| { | |
| "title": "A Micro-Ellipsoid Model for Wet Porous Materials Rendering", | |
| "abstract": "Wet porous materials, like wet ground, moist walls, or wet cloth, are common in the real world. These materials consist of transmittable particles surrounded by liquid, where the individual particle is invisible in the macroscopic view. While modeling wet porous materials is critical for various applications, a physically based model for wet porous materials is still absent. In this paper, we model these appearances in the media domain by extending the anisotropic radiative transfer equation to model porosity and saturation. Then, we introduce a novel particle model- micro-ellipsoid- by treating each particle as a transmittable ellipsoid, analogous to a micro-flake, to statistically characterize the overall optical behavior of the medium. This way, the foundational theory for media with porosity and saturation is established. Building upon this new medium, we further propose a practical bidirectional scattering distribution function (BSDF) model within the position-free framework\u2013WetSpongeCake. As a result, our WetSpongeCake model is able to represent various appearances of wet porous materials using physical parameters (e.g., porosity and saturation), allowing both reflection and transmission. We validated our model through several examples: a piece of wet cloth, sand saturated with different liquids, or damp sculptures, demonstrating its ability to match real-world appearances closely.", | |
| "sections": [ | |
| { | |
| "section_id": "1", | |
| "parent_section_id": null, | |
| "section_name": "1. Introduction", | |
| "text": "Reproducing appearances from the real world with material models is essential in computer graphics. Material models for some common appearances (e.g., plastic, metals, glasses, etc.) have been extensively developed with surface shading models (e.g., microfacet models (Cook and Torrance, 1982 ###reference_b4###; Walter et al., 2007 ###reference_b30###)). Unfortunately, these widely-used shading models have limited capability to represent wet porous materials. Examples of this kind of materials include a damp road or wet sand (see Fig. 7 ###reference_###). Generally, the porous materials are made of many tiny grains. After being filled with liquids, the materials become wet and appear darker. These appearances have become crucial for many applications, like wet roads in the autonomous driving simulation. However, it\u2019s challenging to simulate these materials, as it involves complex optical phenomena.\nIn the literature, wet porous materials can be represented at different levels\u2013geometry, medium or surface materials. While modeling individual particle as a sphere (Peltoniemi and Lumme, 1992 ###reference_b27###) or an prolate spheroid (Kimmel and Baranoski, 2007 ###reference_b18###) is accurate, it leads to extensive memory and time cost. Another group of methods render wet or porous materials with radiative transfer equation (RTE) or its variant. However, they model saturation or porosity of medium in an ad-hoc way, by tweaking the Henyey-Greenstein (HG) phase function parameters (Jensen et al., 1999 ###reference_b17###) or unnormalized porosity-aware transmittance functions (Hapke, 2008 ###reference_b12###). Either way leads to unrealistic rendering. To our knowledge, no foundational medium formulation considering both porosity and saturation exist.\nThe final category of methods represents porous materials using a bidirectional reflectance distribution function (BRDF) (Merillou et al., 2000 ###reference_b24###; Hnat et al., 2006 ###reference_b14###; Lu et al., 2006 ###reference_b20###), incorporating cylinder-shaped holes on the surface. While these approaches are practical, they struggle to accurately reproduce real-world effects and cannot simulate transmission phenomena.\nIn this paper, we aim to model a wet porous appearance with controllable physical parameters that closely replicate real-world effects while maintaining practicality. To this end, we propose a generalized anisotropic radiative transfer equation to model a wet porous medium considering both the porosity and saturation effect. At the core of our wet porous medium is a transmittable micro-ellipsoid model, analogous to the micro-flake model (Jakob et al., 2010 ###reference_b16###), except with different particle shapes.\nIn our model, each particle is represented with an ellipsoid, and the distribution of aggregated particles follows an ellipsoidal distribution. We derive phase function and anisotropic attenuation functions for both isolated and aggregated medium particles. While the generalized anisotropic RTE, along with the transmittable micro-ellipsoid model, allows rendering a wet porous medium, it requires a long convergence time. To address this issue, we further propose a practical surface appearance model within the position-free framework (Guo et al., 2018 ###reference_b10###; Wang et al., 2022 ###reference_b31###)\u2013WetSpongeCake.\nConsequentially, our WetSpongeCake model can faithfully reproduce wet porous materials with physical parameters (porosity and saturation). It can represent a wide range of appearances, including a piece of wet paper, saturated sand, or cloth. To summarize, our contributions include:\na generalized anisotropic radiative transfer equation for wet porous medium, which can capture the effect of surrounding liquid on light propagation through particles,\na micro-ellipsoid model for wet medium particles, which defines its phase function and attenuation functions, and\na practical surface appearance model WetSpongeCake to represent wet porous materials with physical parameters." | |
| }, | |
| { | |
| "section_id": "2", | |
| "parent_section_id": null, | |
| "section_name": "2. Related Work", | |
| "text": "" | |
| }, | |
| { | |
| "section_id": "2.1", | |
| "parent_section_id": "2", | |
| "section_name": "2.1. Wet material models", | |
| "text": "Several groups of approaches have been proposed in the literature for representing wet materials, explicit simulations on randomly generated particles, medium-based models and surface-based BRDFs.\nMethods in the first category represent individual particle as a sphere (Peltoniemi and Lumme, 1992 ###reference_b27###) or a prolate spheroid (Kimmel and Baranoski, 2007 ###reference_b18###), and then perform Monte Carlo simulations on randomly generated particles. These methods are computationally expensive and designed for specific material (e.g., sand).\nIn the second category, Jensen et al. (1999 ###reference_b17###) capture the saturation of materials using a combined surface and subsurface model, leveraging a two-term Henyey-Greenstein (HG) phase function to approximate the scattering of particles. While their method produces convincing results, it lacks physically meaningful parameters, requiring manual adjustments of the HG parameter for control. Meanwhile, other studies (Hapke, 2008 ###reference_b12###, 1999 ###reference_b11###; Shkuratov et al., 1999 ###reference_b28###) introduce a porosity parameter by modifying the original RTE. However, these medium-based approaches rely on volumetric path tracing, which results in significant computational costs. Additionally, several other methods (Moon et al., 2007 ###reference_b25###; Meng et al., 2015 ###reference_b23###; M\u00fcller et al., 2016 ###reference_b26###) have been proposed for simulating light transport in visible discrete particles, but these fall outside the scope of our work.\nIn the third category, several studies represent wet materials using surface-based models, including Lekner and Dorf (1988 ###reference_b19###) and Bajo et al. (2021 ###reference_b2###). The former is based on Angstrom\u2019s model (1925 ###reference_b33###), while the latter improves upon Lekner et al.\u2019s model (1988 ###reference_b19###). Other real-time surface models (Hnat et al., 2006 ###reference_b14###; Merillou et al., 2000 ###reference_b24###; Lu et al., 2006 ###reference_b20###) introduce a cylindrical shape for holes and apply intrinsic roughness to the hole surfaces. These approaches are lightweight and practical. However, they struggle to closely match real-world effects. As pointed out by Twomey et al. (1986 ###reference_b29###), even a minor refractive index (IOR) difference (e.g., water vs. benzene on wet sand (Bohren, 1983 ###reference_b3###)) can cause significantly different appearances, which surface models cannot accurately capture. More recently, d\u2019Eon proposed several shading models (d\u2019Eon, 2021 ###reference_b5###; d\u2019Eon and Weidlich, 2024 ###reference_b6###) to describe porous, diffuse-like BRDFs. While these models provide analytic solutions for porous materials, they do not account for the saturation effect. Lucas et al. (2023 ###reference_b21###; 2024 ###reference_b22###) model porous layers, such as dust or dirt, where grains are distributed on a surface. In contrast, our work models porous materials, such as sand or walls, where particles are distributed throughout the volume." | |
| }, | |
| { | |
| "section_id": "2.2", | |
| "parent_section_id": "2", | |
| "section_name": "2.2. Position-free BSDFs", | |
| "text": "The position-free property was first introduced in volumetric BSDF models by Dupuy et al. (2016 ###reference_b7###), which start the trend of volumetric BSDF models. Guo et al. (2018 ###reference_b10###) reformulate path integral within a medium from the entire spatial dimensions to the depth dimension only, as the incident and exit rays share the same position. This simplification of the path integral leads to a more efficient BSDF evaluation. This idea has been used further by Xia et al. (2020 ###reference_b32###) and Gamboa et al. (2020 ###reference_b8###) for more advanced sampling approaches or modeling pearlescent pigments (Guill\u00e9n et al., 2020 ###reference_b9###).\nWang et al. (2022 ###reference_b31###) further generalize single-scattering to handle finite-thickness slabs with transmission. Their SpongeCake model simplified the computation by treating materials as volume slabs without surface interfaces. They also approximated multiple scattering using single scattering with adjusted parameters, making BSDF evaluations efficient and noise-free. Our surface appearance model also follows the position-free framework as a practical solution for wet porous medium rendering." | |
| }, | |
| { | |
| "section_id": "2.3", | |
| "parent_section_id": "2", | |
| "section_name": "2.3. Micro-flake model", | |
| "text": "The micro-flake model, introduced by Jakob et al. (2010 ###reference_b16###), was created to capture the complex anisotropic behavior of participating media by modeling the distribution of reflective micro-flakes. Later, Heitz et al. (2015 ###reference_b13###) introduced the symmetric GGX (SGGX) representation, which uses positive-definite matrices to efficiently describe micro-flake distributions. This method is flexible, allowing for both surface-like and fiber-like micro-flakes, with easy control over their orientation. However, the micro-flake model treats each particle as an ideal disk, which limits its ability to represent transmittable particles." | |
| }, | |
| { | |
| "section_id": "3", | |
| "parent_section_id": null, | |
| "section_name": "3. Wet porous materials with volume rendering", | |
| "text": "###figure_1### In this section, we first introduce the underlying optical effects of wet porous materials (Sec. 3.1 ###reference_###). Next, we modify the radiative transfer equation to account for porosity and saturation (Sec. 3.2 ###reference_###)." | |
| }, | |
| { | |
| "section_id": "3.1", | |
| "parent_section_id": "3", | |
| "section_name": "3.1. Wet porous materials", | |
| "text": "Wet porous materials are composed of millions of non-spherical particles distributed in air and liquid, forming a complex volume, as illustrated in Fig. 2 ###reference_###. The particles allow both reflection and transmission. These particles are characterized by intrinsic properties such as refractive index and albedo , while their orientations follow a specific distribution. The porosity, , represents the fraction of the volume not occupied by particles. The liquid within the material has a refractive index and an extinction coefficient . The fraction of the liquid occupying the non-particle volume is defined as the saturation . All parameters are summarized in Table 1 ###reference_###.\nWhen a light ray enters the medium, it interacts with particle surfaces, where it is either reflected or refracted. Between interactions, the ray travels through air or liquid, with the liquid absorbing some of its energy. The ray continues to bounce within the material until it exits the volume. Due to the finite thickness () of the volume, the ray can either be reflected (exiting on the same side as its entry) or transmitted (exiting on the opposite side). The refraction between air and liquid occurring within the medium is ignored, following previous work (Kimmel and Baranoski, 2007 ###reference_b18###). We do not consider diffraction effects, as the diffraction patterns in a densely packed particulate media are altered by the proximity of neighboring particles, making diffraction effects negligible, as discussed by Hapke (2008 ###reference_b12###).\n###figure_2###" | |
| }, | |
| { | |
| "section_id": "3.2", | |
| "parent_section_id": "3", | |
| "section_name": "3.2. Generalized RTE considering porosity and saturation", | |
| "text": "The anisotropic RTE (Jakob et al., 2010 ###reference_b16###) is formulated as:\nwhere and are the extinction and scattering coefficients respectively. is the source term. Note that the dependence on the position in Eqn. (1 ###reference_###) is ignored for compactness.\nThe anisotropic radiative transfer equation allows for anisotropic scattering in the sense that the scattered intensity depends on the scattering angle. However, it does not take into account porosity and saturation, which limits its ability to represent wet porous materials. To enhance its representation capabilities, we incorporate both porosity and saturation into the anisotropic RTE.\nPorosity describes the fraction of the medium that is not occupied by particles. Hapke (2008 ###reference_b12###) incorporated porosity into the RTE by treating the discrete medium as a series of discrete layers of particles, resulting in a discontinuous stair-step transmittance. Then, the discontinuous transmittance is transformed into an equivalent continuous function as an approximation:\nwhere is a key factor related to the porosity, derived by considering the that adjusts both the initial value and the extinction rate of the transmittance. When the particles are large compared to the wave-length, is only affected by the porosity , hence:\nthe detailed derivation is shown in the supplementary.\nWhile Eqn. (3 ###reference_###) models porosity, it can not ensure energy conservation, as the transmittance can exceed 1. Inspired by Hapke\u2019s work, we modify the anisotropic RTE (Jakob et al., 2010 ###reference_b16###) to incorporate porosity in a simple yet energy-conserving manner by introducing the porosity-related factor into the coefficients:\nwhere behaves like the density; as porosity decreases, increases, leading to a denser medium. In our formulation, the transmittance function starts at 1, ensuring energy conservation.\nWe compare Hapke\u2019s discontinuous stair-step transmittance, its continuous approximation, traditional exponential transmittance, and our transmittance in Fig. 4 ###reference_###. Clearly, Hapke\u2019s continuous approximation starts with an initial value greater than 1, which violates the principle of energy conservation. In contrast, our transmittance, , not only ensures energy conservation but also also takes porosity effects into account.\n###figure_3### After porous materials are wetted by a liquid, the fraction of the liquid occupying the non-particle volume is defined as saturation . This saturation also influences the anisotropic RTE. We assume that the liquid is an absorption-only medium. When the liquid permeates the medium, the distribution of particles remains unchanged, as the liquid simply replaces the air fraction.\nUnder these assumptions, light extinction in the medium is influenced by both the particles and the liquid. While the attenuation caused by the particles has been modeled in Eqn. (4 ###reference_###), the additional attenuation resulting from the liquid must be considered as well. Therefore, the anisotropic RTE considering porosity and saturation can be formulated:\nwhere is the liquid\u2019s extinction coefficient. The transmittance is obtained by integrating extinction of light along , yielding:\nNote that when and , the above transmittance reduces to the typical exponential transmittance. In Fig. 4 ###reference_### (right), we illustrate the transmittance as a function of distance for various materials with different porosity and saturation.\n###figure_4### ###figure_5###" | |
| }, | |
| { | |
| "section_id": "4", | |
| "parent_section_id": null, | |
| "section_name": "4. Transmittable Micro-ellipsoid model", | |
| "text": "To render a wet porous medium with the generalized anisotropic RTE (Sec. 3.2 ###reference_###), we must establish the characteristics of the particles within the medium. Existing isotropic or anisotropic medium models have focused mainly on particles (micro-flake) that can reflect light and have not addressed transmittable particles. However, transmittability is essential for accurately representing a wet porous medium. In our model, we represent each particle as an ellipsoid, which allows for the modeling of an anisotropic medium while also enabling the transmittability of light. We will first define several key functions for isolated particles (Sec. 4.1 ###reference_###) and then present the statistical functions for aggregated particles (Sec. 4.2 ###reference_###)." | |
| }, | |
| { | |
| "section_id": "4.1", | |
| "parent_section_id": "4", | |
| "section_name": "4.1. Isolated ellipsoidal particle", | |
| "text": "Similar to the non-spherical particle by Jakob et al. (2010 ###reference_b16###), we need to define several functions to characterize a particle: 1) projected area is the area of the particle\u2019s projection onto ; 2) particle phase function represents the probability of the light scattering into , illuminated from direction .\nEach particle is represented as an ellipsoid with normal , as shown in Fig. 5 ###reference_###. By setting orthonormal eigenvectors , where , together with the projected area onto and , an ellipsoid can be established. It can be represented as a symmetric positive definite matrix :\nwhere , and are positive eigenvalues of . The projected area of an ellipsoid with normal onto direction can be computed as:\nIn contrast to the micro-flake model, which defines the particle phase function on a flat disk with a constant flake normal, our particle phase function is defined on an ellipsoid. Here, our particle phase function represents the probability of all light paths exiting in direction , given an entry direction with all possible entry points located on the ellipsoid surface. For a given entry point , the light might have various types of light paths, denoted by , , , \u2026, where stands for reflection and stands for transmission, resulting in a probability:\nHere, indicates the attenuation stemming from the Fresnel term along a specular path of type , where represents the set of all possible combinations of and .\nThen, we need to consider all the possible entry points on the ellipsoid surface. To do this, we equivalently transform it into all light paths starting from the projected area of . This way, the particle phase function can be formulated as\nwhere is the differential area around .\nDeriving a closed-form expression for the particle phase function is quite challenging. To address this, we estimate Eqn. (10 ###reference_###) with Monte Carlo simulation by sampling the projected area of , shooting a ray at that direction, intersecting the ray with the ellipsoid to determine an entry point . From , the following specular paths are also sampled by choosing reflection or refraction at each interaction. When the ray exits the ellipsoid, we record its exit direction. This way, we tabulate the particle phase function as a 3D lookup table. Note that the one-dimension reduction is due to the azimuthal symmetry of the particle shape about the normal. For more detailed information, please refer to supplementary material.\n###figure_6###" | |
| }, | |
| { | |
| "section_id": "4.2", | |
| "parent_section_id": "4", | |
| "section_name": "4.2. Micro-ellipsoid model", | |
| "text": "Built on the isolated particle, we propose our micro-ellipsoid model, which is statistically defined for aggregated ellipsoidal particles. We assume the normal distribution of these aggregated ellipsoidal particles follows an ellipsoidal distribution, similar to the SGGX model (Heitz et al., 2015 ###reference_b13###):\nwhere is the symmetric positive definite matrix to describe the normal distribution, and is the micro-ellipsoid normal.\nNext, we present another key function: the attenuation coefficients and . Jakob et al. (2010 ###reference_b16###) provided a general formulation for these functions:\nBy applying the particle projected area function Eqn. (8 ###reference_###), we have formulation of our attenuation coefficients.\nIt is essential to consider the medium\u2019s saturation, as it influences the IOR, which is a key factor in the phase function. With liquid in the medium, there are two cases of particle interface: air-particle and liquid-particle. We define the phase function for these two cases ( and ), and then interpolate them w.r.t. the saturation:\nWe now have the formulations for both the attenuation coefficients and the phase function, although neither has analytical solutions. Therefore, in practice, we compute these functions using Monte Carlo estimation and store the results in a 3D lookup table for the phase function and a 1D lookup table for the attenuation coefficients, as explained in supplementary material.\nIn Fig. 6 ###reference_###, we visualize two phase functions with different liquid IORs but the same particle IOR. The results show that surrounding the particle with liquid leads to increased forward scattering compared to when it is surrounded by air.\nOur attenuation coefficients and phase function follows the framework by Jakob et al. (2010 ###reference_b16###), ensuring both the system reciprocity and the normalization of the phase function.\nOur micro-ellipsoid model and the micro-flake modelis have different particle shape, where our particle is an ellipsoid, and the micro-flake relies on a flake without any volume. The advantage of the ellipsoid is that it allows for transmittability, although it lacks a closed-form formulation. The micro-flake model can also be considered as a special case of our model, by setting , which makes the ellipsoid into a disk." | |
| }, | |
| { | |
| "section_id": "4.3", | |
| "parent_section_id": "4", | |
| "section_name": "4.3. Isotropic medium", | |
| "text": "In a special case, the particle shape become a sphere, by setting , leading to an isotropic medium. In this scenario, the volume extinction coefficients and become constants:\nThe particle phase function and the micro-ellipsoid phase function are also simplified and only depends on :\nAs the phase function only has one dimension, it can be easily represented by fitting a basis function. In practice, we use two Gaussians to represent the one-dimensional phase function: one for forward scattering and the other for backward scattering:\nwhere and are two one-dimensional Gaussian functions with and as the mean values, and as the variances. and are the weights for two Gaussians. All these parameters are obtained by optimization using the Monte Carlo simulation as GT." | |
| }, | |
| { | |
| "section_id": "5", | |
| "parent_section_id": null, | |
| "section_name": "5. WetSpongeCake: A BSDF model with porosity and saturation", | |
| "text": "We have already defined the medium, which still requires large sampling rates for convergence. To address this, we integrate it into the SpongeCake framework by treating the medium as a layer. We derive an analytical model for single scattering, taking into account both reflection and transmission. Then, we further introduce multiple scattering, delta transmission, and the air-liquid interface." | |
| }, | |
| { | |
| "section_id": "5.1", | |
| "parent_section_id": "5", | |
| "section_name": "5.1. Single scattering", | |
| "text": "The single scattering within a medium with thickness can be computed by the integral over the depth of the single scattering vertex (Wang et al., 2022 ###reference_b31###). At each single scattering vertex, we compute phase function, the extinction of the particles and the liquid, together with cosine terms due to the change of the integration domain. Then, the BRDF and BTDF can be formulated as:\nwhere and . The single scattering can be computed using the precomputed phase function." | |
| }, | |
| { | |
| "section_id": "5.2", | |
| "parent_section_id": "5", | |
| "section_name": "5.2. Multiple scattering", | |
| "text": "To compute multiple scattering efficiently, Wang et al. (2022 ###reference_b31###) estimate the multiple scattering of a BSDF by mapping it into a single-scattering lobe with modified parameters along with a Lambertian term with a simple neural network. Their approach could be applied to our BSDF as well. However, the key challenge is that our phase function is not analytical but represented by a precomputed table, which lacks differentiability. To address this limitation, we represent phase function using a neural network to enable multiple to single scattering mapping.\nSpecifically, we propose a phase function network and a parameter mapping network. The former transforms phase function parameters into a phase function value using a simple multi-layer perceptron (MLP). The parameter mapping network converts multiple scattering parameters into single scattering parameters, together with a Lambertian parameter and their weights. The phase function neural network is trained first. Afterward, the mapping network is trained by rendering the predicted single scattering parameters with the learned phase function network. Once the mapping network is trained, we tabulate the phase function to avoid network inference during rendering. More details are in the supplementary.\nPlease note that we do not utilize the neural representation of the phase function for the single scattering, as it is much sharper, which is beyond the capabilities of the learned network.\nIn the case of an isotropic medium, since the phase function can be represented using two Gaussian distributions, we can directly learn the mapping network without the need to train an additional phase function neural network." | |
| }, | |
| { | |
| "section_id": "5.3", | |
| "parent_section_id": "5", | |
| "section_name": "5.3. The other components", | |
| "text": "For a thin medium, light can pass through without any scattering, such as paper and cloth. This delta function can be determined by analyzing the transmission of light through the medium in the direction of , similar to the SpongeCake model.\nWhen a material is fully saturated (), a thin film of liquid forms on the surface. Our model captures the effects of light reflection and refraction at the interface by layering a dielectric BSDF on top of the medium.\nImportance sampling is a necessary component for a BSDF. For this, we use a simple solution, by sampling the phase function with cumulative distribution function to get the outgoing direction." | |
| }, | |
| { | |
| "section_id": "6", | |
| "parent_section_id": null, | |
| "section_name": "6. Results", | |
| "text": "We have implemented our algorithm inside the Mitsuba renderer (2010 ###reference_b15###).\nAll timings in this section are measured on a 2.20GHz Intel i7 (48 cores) with 32 GB of main memory. We provide the material settings in the supplementary material." | |
| }, | |
| { | |
| "section_id": "6.1", | |
| "parent_section_id": "6", | |
| "section_name": "6.1. Model validation", | |
| "text": "We validate our model through a series of experiments, including comparisons against photographs, equal-time comparison between the surface model and volume model, validation of multiple scattering, and the white furnace test. Please note that our multiple scattering model is applied to all materials, except for the spatially varying anisotropic material, for which we perform a random walk in a position-free framework (Guo et al., 2018 ###reference_b10###).\nTo demonstrate WetSpongeCake\u2019s capability to replicate natural appearances, we compare our model, Jensen et al. (1999 ###reference_b17###), which employs two Henyey-Greenstein (HG) lobes as the phase function, along with a real photograph. The comparison is shown in Fig. 7 ###reference_###. This scene consists of three types of sand (isotropic) in different containers: dry, saturated with water, and saturated with ink, respectively, lit under an environment map and a directional light. We render this scene with both single and multiple scattering, considering global illumination. Our model almost matches the photograp, while Jensen et al.\u2019s method, which adjusts HG parameters manually, can not model liquid absorption.\nIn Fig. 8 ###reference_###, we compare our method, the albedo map method, which achieves darker reflections by adjusting the albedo, along with a real photograph. The scene consists of a wet cloth (anisotropic medium) under two lighting conditions: front-lit and back-lit. We render this scene with both single and multiple scattering. The photographs demonstrate that the wet regions of the cloth exhibit reduced reflection and enhanced transmission, a behavior accurately captured by our model. In contrast, simply modifying the albedo reduces both reflection and transmission, failing to reproduce the desired appearance.\nWe validate the effectiveness of our surface model (both single and multiple scattering), by comparing it to our volume model in Fig. 12 ###reference_###, where the converged volume renderings are treated as GT. We find that the result of our single scattering is almost identical to the GT while having much less noise than the volume rendering with equal time, as it is analytical and does not need a random walk. After introducing the multiple scattering, the result of our surface model still exhibits less noise and lower error than volume rendering with equal time, thanks to our effective way of computing the multiple scattering.\nIn Figs. 13 ###reference_### and 14 ###reference_###, we validate our multiple scattering by comparing its results to the GT obtained by Monte Carlo random walk (Guo et al., 2018 ###reference_b10###) across various materials. The results demonstrate an overall good match, except in cases with a low projected area where the sharp distribution cannot be accurately expressed by the phase function network.\nWe perform a white furnace test for our surface model on materials with no absorption, under a constant environment map with radiance set as one. As shown in Fig. 11 ###reference_###, the delta transmission and the single scattering results in darker pixel values, as expected. When including the multiple scattering using Monte Carlo random walk, a constant image was produced, while our multiple scattering closely approximates a constant image, although there are some inaccuracies due to network bias.\nTo demonstrate the impact of different parameters on appearance, we present renderings of the Sculpture scene under various parameter settings in Fig. 15 ###reference_###. This scene is illuminated by two area light sources and an environment map. As the liquid\u2019s IOR increases, the phase function becomes more forward-scattering, causing a darker appearance. Similarly, higher saturation enhances forward scattering while incorporating liquid absorption, further darkening the sculpture. Furthermore, increased light absorption by the liquid results in a noticeably darker sculpture." | |
| }, | |
| { | |
| "section_id": "6.2", | |
| "parent_section_id": "6", | |
| "section_name": "6.2. More results", | |
| "text": "In Fig. 1 ###reference_###, we show several objects with porous materials lit by an environment map with global illumination, including wet cloths (anisotropic), flowerpots (isotropic), and sculptures (isotropic). Our BSDF can represent a wide range of appearances in the real world with high fidelity, controlled by physical parameters.\nWe design a Paper scene using the photo converted to a grayscale image as a saturation map, as shown in Fig. 10 ###reference_###. The saturation map, displayed in the bottom-right corner of the rendered image, defines the varying saturation across the surface. This scene is illuminated by an environment map, taking global illumination into account. Our model is able to produce the desired effects.\nIn Fig. 9 ###reference_###, we present the appearance effects caused by the air-liquid interface. This scene consists of wood (isotropic), illuminated by an environment map, with global illumination taken into account. The table surface without a liquid film appears diffuse, while the table surface with a liquid film exhibits a specular lobe." | |
| }, | |
| { | |
| "section_id": "6.3", | |
| "parent_section_id": "6", | |
| "section_name": "6.3. Discussion and limitations", | |
| "text": "We have identified several limitations in our method. Our model does not have closed-form formulations and relies on precomputation, which restricts its ability to represent multiple scattering in spatially varying anisotropic materials. Additionally, similar to previous studies (Wang et al., 2022 ###reference_b31###), our multiple scattering network may sometimes struggle to achieve accurate fits." | |
| }, | |
| { | |
| "section_id": "7", | |
| "parent_section_id": null, | |
| "section_name": "7. Conclusion", | |
| "text": "In this paper, we have presented a transmittable micro-ellipsoid model for rendering wet porous materials, applicable for both volume and surface appearance rendering. We generalize the anisotropic radiative transfer equation to account for both porosity and saturation, which enables volume rendering of wet porous materials together with the micro-ellipsoid model. Building on this foundation, we then derive a practical surface appearance model under the position-free framework, termed WetSpongeCake. The WetSpongeCake model effectively represents the diverse appearances of wet porous materials using physically meaningful parameters such as porosity and saturation, and it captures both reflection and transmission effects. The resulting appearances closely mimic real-world observations. To the best of our knowledge, no existing method offers such a lightweight, physically-based shading model for wet porous materials. Finding a closed-form solution for the phase function is crucial for completing the theory and further enhancing the model\u2019s practical applicability. We also believe that our model has potential applications in other areas, such as inverse rendering." | |
| } | |
| ], | |
| "appendix": [], | |
| "tables": { | |
| "1": { | |
| "table_html": "<figure class=\"ltx_table\" id=\"S2.T1\">\n<figcaption class=\"ltx_caption\"><span class=\"ltx_tag ltx_tag_table\">Table 1. </span> Notations.</figcaption>\n<table class=\"ltx_tabular ltx_guessed_headers ltx_align_middle\" id=\"S2.T1.16\">\n<thead class=\"ltx_thead\">\n<tr class=\"ltx_tr\" id=\"S2.T1.16.17.1\">\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t\" colspan=\"2\" id=\"S2.T1.16.17.1.1\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"><span class=\"ltx_text\" id=\"S2.T1.16.17.1.1.1\" style=\"font-size:90%;\">Mathematical notation</span></th>\n</tr>\n</thead>\n<tbody class=\"ltx_tbody\">\n<tr class=\"ltx_tr\" id=\"S2.T1.1.1\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t\" id=\"S2.T1.1.1.1\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S2.T1.1.1.2\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"><span class=\"ltx_text\" id=\"S2.T1.1.1.2.1\" style=\"font-size:90%;\">full spherical domain</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S2.T1.3.3\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r\" id=\"S2.T1.2.2.1\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S2.T1.3.3.2\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\">\n<span class=\"ltx_text\" id=\"S2.T1.3.3.2.1\" style=\"font-size:90%;\">The cosine of the angle between </span><span class=\"ltx_text\" id=\"S2.T1.3.3.2.2\" style=\"font-size:90%;\"> and the z-axis</span>\n</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S2.T1.4.4\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r\" id=\"S2.T1.4.4.1\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S2.T1.4.4.2\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"><span class=\"ltx_text\" id=\"S2.T1.4.4.2.1\" style=\"font-size:90%;\">dot product</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S2.T1.16.18.1\">\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t\" colspan=\"2\" id=\"S2.T1.16.18.1.1\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"><span class=\"ltx_text\" id=\"S2.T1.16.18.1.1.1\" style=\"font-size:90%;\">Physical quantities</span></th>\n</tr>\n<tr class=\"ltx_tr\" id=\"S2.T1.5.5\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t\" id=\"S2.T1.5.5.1\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S2.T1.5.5.2\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"><span class=\"ltx_text\" id=\"S2.T1.5.5.2.1\" style=\"font-size:90%;\">incident direction</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S2.T1.6.6\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r\" id=\"S2.T1.6.6.1\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S2.T1.6.6.2\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"><span class=\"ltx_text\" id=\"S2.T1.6.6.2.1\" style=\"font-size:90%;\">outgoing direction</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S2.T1.7.7\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r\" id=\"S2.T1.7.7.1\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S2.T1.7.7.2\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"><span class=\"ltx_text\" id=\"S2.T1.7.7.2.1\" style=\"font-size:90%;\">phase function</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S2.T1.16.19.2\">\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t\" colspan=\"2\" id=\"S2.T1.16.19.2.1\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"><span class=\"ltx_text\" id=\"S2.T1.16.19.2.1.1\" style=\"font-size:90%;\">Phase function parameters</span></th>\n</tr>\n<tr class=\"ltx_tr\" id=\"S2.T1.8.8\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t\" id=\"S2.T1.8.8.1\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S2.T1.8.8.2\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"><span class=\"ltx_text\" id=\"S2.T1.8.8.2.1\" style=\"font-size:90%;\">refractive index of particle</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S2.T1.9.9\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r\" id=\"S2.T1.9.9.1\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S2.T1.9.9.2\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"><span class=\"ltx_text\" id=\"S2.T1.9.9.2.1\" style=\"font-size:90%;\">particle projected area</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S2.T1.16.20.3\">\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t\" colspan=\"2\" id=\"S2.T1.16.20.3.1\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"><span class=\"ltx_text\" id=\"S2.T1.16.20.3.1.1\" style=\"font-size:90%;\">Medium parameters</span></th>\n</tr>\n<tr class=\"ltx_tr\" id=\"S2.T1.10.10\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t\" id=\"S2.T1.10.10.1\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S2.T1.10.10.2\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"><span class=\"ltx_text\" id=\"S2.T1.10.10.2.1\" style=\"font-size:90%;\">normal projected area</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S2.T1.11.11\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r\" id=\"S2.T1.11.11.1\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S2.T1.11.11.2\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"><span class=\"ltx_text\" id=\"S2.T1.11.11.2.1\" style=\"font-size:90%;\">thickness</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S2.T1.12.12\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r\" id=\"S2.T1.12.12.1\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S2.T1.12.12.2\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"><span class=\"ltx_text\" id=\"S2.T1.12.12.2.1\" style=\"font-size:90%;\">saturation</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S2.T1.13.13\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r\" id=\"S2.T1.13.13.1\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S2.T1.13.13.2\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"><span class=\"ltx_text\" id=\"S2.T1.13.13.2.1\" style=\"font-size:90%;\">porosity</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S2.T1.14.14\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r\" id=\"S2.T1.14.14.1\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S2.T1.14.14.2\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"><span class=\"ltx_text\" id=\"S2.T1.14.14.2.1\" style=\"font-size:90%;\">single-scattering albedo</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S2.T1.15.15\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r\" id=\"S2.T1.15.15.1\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S2.T1.15.15.2\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"><span class=\"ltx_text\" id=\"S2.T1.15.15.2.1\" style=\"font-size:90%;\">refractive index of liquid</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S2.T1.16.16\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_r\" id=\"S2.T1.16.16.1\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"></th>\n<td class=\"ltx_td ltx_align_center ltx_border_b ltx_border_r\" id=\"S2.T1.16.16.2\" style=\"padding-top:0.45pt;padding-bottom:0.45pt;\"><span class=\"ltx_text\" id=\"S2.T1.16.16.2.1\" style=\"font-size:90%;\">liquid extinction coefficient</span></td>\n</tr>\n</tbody>\n</table>\n</figure>", | |
| "capture": "Table 1. Notations." | |
| } | |
| }, | |
| "image_paths": { | |
| "1": { | |
| "figure_path": "2401.15628v3_figure_1.png", | |
| "caption": "Figure 1. In this paper, we present a transmittable micro-ellipsoid model to render wet porous materials with physical parameters. It can produce vivid appearances on a wide range of materials. Examples include the wet cloths, flowerpots and sculptures as shown above.", | |
| "url": "http://arxiv.org/html/2401.15628v3/x1.png" | |
| }, | |
| "2": { | |
| "figure_path": "2401.15628v3_figure_2.png", | |
| "caption": "Figure 2. Wet porous material is made of discrete particles, air, and liquid. After a light ray enters this material, it can be either reflected or refracted by the particle surface and travels in the air or the liquid. The ray is bounced within this material until it leaves the surface.", | |
| "url": "http://arxiv.org/html/2401.15628v3/x2.png" | |
| }, | |
| "3": { | |
| "figure_path": "2401.15628v3_figure_3.png", | |
| "caption": "Figure 3. Several examples of parameters defined on particles and medium.", | |
| "url": "http://arxiv.org/html/2401.15628v3/x3.png" | |
| }, | |
| "4": { | |
| "figure_path": "2401.15628v3_figure_4.png", | |
| "caption": "Figure 4. Left: Comparison of several transmittance functions w.r.t. the traveling distance. Our saturation is set as 0. Right: our transmittance function as a function of the distance across several parameters.", | |
| "url": "http://arxiv.org/html/2401.15628v3/x4.png" | |
| }, | |
| "5(a)": { | |
| "figure_path": "2401.15628v3_figure_5(a).png", | |
| "caption": "(a) ellipsoidal particle\nFigure 5. \n(a) Each particle is modeled as an ellipsoid, defined by eigenvectors (\u03c91subscript\ud835\udf141\\omega_{1}italic_\u03c9 start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, \u03c92subscript\ud835\udf142\\omega_{2}italic_\u03c9 start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and \u03c93subscript\ud835\udf143\\omega_{3}italic_\u03c9 start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT) and eigenvalues, where \u03c93subscript\ud835\udf143\\omega_{3}italic_\u03c9 start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT is aligned with the ellipsoid normal. (b) the particle phase function is defined as the possibility of all paths originating from the projected area of \u03c9isubscript\ud835\udf14\ud835\udc56\\omega_{i}italic_\u03c9 start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, traveling in direction \u03c9isubscript\ud835\udf14\ud835\udc56\\omega_{i}italic_\u03c9 start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, bouncing within the ellipsoid, and finally exiting in direction \u03c9osubscript\ud835\udf14\ud835\udc5c\\omega_{o}italic_\u03c9 start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT.", | |
| "url": "http://arxiv.org/html/2401.15628v3/x5.png" | |
| }, | |
| "5(b)": { | |
| "figure_path": "2401.15628v3_figure_5(b).png", | |
| "caption": "(b) particle phase function\nFigure 5. \n(a) Each particle is modeled as an ellipsoid, defined by eigenvectors (\u03c91subscript\ud835\udf141\\omega_{1}italic_\u03c9 start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, \u03c92subscript\ud835\udf142\\omega_{2}italic_\u03c9 start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and \u03c93subscript\ud835\udf143\\omega_{3}italic_\u03c9 start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT) and eigenvalues, where \u03c93subscript\ud835\udf143\\omega_{3}italic_\u03c9 start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT is aligned with the ellipsoid normal. (b) the particle phase function is defined as the possibility of all paths originating from the projected area of \u03c9isubscript\ud835\udf14\ud835\udc56\\omega_{i}italic_\u03c9 start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, traveling in direction \u03c9isubscript\ud835\udf14\ud835\udc56\\omega_{i}italic_\u03c9 start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, bouncing within the ellipsoid, and finally exiting in direction \u03c9osubscript\ud835\udf14\ud835\udc5c\\omega_{o}italic_\u03c9 start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT.", | |
| "url": "http://arxiv.org/html/2401.15628v3/x6.png" | |
| }, | |
| "6": { | |
| "figure_path": "2401.15628v3_figure_6.png", | |
| "caption": "Figure 6. Visualization of the phase function for particles surrounded by air and water, respectively, with the mean cosine value provided in the bottom-right corner. The remaining parameters are set to \u03b7p=1.8subscript\ud835\udf02\ud835\udc5d1.8\\eta_{p}=1.8italic_\u03b7 start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 1.8, \u03c3=1\ud835\udf0e1\\sigma=1italic_\u03c3 = 1, and \u03c3p=1subscript\ud835\udf0e\ud835\udc5d1\\sigma_{p}=1italic_\u03c3 start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 1. Additional results are provided in the supplementary material.", | |
| "url": "http://arxiv.org/html/2401.15628v3/x7.png" | |
| }, | |
| "7": { | |
| "figure_path": "2401.15628v3_figure_7.png", | |
| "caption": "Figure 7. Comparison between the photograph, our model, and the two-HG method by Jensen et al. (1999). Our model demonstrates a close match with the photograph, whereas the two-HG method by Jensen et al. (1999) captures the change in scattering direction but fails to adequately model liquid absorption.", | |
| "url": "http://arxiv.org/html/2401.15628v3/x8.png" | |
| }, | |
| "8": { | |
| "figure_path": "2401.15628v3_figure_8.png", | |
| "caption": "Figure 8. Comparison between photographs, our model, and an empirical approach which modifies the albedo at two light conditions (front-lit and back-lit). In the photographs, the wet regions of the cloth exhibit reduced reflection and enhanced transmission. Our model replicates these effects accurately, whereas using a darker albedo achieves a similar reduction in reflection but results in incorrect darkening of transmission.", | |
| "url": "http://arxiv.org/html/2401.15628v3/x9.png" | |
| }, | |
| "9": { | |
| "figure_path": "2401.15628v3_figure_9.png", | |
| "caption": "Figure 9. Our model supports a thin liquid film on top of the medium, capturing the reflection induced by the air-liquid interface.", | |
| "url": "http://arxiv.org/html/2401.15628v3/x10.png" | |
| }, | |
| "10": { | |
| "figure_path": "2401.15628v3_figure_10.png", | |
| "caption": "Figure 10. A wet paper saturated by water. Our method is able to characterize the saturation level using a saturation map.", | |
| "url": "http://arxiv.org/html/2401.15628v3/x11.png" | |
| }, | |
| "11": { | |
| "figure_path": "2401.15628v3_figure_11.png", | |
| "caption": "Figure 11. A white furnace test for a WetSpongeCake material with no absorption. While the delta transmission and single scattering result in darker pixel values, our multiple scattering model produces an image that is nearly uniform, with minor inaccuracies. The Monte Carlo multiple scattering successfully passes the white furnace test.", | |
| "url": "http://arxiv.org/html/2401.15628v3/x12.png" | |
| }, | |
| "12": { | |
| "figure_path": "2401.15628v3_figure_12.png", | |
| "caption": "Figure 12. Comparison between our BSDF surface model and the volume rendering with our modified RTE. The references are rendered with our modified RTE at a high sample rate. In both cases, our BSDF produces results with much less noise and lower error than the volume renderings with equal time.", | |
| "url": "http://arxiv.org/html/2401.15628v3/x13.png" | |
| }, | |
| "13": { | |
| "figure_path": "2401.15628v3_figure_13.png", | |
| "caption": "Figure 13. Multiple scattering validation. For each example, we provide the normal projected area \u03c3\ud835\udf0e\\sigmaitalic_\u03c3, particle projected area \u03c3psubscript\ud835\udf0e\ud835\udc5d\\sigma_{p}italic_\u03c3 start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT. The ground truth is obtained through Monte Carlo simulation of multiple scattering. Our results exhibit minor differences from the ground truth in most cases.", | |
| "url": "http://arxiv.org/html/2401.15628v3/x14.png" | |
| }, | |
| "14": { | |
| "figure_path": "2401.15628v3_figure_14.png", | |
| "caption": "Figure 14. Multiple scattering validation for a set of materials, with varying porosity P\ud835\udc43Pitalic_P and saturation S\ud835\udc46Sitalic_S. Monte Carlo simulation of multiple scattering is used as the ground truth. Our results show only minor deviations from the ground truth.", | |
| "url": "http://arxiv.org/html/2401.15628v3/x15.png" | |
| }, | |
| "15": { | |
| "figure_path": "2401.15628v3_figure_15.png", | |
| "caption": "Figure 15. Sculpture scene under different parameter settings. Parameters not explicitly mentioned are set to their default values, as indicated at the top of the figure.", | |
| "url": "http://arxiv.org/html/2401.15628v3/x16.png" | |
| } | |
| }, | |
| "validation": true, | |
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