**Homework 3**
Student name: Cedric Hölzl
Sciper number: 257844
Monte Carlo Sampling (60 pts)
=============================
Tent
----
$$
p(x, y)=(p_1(x)\text{,}
p_1(y))
\\
p_1(t) = \begin{cases}
1-|t|, & -1\le t\le 1\\
0,&\text{otherwise}\\
\end{cases}
\\
P_1(t) = \begin{cases}
1, & t \gt 1\\
\frac{1}{2}(t+1)^2 + 1, & 0 \le t \le 1\\
\frac{1}{2}(t+1)^2, & -1 \le t \lt 0\\
0, & -1 \gt t\\
\end{cases}
\\
P_1^{-1}(t) = \begin{cases}
\sqrt{2t}-1, & 0 \le t \lt 0.5\\
1 - \sqrt{2(1-t)}, & 0.5 \le t \le 1\\
\end{cases}
$$
Uniform disk
------------
$$
p(x, y)= (p_1(\sqrt{x}, 2\pi y)\text{,}
p_2(\sqrt{x}, 2\pi y))
\\
p_i(r, \theta) = \begin{cases}
r * \cos(\theta), & i = 1\\
r * \sin(\theta), & i = 2\\
\end{cases}
\\
P_i^{-1}(x, y) = \begin{cases}
0, & x^2 + y^2 \gt 1\\
\frac{1}{\pi}, & otherwhise\\
\end{cases}
$$
Uniform sphere
--------------
$$
p(x, y)= (p_1(\arccos(1-2x), 2\pi y)\text{,}
p_2(\arccos(1-2x), 2\pi y)\text{,}
p_3(\arccos(1-2x), 2\pi y))
\\
p_i(\theta, \phi) = \begin{cases}
\sin(\theta)\cos(\phi), & i = 1\\
\sin(\theta)\sin(\phi), & i = 2\\
\cos(\theta), & i = 3\\
\end{cases}
\\
P_i^{-1}(x, y, z) = \frac{1}{4\pi}
$$
Uniform hemisphere
------------------
$$
p(x, y) = (p_1(\arccos(1-x), 2\pi y)\text{,}
p_2(\arccos(1-x), 2\pi y)\text{,}
p_3(\arccos(1-x), 2\pi y))
\\
p_i(\theta, \phi) = \begin{cases}
\sin(\theta)\cos(\phi), & i = 1\\
\sin(\theta)\sin(\phi), & i = 2\\
\cos(\theta), & i = 3\\
\end{cases}
\\
P_i^{-1}(x, y, z) = \begin{cases}
\frac{1}{2\pi}, & z \ge 0\\
0, & otherwhise\\
\end{cases}
$$
Cosine hemisphere
-----------------
$$
p(x, y)= (p_1(\sqrt{x}, 2\pi y)\text{,}
p_2(\sqrt{x}, 2\pi y)\text{,}
p_3(\sqrt{x}, 2\pi y))
\\
p_i(r, \theta) = \begin{cases}
r * \cos(\theta), & i = 1\\
r * \sin(\theta), & i = 2\\
\sqrt{1 - r^2(\cos^2(\theta)-\sin^2(\theta))}, & i = 3\\
\end{cases}
\\
P_i^{-1}(x, y, z) = \begin{cases}
\frac{z}{2\pi}, & z \ge 0\\
0, & otherwhise\\
\end{cases}
$$
Beckmann distribution
---------------------
We start with de provided formulas and can build the PDF formula from it (thanks to the mapping hint):
$$
D(\theta, \phi) = \frac{1}{2\pi}\ \cdot\ \frac{2 e^{\frac{-\tan^2{\theta}}{\alpha^2}}}{\alpha^2 \cos^3 \theta}
\\
P_i^{-1}(x, y, z, a) = \frac{e^{\frac{-((x^2 + y^2)/(z^2))}{a^2}}}{\pi a^2 z^3}
$$
To find the warp method, we multiply it by $2\pi$ (equivalent to integrating over $\phi$) and integrate it (as seen in the explanation) to find the following mappings for $\phi$ and $\theta$:
$$
\theta = \arctan(\sqrt{(-a^2 \log(1-y))})
\\
\phi = 2\pi x
$$
So finaly, we can build our warp using those new mappings:
$$
p(x, y, a) = (p_1(\arctan(\sqrt{(-a^2 \log(1-y))}), 2\pi x)\text{,}\\
p_2(\arctan(\sqrt{(-a^2 \log(1-y))}), 2\pi x)\text{,}\\
p_3(\arctan(\sqrt{(-a^2 \log(1-y))}), 2\pi x))
\\
p_i(\theta, \phi) = \begin{cases}
\sin(\theta)\cos(\phi), & i = 1\\
\sin(\theta)\sin(\phi), & i = 2\\
\cos(\theta), & i = 3\\
\end{cases}
$$
Two simple rendering algorithms (40 pts)
========================================
Point lights
------------
Ajax bust illuminated by a point light source:
We can see that the pictures match.
Ambient occlusion
-----------------
Ajax bust rendered using ambient occlusion:
We can see that the pictures match.
Hacker Points
-------------
I tried implementing the Hierarchical Sample Warping, I couldn't find the issue with the PDF function. Also the matrix to be used is currently entered by hand and not parsed from a file. We use a single array containing the Mipmap until we have only a 2x2 image. I couldnt make it function for a 512x512 image(matrix). Unfortunately I didn't have more time to debug and resolve those issues.
Feedback
========
* Time taken: 6h designing, had some trouble with some PDF functions, especialy for the HSW. Coding from this was relativly quick and took 4h. Finaly testing was fast since I just needed to run executables.
* Sources or documents to help with HSW would have been great, I tried to look on the internet and in the book of the course but found not a great amount of information.
* The hardest part was the math. The integrators were easier to implement than expected thanks to the warp functions we made.
* I enjoyed seing the points build a grid with disformation(warp).
Uniform disk
------------
$$
p(x, y)= (p_1(\sqrt{x}, 2\pi y)\text{,}
p_2(\sqrt{x}, 2\pi y))
\\
p_i(r, \theta) = \begin{cases}
r * \cos(\theta), & i = 1\\
r * \sin(\theta), & i = 2\\
\end{cases}
\\
P_i^{-1}(x, y) = \begin{cases}
0, & x^2 + y^2 \gt 1\\
\frac{1}{\pi}, & otherwhise\\
\end{cases}
$$
Uniform sphere
--------------
$$
p(x, y)= (p_1(\arccos(1-2x), 2\pi y)\text{,}
p_2(\arccos(1-2x), 2\pi y)\text{,}
p_3(\arccos(1-2x), 2\pi y))
\\
p_i(\theta, \phi) = \begin{cases}
\sin(\theta)\cos(\phi), & i = 1\\
\sin(\theta)\sin(\phi), & i = 2\\
\cos(\theta), & i = 3\\
\end{cases}
\\
P_i^{-1}(x, y, z) = \frac{1}{4\pi}
$$
Uniform hemisphere
------------------
$$
p(x, y) = (p_1(\arccos(1-x), 2\pi y)\text{,}
p_2(\arccos(1-x), 2\pi y)\text{,}
p_3(\arccos(1-x), 2\pi y))
\\
p_i(\theta, \phi) = \begin{cases}
\sin(\theta)\cos(\phi), & i = 1\\
\sin(\theta)\sin(\phi), & i = 2\\
\cos(\theta), & i = 3\\
\end{cases}
\\
P_i^{-1}(x, y, z) = \begin{cases}
\frac{1}{2\pi}, & z \ge 0\\
0, & otherwhise\\
\end{cases}
$$
Cosine hemisphere
-----------------
$$
p(x, y)= (p_1(\sqrt{x}, 2\pi y)\text{,}
p_2(\sqrt{x}, 2\pi y)\text{,}
p_3(\sqrt{x}, 2\pi y))
\\
p_i(r, \theta) = \begin{cases}
r * \cos(\theta), & i = 1\\
r * \sin(\theta), & i = 2\\
\sqrt{1 - r^2(\cos^2(\theta)-\sin^2(\theta))}, & i = 3\\
\end{cases}
\\
P_i^{-1}(x, y, z) = \begin{cases}
\frac{z}{2\pi}, & z \ge 0\\
0, & otherwhise\\
\end{cases}
$$
Beckmann distribution
---------------------
We start with de provided formulas and can build the PDF formula from it (thanks to the mapping hint):
$$
D(\theta, \phi) = \frac{1}{2\pi}\ \cdot\ \frac{2 e^{\frac{-\tan^2{\theta}}{\alpha^2}}}{\alpha^2 \cos^3 \theta}
\\
P_i^{-1}(x, y, z, a) = \frac{e^{\frac{-((x^2 + y^2)/(z^2))}{a^2}}}{\pi a^2 z^3}
$$
To find the warp method, we multiply it by $2\pi$ (equivalent to integrating over $\phi$) and integrate it (as seen in the explanation) to find the following mappings for $\phi$ and $\theta$:
$$
\theta = \arctan(\sqrt{(-a^2 \log(1-y))})
\\
\phi = 2\pi x
$$
So finaly, we can build our warp using those new mappings:
$$
p(x, y, a) = (p_1(\arctan(\sqrt{(-a^2 \log(1-y))}), 2\pi x)\text{,}\\
p_2(\arctan(\sqrt{(-a^2 \log(1-y))}), 2\pi x)\text{,}\\
p_3(\arctan(\sqrt{(-a^2 \log(1-y))}), 2\pi x))
\\
p_i(\theta, \phi) = \begin{cases}
\sin(\theta)\cos(\phi), & i = 1\\
\sin(\theta)\sin(\phi), & i = 2\\
\cos(\theta), & i = 3\\
\end{cases}
$$
Two simple rendering algorithms (40 pts)
========================================
Point lights
------------
Ajax bust illuminated by a point light source:
Feedback
========
* Time taken: 6h designing, had some trouble with some PDF functions, especialy for the HSW. Coding from this was relativly quick and took 4h. Finaly testing was fast since I just needed to run executables.
* Sources or documents to help with HSW would have been great, I tried to look on the internet and in the book of the course but found not a great amount of information.
* The hardest part was the math. The integrators were easier to implement than expected thanks to the warp functions we made.
* I enjoyed seing the points build a grid with disformation(warp).