# # Series Approximation

$$\newcommand{\dd}{\frac{\partial}{\partial{#1}}} \newcommand{\DD}{\left\langle\!\!\!\left\langle#1\right\rangle\!\!\!\right\rangle} \newcommand{\SA}{\left[\!\left[{#1}\right]\!\right]}$$ Series approximation depends on Perturbation.

The perturbed deltas are polynomial series in $$\DD{c}$$. Define the coeffecients $$\SA{z_n}_m,\SA{\dd{c}z_n}_m$$ of the polynomials for each delta: \begin{aligned} \DD{z_n} &= \sum \SA{z_n}_m \DD{c}^m & \DD{\dd{c}z_n} &= \sum \SA{\dd{c}z_n}_m \DD{c}^m \\ \end{aligned} Some boring algebraic manipulation gives the iterations for the first few coefficients of $$\DD{z_n}$$: \begin{aligned} \SA{z_{n+1}}_1 &= 2 z_n \SA{z_n}_1 + 1 \\ \SA{z_{n+1}}_2 &= 2 z_n \SA{z_n}_2 + \SA{z_n}_1^2 \\ \SA{z_{n+1}}_3 &= 2 z_n \SA{z_n}_3 + 2 \SA{z_n}_1 \SA{z_n}_2 \\ \end{aligned} and similarly for the coefficients of $$\DD{\dd{c}z_n}$$: \begin{aligned} \SA{\dd{c}z_{n+1}}_1 &= 2 \left( \dd{c}z_n \SA{z_n}_1 + z_n \SA{\dd{c}z_n}_1 \right) \\ \SA{\dd{c}z_{n+1}}_2 &= 2 \left( \dd{c}z_n \SA{z_n}_2 + z_n \SA{\dd{c}z_n}_2 + \SA{z_n}_1 \SA{\dd{c}z_n}_1 \right) \\ \SA{\dd{c}z_{n+1}}_3 &= 2 \left( \dd{c}z_n \SA{z_n}_3 + z_n \SA{\dd{c}z_n}_3 + \SA{z_n}_1 \SA{\dd{c}z_n}_2 + \SA{z_n}_2 \SA{\dd{c}z_n}_1 \right) \end{aligned} The coefficients are independent of $$\DD{c}$$ so the same coefficients can be used for many points in an image, and when $$|\DD{c}|$$ is small the sum can be approximated by truncating to the first few terms. However, the coefficients grow quickly as $$n$$ increases, which limits how long the per-reference approximation remains valid, after which we have to switch back to per-point delta iteration.