original:https://zhuanlan.zhihu.com/p/776529221

The historical evolution of hair rendering research

In 1989, Kajiya and Kay extended the Phong model to hair drawing and proposed an empirical model for hair drawing, the Kajiya-Kay model. This model simplifies hair into a series of elongated cylinders and assumes that light is simply reflected on the surface of hair. Specular reflection: highlights. Diffuse reflection: simulates the scattering of light inside the hair and the overall brightness of the hair.

In 2003, Marschner et al. published the paper "Light Scattering from Human Hair Fibers", proposing a physics-based hair reflection model, known as the Marschner model.

Traditional hair rendering models, such as the Marschner model and the Kajiya-Kay model, usually simplify hair into a single-layer cylinder, ignoring the influence of the medulla. In their 2015 paper "Physically-Accurate Fur Reflectance: Modeling, Measurement and Rendering", Yan et al. proposed a more accurate and efficient hair rendering model to address this problem, modeling each hair as two concentric cylinders. The modeling combines the complete hair bidirectional scattering distribution function (BSDF) to accurately describe the multipath propagation and scattering behavior of light in hair. In addition, to ensure physical authenticity, a large amount of physical measurement work was carried out, including nine different hair samples, and finally some reflection parameters of the database were opened for artists to adjust. In order to improve rendering efficiency, Yan [2015] combined the consideration of near-field scattering (R, TT, TRT) of [Zinke and Weber 2007], pre-calculated common scattering paths, stored in Lut, and realized efficient light scattering calculation of single hair fibers. For rendering pipeline optimization, Yan [2015] mainly focused on the dual-cylinder model.

From Marschner[2003] to the extended models of d'Eon[2011] and Chiang[2016], although the continuous increase of hair parameters (such as azimuthal roughness numerical integration of d'Eon[2011] and near-field azimuthal scattering of Chiang[2016]) has increased rendering accuracy, its complex scattering path and large amount of pre-calculation limit its practicality and real-time performance. The double-cylinder hair reflection model proposed by Yan[2015] also has the problems of high computational cost and low practicality. Therefore, Yan[2017] proposed a simplified version, which achieves fast integration through analytical methods and greatly reduces the number of Lobes.

The figure above clearly shows that Marschner [2003] used a longitudinal-azimuthal decomposition representation to simplify the complex three-dimensional light scattering process into two relatively independent dimensions. The longitudinal scattering function describes the propagation and scattering of light along the axis of the hair fiber. The azimuthal scattering function describes the scattering of light in the cross section of the hair fiber (the plane perpendicular to the fiber axis). This model considers T, TT and TRT. The energy conservation problem was corrected in d'Eon [2011]. Yan [2015]'s double cylinder model (hair cuticle and hair medulla) complicated the light interaction and considered R, TrT, TtT and TrRrT. Yan [2017] introduced a unified refractive index (IORs) to simplify the light path propagation and no longer distinguish the refractive indices of different materials, namely R, TT, TRT, TT^s and TRT^s (^s represents the simplified path). Yan[2017] pointed out that unifying IORs does not actually significantly affect the rendering results, and it can still maintain a high degree of realism becauseThe refractive index of the hair cortex and medulla is very close..

In order to solve the problem of high computational complexity in previous hair rendering, Yan[2017] proposedDivision of near field and far field, and introducedLayered Rendering Strategy. The near field is mainly the fine scattering and reflection of light on a single hair. This area requires a high-precision physical model to render hair, such as wave optical phenomena such as interference and diffraction. The far field describes the overall scattering effect of light under the collective action of a large number of hair fibers. In this area, the microscopic structure of a single hair can be averaged, which is more suitable for calculation using statistical/approximate methods to optimize and improve rendering efficiency.Classification criteria: Based on the distance between the light and the hair fiber and the size of the hair fiberThat is, set aThresholdWhen the distance between the light and the hair is less than the threshold, it is classified as near field; otherwise, it is classified as far field.Layered rendering processFirst, ray tracing is used to determine whether it is near field or far field. In the near field, the Mie scattering theory and Fresnel equation are used to calculate reflection and transmission in combination with the pre-calculated scattering table. In the far field, the statistical scattering function + pre-calculated scattering table + MC integral are used to reduce the complexity. Finally, the two are superimposed and normalized. Here is a detailed explanation based on the three points of the paper:

• Simple Reflectance ModelAlthough Yan's model [2015] introduced the medulla and considered more physical details, it still has high computational complexity, especially in the conversion between near field and far field. Yan [2017] proposedSimplified hierarchical reflection model, retaining the key physical phenomena of reflection and reducing unnecessary complex light scattering paths. They describe reflection as three main light scattering paths (R: specular reflection of light on the surface of the hair cuticle, TT: light passes through the hair cuticle and is transmitted from the other side after internal scattering, TRT: light enters the hair, reflects once inside, and finally transmits). Finally, combined withSimplified Bidirectional Scattering Distribution Function (BSDF)The reflection of the capture path reduces the number of lobes required in the calculation (usually used to describe the distribution curve of different scattering directions). Compared with the previous model, the number of lobes in the calculation process is reduced from 9 to 5.
• Improved Accuracy and PracticalityHigh-precision models (such as the Marschner model, Yan [2015], etc.) require complex numerical calculations and a large amount of pre-calculated data, and are therefore difficult to implement in real-time applications. Low-precision empirical models lack sufficient physical reality. Therefore, many improvements were made in Yan [2017]. Although the model simplifies the light scattering path,Combining physical phenomenaThe model is more accurate than the traditional empirical model by reasonably simplifying the longitudinal and azimuthal scattering of light.Transition processing between near field and far fieldTraditional models often fail to smoothly handle the optical transition between the near field and the far field. Yan et al. introducedA near-field-far-field analysisThe solution accurately simulates reflections when the light is close to the hair fiber, while quickly approximating the overall reflection behavior of the light in the far field. This makes the rendering efficient enough for real-time rendering.
• Analytic Near/Far Field SolutionThere is a huge difference in the treatment of the near field (the short-range interaction between light and a single hair fiber, i.e., scattering behavior) and the far field (the long-range collective effect between light and a large number of hair fibers). In order to achieve a seamless transition between the near field and the far field, the authors used an analytical integration method instead of cumbersome numerical integration. The analytical integration can directly calculate the reflection function without the need for complex numerical solutions or pre-calculations, which greatly reduces the calculation time.
• Significant Speed Up
  • Reduce the number of scatter paths used to describeNumber of lobes;
  • Combination of analytical integration and pre-calculation;
  • A simplified BSDF and analytical reflection calculation formula are used to combine ray tracing and reflection calculation inParallelization on GPU, the rendering speed of the model is increased by 6-8 times compared with previous methods.

To summarize briefly, the reflection model proposed by Yan [2017] has good effects and performance. By unifying the IOR of the cortex and medulla, the model only needs 5 lobes to represent the complex scattering of fur, and the tensor approximation is used to minimize the storage overhead. Based on this model, the analytical integration of the far and near fields is proposed to extend the model to multi-scale rendering. It is very simple to implement the BCSDF model in real-time rendering. There are already many implementation methods, and it has been applied to the film and television industry. [The Lion King (HD). 2017 movie] (2019 Oscar Nominee for Best Visual Effects)

XIA[2023] proposed a hair reflection model based on wave optics. Traditional hair rendering models are mostly based on geometric optics approximation. These models work well when processing larger hair fibers, but have poor performance on subtle optical phenomena (such as colored spots on hair, i.e.glints). These scattering effects, including reflection, transmission, and multiple scattering, are difficult to accurately describe with simple geometric optics models. As the diameter of the hair fiber approaches or becomes smaller than the wavelength of light (visible light), wave optics effects become increasingly important, and geometric optics models are unable to capture these effects.

Wave optical effects of hair, such asInterference and diffraction of lightThe computational complexity is very high. Wave optics simulation requires calculating the propagation of electromagnetic fields, not just the path of light. Hair and fur have highly irregular microstructures that further affect the scattering of light. Methods based on geometric optics cannot handle these wave phenomena, and full-wave simulations require high computing resources.

As early as XIA[2020], it was proposed to use wave optics to accurately describe the interaction between light and fibers, and to use the boundary element method (BEM) to simulate the fiber scattering of light at any cross section. In addition, XIA[2020] pointed out that due to the diffraction effect, the fiber exhibits an extremely strong forward scattering effect. Therefore, the wave optics effect should focus the light in the direction of forward scattering. It was also pointed out that the small fiber scattering effect depends significantly on the wavelength of light, resulting in strong wavelength scattering. In addition, the singular softening phenomenon brought by the wave field is also the key to determining the real caustic effect. In order to control the amount of calculation of the BEM simulation, the shape of the fiber is ideally a regular cross-sectional shape. However, Marschner[2003] pointed out that the irregularity of the hair surface has an important influence on the appearance of the hair. Whether such an effect is significant in wave optics is still a problem that needs to be explored and solved.

Traditional geometric optics methods are based on ray tracing, which predicts the propagation path of light by simulating the reflection and refraction of light on hair fibers. However, this method is insufficient when dealing with light waves with wavelengths comparable to the fiber size, and cannot capture the effects of diffraction.Complex optical effects are produced. In actual measurements, fiber scattering shows some sharp optical features, which are caused by the diffraction effect of light. Including the slight color shift in black dog hair, which is also caused by the interference and diffraction of light.

In order to deal with these phenomena that cannot be explained by geometric optics, XIA[2023] developed a 3D wave optics simulator based on physical optics approximation (PO) and used GPU to accelerate computational efficiency. The space is processed through an octree structure. The simulator has a certain degree of versatility and can handle arbitrary 3D geometric shapes, that is, it can handle the microstructure of the fiber surface.

However, XIA[2023] points out that it is unrealistic to directly apply this simulator to the current mainstream rendering framework due to the high computational complexity. Therefore, it is necessary to first migrate the model to the existing hair scattering model and then add aDiffraction lobe of elementary diffraction theoryFinally, aRandom ProcessThe modulation method is used to capture the optical speckle effect. Although it is procedural noise, it is still consistent with the physical simulation result, and the visual effect is close to reality.

XIA[2023] divides the current hair/fiber rendering into two types: traditionalRay-based Fiber Models, the other isWave-based Fiber Models.

Linder[2014] proposed an analytical solution to deal with the scattering behavior of cylindrical fibers, but it is only applicable to perfect circular cross-sections and cannot handle complex hair surface structures. XIA[2020] studied the scattering behavior of fibers with arbitrary cross-sectional shapes through two-dimensional wave optics rendering, showing the manifestation of diffraction effects, but the paper assumes a perfect extrusion structure, that is, the fiber surface is regular. Bennamira&Pattanaik[2021] proposed a hybrid model that uses wave optics to solve the problem of only forward diffraction and traditional geometric optics in other scattering modes. However, XIA[2023] further considered theDependence of longitudinal angle of incidence.

At the end of the paper, the procedural noise is fitted to the speckle pattern in wave optics, and a very realistic effect is produced through statistical property fitting.

XIA[2023] also mentionedComputational Electromagnetics ToolsIt plays an important role in dealing with complex interactions of rays and fibers, especially when using numerical methods such as BEM.Computational ElectromagneticsIt is a computational method used to study electromagnetic phenomena. Since light is an electromagnetic wave, many phenomena in optics can be analyzed using electromagnetic tools. CEM is often used in optics to calculate the interaction between light and the surface of an object (such as hair fibers). Common CEM algorithms include:

• Finite-Difference Time-Domain (FDTD): A numerical method for solving Maxwell's equations by discretizing them in space and time, first proposed by Kane Yee in 1966. It is a direct time-domain method Kane Yee[1966], Taflove[2005].
• Finite Element Method (FEM)：It is used to solve the electromagnetic field distribution in complex geometric structures by dividing the solution area into a finite number of elements Jin[2015].
• Boundary Element Method (BEM): Also known as the Method of Moments (MoM), this is a numerical method that reduces the amount of computation by only dealing with the electromagnetic fields on the surface of an object Gibson[2021], Huddleston[1986], Wu[1977].

Although CEM has many acceleration algorithms, such as Song[1997]'s Multilevel Fast Multipole Algorithm (MLFMA), the improvement is still minimal in hair and fur simulation.

Since full-wave simulations are computationally expensive, XIA [2023] proposed the physical optics approximation (PO) to simplify the reflection and diffraction processes on the surface of, for example, hair fibers.

Physical Optics Plane ModelyesAn application of physical optics (PO) that is specifically designed to simulate the behavior of light on flat or nearly flat surfaces.The scattering and diffraction effects on rough surfaces are effectively calculated by Beckmann-Kirchhoff[1987] and Harvey-Shack[1979]. Gaussian random surfaces by He[1991], Kajiya[1985], periodic static surfaces by Stam[1999], and scratched surfaces by Werner[2017] all use physical optics approximations to deal with surface reflection and diffraction.

For more complex diffraction, Krywonos[2006], Krywonos[2011] proposed improved methods for processing diffraction on rough surfaces. Holzschuch, Pacanowski[2017] proposed a dual-scale microsurface model that combines reflection and diffraction to simulate rough surfaces. Recently, Falster[2020] combined Kirchhoff scalar diffraction theory and path tracing to handle secondary reflection and scattering. Yan[2018] used physical optics to render the mirror microgeometry of rough surfaces.

Unlike a flat surface, the fiber surface is a closed curved surface, and the geometric shape of the hair fiber makes the interaction with light more complex. In addition to reflection and scattering, forward diffraction scattering and large-scale shadow effects need to be dealt with.

XIA[2023] also discussedSpeckle effectandProcedural noiseApplication in hair rendering.

Speckle is a grainy image or diffraction pattern produced when light interacts with a rough surface. Its statistical properties have been extensively studied. When coherent light (such as a laser) is irradiated onto a rough surface or passes through a scattering medium, a random pattern of light and dark spots is produced. In layman's terms, it's like when you shine a laser pointer on a rough wall, you see a granular, flickering pattern instead of a smooth spot of light. Because light is scattered at tiny surface irregularities, light waves from different paths interfere with each other, some are strengthened (forming bright spots) and some cancel each other out (forming dark spots), resulting in this speckled pattern.

Previous studies have explored howMonte Carlo methodto simulate the speckle effect in volume scattering Bar[2019, 2020]. However, these models are mainly applicable toHomogeneous media, which is not applicable to heterogeneous structures such as hair fibers. Steinberg, Yan [2022] studied speckle rendering of planar rough surfaces. However, the authors pointed out thatThe speckle effect on fiber surfaces is different from that on flat surfaces, showing different statistical characteristics.

Therefore, XIA[2023] proposedAccurately capture the statistical characteristics of fiber speckle patternsBy studying the special geometric structure and speckle distribution of the fiber surface, the scattering effect of hair fibers is simulated to provide better speckle effects.

It should be noted that although both thin-film interference and speckle effect are caused by the interference of light, they have significant differences in physical mechanism, visual performance and rendering methods in computer graphics. Monte Carlo methods of thin-film interference, such as random film thickness sampling, can be used to generate random spots of speckle effect to improve the realism of rendering. Approximate algorithms such as hierarchical thickness sampling and pre-calculated interference patterns can also learn from each other. Thin-film interference often involves interference of light waves at different scales, and speckle effect also involves multi-scale scattering of microscopic surface structures.

Between the two, thin film interferometry has a relatively low rendering complexity, and pre-computation can be fully utilized to avoid the burden of real-time calculation. However, the speckle effect has highly random and statistical characteristics, and a large number of random interference paths need to be processed, especially for simulating heterogeneous structures such as hair. Current research such as XIA[2023] is working to improve its efficiency, but there is still a large gap compared to thin film interferometry.

XIA[2023] uses the Wavelet band-limited noise of Cook, DeRose[2005] to control the microscopic geometric changes of hair fibers. This noise is different from conventional procedural noise, such as Perlin[1985], Olano[2002], Perlin, Neyret[2001], etc. A significant advantage of Wavelet noise is that itStatistical distributions can be calculated and controlled.

The advantage of the practical wave optics fiber scattering model of XIA[2023] is its realisticColored highlights (glints)Previous geometric optics models usually assume that the fiber surface is a smooth dielectric cylinder, without considering the complex interaction of light waves on the surface irregular structure. In actual tests, the XIA[2023] model performs well in rendering time, can be used in production environments, and generates more delicate and realistic optical effects than traditional models.

XIA[2023] is an important breakthrough that buildsFirst 3D wave optics fiber scattering simulatorPrevious fiber models (including early wave optics models such as Xia et al. 2020) mostly assumed thatLongitudinal and azimuthal directionsThe scattering onSeparable, which greatly simplifies the calculation. However, the authors' simulation results show that the highlights areInseparable, which is a phenomenon that previous models could not accurately handle. The simulator also predictedSpeckle patternsThis is a phenomenon that has not been captured by all previous fiber scattering models based on geometric optics and wave optics.5-dimensional scattering distributionThe method is to use tabulation, which is very memory intensive. Therefore, procedural noise is used to directly replace a five-dimensional table.

XIA[2023] has only been simulated once so farSpeckle Effect in Reflection Mode, higher order reflection modes are still being studied. And light-colored hair may require higher computational requirements to simulate perfectly. The wave optics fiber scattering model used in this study canEasily combined with previous fiber models.

References

Zotero one-click generated, needs to be corrected.

[1] JT Kajiya and TL Kay, “RENDERING FUR WITIt THREE DIMENSIONAL TEXTURES,” 1989.

[2] SR Marschner, HW Jensen, and M. Cammarano, “Light Scattering from Human Hair Fibers,” 2003.

[3] A. Zinke and A. Weber, “Light Scattering from Filaments,” IEEE Trans. Visual. Comput. Graphics, vol. 13, no. 2, pp. 342–356, Mar. 2007.

[4] L.-Q. Yan, C.-W. Tseng, HW Jensen, and R. Ramamoorthi, “Physically-accurate fur reflectance: modeling, measurement and rendering,” ACM Trans. Graph., vol. 34, no. 6, pp. 1–13, Nov. 2015.

[5] L.-Q. Yan, HW Jensen, and R. Ramamoorthi, “An efficient and practical near and far field fur reflectance model,” ACM Trans. Graph., vol. 36, no. 4, pp. 1–13, Aug. 2017.

[6] M. Xia, B. Walter, C. Hery, O. Maury, E. Michielssen, and S. Marschner, “A Practical Wave Optics Reflection Model for Hair and Fur,” ACM Trans. Graph., vol. 42, no. 4, pp. 1–15, Aug. 2023.

Glints Effect Study

Traditional rendering methods based on geometric optics, such as Yan [2014, 2016], useBidirectional Reflectance Distribution Function (BRDF)To simulate the mirror reflection surface, there are certain limitations.

Yan [2014, 2016] pointed out that traditional BRDF models usually use a smooth normal distribution function (NDF), assuming that the microfacets are infinitely small. But in reality, real surfaces often have obvious geometric features, such as micron-level bumps and flakes in metallic paint, which can cause significant glints under strong directional light sources (such as sunlight). Yan et al. simulated these small-scale surface geometric features more accurately through high-resolution normal maps, and proposed a new method to effectively render these complex specular highlights.

Traditional uniform pixel sampling techniques have too large variance when capturing highlights in these small ranges, resulting in low rendering efficiency and inability to effectively handle the uneven distribution of highlights caused by the complexity of the light path. Therefore, Yan [2014, 2016] introduced a search based on normal distribution and targeted sampling.

In hair renderings, you can observe that the hair and fur will show a shimmering effect of changing color when illuminated by strong directional light sources.

XIA[2023] uses optical speckle theory to simulate highlight noise, and adds the diffraction lobe of basic diffraction theory to process the diffraction effect of light on the surface of fiber structures such as hair, thereby rendering colored highlight effects.

XIA[2023], Chapter 8, states that glints can be easily observed in sunlight. Although subtle when viewed from a distance, these color effects can significantly enhance the appearance of the hair when viewed up close, sometimes causing a slight change in the hue of the fiber.

In Figure 9, the model of XIA[2023] also produces colored shimmer effects on light-colored hair. The shimmer is more subtle on light-colored hair than on dark-colored fibers because multiple scattering averages out the colors, resulting in reduced color contrast. Compared to XIA[2020], XIA[2023] not only handles wavelength-dependent reflections better, but also improves its ability to handle the angle of the hair cuticle, capturing the shift in highlights caused by the tilt of the hair cuticle.

References

[1] L.-Q. Yan, M. Hašan, W. Jakob, J. Lawrence, S. Marschner, and R. Ramamoorthi, “Rendering glints on high-resolution normal-mapped specular surfaces,” ACM Trans. Graph., vol. 33, no. 4, pp. 1–9, Jul. 2014.

[2] L.-Q. Yan, M. Hašan, S. Marschner, and R. Ramamoorthi, “Position-normal distributions for efficient rendering of specular microstructure,” ACM Trans. Graph., vol. 35, no. 4, pp. 1–9, Jul. 2016.

Full Wave Reference Simulator

https://dl.acm.org/doi/10.1145/3592414

1. Introduction

This paper discusses the theoretical basis of the physical wave simulation three-dimensional wave optical fiber scattering simulator used to generate high-precision light scattering simulation data in the rendering black dog hair paper "A Practical Wave Optics Reflection Model for Hair and Fur".

Calculating light reflection from rough surfaces is an important topic. Small-scale geometric structures, such as the tiny features of hair fibers, have a significant impact on the reflection behavior of light. The BRDF describes how a surface reflects light given an incident and outgoing direction. The limitations of geometric optics have been repeated many times. This model that treats light as a straight line propagation fails to capture the wave nature of light when the microstructure is close to the wavelength of light.

Theoretical models that use wave optics to approximate diffraction include the Beckmann-Kirchhoff theory of [Beckmann and Spizzichino 1987] and the Harvey-Shack model of [Krywonos 2006]. The former describes the light reflection behavior of rough surfaces, while the latter is a series of models based on wave optics that more accurately describe the scattering behavior of light on complex surfaces.

Existing models are all aimed at the average reflection behavior of large-area surfaces, ignoring local detail changes. Yan [2016, 2018] is able to capture the changes in light reflection from microstructures in different regions of space. Even models based on electromagnetic wave propagation still require certain approximate processing due to computational complexity. These methods are not actually ground truth.

In order to accurately capture the interference effects, XIA[2023] aims to develop a reference simulation tool that simulates the propagation of light faithfully according to Maxwell's equations. The only approximation is the numerical discretization, which ultimately generates the traditional bidirectional reflectance distribution function (BRDF) as output.This simulator truly achieves ground truth.

That is to say, this simulator can accurately simulate the wave characteristics of light, including interference, diffraction, multiple scattering, etc. The approximations used in the simulator are only meshing and numerical integration errors.

Through high-precision full-wave simulation, it is possible to generateHigh angular and spatial resolution BRDF data.

At the same time, the simulator is able to handle large surface areas (such as 60 × 60 × 10 wavelengths). For example, using visible light with a wavelength of about 500 nanometers, 60 wavelengths is equivalent to 30 microns. In other words, the simulator's calculations are based on the scale of light wavelengths.Real physical sizeIn this case, light of different wavelengths will correspond to different numbers of discretized units.The larger the wavelength(For example, the wavelength of red light is longer than that of blue light). For the same physical size, the required discretization units (such as mesh division) will beRelatively less, so the amount of computation required will beRelatively smaller, processing speed may alsoFaster.

Specifically, the surface is represented as a height field, each grid point corresponds to a height value, and quadrilaterals are used as primitives.

For the scattered field, the boundary integral formulation is used to transform the scattering problem of electromagnetic waves into an integral equation that is solved only on the surface boundary. The key implementation method is the boundary element method (BEM). The adaptive integral method (AIM) based on the three-dimensional fast Fourier transform (3D FFT) is then used to accelerate the calculation process of the boundary integral.

And use GPU to accelerate the parallel processing of large-scale surface scattering problems.

And the paper uses a combined small-scale simulation results to characterize the surface bidirectional scattering behavior.

Related Work

Reflection model based on wave optics

The old-fashioned geometric optics vs. wave optics. This article mainly compares surface scattering models. Classical models of geometric optics include: Cook-Torrance model [Cook and Torrance 1982], Oren-Nayar model [Michael 1994]. In wave optics, physical optics approximations are mainly used to simplify the full wave equation. That is, the first-order approximation (single scattering) in the black dog is used to estimate surface reflection. Classical models includeBeckmann-Kirchoff theoryandHarvey-Shack Model, which use approximate equations in scalar form to model wave optics effects. They are widely used to estimate reflectance on various surface types, such asGaussian random surface,Periodic surfaceEtc. However, the calculation results of these methods are often spatial average results, and it is impossible to perform high-resolution detail reflection.

• Gaussian random surface models of He et al. (1991) and Lanari et al. (2017).
• Periodic surface models by Dhillon et al. (2014), Stam (1999), and Toisoul and Ghosh (2017).
• Multilayer planar surface model of Levin et al. (2013).
• Surface data table model of Dong et al. (2016).
• Study of scratched surfaces by Werner et al. (2017).

In addition, physical optics approximation is also used to estimate theSpace changing appearance,For example:

• Surface data table from Yan et al. (2018)
• Random surface models of Steinberg and Yan (2022).

Some hybrid surface models apply physical optics models to some surface components (such as roughness at small scales), while using geometric optics models for larger scales. Applications of these hybrid models include:

• Surface roughness models by Falster et al. (2020) and Holzschuch and Pacanowski (2017).
• Thin-film interference model of Belcour and Barla (2017).
• Suspended particle model by Guillén et al. (2020).

In addition, physical optics models are used to handle inter-surface effects at longer distances. For example:

• Studies by Cuypers et al. (2012) and Steinberg et al. (2022) explored these long-range effects.

Scattering methods based on wave optics, such asLorenz-Mie theoryandT-Matrix Method, which is also usedVolumetric ScatteringCalculations, for example:

• Theories of Bohren and Huffman (2008) and Mishchenko et al. (2002).
• Application of volume scattering by Frisvad et al. (2007) and Guo et al. (2021).

In addition, complex-valued ray tracing techniques proposed by Sadeghi et al. (2012) and Shimada and Kawaguchi (2005) have been applied to rendering natural phenomena and structural color effects.Long-range effects between surfacesandVolumetric ScatteringSuch issues are currently beyond the scope of this study.

Numerical Methods in Computational Electromagnetism (CEM)

There are many methods for numerical calculations:

• Oskooi et al. (2010) proposed a numerical method based on difference solution of Maxwell's equationsFinite Difference Time Domain (FDTD)FDTD has been used to predict the appearance of wavelength-scale structures (e.g. Auzinger et al. (2018), Musbach et al. (2013)). However, the overhead is considerable as the simulation area increases!
• Finite Element Method (FEM)It is a widely used numerical method for solving partial differential equations, and can also be used in electromagnetic problems. It solves the problem by discretizing the simulation domain in three dimensions. Similar to FDTD, the amount of calculation is too large, not to mention for real-time rendering.
• Gibson (2021) provides a detailedBoundary Element Method (BEM)The main advantage of BEM is that it reduces the dimensionality of discretization by converting the scattering problem into an integral equation on the surface of the object. In FDTD and FEM, the entire three-dimensional space needs to be discretized, while BEM only needs to discretize the surface of the object, which significantly reduces the dimensionality and complexity of the calculation.

The paper chose BEMThe main reason is its scalability, which is conducive to the processing of complex surface structures.

There are many ways to speed up BEM:

• Liu and Nishimura (2006), White and Head-Gordon (1994)Fast Multipole Method (FMM).
• Bleszynski et al. (1996) proposed a three-dimensional fast Fourier transform (3D FFT)Adaptive Integration Method (AIM).
• Liao et al. (2016), Pak et al. (1997)Sparse Matrix Canonical Grid Method (SMCG).

Thesis selected AIMThe reason is that AIM is suitable for processing an area with a relatively small axial size.

result

BRDF value isHemisphereThe standardSpectral data to XYZ to RGB conversion, a colored BRDF map is generated.

That is, as the height field resolution increases, the BRDF output gradually stabilizes.8 samples per wavelengthThe resolution is sufficient to produce accurate results.

By comparing with existing wave optics models,The simulator in this articleIt has the highest accuracy and can handle complex optical phenomena and geometric structures. It is suitable for scenes with high precision requirements and is suitable for multiple reflections, interference, and complex surfaces. However, the computational cost is high, which is a compromise between efficiency and accuracy.

• OHS and GHS ModelsThe calculation is simple and suitable for smooth surfaces and medium roughness surfaces, but the error is large at large incident angles and on complex surfaces. GHS has improved accuracy at large angles compared to OHS.
• Kirchhoff modelThe accuracy is relatively high, but it can only be maintained within the first order range.
• Cutting Plane MethodComputationally efficient, suitable for relatively simple surface geometries. Not so for complex ones.
• The accuracy of this article is the best and is suitable for high-precision scenarios.

Comparison of coherent regions. As the illumination coherence increases, the resolution and detail of the BRDF becomes richer. When the coherence is high, the BRDF contains more high-resolution details.

In addition, the paper also shows the method used to accelerate BRDF calculation.Beam steeringAs shown in Figure 14, the surface isSpecular Reflection, while in the other directionRetroreflective effectThe paper calculates the BRDF values for a series of gradually changing incident angles, as shown in the figure below.

The BRDF image in each incident direction is reduced to a thin line segment. The comparison in Figure 15 shows that Tangent Plane cannot accurately model the surfaceSecond-order reflection(That is, light that is emitted after multiple reflections.) However, if the surface is smooth, using Tangent Plane is still very fast and accurate.

Furthermore, the paper compares the simulator results with BRDF measurements of real surfaces, especially inMultiple reflection effectAs shown in Figure 17, the top is the actual measurement, the middle is the theoretical model of the paper, and the bottom is the Tangent Plane.

You may ask why there is such a big difference? The paper uses an idealized geometric model, while the surface in the experiment may have some slight geometric deviations, which may affect the accuracy of the reflection. In a nutshell, the simulator in the paper can describe higher-order reflections!

Future Work

When dealing with more complexStructured surfaceWhen the light is scattered on the surface, the simulator can more accurately simulate the propagation and scattering behavior of light on the surface.

The future direction of work is of course to reduce computational overhead while ensuring accuracy.

Then the processing area of this BRDF approximation model is expanded.

At the same time, the simulator in this paper can be used as a benchmark reference.

References

A Full-Wave Reference Simulator for Computing Surface Reflectance

Petr Beckmann and Andre Spizzichino. 1987. The scattering of electromagnetic waves from rough surfaces. Artech House.

Andrey Krywonos. 2006. Predicting Surface Scatter using a Linear Systems Formulation of Non-Paraxial Scalar Diffraction. Ph. D. Dissertation. University of Central Florida.

Ling-Qi Yan, Miloš Hašan, Bruce Walter, Steve Marschner, and Ravi Ramamoorthi. 2018. Rendering Specular Microgeometry with Wave Optics. ACM Trans. Graph. 37, 4 (2018).

Ling-Qi Yan, Miloš Hašan, Steve Marschner, and Ravi Ramamoorthi. 2016. Positionnormal distributions for efficient rendering of specular microstructure. ACM Transactions on Graphics (TOG) 35, 4 (2016), 1–9.

RL Cook and KE Torrance. 1982. A Reflectance Model for Computer Graphics. ACM Trans. Graph. 1, 1 (jan 1982). https://doi.org/10.1145/357290.357293

Michael Oren and Shree K. Nayar. 1994. Generalization of Lambert's Reflectance Model (SIGGRAPH '94). https://doi.org/10.1145/192161.192213