Brandon Wang
April 17, 2012
Material adapted from Ravi Ramamoorthi and Pat Hanrahan.

### Logistics

• HW5 due Thursday

### Today

• HW5 Questions

### Acceleration Structures

• Hopefully you have the intuition behind ray tracing now. You shoot rays from the camera, and find what intersects the ray.
• Last time, we looped through every primitive (triangle, etc) to see if they intersected. This is very slow!

Why we care

### Radiometry - What are we trying to describe?

Okay so, we've been shooting rays and computing color values.

But what is the mathematical formalization behind this, and can we describe real scenes?

Our goal is to tie together the brightness and energy bouncing around the scene and eventually hitting our eyes

This ends up with a lot of Physics-like things, and complicated terms. Hopefully, this will help you organize them...

J
J \cdot s^{-1} = W
• Radiant Intensity - Power per unit soid angle
W \cdot sr^{-1}
• Radiance - Power traveling at some point in a specified direction per foreshorteded area per unit solid angle.
W \cdot m^{-2} \cdot sr^{-1}
• Irradiance - Total power per unit area coming into or leaving a location on the surface.
W \cdot m^{-2}

### BRDF

The bidirectional reflectance distribution function

sr^{-1}

\theta_i
\omega_i
and surface normal n

f_r(\omega_i, \omega_o)=\frac{dL_r(\omega_o)}{dE_i(\omega_i)}=\frac{dL_r(\omega_o)}{L_i(\omega_i)\cos\theta_i\,d\omega_i}

wat

### BRDF

The bidirectional reflectance distribution function

How light is reflected at an opaque surface. Takes in the incoming light in

\omega_i
and outgoing light in
\omega_o
, and gives you the ratio of the radiance exiting along
\omega_o
to the irradiance incident on the surface from direction
\omega_i

### BRDF

Wait that picture looks familiar

We defined shading in HW2,3,5(,6) using pictures like these, and we said that it was a combination of lambertian and specular terms.

You have already implemented two BRDFs: the Lambertian model, and the Blinn-Phong model.

Other BRDFs include the Torrance-Sparrow model (accounting for microfacets in specular highlights), Oren-Nayar (accounting for microfacets in diffuse surfaces), and various anisotropic models

### Rendering Equation

L_o(\mathbf x, \omega) = L_e(\mathbf x, \omega) + \int_\Omega f_r(\mathbf x, \omega', \omega) L_i(\mathbf x, \omega') (-\omega' \cdot \mathbf n) d \omega'
• L_o(\mathbf x, \omega)
is the total amount of light directed outward along direction
\omega
, from a particular position
\mathbf x\,\!
• L_e(\mathbf x, \omega)
is emitted light
• \int_\Omega \cdots d\omega'
is an integral over a hemisphere of inward directions
• f_r(\mathbf x, \omega', \omega)
is the BRDF, the proportion of light reflected from
\omega'
to
\omega
at position
\mathbf x\,\!
• L_i(\mathbf x, \omega')
is light coming inward toward
\dmathbf x\,\!
from direction
\omega'
at time
t\,\!
• -\omega' \cdot \mathbf n
is the attenuation of inward light due to incident angle

### Rendering Equation (with wavelength and time)

L_o(\mathbf x, \omega, \lambda, t) = L_e(\mathbf x, \omega, \lambda, t) + \int_\Omega f_r(\mathbf x, \omega', \omega, \lambda, t) L_i(\mathbf x, \omega', \lambda, t) (-\omega' \cdot \mathbf n) d \omega'
• \lambda\,\!
is a particular wavelength of light
• t\,\!
is time
• L_o(\mathbf x, \omega, \lambda, t)
is the total amount of light of wavelength
\lambda\,\!
directed outward along direction
\omega
at time
t\,\!
, from a particular position
\mathbf x\,\!
• L_e(\mathbf x, \omega, \lambda, t)
is emitted light
• \int_\Omega \cdots d\omega'
is an integral over a hemisphere of inward directions
• f_r(\mathbf x, \omega', \omega, \lambda, t)
is the BRDF, the proportion of light reflected from
\omega'
to
\omega
at position
\mathbf x\,\!
, time
t\,\!
, and at wavelength
\lambda\,\!
• L_i(\mathbf x, \omega', \lambda, t)
is light of wavelength
\lambda\,\!
coming inward toward
\mathbf x\,\!
from direction
\omega'
at time
t\,\!
• -\omega' \cdot \mathbf n
is the attenuation of inward light due to incident angle

### Conceptual Questions

1. From a perfectly lambertian/diffuse surface, is there an equal amount of energy radiated in different directions?

No, foreshortened area falls off as cos.

2. Is there an equal amount of radiance in different directions?

Yes

1. Radiance (L) - Power traveling at some point in a specified direction per foreshorteded area per unit solid angle.
W \cdot m^{-2} \cdot sr^{-1}
2. Irradiance (E) - Total power per unit area coming into or leaving a location on the surface.
W \cdot m^{-2}
E(x) = \int_{H^2}L_i(x,\omega)\cos\theta d\omega

### So what does this formalism get us?

We actually get quite a few nice things...

### Light fields

5 or 4 dimensional function of amount of light.

x,y,z,\theta\,\phi

Basis of the Lytro camera

(video)

### Tips for the final

Review your midterm. Remember that the final is cumulative!

Remember that not all topics covered were done in projects.

Review.