epfl-archive/cs440-acg/results/homework-3/report.html

**Homework 3**

Student name: Cedric Hölzl

Sciper number: 257844

Monte Carlo Sampling (60 pts)
=============================

Tent
----
$$
p(x, y)=(p_1(x)\text{,}
		 p_1(y))
\\
p_1(t) = \begin{cases}
		1-|t|, & -1\le t\le 1\\
		0,&\text{otherwise}\\
		\end{cases}
\\
P_1(t) = \begin{cases}
		1, & t \gt 1\\
		\frac{1}{2}(t+1)^2 + 1, & 0 \le t \le 1\\
		\frac{1}{2}(t+1)^2, & -1 \le t \lt 0\\
		0, & -1 \gt t\\
		\end{cases}
\\
P_1^{-1}(t) = \begin{cases}
		\sqrt{2t}-1, & 0 \le t \lt 0.5\\
		1 - \sqrt{2(1-t)}, & 0.5 \le t \le 1\\
		\end{cases}
$$
<img src="tent.png" alt="Tent X2">


Uniform disk
------------
$$
p(x, y)= (p_1(\sqrt{x}, 2\pi y)\text{,}
		  p_2(\sqrt{x}, 2\pi y))
\\
p_i(r, \theta) = \begin{cases}
				r * \cos(\theta), & i = 1\\
				r * \sin(\theta), & i = 2\\
				\end{cases}
\\
P_i^{-1}(x, y) = \begin{cases}
		0, & x^2 + y^2 \gt 1\\
		\frac{1}{\pi}, & otherwhise\\
		\end{cases}
$$
<img src="disk.png" alt="Disk X2">

Uniform sphere
--------------
$$
p(x, y)= (p_1(\arccos(1-2x), 2\pi y)\text{,}
	      p_2(\arccos(1-2x), 2\pi y)\text{,}
		  p_3(\arccos(1-2x), 2\pi y))
\\
p_i(\theta, \phi) = \begin{cases}
				\sin(\theta)\cos(\phi), & i = 1\\
				\sin(\theta)\sin(\phi), & i = 2\\
				\cos(\theta), & i = 3\\
				\end{cases}
\\
P_i^{-1}(x, y, z) = \frac{1}{4\pi}
$$
<img src="sphere.png" alt="Sphere X2">

Uniform hemisphere
------------------
$$
p(x, y) =   (p_1(\arccos(1-x), 2\pi y)\text{,}
			 p_2(\arccos(1-x), 2\pi y)\text{,}
			 p_3(\arccos(1-x), 2\pi y))
\\
p_i(\theta, \phi) = \begin{cases}
				\sin(\theta)\cos(\phi), & i = 1\\
				\sin(\theta)\sin(\phi), & i = 2\\
				\cos(\theta), & i = 3\\
				\end{cases}
\\
P_i^{-1}(x, y, z) =  \begin{cases}
				\frac{1}{2\pi}, & z \ge 0\\
				0, & otherwhise\\
				\end{cases}
$$
<img src="hsphere.png" alt="HemiSphere X2">

Cosine hemisphere
-----------------
$$
p(x, y)= (p_1(\sqrt{x}, 2\pi y)\text{,}
		  p_2(\sqrt{x}, 2\pi y)\text{,}
		  p_3(\sqrt{x}, 2\pi y))
\\
p_i(r, \theta) = \begin{cases}
				r * \cos(\theta), & i = 1\\
				r * \sin(\theta), & i = 2\\
				\sqrt{1 - r^2(\cos^2(\theta)-\sin^2(\theta))}, & i = 3\\
				\end{cases}
\\
P_i^{-1}(x, y, z) =  \begin{cases}
				\frac{z}{2\pi}, & z \ge 0\\
				0, & otherwhise\\
				\end{cases}
$$
<img src="coshsphere.png" alt="Cosine HemiSphere X2">

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}

$$
<img src="beckmann1.png" alt="Beckmann (coef 1.0) X2">
<img src="beckmann05.png" alt="Beckmann (coef 0.5) X2">
<img src="beckmann005.png" alt="Beckmann (coef 0.05) X2">

Two simple rendering algorithms (40 pts)
========================================

Point lights
------------

Ajax bust illuminated by a point light source:
<div class="twentytwenty-container">
    <img src="ajax-simple-ref.png" alt="Reference">
    <img src="ajax-simple.png" alt="Mine">
</div>

We can see that the pictures match.

Ambient occlusion
-----------------

Ajax bust rendered using ambient occlusion:
<div class="twentytwenty-container">
    <img src="ajax-ao-ref.png" alt="Reference">
    <img src="ajax-ao.png" alt="Mine">
</div>

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.

<img src="HSW.png" alt="Cosine HemiSphere X2">


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).

<!-- Slider -->
<script src="https://ajax.googleapis.com/ajax/libs/jquery/3.5.1/jquery.min.js"></script>
<script src="../resources/jquery.event.move.js"></script>
<script src="../resources/jquery.twentytwenty.js"></script>
<link href="../resources/offcanvas.css" rel="stylesheet">
<link href="../resources/twentytwenty.css" rel="stylesheet" type="text/css" />
<script>var markdeepOptions = {onLoad: function() {$(".twentytwenty-container").twentytwenty({default_offset_pct: 0.5, move_slider_on_hover: true});} };</script>
<!-- Markdeep: -->
<script src="https://morgan3d.github.io/markdeep/latest/markdeep.min.js?" charset="utf-8"></script>
<script>window.alreadyProcessedMarkdeep||(document.body.style.visibility="visible")</script>