Difference between revisions of "Physically-based BRDF"
Gongminmin (Talk | contribs) (Created page with "== Introduction == Physically based rendering have been known for many years, but the "ad-hoc" rendering models (such as Phong) are still widely used in game. These "ad-hoc" mo...") |
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The most common used rendering model in game describes only reflectance, not including terms such as SSS. The reflectance equation is: | The most common used rendering model in game describes only reflectance, not including terms such as SSS. The reflectance equation is: | ||
| − | <math>L_0(\mathbf{v})=\int_{\Omega} \rho(\mathbf{l},\mathbf{v}) \otimes L_i(\mathbf{l}) (\mathbf{n} \cdot \mathbf{l}) d\omega_i</math> | + | <center><math>L_0(\mathbf{v})=\int_{\Omega} \rho(\mathbf{l},\mathbf{v}) \otimes L_i(\mathbf{l}) (\mathbf{n} \cdot \mathbf{l}) d\omega_i</math></center> |
Here <math>\rho(\mathbf{l},\mathbf{v})</math> is BRDF, <math>L_i(\mathbf{l})</math> is the contribution from light source, <math>(\mathbf{n} \cdot \mathbf{l})</math> is the angle between light and surface normal. This integration results the sum of all light sources contribute to a surface point. | Here <math>\rho(\mathbf{l},\mathbf{v})</math> is BRDF, <math>L_i(\mathbf{l})</math> is the contribution from light source, <math>(\mathbf{n} \cdot \mathbf{l})</math> is the angle between light and surface normal. This integration results the sum of all light sources contribute to a surface point. | ||
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The simplest BRDF is the Lambert. The well-known Lambertian BRDF in game is present as <math>(\mathbf{n} \cdot \mathbf{l})</math>. However, it is part of reflectance equation, and lambertian term is actually a constant value: | The simplest BRDF is the Lambert. The well-known Lambertian BRDF in game is present as <math>(\mathbf{n} \cdot \mathbf{l})</math>. However, it is part of reflectance equation, and lambertian term is actually a constant value: | ||
| − | <math>\rho_{lambert}(\mathbf{l},\mathbf{v})=\frac{\mathbf{c}_{diff}}{\pi}</math> | + | <center><math>\rho_{lambert}(\mathbf{l},\mathbf{v})=\frac{\mathbf{c}_{diff}}{\pi}</math></center> |
| + | |||
| + | == Punctual Light Sources == | ||
| + | |||
| + | These are classic point, directional, and spot lights in compute games. These local light source can be abstracted to a concept named "''punctual light sources''". They are infinitely small light with centain direction. Since what we want is the lighting reach a surface point, we don't need to consider the attenuation. A punctual light source are parameterized by the light color <math>\mathbf{c}_{light}</math> and the light direction vector <math>\mathbf{l_c}</math>. The light color <math>\mathbf{c}_{light}</math> is specified as the color a white Lambertian surface would have when illuminated by the light from a direction parallel to the surface normal (<math>\mathbf{l_c}=\mathbf{n}</math>). | ||
| + | |||
| + | How to calculate the contribution from this light source to a point? Here we will start by defining a tiny area light source centered on <math>\mathbf{l_c}</math>, with a small angular extent <math>\varepsilon</math>. This tiny area light illuminates a shaded surface point with the incoming radiance function <math>L_{tiny}(\mathbf{l})</math>. The incoming radiance function has the following two properties: | ||
| + | |||
| + | <center><math>\forall\mathbf{l}|\angle(\mathbf{l}, \mathbf{l_c})>\varepsilon, L_{tiny}(\mathbf{l})=0</math></center> | ||
| + | |||
| + | <center><math>if \mathbf{l_c}=\mathbf{n}, \mathbf{c}_{light}=\frac{1}{\pi}\int_{\Omega} L_{tiny}(\mathbf{l}) (\mathbf{n} \cdot \mathbf{l}) d\omega_i</math></center> | ||
| + | |||
| + | The first property says that no light is incoming for any light directions which form an angle greater than <math>\varepsilon</math> with <math>\mathbf{l_c}</math>. The second property follows from the definition of <math>\mathbf{c}_{light}</math>. Since the surface is white, <math>\mathbf{c}_{diff}=1</math>. Applying reflectance equation and Lambert from last article, there's the second property. Because <math>\mathbf{c}_{light}</math> requires <math>\mathbf{l_c}=\mathbf{n}</math>, it is the limit as <math>\varepsilon</math> goes to 0: | ||
| + | |||
| + | <center><math>if \mathbf{l_c}=\mathbf{n}, \mathbf{c}_{light}=\lim_{\varepsilon \to 0}(\frac{1}{\pi}\int_{\Omega} L_{tiny}(\mathbf{l}) (\mathbf{n} \cdot \mathbf{l}) d\omega_i)</math></center> | ||
| + | |||
| + | Since <math>\mathbf{l_c}=\mathbf{n}</math> and <math>\varepsilon \to 0</math>, we can assume <math>\mathbf{n} \cdot \mathbf{l}=1</math>, which gives us: | ||
| + | |||
| + | <center><math>\mathbf{c}_{light}=\lim_{\varepsilon \to 0}(\frac{1}{\pi}\int_{\Omega} L_{tiny}(\mathbf{l}) d\omega_i)</math></center> | ||
| + | |||
| + | That is: | ||
| + | |||
| + | <center><math>\lim_{\varepsilon \to 0}(\int_{\Omega} L_{tiny}(\mathbf{l}) d\omega_i) = \pi \mathbf{c}_{light}</math></center> | ||
| + | |||
| + | Now we shall apply our tiny area light to a general BRDF, and look at its behavior in the limit as goes to 0: | ||
| + | |||
| + | <center><math>L_0(\mathbf{v})=\lim_{\varepsilon \to 0}(\int_{\Omega} \rho(\mathbf{l},\mathbf{v}) \otimes L_{tiny}(\mathbf{l}) (\mathbf{n} \cdot \mathbf{l}) d\omega_i)=\rho(\mathbf{l_c}, \mathbf{v}) \otimes \lim_{\varepsilon \to 0}(\int_{\Omega} L_{tiny}(\mathbf{l}) d\omega_i)(\mathbf{n} \cdot \mathbf{l_c})</math></center> | ||
| + | |||
| + | So | ||
| + | |||
| + | <center><math>L_0(\mathbf{v})=\pi \rho(\mathbf{l_c}, \mathbf{v}) \otimes \mathbf{c}_{light} (\mathbf{n} \cdot \mathbf{l_c})</math></center> | ||
| + | |||
| + | As you see, the tiny area light source term disappears. The remaining of our equation is what we familar with. | ||
[[zh:基于物理的BRDF]] | [[zh:基于物理的BRDF]] | ||
Revision as of 02:08, 29 December 2010
Introduction
Physically based rendering have been known for many years, but the "ad-hoc" rendering models (such as Phong) are still widely used in game. These "ad-hoc" models require laborious tweaking to produce high-quality images. However, physically based, energy-conserving rendering models easily create materials that hold up under a variety of lighting environments.
Surprisingly, physically based models are not more difficult to implement or evaluate than the traditional "ad-hoc" ones.
Reflectance equation
The most common used rendering model in game describes only reflectance, not including terms such as SSS. The reflectance equation is:
Here is BRDF,
is the contribution from light source,
is the angle between light and surface normal. This integration results the sum of all light sources contribute to a surface point.
Diffuse term
The simplest BRDF is the Lambert. The well-known Lambertian BRDF in game is present as . However, it is part of reflectance equation, and lambertian term is actually a constant value:
Punctual Light Sources
These are classic point, directional, and spot lights in compute games. These local light source can be abstracted to a concept named "punctual light sources". They are infinitely small light with centain direction. Since what we want is the lighting reach a surface point, we don't need to consider the attenuation. A punctual light source are parameterized by the light color and the light direction vector
. The light color
is specified as the color a white Lambertian surface would have when illuminated by the light from a direction parallel to the surface normal (
).
How to calculate the contribution from this light source to a point? Here we will start by defining a tiny area light source centered on , with a small angular extent
. This tiny area light illuminates a shaded surface point with the incoming radiance function
. The incoming radiance function has the following two properties:
The first property says that no light is incoming for any light directions which form an angle greater than with
. The second property follows from the definition of
. Since the surface is white,
. Applying reflectance equation and Lambert from last article, there's the second property. Because
requires
, it is the limit as
goes to 0:
Since and
, we can assume
, which gives us:
That is:
Now we shall apply our tiny area light to a general BRDF, and look at its behavior in the limit as goes to 0:
So
As you see, the tiny area light source term disappears. The remaining of our equation is what we familar with.
