sekelle
commented
4 months ago Hi, equation 10 is implemented by this function:
Under the condition that each internal node has 8 children, we already know that for $n$ leaf nodes, there will be $(n-1)/7$ internal nodes. Eq. 10 provides a map or an ordering that indicates where in [0:(n-1)/7] a given internal node should be stored. This is a faster shortcut to the more straightforward way of enumerating the internal node keys as they appear in the Cornerstone leaf cell array, followed by a prefix sum (as described in Karras, 2012). The $\mu$ are the weights of the octal digits of the internal node key to be assigned a storage location. Digits 0 and 7 do not shift that location, while digits 1-3 shift it to lower indices and 4-6 to higher ones. The total sum of these $\mu$ weights determines the storage location.
Eq. 10 can be deduced by induction, starting with a minimal octree, inserting additional leaf cells and determining the resulting effect on the layout of internal nodes computed by the enumeration + prefix-sum method.
UnlimitedR
Hello, I'm reading this paper for personal use. I don't have too much background knowledge and have encountered several questions but tried to figure it out with my own understanding.
However, there is still an equation that I cannot understand at all, which is equation (10) $$𝛿(K_i, d) = \Sigmal^{d/3+1} \mu{k_l},\quad \mu_0, . . . , \mu_7 = {0, −1, −2, −3, 3, 2, 1, 0},$$ May I ask what the parameters such as $\mu$ mean and how it is deduced, and which part of the codes does this equation corresponding to?