9.7 ROUGH DIELECTRIC BSDF
9.7 粗糙介电BSDF

We will now extend the microfacet approach from Section 9.6 to the case of rough dielectrics. This involves two notable changes: since dielectrics are characterized by both reflection and transmission, the model must be aware of these two separate components. In the case of transmission, Snell’s law will furthermore replace the law of reflection in the computation that determines the incident direction.
我们现在将把 9.6 节中的微面方法扩展到粗糙电介质的情况。这涉及两个显着的变化：由于电介质具有反射和透射的特征，因此模型必须了解这两个独立的组件。在透射的情况下，斯涅尔定律将在确定入射方向的计算中进一步取代反射定律。

Figure 9.32 shows the dragon rendered with the Torrance–Sparrow model and both reflection and transmission.
图 9.32 显示了使用 Torrance-Sparrow 模型以及反射和透射渲染的龙。

As before, we will begin by characterizing the probability density of generated samples. The implied BSDF then directly follows from this density and the sequence of events encapsulated by a scattering interaction: visible normal sampling, reflection or refraction, and attenuation by the Fresnel and masking terms.
和以前一样，我们将从表征生成样本的概率密度开始。然后，隐含的 BSDF 直接从该密度和由散射相互作用封装的事件序列得出：可见法线采样、反射或折射，以及菲涅耳和掩蔽项的衰减。

Rough Dielectric PDF
粗糙电介质 PDF

The density evaluation occurs in the following fragment that we previously omitted during the discussion of the smooth dielectric case.
密度评估发生在我们之前在讨论光滑介电情况时省略的以下片段中。

We now turn to the generalized half-direction vector, whose discussion requires a closer look at Snell’s law (Equation (9.2)) relating the elevation and azimuth of the incident and outgoing directions at a refractive interface:
我们现在转向广义半方向矢量，其讨论需要仔细研究斯涅尔定律（方程（9.2）），该定律涉及折射界面处入射和出射方向的仰角和方位角：

η_o sin θ_o = η_i sin θ_i    and    ϕ_o = ϕ_i + π.
η _o sin θ _o = η sin θ 且 phi _o = phi + π。

Since the refraction takes place at the level of the microgeometry, all of these angles are to be understood within a coordinate frame aligned with the microfacet normal ω_m. Recall also that the sines in the first equation refer to the length of the tangential component of ω_i and ω_o perpendicular to ω_m.
由于折射发生在微观几何水平上，因此所有这些角度都应在与微面法线 ω _m 对齐的坐标系内理解。还记得第一个方程中的正弦是指 ω 和垂直于 ω _m 的 ω _o 的切向分量的长度。

A generalized half-direction vector builds on this observation by scaling and adding these directions to cancel out the tangential components, which yields the surface normal responsible for a particular refraction after normalization. It is defined as
广义半方向矢量建立在该观察的基础上，通过缩放和添加这些方向来抵消切向分量，从而在归一化后产生负责特定折射的表面法线。它被定义为

where η = η_i/η_o is the relative index of refraction toward the sampled direction ω_i. The reflective case is trivially subsumed, since η_i = η_o when no refraction takes place. The next fragment implements this computation, including handling of invalid configurations (e.g., perfectly grazing incidence) where both the BSDF and its sampling density evaluate to zero.
其中 η = η/η _o 是朝向采样方向 ω 的相对折射率。反射情况被简单地包含在内，因为当没有发生折射时 η = η _o 。下一个片段实现此计算，包括处理无效配置（例如，完美的掠射发生率），其中 BSDF 及其采样密度评估为零。

The last line reflects an important implementation detail: with the previous definition in Equation (9.34), ω_m always points toward the denser medium, whereas pbrt uses the convention that micro- and macro-normal are consistently oriented (i.e., ω_m · n > 0). In practice, we therefore compute the following modified half-direction vector, where n = (0, 0, 1) in local coordinates:
最后一行反映了一个重要的实现细节：根据方程（9.34）中的先前定义，ω _m 始终指向较致密的介质，而 pbrt 使用微观法线和宏观法线一致定向的约定（即 ω _m · n > 0)。因此，在实践中，我们计算以下修改后的半方向向量，其中局部坐标中的 n = (0, 0, 1)：

Next, microfacets that are backfacing with respect to either the incident or outgoing direction do not contribute and must be discarded.
接下来，相对于入射或出射方向背面的微面没有贡献，必须被丢弃。

Given ω_m, we can evaluate the Fresnel reflection and transmission terms using the specialized dielectric evaluation of the Fresnel equations.
给定 ω _m ，我们可以使用菲涅尔方程的专门介电评估来评估菲涅尔反射和传输项。

We now have the values necessary to compute the PDF for ω_i, which depends on whether it reflects or transmits at the surface.
现在我们有了计算 ω 的 PDF 所需的值，这取决于它是在表面反射还是透射。

As before, the bijective mapping between ω_m and ω_i provides a change of variables whose Jacobian determinant is crucial so that we can correctly deduce the probability density of sampled directions ω_i. The derivation is more involved in the refractive case; see the “Further Reading” section for pointers to its derivation. The final determinant is given by
和之前一样，ω _m 和 ω 之间的双射映射提供了变量的变化，其雅可比行列式至关重要，以便我们可以正确推导采样方向 ω 的概率密度。推导更多地涉及屈光情况；请参阅“进一步阅读”部分以获取其推导的指针。最终的行列式由下式给出

Once more, this relationship makes it possible to evaluate the probability per unit solid angle of the sampled incident directions ω_i obtained through the combination of visible normal sampling and scattering:
再次，这种关系使得可以评估通过可见法向采样和散射相结合获得的采样入射方向 ω 的每单位立体角的概率：

The following fragment implements this computation, while additionally accounting for the discrete probability pt / (pr + pt) of sampling the transmission component.
以下片段实现了此计算，同时另外考虑了对传输分量进行采样的离散概率 pt / (pr + pt)。

Finally, the density of the reflection component agrees with the model used for conductors but for the additional discrete probability pr / (pr + pt) of choosing the reflection component.
最后，反射分量的密度与用于导体的模型一致，但选择反射分量的附加离散概率 pr / (pr + pt) 除外。

Rough Dielectric BSDF
粗糙电介质 BSDF

BSDF evaluation is similarly split into reflective and transmissive components.
BSDF 评估同样分为反射部分和透射部分。

The reflection component follows the approach used for conductors in the fragment 〈Evaluate rough conductor BRDF〉:
反射分量遵循片段<评估粗糙导体 BRDF> 中用于导体的方法：

For the transmission component, we can again derive the effective scattering distribution by equating a single-sample Monte Carlo estimate of the rendering equation with the product of Fresnel transmission, masking, and the incident radiance. This results in the equation
对于透射分量，我们可以通过将渲染方程的单样本蒙特卡罗估计与菲涅尔透射、掩蔽和入射辐射率的乘积等同起来，再次导出有效散射分布。这导致方程

Substituting the PDF from Equation (9.37) and solving for the BTDF f_t(p, ω_o, ω_i) results in
将 PDF 代入方程 (9.37) 并求解 BTDF f _t (p, ω _o , ω) 结果为

Finally, inserting the definition of the visible normal distribution from Equation (9.23) and switching to the more accurate bidirectional masking-shadowing factor G yields
最后，插入方程（9.23）中可见正态分布的定义并切换到更准确的双向掩蔽-阴影因子 G，得出

The next fragment implements this expression. It also incorporates the earlier orientation test and handling of non-symmetric scattering that was previously encountered in the perfect specular case (Section 9.5.2).
下一个片段实现了这个表达式。它还结合了先前在完美镜面反射情况下遇到的非对称散射的定向测试和处理（第 9.5.2 节）。

Rough Dielectric Sampling
粗介电采样

Sampling proceeds by drawing a microfacet normal from the visible normal distribution, computing the Fresnel term, and stochastically selecting between reflection and transmission.
通过从可见正态分布中绘制微面法线、计算菲涅耳项并在反射和透射之间随机选择来进行采样。

Once again, handling of the reflection component is straightforward and mostly matches the case for conductors except for extra factors that arise due to the discrete choice between reflection and transmission components.
再次强调，反射分量的处理非常简单，并且除了由于反射和传输分量之间的离散选择而产生的额外因素之外，大部分与导体的情况匹配。

The transmission case invokes Refract() to determine wi. A subsequent test excludes inconsistencies that can rarely arise due to the approximate nature of floating-point arithmetic. For example, Refract() may sometimes indicate a total internal reflection configuration, which is inconsistent as the transmission component should not have been sampled in this case.
传输案例调用 Refract() 来确定 wi。随后的测试排除了由于浮点运算的近似性质而很少出现的不一致情况。例如，Refract() 有时可能指示全内反射配置，这是不一致的，因为在这种情况下不应对传输分量进行采样。

The last step evaluates the BTDF from Equation (9.40) and packs the sample information into a BSDFSample.
最后一步根据方程 (9.40) 评估 BTDF，并将样本信息打包到 BSDFSample 中。

9.8 MEASURED BSDFs
9.8 测量的 BSDF

The reflection models introduced up to this point represent index of refraction changes at smooth and rough boundaries, which constitute the basic building blocks of surface appearance. More complex materials (e.g., paint on primer, metal under a layer of enamel) can sometimes be approximated using multiple interfaces with participating media between them; the layered material model presented in Section 14.3 is based on that approach.
到目前为止介绍的反射模型代表了光滑和粗糙边界处的折射率变化，它们构成了表面外观的基本构建块。更复杂的材料（例如，底漆上的油漆、搪瓷层下的金属）有时可以使用多个界面以及它们之间的参与介质来近似；第 14.3 节中介绍的分层材料模型基于该方法。

However, many real-world materials are beyond the scope of even such layered models. Examples include:
然而，许多现实世界的材料甚至超出了这种分层模型的范围。示例包括：

• Materials characterized by wave-optical phenomena that produce striking directionally varying coloration. Examples include iridescent paints, insect wings, and holographic paper.
  以波光现象为特征的材料，可产生引人注目的方向变化的颜色。例如虹彩涂料、昆虫翅膀和全息纸。
• Materials with rough interfaces. In pbrt, we have chosen to model such surfaces using microfacet theory and the Trowbridge–Reitz distribution. However, it is important to remember that both of these are models that generally do not match real-world behavior perfectly.
  界面粗糙的材料。在 pbrt 中，我们选择使用微面理论和 Trowbridge-Reitz 分布对此类表面进行建模。然而，重要的是要记住，这两种模型通常都无法完美匹配现实世界的行为。
• Surfaces with non-standard microstructure. For example, a woven fabric composed of two different yarns looks like a surface from a distance, but its directional intensity and color variation are not well-described by any standard BRDF model due to the distinct reflectance properties of fiber-based microgeometry.
  具有非标准微观结构的表面。例如，由两种不同纱线组成的机织织物从远处看就像一个表面，但由于基于纤维的微观几何形状具有独特的反射特性，任何标准 BRDF 模型都无法很好地描述其方向强度和颜色变化。

Instead of developing numerous additional specialized BxDFs, we will now pursue another way of reproducing such challenging materials in a renderer: by interpolating measurements of real-world material samples to create a data-driven reflectance model. The resulting MeasuredBxDF only models surface reflection, though the approach can in principle generalize to transmission as well.
我们现在将不再开发大量额外的专用 BxDF，而是寻求另一种在渲染器中再现此类具有挑战性的材质的方法：通过对现实世界材质样本的测量进行插值来创建数据驱动的反射率模型。所得的 MeasuredBxDF 仅模拟表面反射，但该方法原则上也可以推广到传输。

Measuring reflection in a way that is practical while producing information in a form that is convenient for rendering is a challenging problem. We begin by explaining these challenges for motivation.
以实用的方式测量反射，同时以方便渲染的形式生成信息是一个具有挑战性的问题。我们首先解释这些激励挑战。

Consider the task of measuring the BRDF of a sheet of brushed aluminum: we could illuminate a sample of the material from a set of n incident directions with k = 1, … , n and use some kind of sensor (e.g., a photodiode) to record the reflected light scattered along a set of m outgoing directions with l = 1, … , m. These n × m measurements could be stored on disk and interpolated at runtime to approximate the BRDF at intermediate configurations (θ_i, ϕ_i, θ_o, ϕ_o). However, closer scrutiny of such an approach reveals several problems:
考虑测量拉丝铝板的 BRDF 的任务：我们可以从一组 n 个入射方向 照射材料样本，其中 k = 1, … , n 并使用某种传感器（例如，光电二极管）记录沿一组 m 个出射方向 散射的反射光，其中 l = 1, … , m。这些 n × m 测量值可以存储在磁盘上，并在运行时进行插值，以近似中间配置（θ、 Φ、 θ _o 、 Φ _o ）下的 BRDF。然而，对这种方法的仔细审查揭示了几个问题：

• BSDFs of polished materials are highly directionally peaked. Perturbing the incident or outgoing direction by as little as 1 degree can change the measured reflectance by orders of magnitude. This implies that the set of incident and outgoing directions must be sampled fairly densely.
  抛光材料的 BSDF 具有高度方向性的峰值。只要扰动入射或出射方向 1 度，就能将测量到的反射率改变几个数量级。这意味着必须对入射和出射方向集进行相当密集的采样。
• Accurate positioning in spherical coordinates is difficult to perform by hand and generally requires mechanical aids. For this reason, such measurements are normally performed using a motorized gantry known as a goniophotometer or gonioreflectometer. Figure 9.33 shows two examples of such machines. Light stages consisting of a rigid assembly of hundreds of LEDs around a sample are sometimes used to accelerate measurement, though at the cost of reduced directional resolution.
  球面坐标中的精确定位很难用手执行，通常需要机械辅助。因此，此类测量通常使用称为测角光度计或测角反射计的电动龙门架进行。图 9.33 显示了此类机器的两个示例。有时使用由围绕样品的数百个 LED 刚性组件组成的光台来加速测量，但代价是方向分辨率降低。
• Sampling each direction using a 1 degree spacing in spherical coordinates requires roughly one billion sample points. Storing gigabytes of measurement data is possible but undesirable, yet the time that would be spent for a full measurement is even more problematic: assuming that the goniophotometer can reach a configuration (θ_i, ϕ_i, θ_o, ϕ_o) within 1 second (a reasonable estimate for the devices shown in Figure 9.33), over 34 years of sustained operation would be needed to measure a single material.
  在球坐标中使用 1 度间距对每个方向进行采样大约需要 10 亿个采样点。存储千兆字节的测量数据是可能的，但并不可取，但完整测量所花费的时间甚至更成问题：假设测角光度计可以达到配置 (θ, Φ, θ _o , Φ < b1>）在 1 秒内（对图 9.33 中所示设备的合理估计），需要持续运行超过 34 年才能测量单一材料。

In sum, the combination of high-frequency signals, the 4D domain of the BRDF, and the curse of dimensionality conspire to make standard measurement approaches all but impractical.
总之，高频信号、BRDF 的 4D 域和维数灾难的结合使标准测量方法几乎不切实际。

While there is no general antidote against the curse of dimensionality, the previous example involving a dense discretization of the 4D domain is clearly excessive. For example, peaked BSDFs that concentrate most of their energy into a small set of angles tend to be relatively smooth away from the peak. Figure 9.34 shows how a more specialized sample placement that is informed by the principles of specular reflection can drastically reduce the number of sample points that are needed to obtain a desired level of accuracy. Figure 9.35 shows how the roughness of the surface affects the desired distribution of samples—for example, smooth surfaces allow sparse sampling outside of the specular lobe.
虽然没有针对维数灾难的通用解药，但前面涉及 4D 域密集离散化的示例显然有些过分。例如，将大部分能量集中到一小部分角度的峰值 BSDF 往往会相对平滑地远离峰值。图 9.34 显示了根据镜面反射原理进行的更专业的样本放置如何能够大大减少获得所需精度水平所需的样本点数量。图 9.35 显示了表面的粗糙度如何影响所需的样本分布，例如，光滑的表面允许在镜面波瓣之外进行稀疏采样。

The MeasuredBxDF therefore builds on microfacet theory and the distribution of visible normals to create a more efficient physically informed sampling pattern. The rationale underlying this choice is that while microfacet theory may not perfectly predict the reflectance of a specific material, it can at least approximately represent how energy is (re-)distributed throughout the 4D domain. Applying it enables the use of a relatively coarse set of measurement locations that suffice to capture the function’s behavior.
因此，MeasuredBxDF 基于微面理论和可见法线的分布，以创建更有效的物理信息采样模式。这一选择的基本原理是，虽然微面理论可能无法完美预测特定材料的反射率，但它至少可以近似地表示能量在整个 4D 域中（重新）分布的方式。应用它可以使用一组相对粗糙的测量位置，足以捕获函数的行为。

Concretely, the method works by transforming regular grid points using visible normal sampling (Section 9.6.4) and performing a measurement at each sampled position. If the microfacet sampling routine is given by a function R : S² × [0, 1]² → S² and u^(k) with k = 1, … , n denotes input samples arranged on a grid covering the 2D unit square, then we have a sequence of measurements M^(k):
具体来说，该方法的工作原理是使用可见法线采样（第 9.6.4 节）转换规则网格点，并在每个采样位置执行测量。如果微面采样例程由函数 R 给出： S ² × [0, 1] ² → S ² 和 u ^(k) 其中 k = 1, … , n 表示排列在覆盖 2D 单位正方形的网格上的输入样本，那么我们有一个测量序列 M ^(k) ：

where f_r(ω_o, ω_i) refers to the real-world BRDF of a material sample, as measured by a goniophotometer (or similar device) in directions ω_o and ω_i = R(ω_o, u^(k)). This process must be repeated for different values of ω_o to also capture variation in the other direction argument. Evaluating the BRDF requires the inverse R⁻¹ of the transformation, which yields a position on [0, 1]² that can be used to interpolate the measurements M^(k). Figure 9.36 illustrates both directions of this mapping.
其中 f _r (ω _o , ω) 是指材料样本的现实世界 BRDF，由测角光度计（或类似设备）沿 ω _o , u ^(k) )。必须针对不同的 ω _o 值重复此过程，以捕获另一个方向参数的变化。评估 BRDF 需要变换的逆 R ⁻¹ ，这会产生 [0, 1] ² 上的位置，可用于插值测量值 M ^(k) .图 9.36 说明了该映射的两个方向。

This procedure raises several questions: first, the non-random use of a method designed for Monte Carlo sampling may be unexpected. To see why this works, remember that the inversion method (Section 2.3) evaluates the inverse of a distribution’s cumulative distribution function (CDF). Besides being convenient for sampling, this inverse CDF can also be interpreted as a parameterization of the target domain from the unit square. This parameterization smoothly warps the domain so that regions with a high contribution occupy a correspondingly large amount of the unit square. The MeasuredBxDF then simply measures and stores BRDF values in these “improved” coordinates. Note that the material does not have to agree with microfacet theory for this warping to be valid, though the sampling pattern is much less efficient and requires a denser discretization when the material’s behavior deviates significantly.
这个过程提出了几个问题：首先，非随机使用为蒙特卡洛抽样设计的方法可能会出乎意料。要了解其原理，请记住反演方法（第 2.3 节）计算分布累积分布函数 (CDF) 的逆函数。除了方便采样之外，这种逆 CDF 还可以解释为单位平方对目标域的参数化。这种参数化平滑地扭曲域，使得具有高贡献的区域占据相应大量的单位正方形。然后，MeasuredBxDF 会简单地测量 BRDF 值并将其存储在这些“改进的”坐标中。请注意，材料不必符合微面理论才能使这种扭曲有效，尽管采样模式的效率要低得多，并且当材料的行为显着偏离时需要更密集的离散化。

Another challenge is that parameterization guiding the measurement requires a microfacet approximation of the material, but such an approximation would normally be derived from an existing measurement. We will shortly show how to resolve this chicken-and-egg problem and assume for now that a suitable model is available.
另一个挑战是指导测量的参数化需要材料的微面近似，但这种近似通常是从现有的测量中得出的。我们很快就会展示如何解决这个先有鸡还是先有蛋的问题，并假设现在有一个合适的模型可用。

Measurement Through a Parameterization
通过参数化进行测量

A flaw of the reparameterized measurement sequence in Equation (9.41) is that the values M^(k) may differ by many orders of magnitude, which means that simple linear interpolation between neighboring data points is unlikely to give satisfactory results. We instead use the following representation that transforms measurements prior to storage in an effort to reduce this variation:
等式（9.41）中重新参数化的测量序列的缺陷是值M ^(k) 可能相差多个数量级，这意味着相邻数据点之间的简单线性插值不太可能给出令人满意的结果。相反，我们使用以下表示在存储之前转换测量值，以减少这种变化：

where ω_i^(k) = R(ω_o, u^(k)), and p(ω_i^(k)) denotes the density of direction ω_i^(k) according to visible normal sampling.
其中 ω ^(k) = R(ω _o , u ^(k) )，p(ω ^(k) ) 表示方向 ω < 的密度b4>按照可见正常采样。

If f_r was an analytic BRDF (e.g., a microfacet model) and u^(k) a 2D uniform variate, then Equation (9.42) would simply be the weight of a Monte Carlo importance sampling strategy, typically with a carefully designed mapping R and density p that make this weight near-constant to reduce variance.
如果 f _r 是解析 BRDF（例如微面模型），u ^(k) 是 2D 均匀变量，则方程 (9.42) 只是蒙特卡罗重要性采样的权重策略，通常采用精心设计的映射 R 和密度 p，使权重接近恒定以减少方差。

In the present context, f_r represents real-world data, while p and R encapsulate a microfacet approximation of the material under consideration. We therefore expect M^(k) to take on near-constant values when the material is well-described by a microfacet model, and more marked deviations otherwise. This can roughly be interpreted as measuring the difference (in a multiplicative sense) between the real world and the microfacet simplification. Figure 9.37 visualizes the effect of the transformation in Equation (9.42).
在当前上下文中，f _r 表示真实世界的数据，而 p 和 R 封装了所考虑材料的微面近似值。因此，当材料被微面模型很好地描述时，我们期望 M ^(k) 呈现接近恒定的值，否则会有更明显的偏差。这可以粗略地解释为测量现实世界和微面简化之间的差异（在乘法意义上）。图 9.37 可视化了方程 (9.42) 中变换的效果。

BRDF Evaluation
BRDF评估

Evaluating the data-driven BRDF requires the inverse of these steps. Suppose that M(·) implements an interpolation based on the grid of measurement points M^(k). Furthermore, suppose that we have access to the inverse R⁻¹(ω_o, ω_i) that returns the “random numbers” u that would cause visible normal sampling to generate a particular incident direction (i.e., R(ω_o, u) = ω_i). Accessing M(·) through R⁻¹ then provides a spherical interpolation of the measurement data.
评估数据驱动的 BRDF 需要与这些步骤相反。假设 M(·) 基于测量点网格 M ^(k) 实现插值。此外，假设我们可以访问逆 R ⁻¹ (ω _o , ω)，它返回“随机数”u，这将导致可见法线采样生成特定的入射方向（即，R(ω _o , u) = ω）。然后通过 R ⁻¹ 访问 M(·) 提供测量数据的球形插值。

We must additionally multiply by the density p(ω_i), and divide by the cosine factor⁵ to undo corresponding transformations introduced in Equation (9.42), which yields the final form of the data-driven BRDF:
我们还必须乘以密度 p(ω)，并除以余弦因子 ⁵ ，以撤消方程 (9.42) 中引入的相应变换，从而产生数据驱动的 BRDF 的最终形式：

Generalized Microfacet Model
广义微面模型

A major difference between the microfacet model underlying the ConductorBxDF and the approximation used here is that we replace the Trowbridge–Reitz model with an arbitrary data-driven microfacet distribution. This improves the model’s ability to approximate the material being measured. At the same time, it implies that previously used simplifications and analytic solutions are no longer available and must be revisited.
ConductorBxDF 底层的微面模型与此处使用的近似值之间的主要区别在于，我们用任意数据驱动的微面分布替换了 Trowbridge-Reitz 模型。这提高了模型近似被测量材料的能力。同时，这意味着以前使用的简化和解析解不再可用，必须重新审视。

We begin with the Torrance–Sparrow sampling density from Equation (9.28),
我们从方程 (9.28) 中的 Torrance-Sparrow 采样密度开始，

which references the visible normal sampling density D_ω(ω_m) from Equation (9.23). Substituting the definition of the masking function from Equation (9.18) into and rearranging terms yields
它引用了方程（9.23）中的可见正常采样密度 D _ω (ω _m )。将等式 (9.18) 中掩蔽函数的定义代入 并重新排列项，得到

where
在哪里

provides a direction-dependent normalization of the visible normal distribution. For valid reflection directions (ω_o · ω_m > 0), the PDF of generated samples then simplifies to
提供可见正态分布的方向相关标准化。对于有效的反射方向 (ω _o · ω _m > 0)，生成样本的 PDF 会简化为

Substituting this density into the BRDF from Equation (9.43) produces
将此密度代入方程 (9.43) 中的 BRDF 产生

The MeasuredBxDF implements this expression using data-driven representations of D(·) and σ(·).
MeasuredBxDF 使用 D(·) 和 σ(·) 的数据驱动表示来实现此表达式。

Finding the Initial Microfacet Model
寻找初始微面模型

We finally revisit the chicken-and-egg problem from before: practical measurement using the presented approach requires a suitable microfacet model—specifically, a microfacet distribution D(ω_m). Yet it remains unclear how this distribution could be obtained without access to an already existing BRDF measurement.
我们最后重新审视之前的先有鸡还是先有蛋的问题：使用所提出的方法进行实际测量需要一个合适的微面模型，具体来说，一个微面分布 D(ω _m )。然而，目前尚不清楚如何在不使用现有 BRDF 测量的情况下获得这种分布。

The key idea to resolve this conundrum is that the microfacet distribution D(ω_m) is a 2D quantity, which means that it remains mostly unaffected by the curse of dimensionality. Acquiring this function is therefore substantially cheaper than a measurement of the full 4D BRDF.
解决这个难题的关键思想是微面分布 D(ω _m ) 是一个二维量，这意味着它基本上不受维数灾难的影响。因此，获取此函数比完整 4D BRDF 的测量便宜得多。

Suppose that the material being measured perfectly matches microfacet theory in the sense that it is described by the Torrance–Sparrow BRDF from Equation (9.33). Then we can measure the material’s retroreflection (i.e., ω_i = ω_o = ω), which is given by
假设被测量的材料与微面理论完全匹配，因为它是由方程 (9.33) 中的 Torrance-Sparrow BRDF 描述的。然后我们可以测量材料的逆反射（即 ω = ω _o = ω），其计算公式为

The last step of the above equation removes constant terms including the Fresnel reflectance and introduces the reasonable assumption that shadowing/masking is perfectly correlated given ω_i = ω_o and thus occurs only once. Substituting the definition of G₁ from Equation (9.18) and rearranging yields the following relationship of proportionality:
上述方程的最后一步删除了包括菲涅尔反射率在内的常数项，并引入了合理的假设，即在给定 ω = ω _o 的情况下，阴影/掩蔽完全相关，因此仅发生一次。将 G ₁ 的定义代入方程（9.18）并重新排列，得到以下比例关系：

This integral equation can be solved by measuring f_r(p, ω_j, ω_j) for n directions ω_j and using those measurements for initial guesses of D(ω_j). A more accurate estimate of D can then be found using an iterative solution procedure where the estimated values of D are used to estimate the integrals on the right hand side of Equation (9.46) for all of the ω_js. This process quickly converges within a few iterations.
该积分方程可以通过测量 n 个方向 ω 的 f _r (p, ω, ω) 并使用这些测量值作为 D(ω) 的初始猜测来求解。然后可以使用迭代求解过程找到 D 的更准确估计，其中 D 的估计值用于估计方程（9.46）右侧所有 ω 的积分。这个过程在几次迭代内很快收敛。

^⋆9.9 SCATTERING FROM HAIR
^⋆ 9.9 头发散射

Human hair and animal fur can be modeled with a rough dielectric interface surrounding a pigmented translucent core. Reflectance at the interface is generally the same for all wavelengths and it is therefore wavelength-variation in the amount of light that is absorbed inside the hair that determines its color. While these scattering effects could be modeled using a combination of the DielectricBxDF and the volumetric scattering models from Chapters 11 and 14, not only would doing so be computationally expensive, requiring ray tracing within the hair geometry, but importance sampling the resulting model would be difficult. Therefore, this section introduces a specialized BSDF for hair and fur that encapsulates these lower-level scattering processes into a model that can be efficiently evaluated and sampled.
人类头发和动物毛皮可以用围绕有色半透明核心的粗糙介电界面进行建模。对于所有波长，界面处的反射率通常相同，因此毛发内部吸收的光量的波长变化决定了毛发的颜色。虽然这些散射效应可以使用第 11 章和第 14 章中的 DielectricBxDF 和体积散射模型的组合进行建模，但这样做不仅计算成本高昂，需要在头发几何结构内进行光线追踪，而且对结果模型进行重要性采样也很困难。因此，本节介绍了一种专门针对头发和毛皮的 BSDF，它将这些较低级别的散射过程封装到一个可以有效评估和采样的模型中。

See Figure 9.39 for an example of the visual complexity of scattering in hair and a comparison to rendering using a conventional BRDF model with hair geometry.
请参阅图 9.39，了解头发散射的视觉复杂性示例以及与使用具有头发几何形状的传​​统 BRDF 模型进行渲染的比较。

FURTHER READING
延伸阅读

Hall’s (1989) book collected and described the state of the art in physically based surface reflection models for graphics; it remains a seminal reference. It discusses the physics of surface reflection in detail, with many pointers to the original literature.
Hall（1989）的书收集并描述了基于物理的图形表面反射模型的最新技术；它仍然是一个开创性的参考。它详细讨论了表面反射的物理原理，并引用了许多原始文献。

Microfacet Models
微面模型

Phong (1975) developed an early empirical reflection model for glossy surfaces in computer graphics. Although neither reciprocal nor energy conserving, it was a cornerstone of the first synthetic images of non-Lambertian objects. The Torrance–Sparrow microfacet model was described by Torrance and Sparrow (1967); it was first introduced to graphics by Blinn (1977), and a variant of it was used by Cook and Torrance (1981, 1982).
Phong (1975) 开发了计算机图形学中光泽表面的早期经验反射模型。虽然既不是倒数也不是能量守恒，但它是第一批非朗伯物体合成图像的基石。 Torrance-Sparrow 微面模型由 Torrance 和 Sparrow (1967) 描述； Blinn (1977) 首次将它引入图形学，Cook 和 Torrance (1981, 1982) 使用了它的一个变体。

The papers by Beckmann and Spizzichino (1963) and Trowbridge and Reitz (1975) introduced two widely used microfacet distribution functions. Kurt et al. (2010) introduced an anisotropic variant of the Beckmann–Spizzichino distribution function; see Heitz (2014) for anisotropic variants of many other microfacet distribution functions. (Early anisotropic BRDF models for computer graphics were developed by Kajiya (1985) and Poulin and Fournier (1990).) Ribardière et al. (2017) applied Student’s t-distribution to model microfacet distributions; it provides an additional degree of freedom, which they showed allows a better fit to measured data while subsuming both the Beckmann–Spizzichino and Trowbridge–Reitz distributions. Ribardière et al. (2019) investigated the connection between normal distribution functions (NDFs) and microfacet distributions and also showed how to generate surfaces with distributions described by their model.
Beckmann 和 Spizzichino (1963) 以及 Trowbridge 和 Reitz (1975) 的论文介绍了两种广泛使用的微面分布函数。库尔特等人。 (2010) 引入了 Beckmann-Spizzichino 分布函数的各向异性变体；有关许多其他微面分布函数的各向异性变体，请参阅 Heitz (2014)。 （计算机图形学的早期各向异性 BRDF 模型由 Kajiya (1985) 以及 Poulin 和 Fournier (1990) 开发。）Ribardière 等人。 (2017) 应用学生 t 分布来建模微面分布；它提供了额外的自由度，他们证明这可以更好地拟合测量数据，同时包含 Beckmann-Spizzichino 和 Trowbridge-Reitz 分布。里巴迪埃等人。 (2019) 研究了正态分布函数 (NDF) 和微面分布之间的联系，并展示了如何生成具有模型描述的分布的表面。

The microfacet masking-shadowing function was introduced by Smith (1967), building on the assumption that heights of nearby points on the microfacet surface are uncorrelated. Smith also first derived the normalization constraint in Equation (9.17). Heitz’s paper on microfacet masking-shadowing functions (2014) provides a very well-written introduction to microfacet BSDF models in general, with many useful figures and explanations about details of the topic.
微面掩蔽-阴影函数由 Smith (1967) 引入，其假设是微面表面上附近点的高度不相关。 Smith 还首先导出了方程（9.17）中的归一化约束。 Heitz 关于微面掩蔽-阴影函数的论文（2014 年）对微面 BSDF 模型进行了非常详尽的介绍，其中包含许多有用的图表和有关该主题细节的解释。

The more accurate G(ω_i, ω_o) function for Gaussian microfacet surfaces that better accounts for the effects of correlation between the two directions that we have implemented is due to Ross et al. (2005). Our derivation of the ∧(ω) function, Equation (9.19), follows Heitz (2015).
Ross 等人提出的高斯微面表面更准确的 G(ω, ω _o ) 函数可以更好地解释我们所实现的两个方向之间相关性的影响。 （2005）。我们对 ∧(ω) 函数方程 (9.19) 的推导遵循 Heitz (2015)。

For many decades, Monte Carlo rendering of microfacet models involved generating samples proportional to the microfacet distribution D(ω_h). Heitz and d’Eon (2014) were the first to demonstrate that it was possible to reduce variance by restricting this sampling process to only consider visible microfacets. Our microfacet sampling implementation in Section 9.6 follows Heitz’s improved approach (2018), which showed that sampling the visible area of the Trowbridge–Reitz microfacet distribution corresponds to sampling the projection of a truncated ellipsoid, which in turn can be performed using an approach developed by Walter et al. (2015). See also Heitz (2014) for an overview of traditional techniques that directly sample the microfacet distribution D(ω_h) without considering visibility.
几十年来，微面模型的蒙特卡罗渲染涉及生成与微面分布 D(ω _h ) 成比例的样本。 Heitz 和 d’Eon (2014) 第一个证明可以通过限制采样过程仅考虑可见的微面来减少方差。我们在第 9.6 节中的微面采样实现遵循 Heitz 的改进方法（2018），该方法表明对 Trowbridge-Reitz 微面分布的可见区域进行采样对应于对截断椭球体的投影进行采样，而这又可以使用由沃尔特等人。 （2015）。另请参阅 Heitz (2014)，了解直接对微面分布 D(ω _h ) 进行采样而不考虑可见性的传统技术的概述。

When dealing with refraction through rough dielectrics, a modified change of variables term is needed to account for the mapping from half vectors to outgoing direction. A model based on this approach was originally developed by Stam (2001); Walter et al. (2007) proposed improvements and provided an elegant geometric justification of the half vector mapping of Equation (9.36). The generalized half-direction vector for refraction used in these models and in Equation (9.34) is due to Sommerfeld and Runge (1911).
当处理通过粗糙电介质的折射时，需要修改变量项的变化来考虑从半矢量到出射方向的映射。基于这种方法的模型最初由 Stam (2001) 开发；沃尔特等人。 (2007) 提出了改进，并为方程 (9.36) 的半向量映射提供了优雅的几何论证。这些模型和方程 (9.34) 中使用的广义折射半方向矢量由 Sommerfeld 和 Runge (1911) 提出。

One issue with the specular term of the Torrance–Sparrow BRDF presented in Section 9.6.5 is that it only models a single scattering interaction with the microfacet surface, causing a growing portion of the energy to be lost as the roughness increases. In scenes where many subsequent interactions are crucial (e.g., a complex 3D object made from a translucent rough dielectric material), this energy loss can become so conspicuous that standard microfacet models become effectively unusable.
第 9.6.5 节中介绍的 Torrance-Sparrow BRDF 的镜面反射项的一个问题是，它仅模拟与微面表面的单次散射相互作用，导致随着粗糙度的增加，越来越多的能量损失。在许多后续交互至关重要的场景中（例如，由半透明粗糙介电材料制成的复杂 3D 对象），这种能量损失可能变得非常明显，以至于标准微面模型变得实际上无法使用。

The original model by Torrance and Sparrow (1967) included a diffuse component to simulate light having scattered multiple times. However, a simple diffuse correction is generally unsatisfactory, since the precise amount of energy loss will depend both on the surface roughness and the angle of incidence. Kelemen and Szirmay-Kalos (2001) proposed an improved diffuse-like term that accounts for this dependence. Jakob et al. (2014a) generalized their approach to rough dielectric boundaries in the context of layered structures, where energy losses can be particularly undesirable.
Torrance 和 Sparrow (1967) 的原始模型包含一个漫反射组件来模拟多次散射的光。然而，简单的漫射校正通常不能令人满意，因为能量损失的精确量将取决于表面粗糙度和入射角。 Kelemen 和 Szirmay-Kalos (2001) 提出了一个改进的类扩散项来解释这种依赖性。雅各布等人。 (2014a) 概括了他们在层状结构背景下处理粗糙介电边界的方法，其中能量损失可能特别不受欢迎。

In all of these cases, light is treated as essentially diffuse following scattering by multiple facets. Building on Smith’s uncorrelated height assumption, Heitz et al. (2016b) cast a microfacet BRDF model into a volumetric analogue composed of microflakes—that is, a distribution of mirror facets suspended in a 3D space. With this new interpretation, they are able to simulate an arbitrary number of volumetric scattering interactions to evaluate an effective BRDF that is free of energy loss and arguably closer to physical reality.
在所有这些情况下，光被多个面散射后基本上被视为漫射。基于史密斯的不相关高度假设，Heitz 等人。 (2016b) 将微面 BRDF 模型转换为由微薄片组成的体积模拟，即悬浮在 3D 空间中的镜面分布。通过这种新的解释，他们能够模拟任意数量的体积散射相互作用，以评估有效的 BRDF，该 BRDF 没有能量损失，并且可以说更接近物理现实。

Analytic solutions may sometimes obviate the need for a stochastic simulation of interreflection. For example, Lee et al. (2018) and Xie and Hanrahan (2018) both derived analytic models for multiple scattering under the assumption of microfacets with a v-groove shape. Efficient approximate models for multiple scattering among microfacets were presented by Kulla and Conty Estevez (2017) and by Turquin (2019).
解析解有时可能不需要相互反射的随机模拟。例如，李等人。 (2018) 以及 Xie 和 Hanrahan (2018) 都在假设微面具有 V 形槽形状的情况下导出了多重散射的分析模型。 Kulla 和 Conty Estevez (2017) 以及 Turquin (2019) 提出了微面之间多重散射的高效近似模型。

Microfacet models have provided a foundation for a variety of additional reflection models. Simonot (2009) has developed a model that spans Oren–Nayar’s diffuse microfacet model (1994) and Torrance–Sparrow: microfacets are modeled as Lambertian reflectors with a layer above them that ranges from perfectly transmissive to a perfect specular reflector. Conty Estevez and Kulla (2017) have developed a model for cloth. The halo of a softer and wider secondary highlight is often visible with rough surfaces. Barla et al. (2018) described a model for such surfaces with a focus on perceptually meaningful parameters for it.
Microfacet 模型为各种附加反射模型提供了基础。 Simonot (2009) 开发了一个涵盖 Oren–Nayar 的漫反射微面模型 (1994) 和 Torrance–Sparrow 的模型：微面被建模为朗伯反射器，其上方的层范围从完美透射到完美镜面反射器。 Conty Estevez 和 Kulla (2017) 开发了布料模型。在粗糙的表面上通常可以看到更柔和、更宽的次要高光的光晕。巴拉等人。 （2018）描述了此类表面的模型，重点关注其感知上有意义的参数。

Weyrich et al. (2009) have developed methods to infer a microfacet distribution that matches a measured or desired reflection distribution. Remarkably, they showed that it is possible to manufacture actual physical surfaces that match a desired reflection distribution fairly accurately.
魏里希等人。 (2009) 开发了推断与测量的或期望的反射分布相匹配的微面分布的方法。值得注意的是，他们表明可以制造相当准确地匹配所需反射分布的实际物理表面。

Layered Materials
分层材料

Many materials are naturally composed of multiple layers—for example, a metal base surface tarnished with patina, or wood with a varnish coating. Using a specialized BRDF to represent such structures can be vastly more efficient than resolving internal reflections using standard light transport methods.
许多材料天然地由多层组成，例如，失去光泽的金属基底表面或带有清漆涂层的木材。使用专门的 BRDF 来表示此类结构比使用标准光传输方法解决内反射要高效得多。

Hanrahan and Krueger (1993) modeled the layers of skin, accounting for just a single scattering event in each layer, and Dorsey and Hanrahan (1996) rendered layered materials using the Kubelka–Munk theory, which accounts for multiple scattering within layers but assumes that radiance distribution does not vary as a function of direction.
Hanrahan 和 Krueger (1993) 对皮肤层进行了建模，仅考虑了每层中的单个散射事件，而 Dorsey 和 Hanrahan (1996) 使用 Kubelka-Munk 理论渲染了分层材料，该理论考虑了层内的多次散射，但假设辐射分布不随方向而变化。

Pharr and Hanrahan (2000) showed that Monte Carlo integration could be used to solve the adding equations to efficiently compute BSDFs for layered materials without needing either of these simplifications. The adding equations are integral equations that accurately describe the effect of multiple scattering in layered media; they were derived by van de Hulst (1980) and Twomey et al. (1966).
Pharr 和 Hanrahan (2000) 表明，蒙特卡罗积分可用于求解加法方程，以有效计算层状材料的 BSDF，而无需进行这些简化。加法方程是积分方程，可以准确地描述层状介质中多重散射的影响；它们是由 van de Hulst (1980) 和 Twomey 等人推导出来的。 （1966）。

Weidlich and Wilkie (2007) rendered layered materials more efficiently by making a number of simplifying assumptions. Guo et al. (2018) showed that both unidirectional and bidirectional Monte Carlo random walks through layers led to efficient algorithms for evaluation, sampling, and PDF evaluation. (Their unidirectional approach is implemented in pbrt’s LayeredBxDF in Section 14.3.) Xia et al. (2020a) described an improved importance sampling for this approach and Gamboa et al. (2020) showed that bidirectional sampling was unnecessary and described a more efficient approach for multiple layers.
Weidlich 和 Wilkie (2007) 通过做出一些简化的假设，更有效地渲染了分层材料。郭等人。 (2018) 表明，单向和双向蒙特卡罗随机游走各层都会产生用于评估、采样和 PDF 评估的有效算法。 （他们的单向方法在第 14.3 节中的 pbrt 的 LayeredBxDF 中实现。）Xia 等人。 (2020a) 描述了该方法的改进重要性采样，Gamboa 等人。 （2020）表明双向采样是不必要的，并描述了一种更有效的多层方法。

Another approach for layered materials is to represent the aggregate scattering behavior of a layered surface using a parametric representation. Examples include Jakob et al. (2014a) and Zeltner and Jakob (2018), who applied the adding equations to discretized scattering matrices describing volumetric layers and rough interfaces. Guo et al. (2017) modeled coated surfaces using a modified microfacet scattering model. Belcour (2018) characterized individual layers’ scattering statistically, computed aggregate scattering using the adding equations, and then mapped the result to sums of lobes based on the Trowbridge–Reitz microfacet distribution function. Weier and Belcour (2020) generalized this approach to handle anisotropic reflection from the layer interfaces and Randrianandrasana et al. (2021) further generalized the model to improve accuracy and ensure energy conservation.
层状材料的另一种方法是使用参数表示来表示层状表面的聚集散射行为。例子包括雅各布等人。 (2014a) 以及 Zeltner 和 Jakob (2018)，他们将加法方程应用于描述体积层和粗糙界面的离散散射矩阵。郭等人。 (2017) 使用改进的微面散射模型对涂层表面进行建模。 Belcour (2018) 对各个层的散射进行了统计表征，使用相加方程计算聚合散射，然后根据 Trowbridge-Reitz 微面分布函数将结果映射到波瓣总和。 Weier 和 Belcour（2020）推广了这种方法来处理来自层界面的各向异性反射，Randrianandrasana 等人。 (2021)进一步推广了该模型，以提高精度并确保节能。

It is possible to apply similar approaches to aggregate scattering at other scales. For example, Blumer et al. precomputed the effect of multiple scattering in complex geometry like trees and stored the result in a form that allows for efficient evaluation and sampling (Blumer et al. 2016).
可以将类似的方法应用于其他尺度的聚集散射。例如，布鲁默等人。预先计算了树等复杂几何体中多重散射的影响，并将结果存储为允许有效评估和采样的形式（Blumer 等人，2016）。

BSDF (Re-)Parameterization and Acquisition
BSDF（重新）参数化和采集

Improvements in data-acquisition technology have led to increasing amounts of detailed real-world BRDF data, even including BRDFs that are spatially varying (sometimes called “bidirectional texture functions,” BTFs) (Dana et al. 1999). See Müller et al. (2005) for a survey of work in BRDF measurement until the year 2005 and Guarnera et al. (2016) for a survey through the following decade.
数据采集​​技术的进步导致真实世界的 BRDF 数据的数量不断增加，甚至包括空间变化的 BRDF（有时称为“双向纹理函数”，BTF）（Dana 等人，1999）。参见穆勒等人。 (2005) 对 2005 年之前 BRDF 测量工作的调查，Guarnera 等人。 （2016）对接下来十年的调查。

Fitting measured BRDF data to parametric reflection models is a difficult problem. Rusinkiewicz (1998) made the influential observation that reparameterizing the measured data can make it substantially easier to compress or fit to models. The topic of BRDF parameterizations has also been investigated by Stark et al. (2005) and in Marscher’s Ph.D. dissertation (1998).
将测量的 BRDF 数据拟合到参数反射模型是一个难题。 Rusinkiewicz (1998) 做出了有影响力的观察，即重新参数化测量数据可以使压缩或拟合模型变得更加容易。 Stark 等人也研究了 BRDF 参数化的主题。 (2005) 和 Marscher 的博士学位。论文（1998）。

Building on Rusinkiewicz’s parameterization, Matusik et al. (2003a, 2003b) designed a BRDF representation and an efficient measurement device that repeatedly photographs a spherical sample to simultaneously acquire BRDF evaluations for many directions. They used this device to assemble a collection of isotropic material measurements that is now known as the MERL BRDF database. Baek et al. (2020) extended this approach with additional optics to capture polarimetric BRDFs, whose evaluation yields 4 × 4 Mueller matrices that characterize how reflection changes the polarization state of light. Nielsen et al. (2015) analyzed the manifold of MERL BRDFs to show that as few as 10–20 carefully chosen measurements can produce high-quality BRDF approximations.
Matusik 等人以 Rusinkiewicz 的参数化为基础。 ( 2003a, 2003b) 设计了一种 BRDF 表示方法和一种高效的测量装置，可以重复拍摄球形样本以同时获取多个方向的 BRDF 评估。他们使用该设备组装了各向同性材料测量的集合，现在称为 MERL BRDF 数据库。贝克等人。 (2020) 通过额外的光学器件扩展了这种方法，以捕获偏振 BRDF，其评估产生 4 × 4 Mueller 矩阵，用于表征反射如何改变光的偏振状态。尼尔森等人。 (2015) 分析了 MERL BRDF 的流形，表明只需 10-20 个精心选择的测量就可以产生高质量的 BRDF 近似值。

Dupuy et al. (2015) developed a simple iterative procedure for fitting standard microfacet distributions to measured BRDFs. Dupuy and Jakob (2018) generalized this procedure to arbitrary data-driven microfacet distributions and used the resulting approximation to perform a measurement in reparameterized coordinates, which is the approach underlying the MeasuredBxDF. They then used a motorized goniophotometer to spectroscopically acquire a collection of isotropic and anisotropic material samples that can be loaded into pbrt.
杜普伊等人。 (2015) 开发了一种简单的迭代程序，用于将标准微面分布拟合到测量的 BRDF。 Dupuy 和 Jakob（2018）将此过程推广到任意数据驱动的微面分布，并使用所得的近似值在重新参数化坐标中执行测量，这是 MeasuredBxDF 的基础方法。然后，他们使用电动测角光度计以光谱方式获取一系列各向同性和各向异性材料样本，并将其加载到 pbrt 中。

While the high-dimensional nature of reflectance functions can pose a serious impediment in any acquisition procedure, the resulting data can often be approximated much more compactly. Bagher et al. (2016) decomposed the MERL database into a set of 1D factors requiring 3.2 KiB per material. Vávra and Filip (2016) showed how lower-dimensional slices can inform a sparse measurement procedure for anisotropic materials.
虽然反射函数的高维性质可能对任何采集过程造成严重障碍，但所得数据通常可以更紧凑地近似。巴格尔等人。 (2016) 将 MERL 数据库分解为一组一维因子，每种材料需要 3.2 KiB。 Vávra 和 Filip (2016) 展示了低维切片如何为各向异性材料的稀疏测量程序提供信息。

Hair, Fur, and Fibers
头发、毛皮和纤维

Kajiya and Kay (1989) were the first to develop a reflectance model for hair fibers, observing the characteristic behavior of the underlying cylindrical reflectance geometry. For example, a thin and ideally specular cylinder under parallel illumination will reflect light into a 1D cone of angles. Reflection from a rough cylinder tends to concentrate around the specular 1D cone and decay with increasing angular distance. Kajiya and Kay proposed a phenomenological model combining diffuse and specular terms sharing these properties. For related work, see also the paper by Banks (1994), which discusses basic shading models for 1D primitives like hair. Goldman (1997) developed a probabilistic shading model that models reflection from collections of short hairs. Ward et al.’s survey (2007) has extensive coverage of early research in modeling, animating, and rendering hair.
Kajiya 和 Kay (1989) 第一个开发了头发纤维的反射率模型，观察了底层圆柱形反射率几何形状的特征行为。例如，平行照明下的薄且理想的镜面圆柱体会将光反射成一维锥角。来自粗糙圆柱体的反射往往集中在镜面一维锥体周围，并随着角距离的增加而衰减。 Kajiya 和 Kay 提出了一种现象学模型，结合了具有这些属性的漫反射和镜面反射项。有关相关工作，另请参阅 Banks (1994) 的论文，其中讨论了头发等一维图元的基本着色模型。 Goldman (1997) 开发了一种概率着色模型，用于模拟短毛集合的反射。 Ward 等人的调查（2007 年）广泛涵盖了头发建模、动画和渲染方面的早期研究。

Marschner et al. (2003) investigated the processes underlying scattering from hair and performed a variety of measurements of scattering from actual hair. They introduced the longitudinal/azimuthal decomposition and the use of the modified index of refraction to hair rendering. They then developed a scattering model where the longitudinal component was derived by first considering perfect specular paths and then allowing roughness by centering a Gaussian around them, and their azimuthal model assumed perfect specular reflections. They showed that this model agreed reasonably well with their measurements. Hery and Ramamoorthi (2012) showed how to sample the first term of this model and Pekelis et al. (2015) developed a more efficient approach to sampling all of its terms.
马施纳等人。 (2003) 研究了头发散射的过程，并对实际头发的散射进行了各种测量。他们引入了纵向/方位角分解以及使用修改后的折射率来渲染头发。然后，他们开发了一个散射模型，其中纵向分量是通过首先考虑完美的镜面反射路径，然后通过以它们为中心的高斯分布来允许粗糙度而导出的，并且他们的方位角模型假设完美的镜面反射。他们表明这个模型与他们的测量结果相当吻合。 Hery 和 Ramamoorthi (2012) 展示了如何对该模型的第一项进行采样，而 Pekelis 等人则展示了如何对该模型的第一项进行采样。 （2015）开发了一种更有效的方法来对其所有术语进行采样。

Zinke and Weber (2007) formalized different ways of modeling scattering from hair and clarified the assumptions underlying each of them. Starting with the bidirectional fiber scattering distribution function (BFSDF), which describes reflected differential radiance at a point on a hair as a fraction of incident differential power at another, they showed how assuming homogeneous scattering properties and a far-away viewer and illumination made it possible to simplify the eight-dimensional BFSDF to a four-dimensional bidirectional curve scattering distribution function (BCSDF). (Our implementation of the HairBxDF has glossed over some of these subtleties and opted for the simplicity of considering the scattering model as a BSDF.)
Zinke 和 Weber (2007) 形式化了头发散射建模的不同方法，并阐明了每种方法背后的假设。从双向光纤散射分布函数（BFSDF）开始，该函数将头发上一点的反射差分辐射率描述为另一点的入射差分功率的一部分，他们展示了假设均匀散射特性以及远处观察者和照明是如何实现这一点的可以将八维 BFSDF 简化为四维双向曲线散射分布函数（BCSDF）。 （我们对 HairBxDF 的实现掩盖了其中一些微妙之处，并选择简单地将散射模型视为 BSDF。）

Sadeghi et al. (2010) developed a hair scattering model with artist-friendly controls; Ou et al. (2012) showed how to sample from its distribution. Ogaki et al. (2010) created a tabularized model by explicitly modeling hair microgeometry and following random walks through it.
萨德吉等人。 (2010) 开发了一种具有艺术家友好控件的头发散射模型；欧等人。 (2012) 展示了如何从其分布中采样。大垣等人。 (2010) 通过显式建模头发微观几何形状并遵循随机游走创建了一个表格模型。

D’Eon et al. (2011, 2013) made a number of improvements to Marschner et al.’s model. They showed that their M_p term was not actually energy conserving and derived a new one that was; this is the model from Equation (9.49) that our implementation uses. (See also d’Eon (2013) for a more numerically stable formulation of M_p for low roughness, as well as Jakob (2012) for notes related to sampling their M_p term in a numerically stable way.) They also introduced a Gaussian to the azimuthal term, allowing for varying azimuthal roughness. A 1D quadrature method was used to integrate the model across the width of the hair h.
德翁等人。 (2011,2013)对Marschner等人的模型进行了一些改进。他们证明了他们的 M _p 项实际上并不是能量守恒的，并推导出了一个新的项：这是我们的实现使用的方程 (9.49) 中的模型。 （另请参阅 d'Eon (2013)，了解低粗糙度的 M _p 数值更稳定的公式，以及 Jakob (2012)，了解与采样 M _p 项相关的注释以数值稳定的方式。）他们还在方位角项中引入了高斯函数，允许不同的方位角粗糙度。使用一维求积法对头发 h 宽度上的模型进行积分。

The RGB values used for the hair pigments in HairBxDF::SigmaAFromConcentration() were computed by d’Eon et al. (2011), based on a model by Donner and Jensen (2006). The function implemented in the HairBxDF::SigmaAFromReflectance() method is due to Chiang et al. (2016a), who created a cube of hair and rendered it with a variety of absorption coefficients and roughnesses while it was illuminated with a uniform white dome. They then fit a function that mapped from the hair’s azimuthal roughness and average color at the center of the front face of the cube to an absorption coefficient.
HairBxDF::SigmaAFromConcentration() 中用于头发色素的 RGB 值由 d’Eon 等人计算。 (2011)，基于 Donner 和 Jensen (2006) 的模型。 HairBxDF::SigmaAFromReflectance() 方法中实现的函数归功于Chiang 等人。 (2016a)，他创建了一个头发立方体，并用各种吸收系数和粗糙度对其进行渲染，同时用均匀的白色圆顶进行照明。然后，他们拟合一个函数，该函数将头发的方位粗糙度和立方体正面中心的平均颜色映射到吸收系数。

D’Eon et al. (2014) performed extensive Monte Carlo simulations of scattering from dielectric cylinders with explicitly modeled scales and glossy scattering at the boundary based on a Beckmann microfacet distribution. They showed that separable models did not model all the observed effects and that in particular the specular term modeled by M_p varies over the surface of the cylinder and also depends on ϕ. They developed a non-separable scattering model, where both α and β_m vary as a function of h, and showed that it fit their simulations very accurately.
德翁等人。 （2014）基于贝克曼微面分布，对介电圆柱体的散射进行了广泛的蒙特卡罗模拟，具有明确建模的尺度和边界处的光泽散射。他们表明，可分离模型并没有对所有观察到的效果进行建模，特别是由 M _p 建模的镜面反射项在圆柱体表面上变化，并且还取决于 phi。他们开发了一个不可分离的散射模型，其中 α 和 β _m 都作为 h 的函数变化，并表明该模型非常准确地符合他们的模拟。

All the scattering models we have described so far have been BCSDFs—they represent the overall scattering across the entire width of the hair in a single model. Such “far field” models assume both that the viewer is far away and that incident illumination is uniform across the hair’s width. In practice, both of these assumptions are invalid if one is using path tracing to model multiple scattering inside hair. Two recent models have considered scattering at a single point along the hair’s width, making them more suitable for accurately modeling “near field” scattering.
到目前为止，我们描述的所有散射模型都是 BCSDF，它们代表单个模型中头发整个宽度上的整体散射。这种“远场”模型假设观察者距离很远，并且入射照明在整个头发的宽度上是均匀的。实际上，如果使用路径追踪来模拟头发内部的多重散射，这两种假设都是无效的。最近的两个模型考虑了沿头发宽度的单个点的散射，使它们更适合精确建模“近场”散射。

Yan et al. (2015) generalized d’Eon et al.’s model to account for scattering in the medulla, modeling a scattering cylinder in the interior of fur, and validated their model with a variety of measurements of actual animal fur. Subsequent work developed an efficient model that allows both near- and far-field evaluation (Yan et al. 2017a).
严等人。 (2015) 推广了 d’Eon 等人的模型来解释髓质中的散射，对毛皮内部的散射圆柱体进行建模，并通过对实际动物毛皮的各种测量来验证他们的模型。随后的工作开发了一种有效的模型，可以进行近场和远场评估（Yan et al. 2017a）。

Chiang et al. (2016a) showed that eliminating the integral over width from d’Eon et al.’s model works well in practice and that the sampling rates necessary for path tracing also worked well to integrate scattering over the curve width, giving a much more efficient implementation. They also developed the perceptually uniform parameterization of β_m and β_n that we have implemented in the HairBxDF as well as the inverse mapping from reflectance to σ_a used in our HairBxDF::SigmaAFromReflectance() method.
蒋等人。 (2016a) 表明，从 d'Eon 等人的模型中消除宽度积分在实践中效果很好，并且路径追踪所需的采样率也可以很好地积分曲线宽度上的散射，从而提供更有效的实现。他们还开发了我们在 HairBxDF 中实现的 β _m 和 β _n 的感知均匀参数化，以及使用的从反射率到 σ _a 的逆映射在我们的 HairBxDF::SigmaAFromReflectance() 方法中。

Further recent advances in hair and fur rendering include work by Khungurn and Marschner (2017), who developed a scattering model from elliptical fibers and showed that modeling fibers as elliptical rather than cylindrical gives a closer match to measured data. Benamira and Pattanaik (2021) recently proposed a model that accounts for both elliptical fibers and diffraction effects, which are significant at the scale of human hair.
头发和毛皮渲染方面的最新进展包括 Kungurn 和 Marschner (2017) 的工作，他们开发了椭圆纤维的散射模型，并表明将纤维建模为椭圆形而不是圆柱形可以与测量数据更接近。 Benamira 和 Pattanaik (2021) 最近提出了一个模型，可以解释椭圆纤维和衍射效应，这在人类头发的尺度上非常重要。

Modeling and rendering the individual fibers of fabric is closely related to doing so for hair and fur. Recent work includes Zhao et al. (2016), who fit a procedural yarn model to CT-scanned yarn, and Aliaga et al. (2017), who demonstrated the complexity of scattering from a variety of cloth fibers and developed tabularized scattering functions for them using a precomputed simulation.
对织物的各个纤维进行建模和渲染与对头发和毛皮的建模和渲染密切相关。最近的工作包括赵等人。 (2016)，他将程序纱线模型拟合到 CT 扫描纱线上，Aliaga 等人。 (2017)，他展示了各种布料纤维散射的复杂性，并使用预先计算的模拟为其开发了表格化散射函数。

Glints and Microstructure
闪光和微观结构

The microfacet reflection models in this chapter are all based on the assumption that so many microfacets are visible in a pixel that they can be accurately described by their aggregate statistical behavior. This assumption is not true for many real-world surfaces, where a relatively small number of microfacets may be visible in each pixel; examples of such surfaces include glittery car paint and plastics. Additionally, many types of rough surfaces that aren’t considered glittery (e.g., bead-blasted plastic) are characterized by bright high-frequency glints under directionally peaked illumination (e.g., the sun).
本章中的微面反射模型均基于这样的假设：一个像素中可见如此多的微面，可以通过其聚合统计行为准确地描述它们。对于许多现实世界的表面来说，这种假设并不成立，其中每个像素中可能可见的微面数量相对较少；此类表面的例子包括闪闪发光的汽车油漆和塑料。此外，许多类型的粗糙表面不被认为是闪光的（例如，喷砂塑料），其特征是在定向峰值照明（例如，太阳）下会出现明亮的高频闪烁。

A common characteristic of many glint-rendering techniques is that they replace point evaluations of reflectance functions with a directional and/or spatial average covering a small region (e.g., a ray differential). With such an approach, a single sample suffices to find all glints visible within one pixel, which dramatically accelerates the rendering process.
许多闪烁渲染技术的一个共同特征是，它们用覆盖小区域的方向和/或空间平均值（例如光线微分）代替反射函数的点评估。通过这种方法，单个样本足以找到一个像素内所有可见的闪烁，这极大地加速了渲染过程。

One approach to rendering glints was introduced by Jakob et al. (2014b), who developed a temporally consistent stochastic process that samples glint positions on the fly during evaluation of a spatio-directional average. Wang et al. (2018) showed that the performance of this method could be improved by a separable approximation of the spatial and directional dimensions. These stochastic methods are compact but also very limited in terms of the glint distributions that can be modeled.
Jakob 等人提出了一种渲染闪烁的方法。 (2014b)，他开发了一种时间一致的随机过程，可以在评估空间方向平均值时动态采样闪烁位置。王等人。 (2018)表明，可以通过空间和方向维度的可分离近似来提高该方法的性能。这些随机方法很紧凑，但在可建模的闪烁分布方面也非常有限。

In production rendering systems, fine surface details are often modeled using bump- or normal maps. Glinty surface appearance tends to result when such surfaces have high-resolution detail as well as a specular BRDF, and when they are furthermore subject to sharp (e.g., point or directional) illumination. At the same time, such configurations produce an extremely challenging Monte Carlo integration problem that has motivated numerous specialized methods for rendering normal-mapped specular surfaces.
在生产渲染系统中，精细的表面细节通常使用凹凸贴图或法线贴图进行建模。当此类表面具有高分辨率细节以及镜面 BRDF 时，并且当它们进一步受到锐利（例如，点或定向）照明时，往往会产生闪烁的表面外观。同时，这种配置产生了极具挑战性的蒙特卡罗积分问题，该问题激发了许多渲染法线贴图镜面表面的专门方法。

Yan et al. (2014) proposed a method that organizes the normal maps into a 4D spatiodirectional data structure that can be queried to find reflecting surface regions. Yan et al. (2016) drastically reduced the cost of reflectance queries by converting the normal map into a large superposition of 4D Gaussian functions termed a position-normal distribution. Though image fidelity is excellent, the overheads of these methods can be significant: slow rendering in the former case, and lengthy preprocessing and storage requirements in the second case. Zhu et al. (2019) addressed both of these issues via clustering and runtime synthesis of normal map detail. Wang et al. (2020a) substantially reduced the storage requirements by using a semi-procedural model that matches the statistics of an input texture. Zeltner et al. (2020) proposed a Newton-like equation-solving iteration that stochastically finds glints within texels of a normal map. Atanasov et al. (2021) developed a multi-level data structure for finding glints around a given half vector.
严等人。 (2014) 提出了一种将法线贴图组织成 4D 空间方向数据结构的方法，可以查询该数据结构以查找反射表面区域。严等人。 (2016) 通过将法线贴图转换为称为位置正态分布的 4D 高斯函数的大型叠加，大大降低了反射率查询的成本。尽管图像保真度非常好，但这些方法的开销可能很大：前一种情况下渲染速度较慢，后一种情况下需要冗长的预处理和存储要求。朱等人。 (2019) 通过法线贴图细节的聚类和运行时合成解决了这两个问题。王等人。 (2020a) 通过使用与输入纹理的统计数据相匹配的半过程模型，大大减少了存储需求。泽尔特纳等人。 (2020) 提出了一种类牛顿方程求解迭代，可以随机查找法线贴图纹理像素内的闪烁。阿塔纳索夫等人。 （2021）开发了一种多级数据结构，用于查找给定半向量周围的闪烁。

Other work in this area includes Raymond et al. (2016), who developed methods for rendering scratched surfaces, and Kuznetsov et al., who trained generative adversarial networks to represent microgeometry (Kuznetsov et al. 2019). Chermain et al. (2019) incorporated the effect of multiple scattering among the microstructure facets in such models, and Chermain et al. (2021) proposed a visible normal sampling technique for glint NDF. Loubet et al. recently developed a technique for sampling specular paths that is applicable to rendering caustics as well as rendering glints (2020).
该领域的其他工作包括 Raymond 等人。 (2016)，开发了渲染划痕表面的方法；Kuznetsov 等人，训练生成对抗网络来表示微观几何形状 (Kuznetsov 等人，2019)。切尔曼等人。 (2019) 在此类模型中纳入了微观结构面之间的多重散射效应，Chermain 等人。 (2021) 提出了一种闪烁 NDF 的可见法线采样技术。卢贝特等人。最近开发了一种用于采样镜面反射路径的技术，该技术适用于渲染焦散以及渲染闪烁（2020）。

Wave Optics
波动光学

Essentially all physically based renderers are based on laws that approximate wave-optical behavior geometrically. At a high level, these approximations are sound given the large scale of depicted objects compared to the wavelength of light. At the same time, wave-optical properties tend to make themselves noticeable whenever geometric features occur at scales resembling the wavelength of light, and such features may indeed be present even on objects that are themselves drastically larger.
本质上，所有基于物理的渲染器都基于几何上近似波光学行为的定律。在较高的层面上，考虑到所描绘的物体与光的波长相比规模较大，这些近似值是合理的。与此同时，每当几何特征出现在类似于光波长的尺度上时，波动光学特性往往会变得引人注目，而且这些特征甚至可能确实存在于本身大得多的物体上。

For example, consider a thin film of oil on a puddle, a tiny scratch on an otherwise smooth metallic surface, or an object with micron-scale surface microstructure. These cases can feature striking structural coloration caused by the interference of light, which a purely geometric simulation would not be able to reproduce. It may be tempting to switch to a full wave-optical simulation of light in such cases, though this line of thought quickly runs into fundamental limits: for example, using the Finite Difference Time Domain (FDTD) method, the simulation domain would need to be discretized at resolutions of < 100 nm and simulated using sub-femtosecond timesteps. This can still work when studying local behavior at the micron scale, but it is practically infeasible for scenes measured in centimeters or even meters. These challenges have motivated numerous specialized methods that reintroduce such wave-optical effects within an otherwise geometric simulation.
例如，考虑水坑上的一层薄薄的油膜、光滑金属表面上的微小划痕或具有微米级表面微观结构的物体。这些案例可能具有由光干涉引起的引人注目的结构着色，这是纯粹的几何模拟无法再现的。在这种情况下，切换到光的全波光模拟可能很诱人，尽管这种思路很快就会遇到基本限制：例如，使用时域有限差分 (FDTD) 方法，模拟域需要以 < 100 nm 的分辨率进行离散化，并使用亚飞秒时间步长进行模拟。在研究微米尺度的局部行为时，这仍然有效，但对于以厘米甚至米为单位测量的场景实际上是不可行的。这些挑战激发了许多专门的方法，在其他几何模拟中重新引入这种波光效应。

Moravec (1981) was the first to apply a wave optics model to computer graphics. Other early work in this area includes Bahar and Chakrabarti (1987) and Stam (1999), who modeled diffraction effects from random and periodic structures. Cuypers et al. (2012) modeled multiple diffraction phenomena using signed BSDFs based on Wigner Distribution Functions.
Moravec (1981) 是第一个将波动光学模型应用于计算机图形学的人。该领域的其他早期工作包括 Bahar 和 Chakrabarti (1987) 以及 Stam (1999)，他们对随机和周期性结构的衍射效应进行了建模。库珀斯等人。 (2012) 使用基于维格纳分布函数的带符号 BSDF 对多重衍射现象进行了建模。

Musbach et al. (2013) applied the FDTD to obtain a BRDF of the iridescent microstructure of a Morpho butterfly. Their paper provides extensive references to previous work on this topic. Dhillon et al. (2014) developed a model of diffraction from small-scale biological features such as are present in snake skin. Belcour and Barla (2017) modeled thin film iridescence on a rough microfacet surface and showed the importance of this effect for materials such as leather and how the resulting spectral variation can be efficiently calculated in an RGB-based simulation. Werner et al. (2017) developed a model for rendering surfaces with iridescent scratches. Yan et al. (2018) presented a surface microstructure model that integrates over a coherence area to produce iridescent glints, revealing substantial differences between geometric and wave-based modeling. Toisoul and Ghosh (2017) presented a method for capturing and reproducing the appearance of periodic grating-like structures. Xia et al. (2020b) showed that diffraction and interference are meaningful at the scale of fibers and developed a wave optics-based model for scattering from fibers that they validated with measured data.
穆斯巴赫等人。 (2013) 应用 FDTD 获得了 Morpho 蝴蝶虹彩微结构的 BRDF。他们的论文提供了对该主题先前工作的广泛参考。迪伦等人。 （2014）开发了一种小尺度生物特征（例如蛇皮中存在的特征）的衍射模型。 Belcour 和 Barla (2017) 对粗糙微表面上的薄膜虹彩进行了建模，并展示了这种效应对于皮革等材料的重要性，以及如何在基于 RGB 的模拟中有效计算由此产生的光谱变化。维尔纳等人。 (2017) 开发了一种用于渲染具有虹彩划痕的表面的模型。严等人。 (2018) 提出了一种表面微观结构模型，该模型在相干区域上集成以产生虹彩闪烁，揭示了几何建模和基于波的建模之间的实质性差异。 Toisoul 和 Ghosh (2017) 提出了一种捕获和再现周期性光栅状结构外观的方法。夏等人。 (2020b) 表明衍射和干涉在纤维尺度上是有意义的，并开发了一种基于波动光学的纤维散射模型，并用测量数据进行了验证。

In contrast to the above applications that reproduce dramatic goniochromatic effects, several works have studied how wave-based modeling can improve modeling of common materials (e.g., rough plastic or conductive surfaces). Löw et al. (2012) proposed and compared geometric and wave-based BRDF models in fits to materials to the MERL database. Dong et al. (2015) measured the surface microstructure of metal samples using a profilometer and used it to construct geometric and wave-based models that they then compared to goniophotometric measurements. Holzschuch and Pacanowski (2017) integrated diffraction effects into a microfacet model and showed that this gives a closer fit to measured data.
与上述再现戏剧性的角色效应的应用相比，一些工作研究了基于波的建模如何改进常见材料（例如粗糙的塑料或导电表面）的建模。勒夫等人。 (2012) 提出并比较了适合 MERL 数据库材料的几何和基于波的 BRDF 模型。董等人。 (2015) 使用轮廓仪测量金属样品的表面微观结构，并用它构建几何和基于波的模型，然后将其与测角光度测量进行比较。 Holzschuch 和 Pacanowski (2017) 将衍射效应集成到微面模型中，并表明这更接近测量数据。

Additional Topics
附加主题

The Lambertian BRDF is an idealized model; as noted earlier, it does not match many real-world BRDFs precisely. Oren and Nayar (1994) proposed an improved model based on Lambertian microfacets that allowed the specification of surface roughness. d’Eon has recently (2021) proposed a model based on scattering Lambertian spheres that matches the appearance of many materials well.
朗伯 BRDF 是一个理想化的模型；如前所述，它与现实世界中的许多 BRDF 并不完全匹配。 Oren 和 Nayar (1994) 提出了一种基于朗伯微面的改进模型，该模型允许指定表面粗糙度。 d’Eon 最近（2021）提出了一种基于散射朗伯球的模型，该模型与许多材料的外观很好地匹配。

A number of researchers have investigated BRDFs based on modeling the small-scale geometric features of a reflective surface. This work includes the computation of BRDFs from bump maps by Cabral, Max, and Springmeyer (1987); Fournier’s normal distribution functions (Fournier 1992); and a technique by Westin, Arvo, and Torrance (1992), who applied Monte Carlo ray tracing to statistically model reflection from microgeometry and represented the resulting BRDFs with spherical harmonics. Wu et al. (2011) developed a system that made it possible to model microgeometry and specify its underlying BRDF while interactively previewing the resulting macro-scale BRDF, and Falster et al. (2020) computed BSDFs of microgeometry, accounting for both multiple scattering and diffraction.
许多研究人员基于对反射表面的小尺度几何特征进行建模来研究 BRDF。这项工作包括 Cabral、Max 和 Springmeyer (1987) 根据凹凸贴图计算 BRDF； Fournier 的正态分布函数（Fournier 1992）； Westin、Arvo 和 Torrance (1992) 提出的一项技术，他们应用蒙特卡罗射线追踪对微观几何反射进行统计建模，并用球谐函数表示所得的 BRDF。吴等人。 (2011) 开发了一个系统，可以对微观几何进行建模并指定其底层 BRDF，同时交互式预览生成的宏观尺度 BRDF，Falster 等人。 (2020) 计算了微观几何的 BSDF，同时考虑了多重散射和衍射。

The effect of the polarization of light is not modeled in pbrt, although for some scenes it can be an important effect; see, for example, the paper by Tannenbaum, Tannenbaum, and Wozny (1994) for information about how to extend a renderer to account for this effect. Fluorescence, where light is reflected at different wavelengths than the incident illumination, is also not modeled by pbrt; see Glassner (1994) and Wilkie et al. (2006) for more information on this topic.
光的偏振效应并未在 pbrt 中建模，尽管对于某些场景来说它可能是一个重要的效应；例如，请参阅 Tannenbaum、Tannenbaum 和 Wozny (1994) 的论文，了解有关如何扩展渲染器以解决此效果的信息。荧光（其中光以与入射照明不同的波长反射）也不是由 pbrt 建模的；参见 Glassner (1994) 和 Wilkie 等人。 (2006) 了解有关此主题的更多信息。

Modeling reflection from a variety of specific types of surfaces has received attention from researchers, leading to specialized reflection models. Examples include wood (Marschner, Westin, Arbree, and Moon 2005), car paint (Ergun et al. 2016), paper (Papas et al. 2014), and pearlescent materials (Guillén et al. 2020). Cloth remains a particularly challenging material to render; see the recent survey by Castillo et al. (2019) for comprehensive coverage of work in this area.
对各种特定类型表面的反射进行建模已受到研究人员的关注，从而产生了专门的反射模型。例子包括木材（Marschner、Westin、Arbree 和 Moon 2005）、汽车油漆（Ergun 等人，2016）、纸张（Papas 等人，2014）和珠光材料（Guillén 等人，2020）。布料仍然是一种渲染起来特别具有挑战性的材料；请参阅 Castillo 等人最近的调查。 （2019）全面报道该领域的工作。

Sampling BSDFs well is a key component of efficient image synthesis. Szécsi et al. (2003) evaluated different approaches for sampling BSDFs that are comprised of multiple lobes. It is often only possible to sample some factors of a BSDF (e.g., when sampling the Torrance–Sparrow BRDF using the distribution of visible microfacet normals); Herholz et al. fit parametric sampling distributions to BSDFs in an effort to sample them more effectively (Herholz et al. 2018).
良好的 BSDF 采样是高效图像合成的关键组成部分。塞奇等人。 (2003) 评估了对由多个波瓣组成的 BSDF 进行采样的不同方法。通常只能对 BSDF 的某些因子进行采样（例如，使用可见微面法线的分布对 Torrance-Sparrow BRDF 进行采样时）；赫霍尔茨等人。将参数采样分布拟合到 BSDF，以更有效地对其进行采样（Herholz 等人，2018）。

pbrt’s test suite uses statistical hypothesis tests to verify the correctness of its BSDF sampling routines. The idea of verifying such graphics-related Monte Carlo sampling routines using statistical tests was introduced by Subr and Arvo (2007a). The χ² test variant that is used in pbrt was originally developed as part of the Mitsuba renderer by Jakob (2010).
pbrt 的测试套件使用统计假设检验来验证其 BSDF 采样例程的正确性。 Subr 和 Arvo (2007a) 提出了使用统计测试验证此类与图形相关的蒙特卡洛采样例程的想法。 pbrt 中使用的 χ ² 测试变体最初由 Jakob (2010) 作为 Mitsuba 渲染器的一部分开发。

EXERCISES
练习

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1   It could be argued that a shortcoming of the BxDF sampling interface is that there are not entry points to sample from the 4D distribution of f(ω, ω′). This is a reasonably esoteric case for the applications envisioned for pbrt, however.
1 可以说，BxDF 采样接口的缺点是没有从 f(ω, ω′) 的 4D 分布中采样的入口点。然而，对于 pbrt 设想的应用程序来说，这是一个相当深奥的案例。

2   Strictly speaking, this is not always true: for example, the other method might also involve a delta distribution with matching direction ω′. pbrt does not consider this case, which may lead to a small loss of sampling efficiency in this rare corner case.
2 严格来说，这并不总是正确的：例如，另一种方法也可能涉及具有匹配方向 ω' 的 delta 分布。 pbrt 不考虑这种情况，这可能会导致在这种罕见的极端情况下采样效率略有损失。

3   See the online edition of the book for an additional chapter devoted to bidirectional light transport algorithms.
3 请参阅本书的在线版本，了解专门介绍双向光传输算法的附加章节。

4   The same model was also independently derived by Walter et al. (2007), who dubbed it “GGX.”
4 Walter 等人也独立推导了同一模型。 （2007），他将其称为“GGX”。

5   Both steps use ω_i instead of ω_i^(k) to improve the smoothness of the interpolation.
5 两个步骤均使用 ω 代替 ω ^(k) 以提高插值的平滑度。

6   A repository of compatible files that cover the 360–1000 nm interval at roughly 4 nm resolution is available at http://rgl.epfl.ch/materials.
6 可在 http://rgl.epfl.ch/materials 上找到涵盖 360–1000 nm 间隔、分辨率约为 4 nm 的兼容文件存储库。

7   Other authors generally include A_p in the N_p term, though we find it more clear to keep them separate for the following exposition. Here we also use f for the BSDF, which most hair scattering papers denote by S.
7 其他作者通常将 A _p 包含在 N _p 项中，尽管我们发现在下面的说明中将它们分开会更清楚。这里我们也使用 f 来表示 BSDF，大多数毛发散射论文用 S 表示。

8   Note that this is a different usage of v than in earlier chapters when it was used for the parametric coordinate along the width of a curve.
8 请注意，这是 v 的用法与前面章节中的不同，当时它用于沿曲线宽度的参数坐标。

9   This is due to the Bravais properties of cylindrical scattering. See Appendix B of Marschner et al. (2003) for a derivation and further explanation.
9 这是由于柱面散射的布拉维特性。参见 Marschner 等人的附录 B。 （2003）的推导和进一步解释。

10 This approach is applicable since the BSDF’s definition includes the product of M_p and N_p and so their joint PDF is separable.
10 这种方法是适用的，因为 BSDF 的定义包括 M _p 和 N _p 的乘积，因此它们的联合 PDF 是可分离的。

11 An area light and not a point or directional light is necessary due to subtleties in how lights are seen in specular surfaces. With the light transport algorithms used in pbrt, infinitesimal point sources are never visible in mirrored surfaces. This is a typical limitation of ray-tracing renderers and usually not bothersome in practice.
11 由于在镜面中观察光的方式存在微妙之处，因此需要使用区域光，而不是点光或定向光。通过 pbrt 中使用的光传输算法，无穷小的点源在镜面中永远不可见。这是光线追踪渲染器的典型限制，在实践中通常并不麻烦。