\f$\cos\theta_l\f$ factor appearing is caused by affection of surface area and light direction mutual orientation to the amount of radiance coming to this area.
This is mainly due to geometric laws. The rest part of integral is the key of the whole illumination model.
BRDF defines it's complexity and optical properties of material.
-It has to model all light and material interactions and also has to satisfy some following criteria in order to be physical correct @ref ref_Duvenhage13 "[Duvenhage13]":
+It has to model all light and material interactions and also has to satisfy some following criteria in order to be physical correct [@ref Duvenhage13]:
* Positivity: \f$f(v,l) \geq 0\f$
* Helmholtz reciprocity: \f$f(v,l) = f(l, v)\f$ (follows from 2<sup>nd</sup> Law of Thermodynamics)
* Energy conservation: \f$\displaystyle \forall v \, \int\limits_H f(v,l) \cos\theta_l \, \mathrm{d}l = 1\f$ (in order not to reflect more light than came)
\f[L_o=\int\limits_H (f_d(v,l)+f_s(v, l)) L_i(l) \cos\theta_l\, \mathrm{d}l\f]
-PBR theory is based on **Cook-Torrance specular BRDF** @ref ref_Cook81 "[Cook81]". It imagines surface as set of perfectly reflected micro faces distributed on area in different ways which is pretty good model approximation of real world materials.
+PBR theory is based on **Cook-Torrance specular BRDF** [@ref Cook81]. It imagines surface as set of perfectly reflected micro faces distributed on area in different ways which is pretty good model approximation of real world materials.
If this area is small enough not to be able to recognize separate micro surfaces the results becomes a sort of averaging or mixing of every micro plane illumination contribution.
In that level it allows to work with micro faces in statistical manner manipulating only probabilities distributions of micro surfaces parameters such as normals, height, pattern, orientation etc.
In computer graphics pixels are units of images and it usually covers a relatively large areas of surfaces so that micro planes can be considered to be unrecognizable.
Three parts presented in nominator have its own meaning but can have different implementation with various levels of complexity and physical accuracy.
In that paper only one certain implementation is used. The \f$D\f$ component is responsible for **micro faces normals distribution**.
It is the main instrument that controls reflection's shape and strength according to **roughness** \f$r\f$ parameter.
-The implementation with good visual results is **Trowbridge-Reitz GGX** approach used in Disney's RenderMan and Unreal Engine @ref ref_Karis13 "[Karis13]":
+The implementation with good visual results is **Trowbridge-Reitz GGX** approach used in Disney's RenderMan and Unreal Engine [@ref Karis13]:
\f[D=\frac{\alpha^2}{\pi(\cos^2\theta_h(\alpha^2-1) + 1)^2}\f]
Without it the visual appearance of surface becomes rough too quickly during the parameter's increasing.
The second \f$G\f$ component is called **geometric shadowing** or attenuation factor.
-The point is that micro surfaces form kind of terrain and can cast shadows over each other especially on extreme viewing angles @ref ref_Heitz14 "[Heitz14]".
-**Smith-Schlick model** @ref ref_Heitz14 "[Heitz14]", @ref ref_Schlick94 "[Schlick94]" has been chosen as implementation:
+The point is that micro surfaces form kind of terrain and can cast shadows over each other especially on extreme viewing angles [@ref Heitz14].
+**Smith-Schlick model** [@ref Heitz14], [@ref Schlick94] has been chosen as implementation:
\f[\displaystyle G=\frac{\cos\theta_l \cos\theta_v}{(\cos\theta_l(1-k)+k)(\cos\theta_v(1-k)+k)}\f]
The last component \f$F\f$ shows **how much light is reflected from surface** and is called **Fresnel's factor**.
The rest amount of radiance might be absorbed or refracted by material.
The most accurate expression of it is pretty complicate for calculation so that there is a variety of approximations.
-The good one with less computation efforts is **Schlick's implementation** @ref ref_Schlick94 "[Schlick94]":
+The good one with less computation efforts is **Schlick's implementation** [@ref Schlick94]:
\f[F=F_0+(1-F_0)(1-\cos\theta_{vh})^5\f]
**Index of Refraction** \f$IOR\f$ shows the proportion between light speed in vacuum (or even in air) and in material.
The reference value of \f$IOR\f$ for plastic is **1.5**, and this value can be considered as default for all unknown dielectrics.
In practice this parameter controls reflectance ability of material.
-Also it should be remembered that this approximation produces poor results with large \f$IOR\f$ values so that it is recommended to be kept in range of \f$[1, 3]\f$ in order to get plausible Fresnel's factor @ref ref_Lazanyi05 "[Lazanyi05]", @ref ref_Lagarde13 "[Lagarde13]".
+Also it should be remembered that this approximation produces poor results with large \f$IOR\f$ values so that it is recommended to be kept in range of \f$[1, 3]\f$ in order to get plausible Fresnel's factor [@ref Lazanyi05], [@ref Lagarde13].
This formula might be further propagated onto metals by using \f$F_0\f$ measured specifically for certain metal.
It can be considered as some kind of a 'color' of metal and can be stored as albedo parameter \f$c\f$.
And the final step of defining Fresnel's factor formula is mixing all this \f$F_0\f$ using metallic parameter \f$m\f$ (**metalness**):
Where \f$M\f$ is a number of sources, \f$l_j\f$ is a direction and \f$L_i^{direct}\f$ is an intensity related to this direction.
\f$direct\f$ label means that illumination has been computed directly from sources.
The BRDF can be used directly without any calculation problems.
-The only exception might be \f$k\f$ in \f$G\f$ factor - it is recommended to be equal \f$\frac{(r+1)^2}{8}\f$ in order to get more pleasant results @ref ref_Karis13 "[Karis13]" (that is modification mentioned in previous chapter).
+The only exception might be \f$k\f$ in \f$G\f$ factor - it is recommended to use \f$ k = (r+1)^2 / 8 \f$ in order to get more pleasant results [@ref Karis13] (that is modification mentioned in previous chapter).
And actually it is enough to finally see something.
There will be correct visualization with assumption of complete dark environment and absence of other points influence.
It is called **local illumination**. Based on this name there is also a global or **indirect illumination** and that is the rest of integral:
Anyway even with all optimization techniques this algorithm continues to require too much calculations.
Good visual results require noticeable number of samples and using this approach for every point in real time rendering becomes unrealistic.
The way to avoid these enormous calculations is doing them beforehand somehow.
-The first trick on the way to this is split the sum separating environment light component @ref ref_Karis13 "[Karis13]":
+The first trick on the way to this is split the sum separating environment light component [@ref Karis13]:
\f[L_{indirect}^s \approx \frac{1}{N} \sum_{i=1}^N \frac{f_s(v, l_i) L_i^{indirect}(l_i) \cos\theta_{l_i}}{p(v, l_i)} \approx \left( \frac{1}{N} \sum_{i=1}^N L_i^{indirect}(l_i) \right) \left( \frac{1}{N} \sum_{i=1}^N \frac{f_s(v, l_i) \cos\theta_{l_i}}{p(v, l_i)} \right)\f]
Frankly speaking \f$D(h)\f$ is called normal distribution but cannot be directly used as hemisphere distribution.
Originally it is statistical factor used to define total area of micro faces \f$\mathrm{d}A_h\f$
-whose normals lie withing given infinitesimal solid angle \f$\mathrm{d}h\f$ centered on \f$h\f$ direction using the original small enough area of macro surface \f$\mathrm{d}A\f$ @ref ref_Walter07 "[Walter07]":
+whose normals lie withing given infinitesimal solid angle \f$\mathrm{d}h\f$ centered on \f$h\f$ direction using the original small enough area of macro surface \f$\mathrm{d}A\f$ [@ref Walter07]:
\f[dA_h = D(h)\,\mathrm{d}h\, \mathrm{d}A\f]
\f$p(\phi_h)\f$ is unnecessary to be calculated analytically.
The fact of independency from \f$\phi\f$ is enough to figure out that this coordinate is uniformly distributed.
-Anyway the \f$F(\theta_h)\f$ is next step @ref ref_Cao15 "[Cao15]":
+Anyway the \f$F(\theta_h)\f$ is next step [@ref Cao15]:
\f[F(\theta_h) = \int\limits_0^{\theta_h} \frac{2 \alpha^2 \cos\theta'_h\sin\theta'_h}{(\cos^2\theta'_h(\alpha^2-1) + 1)^2}\, \mathrm{d}\theta'_h = \int\limits_{\theta_h}^0 \frac{2 \alpha^2}{(\cos^2\theta'_h(\alpha^2-1) + 1)^2}\, \mathrm{d}(\cos^2\theta'_h) = \frac{\alpha^2}{\alpha^2-1}\int\limits_0^{\theta_h} \mathrm{d}\frac{1}{\cos^2\theta'_h(\alpha^2-1)+1} =\f]
In this coordinate system reflecting of \f$v\f$ relative to \f$h\f$ is just doubling \f$\theta_{vh}\f$ and Jacobian of it is equal to \f$\frac{1}{2}\f$.
In series of transform the Jacobians are multiplied so that currently \f$|J_T| = \frac{1}{2}\sin\theta_{vh}\f$.
And the final step is inverse transform to Cartesian coordinate system with \f$|J_T| = (\sin\theta_{vl})^{-1} = (\sin2\theta_{vh})^{-1}\f$.
-Combining this all together the following expression is obtained for reflection transform Jacobian @ref ref_Schutte18 "[Schutte18]":
+Combining this all together the following expression is obtained for reflection transform Jacobian [@ref Schutte18]:
\f[|J_T| = \frac{\sin\theta_{vh}}{2\sin2\theta_{vh}} = \frac{\sin\theta_{vh}}{4\sin\theta_{vh}\cos\theta_{vh}} = \frac{1}{4\cos\theta_{vh}}\f]
First of all the \f$\cos\theta_v\f$ and \f$\phi\f$ can be just excluded.
In that way \f$v\f$ is considered to be equal to \f$n\f$.
Of course this approach produces an error and affects the final result.
-It can be fixed more or less by \f$\cos\theta_{l_i}\f$ weighting @ref ref_Karis13 "[Karis13]":
+It can be fixed more or less by \f$\cos\theta_{l_i}\f$ weighting [@ref Karis13]:
\f[\frac{1}{N} \sum_{i=1}^N L_i^{indirect}(l_i) \cos\theta_{l_i}\f]
It can be achieved using mip levels of origin \f$L_i^{indirect}\f$ whose pixels of one level is exact 4 averaged pixels from previous one.
Also mip levels generation is build in most common graphic API so there are no problems with it.
But first of all the area covered by one sample is needed to be found.
-And that can be done as @ref ref_Colbert07 "[Colbert07]":
+And that can be done as [@ref Colbert07]:
\f[\Omega_s = \frac{1}{N\,p(l)} = \frac{4\cos\theta_{vh}}{ND\cos\theta_h}\f]
\f[\int\limits_S y_i^j(l)\, y_{i'}^{j'}(l)\, \mathrm{d}l = \begin{cases} 1 & \quad i,j = i',j' \\ 0 & \quad \mathrm{otherwise}\end{cases}\f]
-Function basis with such traits is known and is described by following formulas @ref ref_Guy18 "[Guy18]":
+Function basis with such traits is known and is described by following formulas [@ref Guy18]:
\f[y_i^{j > 0}(\theta, \phi) = \sqrt{2}K_i^j\cos(j\phi)P_i^j(\cos\theta)\f]
\f[y_i^{j<0}(\theta, \phi) = \sqrt{2}K_i^j\sin(j\phi)P_i^{|j|}(\cos\theta)\f]
Resulted expression can be considered as convolution in terms of spherical functions where \f$L_i^{indirect}(l)\f$ is target and \f$\overline{\cos}\theta_l\f$ is core.
This integral may seem independent but in fact hemisphere is oriented related to \f$n\f$ therefore \f$\overline{\cos}\theta_l\f$ depends on it too and became a kind of 'oriented' version of cosine.
That is pretty tricky and explanation about meaning of convolution on sphere is out of scope of this paper.
-Fact that this is convolution analogue related to \f$n\f$ is enough for now @ref ref_Aguilar17 "[Aguilar17]", @ref ref_Ramamoorthi01 "[Ramamoorthi01]".
+Fact that this is convolution analogue related to \f$n\f$ is enough for now [@ref Aguilar17], [@ref Ramamoorthi01].
The goal of looking at integral from this angle is using of convolution's trait allowing to compute decomposition using just only coefficients of function and core.
\f$\overline{\cos}\theta_l\f$ is independent from \f$\phi_l\f$ and in case of such radial symmetric cores the resulting coefficients boil down to following formula:
Starting from about the third \f$c_i\f$ the coefficients become less and less valuable so that only couple of them is enough in order to approximate \f$\overline{\cos}\theta\f$ with appropriate accuracy.
The same principle is true for convolution too because its coefficients are multiplied by \f$c_i\f$.
So there is no need to use more than even three bands (\f$i = 0, 1, 2\f$) of basis functions.
-Lets write down them all in Cartesian coordinate @ref ref_Ramamoorthi01 "[Ramamoorthi01]":
+Lets write down them all in Cartesian coordinate [@ref Ramamoorthi01]:
\f[y_0^0 = \frac{1}{2}\sqrt{\frac{1}{\pi}} = Y_0^0\f]
The Monte-Carlo algorithm can be applied with just uniform samples distribution without importance sampling at all.
\f$Y_i^j\f$ are used twice: in \f$L_i^j\f$ calculations and in sum after that.
So there is sense to store only squares of it.
-All tables with constants presented below [[14](#(14))]:
+All tables with constants presented below [@ref Ramamoorthi01]:
| |
|-|
@section pbr_references References
-* @anchor ref_Duvenhage13 **[Duvenhage13]**
+<table cellpadding="4">
+<tr><td valign="top">
+@anchor Duvenhage13 **[Duvenhage13]**
+</td><td>
Bernardt Duvenhage, Kadi Bouatouch, D.G. Kourie,
"Numerical Verification of Bidirectional Reflectance Distribution Functions for Physical Plausibility",
*Proceedings of the South African Institute for Computer Scientists and Information Technologists Conference*,
October 2013.
+</td></tr>
-* @anchor ref_Cook81 **[Cook81]**
+<tr><td valign="top">
+@anchor Cook81 **[Cook81]**
+</td><td>
Robert Cook, Kenneth Torrance,
"A Refectance Model for Computer Graphics",
*SIGGRAPH '81: Proceedings of the 8th annual conference on Computer graphics and interactive techniques*,
August 1981, pp. 307-316.
+</td></tr>
-* @anchor ref_Karis13 **[Karis13]**
+<tr><td valign="top">
+@anchor Karis13 **[Karis13]**
+</td><td>
Brian Karis, "Real Shading in Unreal Engine 4", *SIGGRAPH 2013 Presentation Notes*.
+</td></tr>
-* @anchor ref_Heitz14 **[Heitz14]**
+<tr><td valign="top">
+@anchor Heitz14 **[Heitz14]**
+</td><td>
Eric Heitz, "Understanding the Masking-Shadowing Function in Microfacet-Based BRDFs",
*Journal of Computer Graphics Techniques*, Vol. 3, No. 2, 2014.
+</td></tr>
-* @anchor ref_Schlick94 **[Schlick94]**
+<tr><td valign="top">
+@anchor Schlick94 **[Schlick94]**
+</td><td>
Christophe Schlick, "An inexpensive brdf model for physically-based rendering",
*Computer Graphics Forum 13*, 1994, pp. 233-246.
+</td></tr>
-* @anchor ref_Lazanyi05 **[Lazanyi05]**
+<tr><td valign="top">
+@anchor Lazanyi05 **[Lazanyi05]**
+</td><td>
Istvan Lazanyi, Lazslo Szirmay-Kalos, "Fresnel term approximations for Metals", January 2005.
+</td></tr>
-* @anchor ref_Lagarde13 **[Lagarde13]**
+<tr><td valign="top">
+@anchor Lagarde13 **[Lagarde13]**
+</td><td>
Sebastien Lagarde, "Memo on Fresnel equations",
*Blog post*: [https://seblagarde.wordpress.com/2013/04/29/memo-on-fresnel-equations/](https://seblagarde.wordpress.com/2013/04/29/memo-on-fresnel-equations/).
+</td></tr>
-* @anchor ref_Walter07 **[Walter07]**
+<tr><td valign="top">
+@anchor Walter07 **[Walter07]**
+</td><td>
Bruce Walter, Stephen Marschner, Hongsong Li, Kenneth Torrance,
"Microfacet Models for Refraction through Rough Surfaces", *Proceedings of Eurographics Symposium on Rendering*, 2007.
+</td></tr>
-* @anchor ref_Cao15 **[Cao15]**
+<tr><td valign="top">
+@anchor Cao15 **[Cao15]**
+</td><td>
Jiayin Cao, "Sampling microfacet BRDF", November 1, 2015,
*Blog post*: [https://agraphicsguy.wordpress.com/2015/11/01/sampling-microfacet-brdf/](https://agraphicsguy.wordpress.com/2015/11/01/sampling-microfacet-brdf/).
+</td></tr>
-* @anchor ref_Schutte18 **[Schutte18]**
+<tr><td valign="top">
+@anchor Schutte18 **[Schutte18]**
+</td><td>
Joe Schutte, "Sampling techniques for GGX with Smith Masking-Shadowing: Part 1", March 7, 2018,
*Blog post*: [https://schuttejoe.github.io/post/ggximportancesamplingpart1/](https://schuttejoe.github.io/post/ggximportancesamplingpart1/).
+</td></tr>
-* @anchor ref_Colbert07 **[Colbert07]**
+<tr><td valign="top">
+@anchor Colbert07 **[Colbert07]**
+</td><td>
Mark Colbert, Jaroslav Krivanek, "GPU-Based Importance Sampling", *NVIDIA GPU Gems 3*, Chapter 20, 2007.
+</td></tr>
-* @anchor ref_Guy18 **[Guy18]**
+<tr><td valign="top">
+@anchor Guy18 **[Guy18]**
+</td><td>
Romain Guy, Mathias Agopian, "Physically Based Rendering in Filament", *Part of Google's Filament project documentation*:
[https://google.github.io/filament/](https://google.github.io/filament/Filament.md.html)
+</td></tr>
-* @anchor ref_Aguilar17 **[Aguilar17]**
+<tr><td valign="top">
+@anchor Aguilar17 **[Aguilar17]**
+</td><td>
Orlando Aguilar, "Spherical Harmonics", *Blog post*:
[http://orlandoaguilar.github.io/sh/spherical/harmonics/irradiance/map/2017/02/12/SphericalHarmonics.html](http://orlandoaguilar.github.io/sh/spherical/harmonics/irradiance/map/2017/02/12/SphericalHarmonics.html)
+</td></tr>
-* @anchor ref_Ramamoorthi01 **[Ramamoorthi01]**
+<tr><td valign="top">
+@anchor Ramamoorthi01 **[Ramamoorthi01]**
+</td><td>
Ravi Ramamoorthi, Pat Hanrahan, "An Efficient Representation for Irradiance Environment Maps",
*SIGGRAPH '01: Proceedings of the 28th annual conference on Computer graphics and interactive techniques*, August 2001, pp. 497-500
+</td></tr>
+
+</table>
projects / occt.git / blobdiff