Zernike by one Pascal triangle: A high performance, low memory cost and flexible computation scheme for Zernike polynomials | Journal of the European Optical Society-Rapid Publications

EOSAM 2025

Open Access

Table 3

Counts of arithmetic instructions for a Zernike component $z_{n}^{\pm m}$ $Mathematical equation: $ {z}_n^{\pm m}$$ , where sin and cos components of even and odd orders are considered separately.

Computations	Count of Multiplication (×)	Count of addition (+)
${[x^{2 j - 2 l - 1} y^{2 l + 1}]}_{j \geq 1,0 \leq l \leq j}$ $Mathematical equation: $ {\left[{x}^{2j-2l-1}{y}^{2l+1}\right]}_{j\ge \mathrm{1,0}\le l\le j}$$	$\frac{j (j + 1)}{2}$ $Mathematical equation: $ \frac{j(j+1)}{2}$$	0
${[x^{2 j - 2 l} y^{2 l}]}_{j \geq 0,0 \leq l \leq j}$ $Mathematical equation: $ {\left[{x}^{2j-2l}{y}^{2l}\right]}_{j\ge \mathrm{0,0}\le l\le j}$$	$\frac{(j + 1) (j + 2)}{2} - 2$ $Mathematical equation: $ \frac{(j+1)(j+2)}{2}-2$$	0
${[x^{2 j - 2 l} y^{2 l + 1}]}_{j \geq 0,0 \leq l \leq j}$ $Mathematical equation: $ {\left[{x}^{2j-2l}{y}^{2l+1}\right]}_{j\ge \mathrm{0,0}\le l\le j}$$	$\frac{(j + 1) (j + 2)}{2} - 1$ $Mathematical equation: $ \frac{(j+1)(j+2)}{2}-1$$	0
${[x^{2 j - 2 l + 1} y^{2 l}]}_{j \geq 0,0 \leq l \leq j}$ $Mathematical equation: $ {\left[{x}^{2j-2l+1}{y}^{2l}\right]}_{j\ge \mathrm{0,0}\le l\le j}$$	$\frac{(j + 1) (j + 2)}{2} - 1$ $Mathematical equation: $ \frac{(j+1)(j+2)}{2}-1$$	0
$z_{2 j}^{- 2 l}$ $Mathematical equation: $ {z}_{2j}^{-2l}$$	$\frac{(j + l) (j - l + 1)}{2}$ $Mathematical equation: $ \frac{(j+l)(j-l+1)}{2}$$	$\frac{(j + l) (j - l + 1)}{2} - 1$ $Mathematical equation: $ \frac{(j+l)(j-l+1)}{2}-1$$
$z_{2 j}^{+ 2 l}$ $Mathematical equation: $ {z}_{2j}^{+2l}$$	$\frac{(j + l + 1) (j - l) + 2 (j + 1)}{2}$ $Mathematical equation: $ \frac{(j+l+1)(j-l)+2(j+1)}{2}$$	$\frac{(j + l + 1) (j - l) + 2 (j + 1)}{2} - 1$ $Mathematical equation: $ \frac{(j+l+1)(j-l)+2(j+1)}{2}-1$$
$z_{2 j + 1}^{- (2 l + 1)}$ $Mathematical equation: $ {z}_{2j+1}^{-(2l+1)}$$	$\frac{(j + l + 1) (j - l) + 2 (j + 1)}{2}$ $Mathematical equation: $ \frac{(j+l+1)(j-l)+2(j+1)}{2}$$	$\frac{(j + l + 1) (j - l) + 2 (j + 1)}{2} - 1$ $Mathematical equation: $ \frac{(j+l+1)(j-l)+2(j+1)}{2}-1$$
$z_{2 j + 1}^{+ (2 l + 1)}$ $Mathematical equation: $ {z}_{2j+1}^{+(2l+1)}$$	$\frac{(j + l + 1) (j - l) + 2 (j + 1)}{2}$ $Mathematical equation: $ \frac{(j+l+1)(j-l)+2(j+1)}{2}$$	$\frac{(j + l + 1) (j - l) + 2 (j + 1)}{2} - 1$ $Mathematical equation: $ \frac{(j+l+1)(j-l)+2(j+1)}{2}-1$$

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