WHAT IS BRDF?

“Bidirectional Reflectance Distribution Function”

In the past, there were no such fancy words or concepts in our lives. The only thing that we did, were playing with the options of the reflection channel. Whenever the physical render engines included our lives, then we changed our point of view to the world; and now we started looking at a metallic surface or even a table surface with very careful eyes. Because of these realistic render engines, we decided that we need to know the laws behind the appearance. We managed well with these realistic engines for a while. But when we decided to swim in the water we did not know, and when we dived deep into that water, the mathematical complexity of the physical world appeared. What we previously called Reflection suddenly became a "Reflectance". For years, the reflection we know as a friend turned into a "mathematical monster." We have to face such these mathematical monsters. Our adventure has just begun.

Now let's try to scare you a little bit before we get into the matter: for example, look at the figure below:

什么是BRDF？

“双向反射分布函数”

过去，我们的生活中没有如此花哨的词语或概念。我们所做的唯一一件事就是使用反射通道的选项。每当物理渲染引擎包括我们的生活时，我们便将我们的观点改变为世界。现在我们开始用非常细心的眼睛观察金属表面甚至桌子表面。由于这些逼真的渲染引擎，我们决定需要了解外观背后的规律。我们使用这些逼真的引擎进行了一段时间的良好管理。但是当我们决定在水中游泳时，我们并不知道，当我们深入到水中时，物理世界的数学复杂性就出现了。我们之前称为反射的内容突然变成了“反射”。多年以来，我们作为朋友认识的反思变成了“数学怪物”。我们必须面对这样的数学怪兽。我们的冒险才刚刚开始。

现在，让我们在涉及此问题之前先吓一跳：例如，看下图：

or look at this:

或看看这个：

If you were not afraid enough, we kept the worst at the end: Now look at the formula below:

如果您还不够害怕，那么我们将把最坏的情况保留在最后：现在看下面的公式：

It's all about BRDF and reflection that will come out if you go deep enough. It's one of the "Mathematical Monster" that we mentioned. Now, beside the fearless mathematicians who are not afraid to see this, let's try to make things a little easier for 3D artists with mortal eyes.

如果深入的话，所有关于BRDF和反射的问题都会出现。 这是我们提到的“数学怪兽”之一。 现在，除了不惧怕数学家的无所畏惧的数学家之外，让我们尝试使具有凡人眼光的3D画家更轻松一些。

Well let's ask again: What is BRDF?

BRDF is actually a formulated function that you often see in reflection models we have used for years. Like the above formula. So, in fact BRDF is a function and this function is defined as the ratio of the reflected light (radiance energy) to the incoming light (irradiance energy) at the point "x" on the surface.

好，让我们再问一次：什么是BRDF？

BRDF实际上是一个公式化函数，您经常在我们使用了多年的反射模型中可以看到。 就像上面的公式。 因此，实际上BRDF是一个函数，此函数定义为在表面上的点“ x”处反射光（辐射能量）与入射光（辐射能量）之比。

BRDF is one of the most useful models of reflection and was first described in the field of Radiometry. It deals with the directional distribution reflection of light from any surface. It does not deal with the part of the light that travels into the surface, it only deals with the reflected part. In fact, BRDF is a simplified version of the "Bidirectional Scattering Surface Reflection Distribution Function (BSSRDF)". In BSSRDF, it is also taken into account which point the light enter into the surface and where it left off from the surface. In BRDF the assumption is that the surface is homogeneous. Thus, the BSSRDF is reduced to BRDF.

The fact that we have known for many years as "reflection" has already been described as a function. Only these reflection models have changed over the years. We used this function for years, but the name was "Phong" or "Lambertian" or "Blinn-Phong" before. These were relatively simple models that did not produce physical results. We are now using BRDF models, such as Torrence-Sparrow or Cook-Torrence, which are based on years ago and produce satisfactory results on physical renders.

A BRDF must obey with the basic laws of physics in order to give realistic results. Let's look at these briefly:

BRDF是最有用的反射模型之一，最早是在辐射测量领域描述的。它处理来自任何表面的光的定向分布反射。它不处理进入表面的光的一部分，仅处理反射的部分。实际上，BRDF是“双向散射表面反射分布函数（BSSRDF）”的简化版本。在BSSRDF中，也要考虑到光线进入表面的点和离开表面的位置。在BRDF中，假设表面是均匀的。因此，BSSRDF减少为BRDF。

多年来我们将“反射”这一事实称为功能。这些年来只有这些反射模型发生了变化。我们使用此功能已有多年，但以前的名称是“ Phong”或“ Lambertian”或“ Blinn-Phong”。这些是相对简单的模型，不会产生物理结果。我们现在使用的是BRDF模型，例如Torrence-Sparrow或Cook-Torrence，它们基于几年前，并且在物理渲染上产生令人满意的结果。

Energy conservation

Energy conservation must be ensured in any system. Therefore, a good and realistic BDRF must obey with the law of conservation of energy. Accordingly, for all possible directions, the total energy of the light reflected from a surface can never be more than the total energy of the incoming light. We will show this later using the new Octane BDRF models.

能力守恒

任何系统中都必须确保能量守恒。因此，良好而现实的BDRF必须遵守能量守恒定律。因此，对于所有可能的方向，从表面反射的光的总能量永远不会大于入射光的总能量。我们稍后将使用新的Octane BDRF模型进行演示。

Non-negativity

The reflection value must be between 0 and 1 because of the conservation of the energy. Therefore, the ratio of the reflected light (radiance energy) to the incoming light (irradiance energy) must be between 0 and 1. BRDF also includes this ratio with the cosine term. That is, the range of BRDF is 0 to infinity. Also both radiance and irradiance values can not be negative. Therefore BRDF can not take negative value either.

非负性

由于能量守恒，反射值必须在0到1之间。因此，反射光（辐射能量）与入射光（辐射能量）之比必须在0到1之间。BRDF在余弦项中也包含该比率。也就是说，BRDF的范围是0到无穷大。同样，辐射度和辐照度值都不能为负。因此，BRDF也不能取负值。

Reciprocity

This feature is also known as Helmotz's reciprocity law. This law is caused by another physical characteristic of the light. According to this law; the BRDF value must not change when the incoming vector and the outgoing (view) vector are interchanged. In other words, when the travel of the light reflected from the surface is reversed, the light must follow the same path and the BRDF must be the same in both cases.

互惠

此功能也称为Helmotz的对等定律。该定律是由光的另一物理特性引起的。根据这条法律；当传入向量和传出（视图）向量互换时，BRDF值不得更改。换句话说，当从表面反射的光的传播反向时，在两种情况下，光必须遵循相同的路径，并且BRDF必须相同。

MICROFACETS THEORY

Microfacet theory is a technique developed on the basis of roughness or imperfection of every surface in the real world (let's call it Roughness). Works in cooperation with shadowing and masking. In realistic BRDF models, the surface is considered to be formed from uneven grooves (microfacet). Microfacet is calculated by how much of the ratio is due to masking and shadowing of interaction with the light.

The masking state is, when the light is reflected by one surface of the microfacet, then the reflected portion is blocked by the other "microfacet". Shadowing is the case where the light is blocked by the other "microfacet" before it reaches the surface of the "microfacet".

微面理论

微面理论是一种基于现实世界中每个表面的粗糙度或不完美性（称为“粗糙度”）而开发的技术。与阴影和遮罩配合使用。在实际的BRDF模型中，表面被认为是由凹凸不平的凹槽（微刻面）形成的。通过与光交互作用的遮罩和阴影产生多少比例来计算微面。

遮蔽状态是，当光被微面的一个表面反射时，反射部分被另一“微面”阻挡。阴影是指光线在到达“微面”的表面之前被另一个“微面”遮挡的情况。

FRESNEL EFFECT

A good BRDF should also present the Fresnel Equation. Fresnel is the name given to the fact that the surface reflection varies according to the observed angle and the surface IOR (Index of Refraction) values. The Fresnel effect is, which reflects less light from the direct-facing surfaces than other angles (reflection changes when you look at different angles other then grazing angle). For example, if you look at the plant pot on a reflective table in orthogonal direction (steep angles), you will see a weak reflection. But if you move closer to the Shallow Angle (grazing angle), you will see stronger reflections. So Grazing Angle explains why most materials are actually "The Reflectors". The Fresnel effect takes an important place in creating realistic material.

菲涅耳效应

良好的BRDF还应显示菲涅耳方程。 菲涅耳的名称是表面反射随观察到的角度和表面IOR（折射率）值而变化的事实。 菲涅耳效果是，与其他角度相比，它从直接朝向的表面反射的光更少（当您查看与掠射角度不同的其他角度时，反射会改变）。 例如，如果您在反射台上以正交方向（陡角）观察花盆，则会看到微弱的反射。 但是，如果您靠近“浅角”（掠射角），则会看到较强的反射。 因此，Grazing Angle解释了为什么大多数材质实际上是“反射器”。 菲涅耳效果在创建逼真的材质中占有重要地位。

isotropy and anisotropy BRDF

Isotropic and Anisotropic BRDFs are an important subclass of BRDF. For many materials in the nature, if the direction of the light and the view direction are fixed, the reflection does not change as the material is rotated around the surface normal. These properties are called isotropic materials. Examples of isotropic materials include metals, plastics and painted surfaces.

Anisotropic surfaces, on the other hand, change their reflectance values by rotating the surface around its normal, in contrast to isotropic surfaces. Examples of anisotropic surfaces are metal brushes, polished metal, human hair, fur, velvet and wood.

各向同性和各向异性

各向同性和各向异性BRDF是BRDF的重要子类。 对于自然界中的许多材质，如果光的方向和视图的方向固定，则反射不会随着材质绕表面法线旋转而改变。 这些特性称为各向同性材质。 各向同性材质的示例包括金属，塑料和喷漆表面。

另一方面，与各向同性表面相反，各向异性表面通过绕其法线旋转表面来更改其反射率值。 各向异性表面的例子是金属刷，抛光金属，人发，毛皮，天鹅绒和木材。

来源：

我们试图在这里简单地谈论BRDF，但这是一个非常复杂且漫长的话题，您可以通过以下链接获得更多信息。

BRDF模型概述-此处

有关BRDF的维基百科页面-此处

关于微刻方法-这里

用于计算机图形学的BRDF-在这里

菲涅耳方程的Wikipedia页面-此处