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Classnotes for CMSC740 taught by Matthias Zwicker
Classnotes for CMSC740 taught by Matthias Zwicker (Spring 2020).<br>
This first portion of the class focuses on ray tracing (specifically, path tracing) and is based on the [https://www.pbrt.org/ PBRT book]<br>
The second portion of the class introduces deep learning approaches to computer graphics.


==Acceleration==
==Acceleration==
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==Radiometry==
==Radiometry==
{{main | Wikipedia: Radiometry}}
===Geometrical Optics===
===Geometrical Optics===
[[Wikipedia: Geometrical Optics]]
{{main | Wikipedia: Geometrical Optics}}
* Light are rays which reflect, refract, and scatter
* Light are rays which reflect, refract, and scatter
===Solid Angle===
===Solid Angle===
* Solid angle = area / radius^2 on a sphere
* Solid angle = area / radius^2 on a sphere
===Spectral Radiance===
{{main | Wikipedia: Radiance}}
* Spectral radiance is energy per time per wavelength per solid angle per area
* <math>L(t, \lambda, \omega, \mathbf{x})=\frac{d^4 Q(t, \lambda, \omega, \mathbf{x})}{dt d\lambda d\omega dA^\perp}</math>
: where energy is <math>Q(t, \lambda, \omega, \mathbf{x})</math>
===Radiance===
{{main | Wikipedia: Radiance}}
* power per solid angle per area
* <math>L(\omega, \mathbf{x}) = \frac{d^2 \Phi(\omega, \mathbf{x})}{d\omega dA^\perp}</math>
===Irradiance===
===Radiant Intensity===


==BRDF, Reflection Integral==
==BRDF, Reflection Integral==
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This is because <math>E\left[\frac{b-a}{N}\sum f(X_i)\right] = \frac{b-a}{N}\sum E[f(X_i)] = \frac{1}{N}\sum \int_{a}^{b}(b-a)f(x)(1/(b-a))dx = \int_{a}^{b}f(x)dx</math><br>
This is because <math>E\left[\frac{b-a}{N}\sum f(X_i)\right] = \frac{b-a}{N}\sum E[f(X_i)] = \frac{1}{N}\sum \int_{a}^{b}(b-a)f(x)(1/(b-a))dx = \int_{a}^{b}f(x)dx</math><br>
Note that in general, if we can sample from some distribution with pdf <math>p(x)</math> then we use the estimator:
Note that in general, if we can sample from some distribution with pdf <math>p(x)</math> then we use the estimator:
* <math>E\left[ \frac{1}{N} \sum \frac{f(X_i)}{p(X_i)} \right]</math>
* <math>\hat{I} = \frac{1}{N} \sum \frac{f(X_i)}{p(X_i)}</math>


===Importance Sampling===
===Importance Sampling===
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Then apply the above equation.
Then apply the above equation.


===Integration===
Some notes about integration
* Hemisphere: <math>d\omega = \sin \theta d\theta d\phi</math> where <math>\theta \in [0, \pi/2)</math> and <math>\phi \in [0, 2\pi)</math>
==Path Tracing==
===PDF for sampling light sources===
<math>p_{\omega}(x) = \frac{1}{\# lights} * \frac{1}{area of light} * conversion</math>
===Refractive objects===
;For mirrors:
In practice, refractive objects are handled as a distinct case.<br>
You do not need to sample a direction.
;For glass:
* Randomly sample either reflected or refracted ray with given probability.
* Typically produces a lot of noice
===Emitting surfaces===
* If a ray accidentally hits emitting surface, don't add emission
** Exception: If eye ray hits emitting surface
** Exception: If ray is generated from a refractive surface
==Advanced Sampling Techniques==
===Multiple Importance Sampling===
Weighted sampling between <math>F_a</math> <math>F_d</math>
* Take <math>N</math> samples from each technique (j=1,...,N)
* <math>F=\frac{1}{N} \sum_{j=1}^{N} \sum_{i=1}^{n} w_i(X_{i,j}) \frac{f(X_{i,j})}{p_i(X_{i,j})}</math>
** Make sure <math>\sum_{i=1}^{n}w_i(x) = 1</math>
* Weights for provable variance reduction
** Balance heuristics: <math>w_i(x)=\frac{p_i(x)}{\sum_{k=1}^{n}p_k(x)}</math>
** Power heuristics: <math>w_i(x)=\frac{p_i^2(x)}{\sum_{k=1}^{n}p_k^2(x)}</math>
===Stratified Sampling===
* Intuition: clumping of samples is bad
* Instead of canonic uniform random variables, generate variables in strata
* Also known as "Jittered sampling"
Other stratified sampling patterns
* N-rooks (Latin hypercube)
* Quasi Monte Carlo
===Bidirectional Path Tracing===
Trace path from eye and light<br>
Example:
* from eye we get path <math>z_0, z_1, z_2</math><br>
* from light we get <math>y_0, y_1</math><br>
* Then make shadow rays from every pair of z, y<br>
Path of length k with k+1 vertices
* s vertices from light, t from eye
* Path denoted <math>\bar{X}^{s,t}</math> (e.g. <math>\bar{X}^{2,3}</math>)
* We also get probability density for this path <math>p_{s,t}</math>
==Participating Media==
===Transmittance===
* Multiplicative property
** <math>T(s)=T(s_0) * T(s_1)</math>
* Beer's law <math>T(s)=e^{-sigma_t s}</math>
** For homogenous media where <math>\sigma(x) = \sigma</math> is constant
===Phase Functions===
====Henyey-Greenstein phase function====
* <math>p(\cos \theta) = \frac{1-g^2}{4\pi(1+g^2-2g\cos \theta)^{1.5}}</math>
====Properties====
* Unitless
* Reciprocity
** <math>p(\omega' \rightarrow \omega) = p(\omega \rightarrow \omega')</math>
* Energy conservation
** Integrates to 1
* Average phase angle determined by g
===Volume Rendering Equation===
====Integro-integral form====
* <math>L(\mathbf{x}, \omega) = \int_{0}^{\infty}\exp(-\int_{0}^{s'}\sigma_t(\mathbf{x}-s''\omega)ds'')S(\mathbf{x}-s'\omega, \omega)ds</math>
** <math>\exp(-\int_{0}^{s'}\sigma_t(\mathbf{x}-s''\omega)ds'')</math> is Transmittance <math>T(s')</math> due to extinction
** <math>S(\mathbf{x}-s'\omega, \omega)</math> is source (emission, in-scattering)
===Subsurface Scattering===
====BSSRDF====
bidirectional surface scattering reflectance distribution function
* <math>S(\mathbf{x}_i, \omega_i, \mathbf{x}_o, \omega_o)</math>


==Surface Reconstruction==


[[Visible to::users]]
===Crust Technique===
[https://www.cs.ubc.ca/~sheffa/dgp/ppts/crust.pdf Crust Slides from Univ. BC]
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