A General BRDF Representation Based on Tensor Decomposition

Generating photo‐realistic images through Monte Carlo rendering requires efficient representation of light–surface interaction and techniques for importance sampling. Various models with good representation abilities have been developed but only a few of them have their importance sampling procedure. In this paper, we propose a method which provides a good bidirectional reflectance distribution function (BRDF) representation and efficient importance sampling procedure. Our method is based on representing BRDF as a function of tensor products. Four‐dimensional measured BRDF tensor data are factorized using Tucker decomposition. A large data set is used for comparing the proposed BRDF model with a number of well‐known BRDF models. It is shown that the underlying model provides good approximation to BRDFs.     

A wavelet-based framework for acquired radiometric quantity representation and accurate physical rendering

In this paper, we present a framework based on a generic representation, which is able to handle most of the radiometric quantities required by global illumination software. A sparse representation in the wavelet space is built using the separation between the directional and the wavelength dependencies of such radiometric quantities. Particularly, we show how to use this representation for spectral power distribution, spectral reflectance and phase function measurements modeling. Then, we explain how the representation is useful for performing spectral rendering. On the one hand, it speeds up spectral path tracing by importance sampling to generate reflected directions and by avoiding expensive computations usually done on-the-fly. On the other hand, it allows efficient spectral photon mapping, both in terms of memory and speed. We also show how complex light emission from real luminaires can be efficiently sampled to emit photons with our numerical model.     

Barycentric parameterizations for isotropic BRDFs

A bidirectional reflectance distribution function (BRDF) is often expressed as a function of four real variables: two spherical coordinates in each of the "incoming" and "outgoing" directions. However, many BRDFs reduce to functions of fewer variables. For example, isotropic reflection can be represented by a function of three variables. Some BRDF models can be reduced further. In This work, we introduce new sets of coordinates which we use to reduce the dimensionality of several well-known analytic BRDFs as well as empirically measured BRDF data. The proposed coordinate systems are barycentric with respect to a triangular support with a direct physical interpretation. One coordinate set is based on the BRDF mode) proposed by Lafortune. Another set, based on a model of Ward, is associated with the "halfway" vector common in analytical BRDF formulas. Through these coordinate sets we establish lower bounds on the approximation error inherent in the models on which they are based. We present a third set of coordinates, not based on any analytical model, that performs well in approximating measured data. Finally, our proposed variables suggest novel ways of constructing and visualizing BRDFs.     

Template‐Based Sampling of Anisotropic BRDFs

BRDFs are commonly used to represent given materials’ appearance in computer graphics and related fields. Although, in the recent past, BRDFs have been extensively measured, compressed, and fitted by a variety of analytical models, most research has been primarily focused on simplified isotropic BRDFs. In this paper, we present a unique database of 150 BRDFs representing a wide range of materials; the majority exhibiting anisotropic behavior. Since time‐consuming BRDF measurement represents a major obstacle in the digital material appearance reproduction pipeline, we tested several approaches estimating a very limited set of samples capable of high quality appearance reconstruction. Initially, we aligned all measured BRDFs according to the location of the anisotropic highlights. Then we propose an adaptive sampling method based on analysis of the measured BRDFs. For each BRDF, a unique sampling pattern was computed, given a predefined count of samples. Further, template‐based methods are introduced based on reusing of the precomputed sampling patterns. This approach enables a more efficient measurement of unknown BRDFs while preserving the visual fidelity for the majority of tested materials. Our method exhibits better performance and stability than competing sparse sampling approaches; especially for higher numbers of samples.     

Interactive Editing of Lighting and Materials using a Bivariate BRDF Representation

We present a new Precomputed Radiance Transfer (PRT) algorithm based on a two dimensional representation of isotropic BRDFs. Our approach involves precomputing matrices that allow quickly mapping environment lighting, which is represented in the global coordinate system, and the surface BRDFs, which are represented in a bivariate domain, to the local hemisphere at a surface location where the reflection integral is evaluated. When the lighting and BRDFs are represented in a wavelet basis, these rotation matrices are sparse and can be efficiently stored and combined with pre‐computed visibility at run‐time. Compared to prior techniques that also precompute wavelet rotation matrices, our method allows full control over the lighting and materials due to the way the BRDF is represented. Furthermore, this bivariate parameterization preserves sharp specular peaks and grazing effects that are attenuated in conventional parameterizations. We demonstrate a prototype rendering system that achieves real‐time framerates while lighting and materials are edited.     

g‐BRDFs: An Intuitive and Editable BTF Representation

Measured reflection data such as the bidirectional texture function (BTF) represent spatial variation under the full hemisphere of view and light directions and offer a very realistic visual appearance. Despite its high‐dimensional nature, recent compression techniques allow rendering of BTFs in real time. Nevertheless, a still unsolved problem is that there is no representation suited for real‐time rendering that can be used by designers to modify the BTF's appearance. For intuitive editing, a set of low‐dimensional comprehensible parameters, stored as scalars, colour values or texture maps, is required. In this paper we present a novel way to represent BTF data by introducing the geometric BRDF (g‐BRDF), which describes both the underlying meso‐ and micro‐scale structure in a very compact way. Both are stored in texture maps with only a few additional scalar parameters that can all be modified at runtime and thus give the designer full control over the material's appearance in the final real‐time application. The g‐BRDF does not only allow intuitive editing, but also reduces the measured data into a small set of textures, yielding a very effective compression method. In contrast to common material representation combining heightfields and BRDFs, our g‐BRDF is physically based and derived from direct measurement, thus representing real‐world surface appearance. In addition, we propose an algorithm for fully automatic decomposition of a given measured BTF into the g‐BRDF representation.     

A PCA Decomposition for Real‐time BRDF Editing and Relighting with Global Illumination

We propose a novel rendering method which supports interactive BRDF editing as well as relighting on a 3D scene. For interactive BRDF editing, we linearize an analytic BRDF model with basis BRDFs obtained from a principal component analysis. For each basis BRDF, the radiance transfer is precomputed and stored in vector form. In rendering time, illumination of a point is computed by multiplying the radiance transfer vectors of the basis BRDFs by the incoming radiance from gather samples and then linearly combining the results weighted by user‐controlled parameters. To improve the level of accuracy, a set of sub‐area samples associated with a gather sample refines the glossy reflection of the geometric details without increasing the precomputation time. We demonstrate this program with a number of examples to verify the real‐time performance of relighting and BRDF editing on 3D scenes with complex lighting and geometry.     

Accurate fitting of measured reflectances using a Shifted Gamma micro‐facet distribution

Material models are essential to the production of photo‐realistic images. Measured BRDFs provide accurate representation with complex visual appearance, but have larger storage cost. Analytical BRDFs such as Cook‐Torrance provide a compact representation but fail to represent the effects we observe with measured appearance. Accurately fitting an analytical BRDF to measured data remains a challenging problem. In this paper we introduce the SGD micro‐facet distribution for Cook‐Torrance BRDF. This distribution accurately models the behavior of most materials. As a consequence, we accurately represent all measured BRDFs using a single lobe. Our fitting procedure is stable and robust, and does not require manual tweaking of the parameters.     

Practical Product Importance Sampling for Direct Illumination

We present a practical algorithm for sampling the product of environment map lighting and surface reflectance. Our method builds on wavelet‐based importance sampling, but has a number of important advantages over previous methods. Most importantly, we avoid using precomputed reflectance functions by sampling the BRDF on‐the‐fly. Hence, all types of materials can be handled, including anisotropic and spatially varying BRDFs, as well as procedural shaders. This also opens up for using very high resolution, uncompressed, environment maps. Our results show that this gives a significant reduction of variance compared to using lower resolution approximations. In addition, we study the wavelet product, and present a faster algorithm geared for sampling purposes. For our application, the computations are reduced to a simple quadtree‐based multiplication. We build the BRDF approximation and evaluate the product in a single tree traversal, which makes the algorithm both faster and more flexible than previous methods.     

Analysis of broadband surface BRDFs derived from TOA SW CERES measurements for surfaces classified by the IGBP land cover

Most studies on the reflectance properties of the Earth's surface are addressed estimating the bidirectional reflectance distribution function (BRDF) of high spatial resolution and high spectral resolution satellite measurements. This article assesses the development of broadband (BB) BRDFs from radiances corresponding to large footprints classified according to the International Geosphere-Biosphere Programme (IGBP) land-cover classification. Top-of-atmosphere (TOA) shortwave (SW) CERES (Clouds and the Earth's Radiant Energy System) measurements are employed to invert the bidirectional reflectance factor (BRF) Rahman–Pinty–Verstraete (RPV) model for regions identified with the same IGBP type. The inversion of this non-linear parametric model is optimized to improve the computation efficiency and merged into a radiative transfer model to correct the surface radiances for the atmospheric effect. Analysis of the nature of the reflectance field simulated for several regions selected for every IGBP type determines whether the creation of general BRF models for surfaces defined by the same IGBP land cover is feasible. According to the results gathered in this study, the BB BRDFs for regions classified by the IGBP classification show values for the coefficient of variation (CV) between 3.5% and 44.1%. Consequently, the high differences achieved in the reflectance fields discourage the creation of BRDFs based on the IGBP land types.     

Surface scratches: measuring, modeling and rendering

  One of the most important criticisms that can be made concerning synthesized images is the brand new and too clean aspect of objects. Surface color modifications can be used to introduce dirtiness or other aging-linked characteristics. Also, techniques such as bump or displacement mapping allow users to improve surface aspects by introducing geometrical perturbations. In parallel, the bidirectional reflectance distribution function (BRDF) is a crucial factor in achieving a high degree of realism. It turns out that surfaces are very often covered by defects such as scratches that are related to both textures and BRDFs due to their size. Scratches do not always affect the apparent geometry but nevertheless can remain strongly visible. None of the previously mentioned methods is suited for rendering these defects efficiently. We propose a new method, based on extensions to existing BRDFs and classical 2D texture mapping techniques, to render efficiently individually visible scratches. We use physical measurements on "real objects" to derive an accurate geometric model of scratches at small scale range (roughness scale), and we introduce a new geometric level between bump mapping and BRDFs. Beyond providing graphical results closely matching real cases, our method opens the way to a new class of considerations in computer graphics based on defects that require the coupling of both BRDFs and texturing techniques.     

An Analysis of the In‐Out BRDF Factorization for View‐Dependent Relighting

Interactive rendering with dynamic natural lighting and changing view is a long‐standing goal in computer graphics. Recently, precomputation‐based methods for all‐frequency relighting have made substantial progress in this direction. Many of the most successful algorithms are based on a factorization of the BRDF into incident and outgoing directions, enabling each term to be precomputed independent of viewing direction, and re‐combined at run‐time. However, there has so far been no theoretical understanding of the accuracy of this factorization, nor the number of terms needed. In this paper, we conduct a theoretical and empirical analysis of the BRDF in‐out factorization. For Phong BRDFs, we obtain analytic results, showing that the number of terms needed grows linearly with the Phong exponent, while the factors correspond closely to spherical harmonic basis functions. More generally, the number of terms is quadratic in the frequency content of the BRDF along the reflected or half‐angle direction. This analysis gives clear practical guidance on the number of factors needed for a given material. Different objects in a scene can each be represented with the correct number of terms needed for that particular BRDF, enabling both accuracy and interactivity.     

Time-varying BRDFs

The properties of virtually all real-world materials change with time, causing their bidirectional reflectance distribution functions (BRDFs) to be time varying. However, none of the existing BRDF models and databases take time variation into consideration; they represent the appearance of a material at a single time instance. In this paper, we address the acquisition, analysis, modeling, and rendering of a wide range of time-varying BRDFs (TVBRDFs). We have developed an acquisition system that is capable of sampling a material's BRDF at multiple time instances, with each time sample acquired within 36 sec. We have used this acquisition system to measure the BRDFs of a wide range of time-varying phenomena, which include the drying of various types of paints (watercolor, spray, and oil), the drying of wet rough surfaces (cement, plaster, and fabrics), the accumulation of dusts (household and joint compound) on surfaces, and the melting of materials (chocolate). Analytic BRDF functions are fit to these measurements and the model parameters' variations with time are analyzed. Each category exhibits interesting and sometimes nonintuitive parameter trends. These parameter trends are then used to develop analytic TVBRDF models. The analytic TVBRDF models enable us to apply effects such as paint drying and dust accumulation to arbitrary surfaces and novel materials.     

DeepBRDF: A Deep Representation for Manipulating Measured BRDF

Effective compression of densely sampled BRDF measurements is critical for many graphical or vision applications. In this paper, we present DeepBRDF, a deep-learning-based representation that can significantly reduce the dimensionality of measured BRDFs while enjoying high quality of recovery. We consider each measured BRDF as a sequence of image slices and design a deep autoencoder with a masked L₂ loss to discover a nonlinear low-dimensional latent space of the high-dimensional input data. Thorough experiments verify that the proposed method clearly outperforms PCA-based strategies in BRDF data compression and is more robust. We demonstrate the effectiveness of DeepBRDF with two applications. For BRDF editing, we can easily create a new BRDF by navigating on the low-dimensional manifold of DeepBRDF, guaranteeing smooth transitions and high physical plausibility. For BRDF recovery, we design another deep neural network to automatically generate the full BRDF data from a single input image. Aided by our DeepBRDF learned from real-world materials, a wide range of reflectance behaviors can be recovered with high accuracy.     

Photo-realistic Rendering of Metallic Car Paint from Image-Based Measurements

State‐of‐the‐art car paint shows not only interesting and subtle angular dependency but also significant spatial variation. Especially in sunlight these variations remain visible even for distances up to a few meters and give the coating a strong impression of depth which cannot be reproduced by a single BRDF model and the kind of procedural noise textures typically used. Instead of explicitly modeling the responsible effect particles we propose to use image‐based reflectance measurements of real paint samples and represent their spatial varying part by Bidirectional Texture Functions (BTF). We use classical BRDF models like Cook‐Torrance to represent the reflection behavior of the base paint and the highly specular finish and demonstrate how the parameters of these models can be derived from the BTF measurements. For rendering, the image‐based spatially varying part is compressed and efficiently synthesized. This paper introduces the first hybrid analytical and image‐based representation for car paint and enables the photo‐realistic rendering of all significant effects of highly complex coatings.