TU Wien Rendering #3 - BRDF models, The Rendering Equation

now there's another fundamental question which is uh what makes the difference between different materials and the the other question is how do we model it well different materials reflect incoming light to different directions and they absorb different amounts of it in different wavelengths uh that's the answer we are going to talk a lot about this but this is an example these are different material models so the specular uh in the specular case there is one incoming Direction and there's one possible outgoing Direction that's it this is what always happen this is for instance a mirror because I see exactly the reflection of myself there's no other thing that I see in the mirror but for a diffused surface for one incoming Direction there is many possible outcomes in many possible directions and this gives a diffused surface we're going to see examples of that and it WR uh it writes spread please forget this term let's call this glossy instead because this is what it is this is like the mixture of these two so these are some basic material models that we are going to see in our renderers later on uh now to formalize this in a way let's create a function that's a probability density function with three parameters so this is a three-dimensional function uh one variable is the incoming light Direction the other variable is a point on the surface and what I'm interested in is how much light is Flowing out from this point in different directions now uh a bit more formalized this F is going to be this function I'm interested in the incoming Direction and the point in space this two is what I have and I would be interested in the outgoing directions what is the probability of different outgoing directions and this is how we will write it formally Omega is an incoming direction X is a point in space that we choose and Omega Prime is the outgoing Direction and this we call the brdf or uh bir directional reflectance distribution function so this is a very complicated name for something that's very simple brdf now uh what about materials that don't reflect all incoming light there are some materials that transmit some of it so for instance glass water uh gemstones and such well it could look like that and here above you can see some vdfs and Below things because it's not reflected it's transmitted there are some materials that let them through so here's an example well everyone had seen windows and things like that uh well the question is why like just a physical question why are these objects transparent sorry they transmit the light yes but what is happening here exactly so just some uh physical intricacy that uh the most fundamental question you know what is inside of an at and the best answer is nothing because an atom is 99% empty space there is the nucleus which is uh the whole atom is the size for instance of a football field if you imagine that then the nucleus is a small piece of rice in the middle of the football field that's the nucleus and the electrons are also very small things like small Rises which are orbiting this nucleus from very far away like the side sides of the football field and in between there's nothing absolutely nothing so the more interesting question would be why is not everything transparent I mean there's absolutely nothing in there that would divert the or absorb the light right everything just everything should go through why is not everything transparent not only glass but everything and the reason is uh absorption so these electrons are orbiting the nucleus and what essentially is happening is that electrons uh can absorb photons are if you imagine uh light is not Rays or not waves but particles then photon is the basic particle of life so electrons they absorb photons and if they do they go from an inner uh orbit like a lower energy level they jump to a higher energy level because it's basically you after lunch you eat something you get more energetic you get more jumpy so it jumps to an outer orbit uh from the nucleus it's a bit further away so it absorbs the light so the light doesn't go through so this is why most things are not transparent but the question is why is then glass transparent and the answer is that these orbits these different uh places around the nucleus they are so far apart that in the visible light spectrum if the electrons absorb a photon they don't get enough energy to jump to the next orbit and this is why all this is why most of the light is going through these glass materials and the interesting thing is that this is not always the case this is the case for visible light spectrum there's a another Spectrum which is absorbed so if you have a spectrum that give that is a higher energy Spectrum then it may give enough energy for this electron to jump to a different orbit and uh we can easily find out uh what spectrum it is because for instance we use glass for a number of different beneficial things well for instance you cannot get sunburn if you are inside of the house and you have your windows closed and we are wearing sunglasses in order to protect our eyes from something so uh is there someone who tells me uh what this spectrum is that exactly uh just a bit louder ultraviolet exactly so ultraviolet is a is a spectrum with a higher amount of energy yes uh and if you absorb it then this jump is possible so this is why it is absorbed so just some uh physical intricacies so lights may get reflected if we have a material that most of the time reflects light then we call it the brdf the r is the interesting part that's the reflection and if it transmittance is possible with the material model we have the btdf which is the bir directional transmittance distribution function and uh as an umbrella term for both of these this is basically the whatever term is bsdf so by direction of scattering distribution function I'm not saying this because this is lots of fun I'm saying this because you're going to find this these terms in the literature all the time so bsdf is basically things that reflect and things that transmit okay what are the properties of brds and after this we will suddenly uh put together something beautiful very rapidly so there is ham Hol reciprocity

it means that the direction of the ray of light can be reversed what it means mathematically is that I can swap the incoming and outgoing directions and I'm going to get the same probabilities so the probability of going here to there is the same probability is coming from there to here if I look at things uh from both sides I will get the same probabilities so that's often useful uh in physics positivity this is uh saf explanatory well uh it cannot be less than zero a probability uh for every outgoing Direction there is some positive probability or there is zero that's it nothing else is really possible so formally this is how it looks like and it makes some mathematicians awfully

happy and there's energy conservation perhaps the most important property uh an object May reflect or absorb incoming light but it is impossible that more is coming out than the incoming amount well uh obviously we have light sources and things like that but we are talking about strictly material models uh so this means that if I integrate this function for all possible incoming directions uh then I get if I take into

consideration light attenuation that we have talked about this is uh why it is so hot at noon and cold at night then I'm going to get one or less and this is because if it equals one then this means that this kind of material uh reflects all light that comes in and if it's less than one then this means that some amount of light is absorbed okay we are almost there at the

rendering equation generally what we are going to do is that we pick a point x and this uh direction is going to point towards the camera or my eye this is basically means the same thing it's just an abstraction and what I'm going to be doing is I'm going to sum up all the possible incoming directions where light can come to this point and I'm interested in how much is reflected towards my direction and let's not forget that objects can emit light themselves and we will also compute this reflected amount of light so uh just intuition light exiting the surface towards my eye is the amount that it emits itself if it's a light source and the amount that it reflects from the incoming light that comes from its surroundings and this is how we can formally write this with this beautiful uh integral equation let's see let's tear it uh a and see uh what means what this is the emitted light so this is light from point x going towards my eye how much is it well the amount that is in point x emitted towards Mya if it's a light source like that one then I definitely have this amount and there is an amount that I reflect that is reflected let's see what's going on this is what I just told you and again and this is the integration this is the interesting part so I am in I'm integrating this Omega Prime so all possible incoming directions what you have seen the hemisphere on the previous image hemisphere uh means basically half the one half of a sphere uh we are integrating over a hemisphere not over a full sphere because if we take into consideration the cosine if the light comes from above that cosine 0° is one and as I rotate this light source around this point then this cosine will get to 90° so from here to there and the cosine of 90° is zero therefore there's going to be no throughput if it comes from that direction and if I have something that's higher that would be negative we don't deal with these cases so this is why I'm integrating over a hemisphere so some light is coming to this point in different directions and what I'm interesting interested in is how much is this of this light is reflected towards my eye this is multiplied by uh the incoming radians there is the brdf and light attenuation that's it uh this is still a bit difficult elusive so first we are going to train ourselves like bodybuilders on smaller weights so we're going to create an easier version of this uh because apparently this is terribly difficult to solve if you take a look and if you would sit down and try to solve it for a difficult scene where you have uh objects and geometries and different brdf different material models you will find that this is impossible to solve analytically and one of the first problems is yes uh this is equation is just for one point right so we are looking at one point and then we want to calculate exactly yes and here comes the catch so I'm interested in how much light is going towards my air from this point how much is it well it depends if I turn on other light sources then this point is going to be brighter because the radiance coming out of this point depends on its surroundings is the window open are the curtain pulled or not so X this point x depends on this other point Y for instance all other points then we can say okay let's not compute this x first let's compute this y Point instead first because then I will know X okay but this y also depends on X because how bright uh light is on the other side of the room also depends on how bright it is in this side of the room so there is some uh recursion in there and it's if you don't think out of the box this is impossible to solve because you don't know where to start this integral is hopeless to uh compute in closed form because there may be shapes and different objects in there and this will make integration immensely difficult the integral is infinite dimensional later you will see that if I compute one bounce this x that I have been talking about that's okay but I need to compute multiple bounces I need to start tracing Rays from the camera and see how much light is entering the lens of the camera but one bounce is not enough is two bounces enough so after the X I continue somewhere else is this enough say something it's not enough okay but I think maybe three is enough is three enough it's not enough okay well you guys are very picky okay uh is 10 bounces enough okay why not because there's still some amount of energy left that if I would continue this light path I would encounter other objects and I don't have any knowledge of that we need to compute an infinite amount of bounces even 1,000 is not enough so and this rendering equation is going to be 1. 1 bounds and if I want to compute the second bounce that's going to be embedded there's another in integral equation another rendering equation and this goes on infinitely this is the biggest equation in the whole universe it's impossible to solve and it is often singular I will later show you why so even if you would want to integrate it you couldn't so uh this is horribly difficult this seems impossible and apparently at this point we cannot solve it so this is the end of the course and we have an impossible problem there is no reason even to try and goodbye see you never because there's never going to be any more lectures but in order to understand what's going on uh first we're going to put together a simpler uh version of this equation that we can understand and we can work our way up uh there there's another formulation of the rendering equation I'm not going to deal with this too much uh you can imagine uh this other version as moving points around so uh there is a light source in P3 and there is the sensor at P0 and this is one example light path and what I'm doing is I'm not stopping at one point and integrating all possible incoming directions because this is what I did with the original formulation what I do is I compute one light path I compute how much light is going through I add that to the sensor and then I move this P2 around I move it a bit I compute the new light path how much is going through I move this P2 around again so imagine this moving everywhere and imagine also P1 moving everywhere so all these points are moving everywhere and I compute the contribution of this light source to the sensor so this is another kind of integration I'm not going to go through this uh what is interesting is that there is a geometry term in there and uh this describ the geometric relation of different points and light attenuation between them uh I'm not going to deal with this too much I just put it here because uh if you're interested then chew your way through it uh in lature they often write it this way