A personal annotations of Veach's thesis (21)

Bidirectional path tracing

Bidirectional path tracing connects vertices. The vertices are generated from 1. a light source, 2. an eye. My question is how these vertices are connected, this is rather an implementation question. If we connect all the generated vertices, this is combinatorial explosion and we can not handle this. Daniel answered me that 1. make two paths a pair (one comes from a light, the other comes from an eye), 2. connect vertices only from the paired paths. This is an efficient implementation.

Figure 1. Connect vertices in Bidirectional path tracing

The number of generated rays are the computation cost, but how many contributions we can get differs depends on how to connect them as shown in Figure 1. This figure is a simplified version (no occlusion at all). But, we could have many contribution by generating smaller number of rays.

Overview idea of multiple importance sampling method

This paper has a lot of interesting ideas. I will summarize them after Chapter 9.

If we could sample infinitely in all the directions in the environment, but our computational resource is usually limited. Therefore, we use several different sampling methods. Each sampling method has advantages and disadvantages. Multiple importance sampling method is: how to combine them and exploit all the benefits. After the combination, Veach showed it is possible to stay unbias. This is great. Also this is not only for the light transport method, but this is general.

The proposed sampling methods are: sampling method that awares the light source direction, sampling method based on BRDF. If we know where the light comes from, or which direction has no light, we could sample them more efficiently. For example, it is useless to sample the direction that has no light. These sample doesn't affect the result. BRDF gives us a hint, most of the light comes in a specific direction. Another possibility is visibility.

Thanks to Daniel, Leo, and Carsten.

Comments

Why A^{T}A is invertible? (2) Linear Algebra

Why A^{T}A has the inverse Let me explain why A^{T}A has the inverse, if the columns of A are independent. First, if a matrix is n by n, and all the columns are independent, then this is a square full rank matrix. Therefore, there is the inverse. So, the problem is when A is a m by n, rectangle matrix. Strang's explanation is based on null space. Null space and column space are the fundamental of the linear algebra. This explanation is simple and clear. However, when I was a University student, I did not recall the explanation of the null space in my linear algebra class. Maybe I was careless. I regret that... Explanation based on null space This explanation is based on Strang's book. Column space and null space are the main characters. Let's start with this explanation. Assume x where x is in the null space of A . The matrices ( A^{T} A ) and A share the null space as the following: This means, if x is in the null space of A , x is also in the n...

Gauss's quote for positive, negative, and imaginary number

Recently I watched the following great videos about imaginary numbers by Welch Labs. https://youtu.be/T647CGsuOVU?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF I like this article about naming of math by Kalid Azad. https://betterexplained.com/articles/learning-tip-idea-name/ Both articles mentioned about Gauss, who suggested to use other names of positive, negative, and imaginary numbers. Gauss wrote these names are wrong and that is one of the reason people didn't get why negative times negative is positive, or, pure positive imaginary times pure positive imaginary is negative real number. I made a few videos about explaining why -1 * -1 = +1, too. Explanation: why -1 * -1 = +1 by pattern https://youtu.be/uD7JRdAzKP8 Explanation: why -1 * -1 = +1 by climbing a mountain https://youtu.be/uD7JRdAzKP8 But actually Gauss's insight is much powerful. The original is in the Gauß, Werke, Bd. 2, S. 178 . Hätte man +1, -1, √-1) nicht positiv, negative, imaginäre (oder gar um...

Why parallelogram area is |ad-bc|?

Here is my question. The area of parallelogram is the difference of these two rectangles (red rectangle - blue rectangle). This is not intuitive for me. If you also think it is not so intuitive, you might interested in my slides. I try to explain this for hight school students. Slides: A bit intuitive (for me) explanation of area of parallelogram (to my site, external link) .

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