A personal annotations of Veach's thesis (20) pp.310-321

p.310 Special cases for short subpaths

There are zero subpaths vertices and one path vertices to generate subpaths. If I draw a picture, there is one line from the lens to the light source. So, I could not distinguish the difference. Again, I asked my friend/specialist. It is quite convenient to have such friends, but, I should study more, otherwise, these friends will be bothered by me. The correct picture is shown in Figure 1.

Figure 1. Short subpath

The differences are:

Zero subpath vertices: The sample is done from the lens only, the probability is only related with lens, and it coincidentally hit to a light source.
One subpath vertices: The sample is done on the light source with sample density probability only. Then this vertex is connected to the lens.

The path's generation probability is not the same, therefore, the contribution is also not the same.

Acknowledgements
Thanks to Leo and Carsten.

p.321 Implementation

It's a bit details, but I have a question in the following equation.

I computed this as follows.

I may have a mistake. Actually, today I write 3 - 4 = 1 and my matrix becomes unsolvable when I try to get a row reduced echelon form. I was astonished when I compare my calculation and the output of the octave. But, still where is N_0?

Comments

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) .

Geometric Multiplicity: eignvectors (2)

If eigenvectors of a matrix A are independent, it is a happy property. Because the matrix A can be diagonalized with a matrix S that column vectors are eigenvectors of A . For example, Why this is a happy property of A? Because I can find A's power easily. A^{10} is not a big deal. Because Λ is a diagonal matrix and power of a diagonal matrix is quite simple. A^{10} = SΛ^{10} S^{-1} Then, why if I want to compute power of A ? That is the same reason to find eigenvectors. Eigenvectors are a basis of a matrix. A matrix can be represented by a single scalar. I repeat this again. This is the happy point, a matrix becomes a scalar. What can be simpler than a scalar value. But, this is only possible when the matrix S's columns are independent. Because S^{-1} must be exist. Now I come back to my first question. Is the λ's multiplicity related with the number of eigenvectors? This time I found this has the name. Geometric multiplicity (GM): the number of in...

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...

Lx=d: SundayResearch

Search This Blog