we developed a decoding which runs entirely on the GPU. As we use only standard WebGL for our implementation, our compression is suitable for every modern browser.
Spherical Fibonacci point sets yield nearly uniform point distributions on the unit sphere S² ⊂ R³. The forward generation of these point sets has been widely researched and is easy to implement, such that they have been used in various applications.
Unfortunately, the lack of an efficient mapping from points on the unit sphere to their closest spherical Fibonacci point set neighbors rendered them impractical for a wide range of applications, especially in computer graphics. Therefore, we introduce an inverse mapping from points on the unit sphere which yields the nearest neighbor in an arbitrarily sized spherical Fibonacci point set in constant time, without requiring any precomputations or table lookups.
We show how to implement this inverse mapping on GPUs while addressing arising floating point precision problems. Further, we demonstrate the use of this mapping and its variants, and show how to apply it to fast unit vector quantization. Finally, we illustrate the means by which to modify this inverse mapping for texture mapping with smooth filter kernels and showcase its use in the field of procedural modeling.
Voxel-based approaches are today's standard to encode volume data. Recently, directed acyclic graphs (DAGs) were successfully used for compressing sparse voxel scenes as well, but they are restricted to a single bit of (geometry) information per voxel. We present a method to compress arbitrary data, such as colors, normals, or reflectance information. By decoupling geometry and voxel data via a novel mapping scheme, we are able to apply the DAG principle to encode the topology, while using a palette-based compression for the voxel attributes, leading to a drastic memory reduction. Our method outperforms existing state-of-the-art techniques and is well-suited for GPU architectures. We achieve real-time performance on commodity hardware for colored scenes with up to 17 hierarchical levels (a 128K3voxel resolution), which are stored fully in core.
In this paper we present a novel GPU-friendly real-time voxelization technique for rendering homogeneous media that is defined by particles, e.g. fluids obtained from particle-based simulations such as Smoothed Particle Hydrodynamics (SPH). Our method computes view-adaptive binary voxelizations with on-the-fly compression of a tiled perspective voxel grid, achieving higher resolutions than previous approaches. It allows for interactive generation of realistic images, enabling advanced rendering techniques such as ray casting-based refraction and reflection, light scattering and absorption, and ambient occlusion. In contrast to previous methods, it does not rely on preprocessing such as expensive, and often coarse, scalar field conversion or mesh generation steps. Our method directly takes unsorted particle data as input. It can be further accelerated by identifying fully populated simulation cells during simulation. The extracted surface can be filtered to achieve smooth surface appearance. Finally, we provide a new scheme for accelerated ray casting inside the voxelization.
Most graphics hardware features memory to store textures and vertex data for rendering. However, because of the irreversible trend of increasing complexity of scenes, rendering a scene can easily reach the limit of memory resources. Thus, vertex data are preferably compressed, with a requirement that they can be decompressed during rendering. In this paper, we present a novel method to exploit existing hardware texture compression circuits to facilitate the decompression of vertex data in graphics processing unit (GPUs). This built-in hardware allows real-time, random-order decoding of data. However, vertex data must be packed into textures, and careless packing arrangements can easily disrupt data coherence. Hence, we propose an optimization approach for the best vertex data permutation that minimizes compression error. All of these result in fast and high-quality vertex data decompression for real-time rendering. To further improve the visual quality, we introduce vertex clustering to reduce the dynamic range of data during quantization. Our experiments demonstrate the effectiveness of our method for various vertex data of 3D models during rendering with the advantages of a minimized memory footprint and high frame rate.
Motion Blur is an important effect of photo-realistic rendering. Distribution ray tracing can simulate motion blur very well by integrating light, both over the spatial and the temporal domain. However, increasing the problem by the temporal dimension entails many challenges, particularly in cinematic multi-bounce path tracing of complex scenes, where heavy-weight geometry with complex lighting and even offscreen elements contribute to the final image. In this paper, we propose the Motion DAG (Directed Acyclic Graph), a novel data structure that filters an entire animation sequence of an object, both in the spatial and temporal domain. These Motion DAGs interleave a temporal interval binary tree for filtering time consecutive data and a sparse voxel octree (SVO), which simplifies spatially nearby data. Motion DAGs are generated in a pre-process and can be easily integrated in a conventional physically based path tracer. Our technique is designed to target motion blur of small objects, where coarse representations are sufficient. Specifically, in this scenario our results show that it is possible to significantly reduce both, memory consumption and render time.
In this paper we present a method for fast screen-space ray tracing. Single-layer screen-space ray marching is an established tool in high-performance applications, such as games, where plausible and appealing results are more important than strictly correct ones. However, even in such tightly controlled environments, missing scene information can cause visible artifacts. This can be tackled by keeping multiple layers of screen-space information, but might not be afforable on severely limited time-budgets. Traversal speed of single-layer ray marching is commonly improved by multi-resolution schemes, from sub-sampling to stepping through mip-maps to achieve faster frame rates. We show that by combining these approaches, keeping multiple layers and tracing on multiple resolutions, images of higher quality can be computed rapidly. Figure 1 shows this for two scenes with multi-bounce reflections that would show strong artifacts when using only a single layer.
Most state-of-the-art compression algorithms use complex connectivity traversal and prediction schemes, which are not efficient enough for online compression of large meshes. In this paper we propose a scalable massively parallel approach for compression and decompression of large triangle meshes using the GPU. Our method traverses the input mesh in a parallel breadth-first manner and encodes the connectivity data similarly to the well known cut-border machine. Geometry data is compressed using a local prediction strategy. In contrast to the original cut-border machine, we can additionally handle triangle meshes with inconsistently oriented faces. Our approach is more than one order of magnitude faster than currently used methods and achieves competitive compression rates. © 2017 The Author(s) Computer Graphics Forum © 2017 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd.
The ubiquity of large virtual worlds and their growing complexity in computer graphics require efficient representations. This means that we need smart solutions for the underlying storage of these complex environments, but also for their visualization. How the virtual world is best stored and how it is subsequently shown to the user in an optimal way, depends on the goal of the application. In this respect, we identify the following three visual representations, which form orthogonal directions, but are not mutually exclusive. Realistic representations aim for physical correctness, while illustrative display techniques, on the other hand, facilitate user tasks, often relating to improved understanding. Finally, artistic approaches enable a high level of expressiveness for aesthetic applications. Each of these directions offers a wide array of possibilities. In this dissertation, our goal is to provide solutions for strategically selected challenges for all three visual directions, as well as for the underlying representation of the virtual world.
This paper investigates a first order generalization of signed distance fields. We show that we can improve accuracy and storage efficiency by incorporating the spatial derivatives of the signed distance function into the distance field samples. We show that a representation in power basis remains invariant under barycentric combination, as such, it is interpolated exactly by the GPU. Our construction is applicable in any geometric setting where point-surface distances can be queried. To emphasize the practical advantages of this approach, we apply our results to signed distance field generation from triangular meshes. We propose storage optimization approaches and offer a theoretical and empirical accuracy analysis of our proposed distance field type in relation to traditional, zero order distance fields. We show that the proposed representation may offer an order of magnitude improvement in storage while retaining the same precision as a higher resolution distance field.
We describe a method to use Spherical Gaussians with free directions and arbitrary sharpness and amplitude to approximate the precomputed local light field for any point on a surface in a scene. This allows for a high‐quality reconstruction of these light fields in a manner that can be used to render the surfaces with precomputed global illumination in real‐time with very low cost both in memory and performance. We also extend this concept to represent the illumination‐weighted environment visibility, allowing for high‐quality reflections of the distant environment with both surface‐material properties and visibility taken into account. We treat obtaining the Spherical Gaussians as an optimization problem for which we train a Convolutional Neural Network to produce appropriate values for each of the Spherical Gaussians' parameters. We define this CNN in such a way that the produced parameters can be interpolated between adjacent local light fields while keeping the illumination in the intermediate points coherent.
Video memory is a valuable resource that has grown much slower than the rendering power of GPUs over the last years. Today, video memory is often the limiting factor in interactive high-quality rendering applications. The most often used solution to reduce memory consumption is to apply level-of-detail (LOD) methods: only a simplified version of the mesh with less vertices and triangles is kept in memory. In this paper we examine a simple orthogonal compression approach that is mostly neglected: adapting the level-of-precision (LOP) of vertex data. The main idea is to quantize vertex positions according to the current view distance, and adapt precision by adding or removing single bit planes. We provide an analysis of the resulting image error, and show that visual artifacts can be avoided by simply constraining the quantization for critical vertices. Our approach allows both random access on vertex data as well as quickly switching between LOP. In experiments we found that our approach compresses vertex positions by about 70% on average without loss in rendering performance or image quality.
We present two methods for lossy compression of normal vectors through quantization using "base" polyhedra. The first revisits subdivision-based quantization. The second uses fixed-precision barycentric coordinates. For both, we provide fast (de)compression algorithms and a rigorous upper bound on compression error. We discuss the effects of base polyhedra on the error bound and suggest polyhedra derived from spherical coverings. Finally, we present compression and decompression results, and we compare our methods to others from the literature.
3D models of millions of triangles invariably repeatedly use the same 12-byte unit normals. Several bit-wise compression algorithms exist for efficient storage and progressive transmission and visualization of normal vectors. However such methods often incur a reconstruction time penalty which, in the absence of dedicated hardware acceleration, make real-time rendering with such compression/reconstruction methods prohibitive. In particular, several methods use a subdivided octahedron to create look-up normals, where the bit length of normal indices varies according to the number of subdivisions used. Not much attention has been given to the error in the normals using such schemes.
HEALPix -- the Hierarchical Equal Area iso-Latitude Pixelization -- is a
versatile data structure with an associated library of computational algorithms
and visualization software that supports fast scientific applications
executable directly on very large volumes of astronomical data and large area
surveys in the form of discretized spherical maps. Originally developed to
address the data processing and analysis needs of the present generation of
cosmic microwave background (CMB) experiments (e.g. BOOMERanG, WMAP), HEALPix
can be expanded to meet many of the profound challenges that will arise in
confrontation with the observational output of future missions and experiments,
including e.g. Planck, Herschel, SAFIR, and the Beyond Einstein CMB
polarization probe. In this paper we consider the requirements and constraints
to be met in order to implement a sufficient framework for the efficient
discretization and fast analysis/synthesis of functions defined on the sphere,
and summarise how they are satisfied by HEALPix.
An internal standard for floating-point formats and arithmetic is described that the author believes could be adopted for microprocessors in general.
Several years ago the Microprocessor Standards Committee of the IEEE Computer Society established a Floating-Point Working Group to draft a standard binary floating-point arithmetic on 32-bit microprocessors. As that task neared completion, a second working group was established to generalize the proposed binary standard for other radices and wordlengths. We discuss the emerging generalization, its influence on final deliberations on the proposed binary standard, and the implications for numerical computation.
This paper introduces the concept of Geometry Compression, al- lowing 3D triangle data to be represented with a factor of 6 to 10 times fewer bits than conventional techniques, with only slight loss- es in object quality. The technique is amenable to rapid decompres- sion in both software and hardware implementations; if 3D render- ing hardware contains a geometry decompression unit, application geometry can be stored in memory in compressed format. Geome- try is first represented as a generalized triangle mesh, a data struc- ture that allows each instance of a vertex in a linear stream to spec- ify an average of two triangles. Then a variable length compression is applied to individual positions, colors, and normals. Delta com- pression followed by a modified Huffman compression is used for positions and colors; a novel table-based approach is used for nor- mals. The table allows any useful normal to be represented by an 18-bit index, many normals can be represented with index deltas of 8 bits or less. Geometry compression is a general space-time trade- off, and offers advantages at every level of the memory/intercon- nect hierarchy: less storage space is needed on disk, less transmis- sion time is needed on networks.
The traditional approach for parametrizing a surface involves cutting it into charts and mapping these piecewise onto a planar domain. We introduce a robust technique for directly parametrizing a genus-zero surface onto a spherical domain. A key ingredient for making such a parametrization practical is the minimization of a stretch-based measure, to reduce scale-distortion and thereby prevent undersampling. Our second contribution is a scheme for sampling the spherical domain using uniformly subdivided polyhedral domains, namely the tetrahedron, octahedron, and cube. We show that these particular semi-regular samplings can be conveniently represented as completely regular 2D grids, i.e. geometry images. Moreover, these images have simple boundary extension rules that aid many processing operations. Applications include geometry remeshing, level-of-detail, morphing, compression, and smooth surface subdivision.
Quantization Errors of Popular Normal-Vector Representations. Tech. rep., Inf. 9, Univ. of Er-langen PNORMS: Platonic De-rived Normals for Error Bound Compression
- [ Mss
- Meyer Q Süssmuth
- J Sussner
- G Stam-Minger M
- G Greiner
- F Ob06 ] Oliveira J
- B F Buxton
[MSS * 10] MEYER Q., SÜSSMUTH J., SUSSNER G., STAM-MINGER M., GREINER G.: Quantization Errors of Popular Normal-Vector Representations. Tech. rep., Inf. 9, Univ. of Er-langen, 2010. In Preparation, Available at www9.cs.fau.de. [OB06] OLIVEIRA J. F., BUXTON B. F.: PNORMS: Platonic De-rived Normals for Error Bound Compression. In Proc. of VRST '06 (2006), pp. 324–333.
Quantization Errors of Popular Normal-Vector Representations
- Meyer Q Süssmuth
- J Sussner
- G Stam-Minger M
- Greiner G
[MSS * 10] MEYER Q., SÜSSMUTH J., SUSSNER G., STAM-MINGER M., GREINER G.: Quantization Errors of Popular
Normal-Vector Representations. Tech. rep., Inf. 9, Univ. of Erlangen, 2010. In Preparation, Available at www9.cs.fau.de.
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Standard for Floating-Point Arithmetic
[IEE08] IEEE: IEEE Standard for Floating-Point Arithmetic.
IEEE Std 754-2008 (2008), 1-58. 2