pbr

Simple implementation of a PBR Renderer using opengl 4.3. This project aims to get an understanding of the Cook-Torrance BRDF.

Requirements

This project requires a GPU with OpenGL 4.3+ support.

The following dependencies depend on cgo. To make them work under Windows a compatible version of mingw is necessary. Information can be found here. In my case I used x86_64-7.2.0-posix-seh-rt_v5-rev1. After installing the right version of mingw you can continue by installing the dependencies that follow next.

This project depends on glfw for creating a window and providing a rendering context, go-gl/gl for providing bindings to OpenGL, go-gl/mathgl provides vector and matrix math for OpenGL, nuklear is a gui framework on top of opengl (here using my fork that specifically supports glfw 3.2 with OpenGL 4.3), hdr is a loader for high dynamic range images that can have light intensities greater than 1 for all color channels.

go get -u github.com/go-gl/glfw/v3.2/glfw       // window and event handling
go get -u github.com/go-gl/gl/v4.3-core/gl      // opengl api
go get -u github.com/go-gl/mathgl/mgl32         // opengl vector math
go get -u github.com/adrianderstroff/nuklear/nk // gui framework
go get -u github.com/mdouchement/hdr            // hdr image loader

After getting all dependencies the project should work without any errors.

Theory

PBR

PBR stands for Physically Based Rendering and describes rendering techniques that model how light interacts with a material or medium in a physically plausible manner.

An important component of PBR is the microfacet model. Basically each surface can be described as microscopic perfect mirrors (microfacets). Depending on how the microfacets are aligned to each other, different kinds of surfaces can be modelled. A microfacet is described by its normal h. Are the normals of all microfacets parallel to each other, then the surface is completetly smooth and behaves like a perfect mirror. Are however the microfacet normals distributed in all directions then the surface is really rough as light is scattered in all directions.

Another important property of PBR techniques is that materials have to obey the conservation of energy, meaning that reflected light has to have the same or less energy than prior to the intersection with the material.

Now one possible model to render PBR is the reflectance equation:

Here L₀ is the sum of reflected radiance of a small patch of the surface in direction of the viewer v.

The Ω represents the hemisphere from where light can hit the surface patch. The hemisphere can be thought of as a unit half sphere and light coming from any point of that half sphere towards the surface patch.

The function f(l, v) is called the bidirectional reflectance distribution function (short BRDF). It returns a value between 0 and 1 and describes how much of the light's radiance L_i coming from l is reflected in the viewer's direction v. As the BRDF is a distribution, the sum of all function values for all combinations of l and v have to add up to 1.

The dot product (n ⋅ l) describes how much of the light illuminates the surface patch. If the light rays hit the surface patch from above, where the direction of the light rays is parallel to the surface patch normal, the patch is most lit. The more the direction of the light rays is tilted the less illuminated the surface patch will be with no light hitting the surface patch when the light direction is perpendicular to the surface patch normal.

Lastly the dω_i describes a small patch on the hemisphere from where the light ray is coming.

Radiometric Quantities

The following information is taken from the pbr-book, for more in-depth information check their website.

PBR bases it's theory on radiometric quantities to describe the "brightness" of a light when interacting with different materials. As brightness is not a property that can be physically described radiance and irradiance are used instead.

Let's start with Light Energy first. A light source emits photons of different wavelength. Each wavelength λ has a specific energy Q = hc / λ, with h being Planck's constant and c the speed of light.

Next based on the Light Energy, we are now interested how much energy is emited per time. This can be done by measuring how much energy passes through a region. This quantity is called the Light Flux Φ and is simply the Light Energy differentiated by time t, or in formula Φ = dQ / dt.

If we now also take area that is illuminated by the light source we can calculate how much photons per time instance hit or path through this area A. This quantity is called Irradiance E and describes the density of the Light Flux. The irradiance is calculated as E = dΦ / dA or E = d² Q / (dt dA). The density of Light Flux leaving an area A is sometimes referred to as Radiant Exitance.

To define Radiance we first have to define the Solid Angle. In 2D we can take an object and project it onto a unit circle by tracing two lines from the center of the circle to both "ends" of the shape. We then measure the arc length of the shapes projection which is the angle (in radians). If the whole circle is covered we have an angle of 2π. The Solid Angle is an extension to a unit sphere. Now we project an object onto the surface of the unit sphere getting a specific area covering the sphere. The Solid angle is measured in steradians (sr). The full sphere has a Solid Angle of 4π while a hemisphere, which is a half sphere, has a Solid Angle of 2π. The Solid Angle can be represented by a vector with a direction and a magnitude that represents the area. We will assume that we will observe an infinitisimally small area so the Solid Angle will simply be a normalized vector ω indicating a direction relative to the center of the unit sphere.

With the concept we can describe Radiance as the Irradiance with respect to a solid angle ω. Thus the Radiance L can be calculated as L = dE' / dω or L = d³ Q / (dt dA' dω). It's important to note, that Radiance measures the Irradiance with respect to the area A' which is A projected on the plane P. The plane P is orthogonal to the Solid Angle ω, or in other words ω coincides with the normal of P.

Cook-Torrance BRDF

Most of this part is taken from https://learnopengl.com/PBR/Theory. This tutorial and the following ones on the website focus on an approach that simplifies to evaluates the integral of the Reflectance Equation in a precomputation step to get decent fps.

However the idea of this project is to have an implementation that helps understanding the Cook-Torrance BRDF and also is easy to translate into my raytracing project, where it won't be possible to evaluate the integral ahead of time. Thus all computations will be carried out as is while still trying to follow the theory of the learnopengl website.

The BRDF

The Cook-Torrance BRDF consists of two terms, a diffuse Lambert term and a specular Cook-Torrance term f(l,v) = k_d f_lambert + k_s f_{cook-torrance}. The coefficients k_d and k_s have to add up to 1 to obey the law of conservation of energy. Here the coefficents are vectors, so they have to add up 1 componentwise.

Diffuse Term

The diffuse part is constant and describes ideal diffuse material that scatters light in all directions of the hemisphere evenly. Thus the diffuse part is f_lambert = c / π. Here c is the color the surface or sometimes called the albedo.

The division of π comes from the fact, that we have to adhere to the conservation of energy. A detailed explanation can be found in Rory's blogpost http://www.rorydriscoll.com/2009/01/25/energy-conservation-in-games/. In short we want to make sure that the inequality

holds true. To make things easier, we are only taking the diffuse part of the BRDF into account and also keeping the view vector v constant, we only integrate over the outgoing rays l. As c_d and L_i are constant in terms of the outgoing ray l they can be written in front of the integral. Then L_i can be divided on both sides of the inequality. So now we just need to integrate the cosine term over the hemisphere. Since its hard to integrate over the hemisphere we can instead integrate over the halfsphere in polar coordinates using two integrals with φ = [0,2π] and θ = [0,π/2]. After solving the integral we end up with π c_d <= 1. The surface color c is defined in the range (0,0,0) to (1,1,1) thus we need to divide by π to fullfill the inequality and thus to obey the conservation of energy.

Specular Term

The specular term is a bit more complicated. It consists of three functions D, F, G and a normalization factor as shown below:

TODO: understand the normalization factor. Check the link http://www.pbr-book.org/3ed-2018/Reflection_Models/Microfacet_Models.html for potential explanation.

In the Disney paper, they mention that the normalization comes from microfacet derivation (https://disney-animation.s3.amazonaws.com/library/s2012_pbs_disney_brdf_notes_v2.pdf).

In the following all three functions are being discussed.

Normal Distribution Function

The normal distribution function statistically approximates the distribution of microfacet normals with respect to the surface normal depending on the surface's roughness α. For a completely rough surface (α = 1.0) the microfacet normals are completely randomly displaced in all directions making the surface completely diffuse. On the other hand, a completetly smooth surface (α = 0.0) has all microfacet normals aligned to the surface normal, making it a perfect mirror. A property of the normal distribution function is that all function values have to add up to 1, since there cannot be more light reflected than received.

The used normal distribution function is called the Trowbridge-Reitz GGX. It takes the normal n of the surface, the halfway vector h that is calculated as h=(l+v)/||(l+v)||, so the vector between the normalized direction to the light l and the normalized direction to the viewer v. The parameter α describes the roughness of the surface, with α=0 being completely smooth and α=1 being completely rough.

Below are results of the simple pbr model, that uses a direct light source and the same parameters for the whole mesh. Only the roughness was varied while the other parameters were kept static. For smaller values of α the surface shows a small highlight while the rest of the surface is black. This is because the microfacets are more aligned towards the surface normal behaving more like a mirror.

Higher values of α however are randomly distributed with respect to the surface normal thus the light is reflected in all directions. Because the light is reflected in different directions, the overall luminance is lower compared to a smoother surface resulting in the grayish surface. For α=1 the light is evenly scattered in all directions resulting in a perfect diffuse color.

http://www.reedbeta.com/blog/hows-the-ndf-really-defined/

To results can be found in the project cmd/pbr-test/ to play around with all parameters.

Geometry Function G

The geometry function also utilizes the microfacet model to statistically determine if the light reaches the viewer. Thereby two cases have to be taken into consideration. The first case determines if the light gets trapped in the surface, the second case determines if the reflected ray gets obstructed by a microfacet and thus not reaches the viewer. In both cases the function dependends on the angle between the surface normal n and the normalized light direction l and the angle between n and the normalized view direction v respectively as well as a roughness k.

Depending on which approach (either direct lighting or image based lighting) different values of the roughness k are being used. To model the geometry term a combination of GGX and Schlick-Beckmann was employed. For this model the following roughness values k_direct for direct lighting and k_ibl for image based lighting were used.

test http://www.codinglabs.net/article_physically_based_rendering_cook_torrance.aspx

Fresnel Reflection F

Acknowledgement

Obj Models

The models bunny.obj and dragon.obj are popular reconstructed models of scans taken from the Stanford 3D Scanning repository.

Rusted Iron Texture

The textures in assets/textures/material1/ are all except for ao.png taken from freepbr.

Gun Model and Textures

The gun.obj and textures in assets/images/textures/material/gun are the work of Andrew Maximum and are taken from https://www.artstation.com/artwork/3k2.