Mtg 20/26: Thu-20-Mar-2025

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Reflection Models

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Hello.
So we're at meeting 20 today, of 26
any questions or concerns You
okay, I'm
in the base color parameter, therefore we perform linear
interpolation between The column F zero for dielectrics and the
base color for metals. Using the metallic parameter as weight, F
zero equals mix F zero base color, metallic. For a metallic
value of zero, we get the dielectric F zero. And for a
metallic value we
Okay, so I want to talk about reflection models. So we asked
you look at the First two sections of chapter Nine And
so in pbrt, where Are reflection computations Performed in
so this the answer that I was Looking for from the Book,
reflection coordinate system you
is So if you find that you're not happy with the way things
got marked by The system. Let me know when I'll review them. You
so why is cosine weighted hemisphere sampling useful in
diffuse reflection computations?
Was it because they could be directly used? Was that the
answer? I wasn't sure if that's the answer you're looking for or
not. They could
also, like, improve the error without
additional competition.
Yeah. I so
I was looking for something Like it reduces error without
additional computation so
it, so reduces error without increased computational cost.
So you can see in there's an interactive
there's, well, it's not interactive so much. There's one
with and without the cosine weighted sampling, and you can
see that the one with looks better.
That's that weird gargoyle looking thing. Yes, I
How would you describe material with the property if you choose
a point on the surface and rotate it around its normal axis
at that point, the distribution of light reflected at that
if you choose a point on the surface and rotate this surface
around its normal access axis at that point, distribution of
light reflected at that point does not change.
Isotropic, Yes, I
so What's the other way we could describe a surface that doesn't
have that property?
Pitch Black. No what the distribution would not changed.
There's no lighting. I
don't know if I'm saying it right, but anisotropic
somewhere.
Yeah, I
so here is an example of an isotropic i
So an isotropic surface,
something with a matte paint and maybe A black chalkboard. I
uh, velvet And the moon surface. Yeah. So
another one is Cat size.
So these were things that reflected, like right back at
the light or the camera. Was that the definite or retro
reflective?
Yeah.
So would a mirror be an example, or is that too much scatter? And
it's not really directed or vocal. It's
a mirror, just as it is,
would be
a different type of reflective surface,
if it's if it's a perfect Mirror, then, then we've got
the light going off at an angle equal to the angle of incidence
and
does that make sense?
Yeah, because it compares to these ones, will send the light
right back to the surface right during rebounding. Is that
correct?
Yeah, it's collecting the incoming rays and directing them
back
so we can Have like
reflective material or reflectors on a bike or
so I have a couple videos today. I
that may be
provide a bit of context for some of the other things we've
been discussing. I
it so part of the part of them are implementation, which we
won't look at, we'll just look at the theory stuff,
and you can let me know if you think they're valuable. I
so I was looking at
blin fun reflectance reflection model,
because
Now in the section on diffuse reflection and talked about
micro facet
models.
Anyway,
that was a bit of my thinking, but let's get started on the
videos. Okay,
do you look at the gorgeous woman? And immediately you think
to yourself, no, this woman is out of my league. She's
unattainable. The average middle aged man
d squared for the radius we choose the distance you
there are three simple changes I tell all of my arthritis
patients to make. Number one, you have to start drinking more
water. You wouldn't believe happy, huh?
Hello everybody. Welcome to shaders monthly. Today we talk
about the form and bling form shading models. Both models are
reflection models. This means they describe mathematically how
light is reflected at an opaque surface. Here you see the result
of the shader implementation that we will have achieved at
the end of this episode, the Fung model was published by
Bucha Fong in 1975 and later refined to the blinnfong model
by Jim Blinn two years later. Blinn Fung shading used to be
the standard in real time graphics for quite a while. For
example, it was the only model available in the OpenGL fixed
function pipeline before the introduction of programmable
shaders in 2003 from today's point of view, these models are
quite simple. Consequently, I will not only talk about the
form and bling form models in this video, but I will present
these models in a general theoretical framework that can
be easily extended to more complex reflection models. In
particular, I will introduce the rendering equation and the B
directional reflection distribution function, BRDF,
which are the fundamental tools in computer graphics to describe
what happens when light is reflected at an opaque surface.
If you are not interested in the theory, you can jump directly to
the implementation at the end of the video. To perform shading,
we need to simulate how light is emitted from light sources, how
it interacts with 3d objects that are made of different
materials, and how it is finally observed by our eyes or the
sensor of a camera and creates an image of the world. So what
is light? Light is a quantized electromagnetic wave. It is
quantized because it is made of particles called photons, which
are packets of energy that travel as waves to space. And
light travels extremely fast, with about 300 million meters
per second. The wavelength of the visible range is
approximately between 380 and 770 nanometers. Visible light
that is monochromatic, which means that it only contains a
single wavelength, corresponds to a speckled color from violet,
starting at 380 nanometers over blue, cyan, green, yellow,
orange, and ending at Red, is 770 nanometers. Visible light is
just one form of electromagnetic radiation. Gamma radiation has
the shortest wavelength and carries the highest energy per
photon. Gamma radiation is produced, for example, in a
nuclear explosion. Then we have X rays, which also pass through
a material. For example, we all know X rays from medical
applications, ultraviolet, infrared, microwaves, which we
all know well because they heat our food in the microwave oven,
and finally, radio waves with the longest wavelengths, which
we use for radio and TV broadcast and wireless
communication, light Sources often emit a wide spectrum of
different wavelengths. In particular, white light is a
superposition of many wavelengths. Here you see the
spectral power distribution of daylight. This figure shows the
spectrum of the standardized CIE Illumina P 65 that can be used
as a reference for the so called White Point, which we will see
later. So daylight is composed out of all these spectral
colors. We can witness this in nature when we observe rainbows.
Rainbows are created when rain drops split the sunlight into
its individual spectral colors. What happens when light
interacts with objects? Let's assume light is emitted from a
light source in a homogeneous medium, such as air. Light is
traveling along straight lines through the world. Some light
rays may hit the eye directly. Others are reflected from an
object surface towards the eye. When a light ray hits the
surface, the photons are either reflected, transmitted or
absorbed. The energy of absorbed photons is converted into other
forms of energy. Typically, this is heat dependent on the
material. Photons with certain wavelengths are absorbed, which
means that light changes its spectrum and consequently its
perceived color. For example, this tea port appears red under
white light because most photons that are not in the range of the
red spectrum are absorbed. The process of reflection can repeat
many times. In a scene, for example, this ray hits the Blue
cup and changes its color because of absorption. The
remaining light is reflected and hits the red teapot. The
spectrum is further reduced at this surface, and the remaining
part is reflected in the direction of the eye. If the
light rays reach the eye, receptors on the retina are
activated and an image is formed in the brain, there are two
systems of light sensory cells in humans. The first system
consists of rods that react very sensitive to light, but do not
produce color vision. The second system consists of cones, which
are the color receptors. There are L, M and S cones, which
react to light at long, medium and short wavelengths. The
diagram shows the curves for normalized absorption of the
cones over the wavelengths. The Center for blue, sensitive s
cones is at 420 nanometers. For Green, sensitive M, cones at 534
nanometers, and for red, sensitive L, cones at 564
nanometers. Because there are only three types of cones, the
human visual system cannot resolve the real spectral power
distribution of light. Instead, the true distribution is sampled
very sparsely by the response of the cones. Consequently, human
observers can be tricked to some extent, different spectral power
distributions can produce the same perceived color in low
light situations, the cones are not sensitive enough, and only
the rods are working. That is why we do not perceive color at
night, as you have already seen in previous episodes of shaders
monthly. We typically use the RGB color model as the output of
our fragment shaders. This means different colors are created by
the additive mixture of three primary colors, red, green and
blue. So for red, we have the weighting factors, 100, which is
the first dimension of this color space. For Green, the
weights are 010, which is the second dimension. And for blue,
the weights are 001, which is the third dimension. The
additive mixture of red and green gives 110, resulting in
yellow. The additive mixture of red and blue gives 101,
resulting in magenta. The additive mixture of green and
blue gives 011, resulting inside if all three weights are equal,
this results in a shade of gray. When using arbitrary RGB
weighting factors in the range from zero to one, many different
colors can be created. The use of three primary colors is to
some extent, motivated by the three code types in the human
visual system. However, until now, these RGB weights have no
reconnection to quantitative values in the physical world. To
establish this connection, we have to talk about the CIE RGB
color space. This color space was defined by the CIE in 1931
and is based on experiments with human participants. The
participants were shown a target color shown on the left here,
and had the task to recreate the same color perception by
shooting the additive mixture of three lines of 700 nanometers,
541.1 nanometers. And 435.8 nanometers. So the CIE
experiments used real, physical red, green and blue,
monochromatic light sources of a defined wavelength. The question
was, if all possible colors could be reproduced by additive
mixture of these three lights. The experiment showed, yes,
three colors are sufficient, but some weights had to be negative.
So what does this mean? How are negative weights created? The
trick is simple. Light can be subtracted from the right side
by adding light to the left side. So if the participants use
additive light on the left side to match the two colors. This
was counted as a negative weight. Here you see the
resulting color matching functions for red, green and
blue. So for example, if we want to create a monochromatic bluish
sign, spectral color with 480 nanometers, we have to use the
RGB weights of minus 0.05, for red, 0.04 for green and 0.15 for
blue. To represent any spectral power distribution in the CIE
RGB color space, we can multiply it with the color matching
functions and integrate over the complete spectrum. This results
in three scalar RGB values. So the CIE RGB color space defines
a three dimensional space of all perceivable colors. This space
is visualized here in a special two dimension called the CIE
chromaticity diagram. Here on the border, we see the
monochromatic spectral colors which enclose the space of all
possible colors. The colors that can be created with positive
weights are always in this triangle, spared by the three
primary colors at 700 nanometers, 546.1
nanometers and 435.8 nanometers. For colors outside of this
triangle, negative weights must be used. Next, we introduce the
sRGB color space, which is different from CIE RGB. SRGB is
the current standard for monitors, websites and images
without an explicit color profile. Therefore, if we write
color values from our fragment shader into the frame buffer and
display the result in a standard consumer monitor. The expected
shader output is sRGB. The RGB values are in the range from
zero to one. SRGB uses different primary colors. Here you see the
color gamut of sRGB. Consequently, the range of
displayable colors is smaller compared to CIE RGB, the
standardized CIE illuminant D 65 defines a white point for sRGB.
Fortunately, we do not lose a connection to physical
quantities when using sRGB instead of CIE RGB. If gamma
correction is performed beforehand, there is a linear
transformation from s RGB to CIE RGB. What do I mean by gamma
correction? Here you see the function to decode a color
channel from s RGB to radiometric linear s RGB values
in the figure, this function is plotted as a black curve. The
colors that you see down here are s RGB values. So these color
steps are linear in s RGB space. Interestingly, as you are
probably watching this video in something that is close to s
RGB, these values should appear approximately linear to you.
However, if you measured the brightness with the radiometric
light meter, it would not be linear. The reason is that our
visual perception is not linear. The human visual system is
better at distinguishing darker intensities than lighter ones,
which is considered in CS RGB encoding. However, in rendering,
it is important to work in a linear space, because otherwise
we cannot simply add contributions from different
light sources. Therefore, it is important that we apply gamma
decoding and encoding in our shader. We could use the exact
function that is denoted here, or we can approximate it. In the
figure, the red curve is a simpler gamma decoding function
which decodes the color value by raising it to the power of 2.2
ok. This concludes what we need to know about representing
colors in our shader. The main take home message is that we do
not need to simulate the Light Transport for every wavelength.
Computing the Light Transport for three primary colors is
sufficient for most applications. Next we talk about
the rendering equation. The rendering equation was
introduced in a SIGGRAPH paper by James Cargill in 1986 this is
more than 10 years after the form model was published. The
rendering equation describes what happens when light is
reflected at an opaque surface. Solving the rendering equation
is very challenging, and many different approaches try to
approximate the exact solution. As you see on this slide, there
are several terms, such as a solid angle, irradiance and
radians, that we need to introduce before coming back to
the rendering equation, let's start with a solid angle. In
school, we all learned about angles in a 2d plane. In 2d if
we want to know the angle that is covered by an object seen
from a certain point, we place a circle around this point and
project the 2d object onto the circle. Then the angle is
defined as the arc length s over the radius r of the circle.
Although the angle is a dimensionless quantity, it is
specified in the unit radian. Obviously, for unit circle with
a radius of one, we can drop the division by the radius for a
half revolution in the unit circle, the arc length is pi,
which is in fact, the definition of pi. Consequently, for a full
revolution in the unit circle, the angle is two pi, the
equivalent of an angle in 3d space is called a solid angle.
If we want to know the solid angle that is covered by an
object seen from a certain point, we place a sphere around
this point and project a 3d object onto the sphere, then the
solid angle is defined as the size of the projected area s
over the radius squared. The solid angle is a dimensionless
quantity, but it is specified in the unit steer radian. If we
want to compute the solid angle, we can split the whole solid
angle into infinitesimal solid angle pieces denoted by d omega
here, and integrate over the whole domain. One way to solve
this integral is to parameterize the solid angle in spherical
coordinates. These coordinates are the polar angle, theta and
the other muscle angle, phi, d omega is equal to an
infinitesimal surface patch ds over the radius squared. We set
the Radius to one and compute the area of a surface patch, ds
by the multiplication of d theta and d phi now we see in the
figure that the size of a d phi step depends on theta. When
theta equals zero, the contribution is zero, and for
theta equal to pi over two, the contribution is a full D phi
step. We can take this into account and use a factor sinus
theta, to scale the d phi step accordingly. Rearranging the
factors gives a final result of the root. Two quick examples, if
we want to compute the solid angle of an entire sphere, the
integration limits of phi go from zero to two pi, and of
theta from zero to pi, we can solve this interval
analytically, and the result is a solid angle of four pi. Is the
radians. If we do the same for a hemisphere, the result is two
pi. Okay, let's define the other terms that we need for the
rendering equation. First we have the radial flux, which is a
measure of power. It is equal to radian energy per time. The unit
is watt, which is joule per second. We can think of it this
way, every photon carries energy that is equal to Planck's
constant h times the speed of light, C, divided by the
wavelength lambda. So the radian flux is the sum of the photon
energies that are emitted per time in graphics. The radian
flux can be used as a typical parameter for a point light
source that emits light uniformly in all directions.
Another quantity that is often used in graphics is radian
intensity. It is equal to radian flux per solid angle, the unit
is one plus the radian in graphics, the radian intensity
is often used if a light source does not radiate equally in all
directions. For example, a spotlight, radian intensity is
the sum of the photon energies that are emitted per time in
solid angle, as I have tried to illustrate in this figure here,
let's consider an example that relates the two quantities,
gradient flux and radiant intensity that we have already
introduced. If we have a point light that emits light uniformly
in all directions with a certain flux, then the radiant intensity
can be computed by dividing the total flux by the total solid
angle. We know from the previous slide that the total solid angle
for a sphere around the point light is four pi irradiance. So
for a point light, we get radiant intensity is equal to
radian flux divided by four pi
another important quantity is irradiance. It is equal to
radian flux per area. In other words, irradiance is the sum of
the photon energies that are received by a surface per time
and per area. The unit is watt per square meter. The received
flux can come from all directions within the hemisphere
above the surface. As an example, we compute the
irradiance produced by a point light source for this small blue
surface, EA of infinitesimal size, irradiance is flux per
area. We know the total amount of flux that is emitted by the
point light in all directions, but how much of this flux is
received by the blue surface? Well, we have already computed
the rated intensity, which was flux per solid angle. We put
this result in the equation for the irradiance. Now we need to
compute the solid angle. The solid angle d omega is equal to
the projected area, ds divided by radius squared. For the
radius, we choose the distance from the blue surface to the
point light, then we don't have to project the surface onto the
sphere, at least not if we assume that it is of
infinitesimal size, but we have to consider that the surface
area that is seen from the direction of the point light
appears shorter if theta is the angle between the surface normal
and the incident light direction. Then the required
foreshortening factor is cosine theta. This means, if the
incident light direction is perpendicular to the surface,
theta is zero, cosine theta is one and the visible area is
largest. But if the light comes from the side, we have to
consider the cosine theta for shortening factor. If we put
this result into the equation for the irradiance, we see that
the factor DA is canceled the irradiance caused by a point
light is the radiant intensity of the light times cosine theta
divided by the squared distance between the surface and the
light. In the case of a point light, we can further replace
the radiant intensity with the radiant flux divided by four pi,
as we have computed in the last example. To summarize this
result, firstly, the irradiance caused by a point light
decreases quadratically with distance. Secondly, the
irradiance is largest for a light direction perpendicular to
the surface, and decreases with the cosine of the angle of
incidence. The last quantity that we need for the rendering
equation is radians. If we observe a surface patch and want
to measure its brightness, we are not interested in the total
flux per area, but only in the fraction that is going in a
certain direction. For example, we are only interested in the
part that is going in the direction of our eye or camera.
So we are interested in the flux per area per solid angle.
However, when we say area here, we mean the area in flux
direction. If a surface area is not oriented in flux direction.
Its contribution in terms of area, is smaller. We have to
multiply the surface area with cosine theta to get the
contributing area in the flux direction. Radians is flux per
solid angle per projected area. An important property of radians
is that it does not change with viewing distance, at least if we
do not consider participating media such as fog or smoke. When
we move a camera or our eye towards the surface, the solid
angle per pixel gets larger. But on the other hand, the area that
we observe per pixel gets smaller. These two effects
cancel each other out, and the radiance does not change. This
is something that we observe also in our everyday lives.
Let's say we look at a wall and move towards it, then the
brightness does not change. How can we compute the irradiance
produced by incoming radians? Irradiance is flux per area. So
received flux can come from all directions within the hemisphere
above the surface. Radians is flux per solid angle per
projected area. If we put the definition of irradiance in this
equation, we get radians is equal to de over D, omega times
cosine, theta. Rearranging the equation gives de is equal to
radians times cosine, theta times d, omega. De is a
contribution to the irradiance by the radians received from
certain solid angle, D, omega. To compute the complete
irradiance, E, we have to integrate over the hemisphere.
The irradiance for the surface patch is given by the integral
over the incoming radians, l times cosine theta, where theta
is the angle of incidence. Let's remember this result and go back
to the rendering equation.
The rendering equation calculates the outgoing
radiance, l, o, in the direction v, for surface patch at location
X with normal n, the outgoing radiance is the sum of two
terms. The first term is the emitted radiance, L, E in the
direction v. This term is only larger than zero if the surface
is a light source that produces radian flux itself, the second
term is integral over the hemisphere. It is integrating
over the contributions of all incoming radiances, Li from the
hemisphere above the surface patch each incoming radiance Li
from direction l generates an irradiance contribution. De This
is the same relationship between radians and the resulting
irradiance contribution that we have discussed a moment ago on
the previous slide. Again, we have the cosine theta factor,
where theta is the angle between the surface normal and the
direction of the incoming radians. Consequently, radians
with the larger angle theta contributes less to the
irradiance of the surface patch. Each infinitesimal irradiance
contribution de is now multiplied with the BRDF, which
is shown for B directional reflection distribution
function. This is a function that describes the material
property of the surface patch. It defines how much radiance is
emitted by the patch in direction v, if it receives an
irradiance contribution from direction L. Of course, the vrdf
and the radiance values are also dependent on the wavelengths, so
we have to evaluate the rendering equation for each one
of our three RGB primary colors. The rendering equation describes
the full light transport in a, c. This means that an outgoing
radiance, l, o of a surface patch contributes this radians
as incoming radiance, Li and another surface patch. So all
surface patches in the scene are connected via the radians. They
are contributing to each other. This is why the rendering
equation is very hard to solve. In practice, the BRDF describes
the angle dependent spectral reflection factor for a surface
by the ratio of the outgoing radians, lo and the incoming
irradiance, E, both the incoming direction L and outgoing
direction d could be parameterized with spherical
coordinates, theta and phi. Consequently, the BRDF is a four
dimensional function, if you do not consider the dependency on
the wavelength by specifying this 40 function, the reflection
properties of a surface are described precisely. The BRDF of
a material could be measured and stored in a 40 table. However,
this would require a significant amount of memory. Furthermore,
we cannot edit materials easily with such a representation.
Consequently, for most cases, parametric BRDF models are used.
For example, the form model and the bling form model, which we
want to implement today in our shaders, are parametric BRDF
models. Here you see different materials. The metal material on
the left is almost a perfect mirror. For such a material, the
angle of incidence is equal to the angle of reflection for the
second metal sphere, the material is not perfectly
smooth. For such a material, we assume that there is some
variation in the orientation of the micro facets in graphics,
the term micro facets is used for tiny surface patches that
are much smaller than a pixel. Each micro facet behaves like a
perfect mirror. It is assumed that the orientation of the
micro facets follow some distribution around the
macroscopic surface normal so statistically, these small
variations in orientation cause the reflected light to be
distributed around the macroscopic reflection
direction, if the roughness of the surface increases, the
scattering of light around the reflection direction becomes
larger. For metals, photons are either reflected or they enter
the material and are absorbed. This is dependent on the
wavelength, therefore, the reflected light might be
colored. Examples are copper or gold. The behavior that I have
just described is true for metals, for dielectric
materials, the reflection model is a bit different as for
metals, a certain fraction of photons are reflected at the
surface, and we call this part the specular reflection. For
dielectric materials, the specular reflection does not
depend on the wavelengths, so it is not colored. Photons that are
not reflected enter the material here, they are either absorbed
or they are scattered in random directions below the surface. At
some point, they might come out of the surface again and exit in
random directions. We call this part the diffuse reflection. The
diffuse reflection depends on the wavelength and consequently
is colored for dielectrics. Okay. Now we are ready for the
form BRDF. The form BRDF has two parts, one for diffuse and one
for specular reflection. The Diffuse part is a simple
constant denoted by KD. It does not depend on the light
direction L or the view direction v. Note that the light
direction L is defined to go towards the light. So the yellow
arrow that goes from the light towards the surface is minus l.
In this figure, the reflected direction R has the same angle
to the surface normal n as the light direction. So if I change
the light direction here, the reflection direction changes
accordingly. The specular part is a dot product of the view
direction v and the reflection direction r, raised to the power
of NS and weighted with a constant Ks. If I increase Ks,
here you see that the specular part is appearing as a lobe
around the reflected direction, the dot product of the view
direction v and the reflection direction r is equal to the
cosine of the angle between these vectors. If the view
direction matches the reflection direction, the contribution of
the specular lobe is largest and reduces when the angle gets
larger, the exponent NS controls the fall off of the specular
lobe. For a higher exponent, the fall off is faster, as we have
learned. A higher exponent corresponds to a smoother
surface. The exponent Ns is typically referred to as
shininess. Okay, great. Let's implement the form blef, as we
have seen, the exact evaluation of the rendering equation is
difficult in computationally demanding the equation becomes
much simpler if the light exchange between surfaces is
neglected and only the so called direct light is taken into
account, which is emitted from a number of light sources. For
each light source, we compute the irradiance and multiply it
with the BRDF. Then we can sum up the individual light source
contributions. Or if the emitted light is zero and we have only
one light, we simply have outgoing radians equals the BRDF
multiplied with the irradiance. Let's put in the form BRDF and
the irradiance of a point light, the cosine theta term can be
replaced with the dot product of the light direction and the
surface normal. Our implementation starts from the
total stock example from episode number three. The link is
provided in the video description. I will use the GS
and composer at GSN minus lip.org as a shader editor here,
but you can use any other shader editor as well. Links to XM i
What'd you think of that? I
thought it was cool.
Yeah,
it's very thorough. I kind of want to go through the video
again on my own time, just to sort of go over it a little bit
slower. But like some of his things, like when he
demonstrated how, like the palm shooters work, like that last
Demonstration he did, I thought that was really interesting.
I You didn't mention the garage here, though.
This is your brain when you're struggling to focus, and this is
your brain when you're in the zone, but getting in the zone
and staying there,
therefore we perform linear interpolation between the
current F Zero for dielectrics and the base color for metal.
Now.
Music.
Hello, everybody. Welcome to Sheila's Mansfield Today we talk
about the cook Torrance micro facet, BRDF. This is a
reflection model that was introduced by Robert Cook and
Kenneth Torrance in 1981 micro facets are imaginary, tiny
surface patches that are much smaller than an output pixel.
This model allows describing the behavior of the macroscopic
surface patch Mathematically, the micro facet model
approximates the real world physical material properties
more closely than the form or plane form reflection model,
which we have discussed in episode number four, blin form
shading used to be the standard in real time rendering because
it is fast to compute. Nowadays, the computing power has
increased, and the computer graphics industry has almost
completely moved to Physically Based Rendering, or PBR for
short. Today, the my professor model is used everywhere in
offline as well as real time rendering. Therefore, it is
important for us to know the underlying theory. In a tutorial
presented at C graph in 2012 brand Burley, who works for Walt
Disney Animation Studios, presented a user friendly
interface to the micro facet, BRDF. In this interface, all
parameters are in the interval from zero to one. This approach
is a current de facto industry standard to represent a
material, and is known as the metallic roughness workflow. As
the name suggests, the two most important parameters are the
metallic and the roughness parameters for metals such as
gold, silver, copper and so on. The metallic parameter is one
for dielectric materials, which are also called non metals, such
as plastic, wood and rubber. The metallic parameter is zero. The
roughness parameter controls the distribution of the orientations
of the micro facets. The micro facets are not real geometry,
but are only evaluated statistically. In the micro
facet model, each tiny micro facet surface behaves like a
perfect mirror for very smooth polished surfaces, the roughness
is zero. All micro facets point in the same direction, and the
material behaves like a perfect mirror. If the roughness
parameter is increased, the orientations of the micro facets
are more random, and light is collected from larger range of
incoming light directions in Disney's model, the metallic
parameter could also be set to a fractional value, for example,
0.5 This would mean that the material behaves with 50% like
metal and with the other 50% like a dielectric material,
which is physically not plausible. Therefore we should
not use values other than zero and one for the metallic
parameter. If we stay within the physical world, it is much
easier to produce a realistic material that works well in
different lighting conditions. However, in textures or mid
maps, the resolution is limited, and we might have a mix of
materials within a single pixel. In these situations, a
fractional value for the metallic parameter makes perfect
sense. Another parameter is called base color. It is an RGB
vector for dielectrics. It contains the RGB values for the
albedo, which is the amount of diffuse reflection in the
interval from zero to one for metal this parameter contains
the RGB values for the Fresnel reflectance, which we have not
talked about yet, but it will be introduced in a minute. Then
there is an additional parameter called reflectance, which is
used only for die electrics. It also contains a Fresnel
reflectance, but this time for dielectric materials. This
parameter is called specular in Disney's original tutorial
notes. But I like the name reflectance much more. Whatever
it is called, the parameter, controls the amount of
dielectric specular reflection. In contrast to metals, for which
the specular reflection at a perpendicular incident angle can
reach up to 100% for die electrics, it is much less. It
ranges approximately from zero to 16% the quadratic remapping
function is used to map zero to 16% to a user friendly interval
from zero to one. These very few parameters are sufficient to
control our basic micro facet. BRDF, Disney's full model has
several other parameters for subsurface scattering, an
additional clear coat layer and other advanced features that we
are not discussing today. Now let's dive more deeply into the
theory. Our microface BRDF is used in combination with the
rendering equation. As a refresher, here is a slide from
episode number four. The rendering equation calculates
the outgoing radians, l, o, in the direction v, for surface
patch application x, with normal n, the outgoing radiance is a
sum of two terms. The first term is the emitted radiance, l, e,
in the direction v. This term is only larger than zero. As a
surface is a light source that produces some radian flux
itself. The second term is the integral over the complete solid
angle of the hemisphere above the surface. We integrate over
the contributions from all incoming radiances, Li from the
hemisphere above the surface. Patch each incoming radiance, Li
from direction l generates an irradiance contribution de, each
infinitesimal irradiance contribution de, is multiplied
by the BRDF, which is short for V directional reflection
distribution function. The BRDF defines how much radiance is
emitted by the surface patch in direction v, if it receives an
irradiance contribution from direction L. This way the BRDF
describes the material properties of the surface patch.
This is where we need to insert the microface BRDF that we
discussed today. Here you see the equation for the complete
BRDF. A quick introduction of the use notation before we talk
about it, V is the outgoing view direction. L is the light
direction it points from the surface location towards the
lot. Consequently, minus L is the incoming vector from the
light to the surface. N is the surface normal. H is the halfway
vector. It has a unit vector at the half angle between the view
and the light direction. The vrdf is a function of the view
and the light direction. It has a diffuse part and a specular
part. The constant diffuse part, rho, d, divided by pi, is known
from the normalized form vrdf that we have introduced in
episode number four. For the specular part, we use the cook
tolerance microfacet model, the cook Torres microface model has
three terms, Fresnel reflectance, the normal
distribution function and the geometry term. We will introduce
these terms one by one in the following slides. Let's start
with Pranav reflectance. Here you see the image of a swimming
pool with a very flat water surface. In the bottom part of
the image, we can see the ground of the pool, but here in the
upper part, we don't see the ground, but only the reflected
sky. Why do we observe this effect? The effect occurs
because the ratio of transmitted and reflected light is not
constant. It depends on the angle of incidence and the
refractive indices of the involved materials. If we look
into the water from above, only a small part of the light from
the sky is reflected in our view direction at the water surface,
most of the light is transmitted and we see the ground of the
pool, the more we look from the side and observe the surface at
increasingly grazing angles, the larger becomes the reflected
part, the more light is reflected, the less light is
transmitted. The setup is illustrated in this figure. We
have the interface between two materials with two different
indices of refraction, eta one and eta two. In the swimming
pool example, we observed the interface between water and air
because water is an optically denser medium than air, its
index of refraction is larger. The angle of incidence of the
light is defined relative to the surface normal, and is denoted
by theta one here for a perfectly flat surface, the law
of reflection tells us that the reflected direction has the same
angle as the incoming Ray. Therefore this angle here
between the normal and the reflected direction is equal to
theta one as well. Let's have a look at the angle of the
transmitted ray, theta two, during the transition of the
light from an optically thinner to an optically denser medium,
the light ray deviates towards the normal consequently, theta
two is smaller than theta one for such a situation,
mathematically, the relationship between the angle of incidence
theta one and the angle of the transmitted ray. Theta two is
given by Snells law, eta one times sine of theta equals eta
two times sine of theta two. Given eta one, eta two and theta
one, we can solve for theta two. With theta two, we can compute
the fraction of the light that is reflected using the Fresnel
equation shown here, the transmitted fraction is then
given by one minus the reflected part.
Let's look at Fresnel reflection for dielectrics light photons
are either reflected, transmitted or absorbed with an
angle dependent relative frequency. The reflected part is
scattered on the micro face. Macroscopically, this creates a
specular part of the reflection in opaque materials, the
transmitted part is randomly deflected below the surface,
partially absorbed and emitted into random directions.
Macroscopically, this creates a diffuse part of the reflection.
The more light is reflected, the less light is transmitted, and
the diffused part becomes smaller. The likelihood of
absorption inside the medium is wavelength dependent. Therefore
the diffuse part is colored and the specular part is not. For
metals, the complete transmitted fraction is absorbed. This means
there is no diffuse reflection. For metals, the reflected part
depends on the wavelength, so the specular part is covered. On
this slide, we see the index of refraction for several
dielectric materials. The table also contains the F zero values
in percent. F zero is a reflectance for perpendicular
incidence of light when the incident angle, theta one is
equal to zero. If you insert theta one equals zero into the
Fresnel equations, all the cosine terms evaluate into one,
and the equations become much simpler, as you see here. If we
then put in 1.0 for eta one for vacuum or air, and set eta two
to 1.5 for glass, we can compute a value of 4% for F zero. This
means that at the transition from air to glass at normal
incidence, only 4% of the light is reflected and 96% is
transmitted. On this slide, we have listed the F zero values
for several metals as discussed. These are wavelengths dependent.
So we have three values for the red, green and blue channel in
the first column, the reflectance values are given in
linear space, which we can use directly in our shader code. The
second column gives the corresponding sRGB values, as we
have discussed in episode number four, it is important to perform
gamma correction when we convert between these two
representations, when we want to apply the Fresnel reflectance in
the context of the micro facet model, we need to be careful
which angle we use for theta one for a perfectly flat surface,
theta one is the angle between the incident light direction and
the normal of the surface, or because of the law of
reflection, theta one is also The angle between the view
direction and the surface normal. However, in the micro
facet model, we assume that the surface is built from tiny
surface patches that are much smaller than an output pixel.
Each micro facet behaves like a perfectly flat mirror surface.
This means we only get a light contribution in the view
direction if the normal of the micro facet is pointing exactly
in the direction of the macroscopic half wave vector.
Consequently, the Fresnel reflectance must be computed for
those micro facets which have the macroscopic halfway vector
as surface normals. Therefore, for the micro facet model, theta
one is the angle between the incident light direction and the
macroscopic halfway vector, or because of the law of
reflection, the same angle can be found between the view
direction and the macroscopic halfway vector in the paper by
cook and Torres from 1981 the Fresnel reflectance for eta one
equals one is calculated by the shown equations. We see that the
scalar product of the view direction and the macroscopic
half wave vector is used. Though these equations do not use
trigonometric functions like sine or cosine. They are
nevertheless computationally expensive. Therefore the faster
Schlick approximation is used in many implementations. The figure
shows a comparison between schliep s approximation and the
true equations for glass, for theta, one equals zero, we get
4% reflectance at a grazing angle of incidence of 90
degrees. The reflectance is 100% we see that the approximation is
not perfect for the in between angles, but it follows the true
graph reasonably well. This was the Fresnel reflectance term.
The next term is a normal distribution function. The
distribution of the micro faced normals changes depending on the
roughness of the surface. For a perfectly smooth, polished
surface, all the micro facet normals point in the same
direction as a macroscopic normal. For a rougher surface,
the micro facet model assumes that the orientations of the
micro facets follow some distribution around the
macroscopic surface normal. So statistically, these small
variations in orientation cause a reflected light to be
distributed around the macroscopic reflection
direction. If the roughness of the surface increases, the
scattering of light around the reflection direction becomes
larger. There are several suggestions for the normal
distribution function in the literature, all three options
that are shown here are parameterized by alpha, where
alpha is equal to Rp squared and RP is our user friendly
perceived roughness value in the interval from zero to one.
Furthermore, we see that all three options for the normal
distribution function depend on the scalar product of the normal
and the halfway vector, which we have also used for the specular
lobe in the blend form model in episode number four. In the
figure on the right, we compare the three options for the same
roughness value. The x axis represents the angle between the
normal and the halfway vector. The yellow curve is the GG X
distribution proposed by Walter and co authors in 2007 nowadays,
it is a predominant model because of the softer tails that
can also be observed in brdfs captured from real world
materials. The last term of the microfacet model is the geometry
term depending on the light's incident direction and the
observer's viewing direction. Shadowing and masking effects
occur at the micro facets. Blinn as well as cook and tolerance.
Assume the micro facets are V shaped and derive a geometry
factor from this model. There can be either no interference, a
masking effect for grazing view directions, or a shadowing
effect for grazing light directions. Another well known
model for the geometry term was proposed by B J Smith in 1967
the geometry term is created by multiplying two factors that are
computed with the same function, g1 the g1 function is evaluated
for the light direction and the view direction. Different g1
functions that are mentioned in the paper by Walter and co
authors are given here. The first approximation by Schlick
for the GG X variant is given by this equation. Use the graphs
for the mentioned g1 functions, bag, man, GG X and slip GG X for
different roughness values, for a roughness value of Z row, the
geometry term is one, which means it has no effect. For
larger roughness values, we get a stronger influence of this
term, which is the expected result, because masking and
shadowing becomes more likely for rougher surfaces, let's
implement the Cooke tolerance, microfacet BRDF, as we have seen
the Cooke Torres microface BRDF that we use in our specular part
is quite general. We can use different options for the three
terms. Let's take the shake approximation for the Fresnel
reflectance, the GG X normal distribution function and the
Smith geometry term with the Schlick GG x, g1 function. I use
the GSN composer and GSN minus lip.org as a shader editor, but
you can use any other shader editor as well. I
now right on time.
Any questions or thoughts about
those two videos, does He say,
like, roughness is between zero and one don't exist. Like at the
start of that second video, that part, I didn't click
it. He said, We
it's either metallic or it's not.
So we don't have objects that are half metallic and half
dielectric.
I think that's the roughness can vary continuously.
Thanks for today.

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