Advanced Computer Graphics: Difference between revisions

Retrieved from "https://wiki.davidl.me/view/Advanced_Computer_Graphics"

@@ Line 1: / Line 1: @@
-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==
+How to speed up intersection calculations during ray tracing.
+===Object Subdivision===
+Bounding volume hierarchy
+* Create a tree where objects are placed together.
+* Each node corresponds to a region covering all the objects it has
+* You want to insert in some greedy way:
+** Minimize the size of each region
+* Subtrees may overlap and are unorder
-[[Visible to::users]]
+===Spacial Subdivision===
+Octtree, kd tree
+==Radiometry==
+{{main | Wikipedia: Radiometry}}
+===Geometrical Optics===
+{{main | Wikipedia: Geometrical Optics}}
+* Light are rays which reflect, refract, and scatter
+===Solid Angle===
+* 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==
+==Monte Carlo Integration==
+Suppose we want to estimate <math>I_1 = \int_{a}^{b}f(x)dx</math>.<br>
+Then we can use <math>\hat{I_1} = \frac{b-a}{N}\sum f(X_i)</math> where <math>X_1,...,X_n \sim Uniform(a,b)</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:
+* <math>\hat{I} = \frac{1}{N} \sum \frac{f(X_i)}{p(X_i)}</math>
+===Importance Sampling===
+Suppose we can only sample from pdf <math>g(x)</math> but we want to sample from pdf <math>p(x)</math> to yield a more reliable (less variance) estimate.<br>
+Then we can sample from <math>p(x)</math> using <math>Y = F_{p}^{-1}(F_{g}(X))</math>.<br>
+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==
+===Crust Technique===
+[https://www.cs.ubc.ca/~sheffa/dgp/ppts/crust.pdf Crust Slides from Univ. BC]