CS184 Section 11
Radiometry

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

Logistics

Today

Acceleration Structures

Radiosity

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...

Radiosity - Terms

BRDF

The bidirectional reflectance distribution function

sr^{-1}

L is radiance, E is irradiance,

\theta_i
is the angle made between
\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'

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'

Time?

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

Relating Radiance and Irradiance

  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

Rendering Stuff Accurately

Making stuff

(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.