warning: there will be equations

do not be afraid of them

goal of photo-realism: produce a computer-generated image that is indistinguishable from an actual photograph

today we use the same principles behind the earliest cg

rendering system : light simulation

emit from source, enter camera, bounce around scene

modeling quality \(\quad\Rightarrow\quad\) render quality

(light source, camera, materials, surfaces)

modeling light sources is relatively easy

modeling camera lens system can be complex

pinhole works surprisingly well, much easier to control

modeling materials is not difficult

light reflects off surface in all directions at different amounts

materials describe how much light bounces

we can faithfully describe almost any material as a
math function (\(\rho\)) with just a few parameters

\[\begin{array}{ll} \text{diffuse} & \rho(...) = \frac{\red R_d}{\pi} \\ \text{specular} & \rho(...) = {\red R_s} (\mathbf{h} \cdot \mathbf{n})^{\red p} \\ \end{array}\]

\[\begin{array}{rcl} \rho(...) & = & a \cdot b \cdot F(\omega_i \cdot \mathbf{h}) \\ a & = & \frac{\sqrt{({\red n_u}+1)({\red n_v}+1)}}{8\pi} \\ b & = & \frac{(\mathbf{n}\cdot\mathbf{h})^{{\red n_u}\cos^2\phi+{\red n_v}\sin^2\phi}}{(\mathbf{h}\cdot\omega_i)\max(\cos\theta_i,\cos\theta_o)} \\ F(\omega_i \cdot \mathbf{h}) & = & {\red R_s} + (1-{\red R_s})(1-(\omega_i \cdot \mathbf{h}))^5 \end{array}\]

(schlick's approximation to the fresnel equation)

modeling surfaces is straightforward

use triangles to describe any surface to arbitrary precision

using ray for light and triangle for surface,
simulation involves simple algebra

\[\begin{array}{ll} \text{ray} & P = P_o + tV \\ \text{plane} & P \cdot N + d = 0 \\ \text{ray-plane} & t = -(P_o \cdot N + d) / (V \cdot N) \\ \end{array}\]

the rendering equation puts the parts together

all photo-realistic graphics engines solve this equation

\[L_o (\mathbf{x}, \omega_o) = L_e(\mathbf{x}, \omega_o) + \int_\Omega \rho(\mathbf{x}, \omega_i, \omega_o) L_i(\mathbf{x}, \omega_i) (\omega_i \cdot \mathbf{n}) d\omega_i\]

emit (\(L_e\)), camera (\(L_o\)), bounce (\(\rho\), \(L_i\), \(L_o\))

the only thing remaining is doing the light simulation

solve the rendering equation

"what do we do now sheriff?"

"now, we render"

light simulation isn't cheap

steps toward photo-realism causes computation time to grow

for example, let's look at rendering an image
two spheres on a plane with two lights

note: following simulations are already "smart"

simple
< 1 second

shadows
1 second

mirror reflection
7 seconds

soft shadows
30 seconds

multiple bounces
32 seconds

glossy reflection
258 seconds

translucency
681 seconds

environment
776 seconds

and it only gets worse from here

adding more realism involves computing more light bounces

we could use approximations for the rendering equation
but the final image will not be accurate

the following are a few different ways
to improve render times
without changing final image

put triangles into boxes

throw darts

roll multiple unfair dice