Average gradient of Zernike polynomials over polygons

Vyas Akondi; Alfredo Dubra

doi:10.1364/OE.393223

1. Introduction

Slope wavefront sensors, such as the Shack-Hartmann [1–6], are widely used in medical, scientific, and industrial applications, including autorefraction and clinical aberrometry [7–13], refractive surgery [14–23], adaptive optics retinal imaging [24–29], visual psychophysics [30–34], vision simulators [35–38], microscopy [39–41], astronomy [42–44], line-of-sight communications [45], high power lasers [46–49] and metrology [50]. These sensors measure wavefront slopes averaged over a set of non-overlapping discrete regions or subapertures. The wavefront is then estimated either by solving a set of discrete difference equations (zonal reconstruction) [51–56] or by fitting the data to a linear combination of basis functions (modal reconstruction) [54–66]. The latter, often used due to its robustness to noise [54,66], requires the estimation of average slopes of the basis functions over the sampling subapertures, which is the focus of this work. The Zernike polynomials are the most commonly used basis functions due to their orthogonality over the unit circle which, by design, makes them convenient to describe rotationally symmetric optical systems [65–71]. When this approach is pursued, the fidelity with which a wavefront derived from slope sensor data can be described depends on the number of samples [54], the number of Zernike polynomials [60,72], and the errors in the calculation of their average slopes [73–78]. Here we derive formulae to evaluate the average Zernike gradients over polygonal subapertures, such as those found in the lenslet arrays of Shack-Hartmann wavefront sensors (SHWSs), which are typically square or hexagonal.

This paper is structured as follows. First, we show that the double integration needed to estimate the average Zernike slope over a subaperture can be reduced to a single integral of the Zernike itself along the subaperture perimeter. Then, a Cartesian expression for Zernike polynomials [70,71], modified from that by Carpio and Malacara [79], is integrated along line segments to derive a closed-form expression for the average slopes over polygonal areas. Finally, and as a practical example, we use these formulae in a simulated SHWS with square lenslets [8,26,29,39,44,47,75]. The computing time and accuracy of the derived formulae are compared to the following alternative methods. First, the average Zernike slope over the entire aperture is approximated as the value at its center [73–78]. Second, the averaging of a discrete number of Zernike slope samples spread over the subaperture [73,74]. Then, we use two-dimensional numerical integration of the Zernike slope over the subaperture areas [39,75,77], the one-dimensional numerical integration of the Zernike polynomials along the perimeter of the subapertures, and symbolic integration of Zernike polynomials along the perimeter of the subapertures. Finally, we show that the expressions derived here provide fastest calculation time and highest accuracy.

2. Average gradient of a two-dimensional function

The average slope of a wavefront $W$ along the x-direction over a subaperture $\Omega $, such as that of a lenslet in a SHWS, is defined as,

(1)$${\left\langle {\frac{{\partial W(x,y)}}{{\partial x}}} \right\rangle _\Omega } = {{\iint\limits_\Omega {\frac{{\partial W(x,y)}}{{\partial x}}\textrm{d}x\textrm{d}y} } \mathord{\left/ {\vphantom {{\iint\limits_\Omega {\frac{{\partial W(x,y)}}{{\partial x}}\textrm{d}x\textrm{d}y} } {\iint\limits_\Omega {\textrm{d}x\textrm{d}y} }}} \right. } {\iint\limits_\Omega {\textrm{d}x\textrm{d}y} }}. $$

When the area is convex and the function is well-behaved, the gradient theorem, a generalization of the fundamental theorem of calculus for line integrals (see page 191 of [80]), can be used to reduce the calculation to a difference between wavefront evaluated along the “left” and “right” portions of the perimeter ($\wp {}_L$ and ${\wp _R}$ respectively, in Fig. 1), starting at the point with the lowest y-value and ending at the point with the highest y-value,

(2)$${\left\langle {\frac{{\partial W(x,y)}}{{\partial x}}} \right\rangle _\Omega } = {{\left[ {\int\limits_B^A {W({{\wp_R}(y ),y} )\textrm{d}y} - \int\limits_B^A {W({{\wp_L}(y ),y} )\textrm{d}y} } \right]} \mathord{\left/ {\vphantom {{\left[ {\int\limits_B^A {W({{\wp_R}(y ),y} )\textrm{d}y} - \int\limits_B^A {W({{\wp_L}(y ),y} )\textrm{d}y} } \right]} {\iint\limits_\Omega {\textrm{d}x\textrm{d}y} }}} \right. } {\iint\limits_\Omega {\textrm{d}x\textrm{d}y} }}.$$

Fig. 1. Separation of the “left” and “right’ perimeters of a convex subaperture region Ω used for calculating the average wavefront slope along the x-axis.

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Reversing the integration limits of the “left” perimeter (lower x-coordinate) makes the numerator of the average slope calculation a single counterclockwise integral along the entire perimeter $\wp$ (see Fig. 1),

(3)$${\left\langle {\frac{{\partial W(x,y)}}{{\partial x}}} \right\rangle _\Omega } = {{\int\limits_\wp {W({{x_{\textrm{perimeter}}}(y ),y} )\textrm{d}y} } \mathord{\left/ {\vphantom {{\int\limits_\wp {W({{x_{\textrm{perimeter}}}(y ),y} )\textrm{d}y} } {\iint\limits_\Omega {\textrm{d}x\textrm{d}y} }}} \right. } {\iint\limits_\Omega {\textrm{d}x\textrm{d}y} }}$$

This reasoning can be extended to non-convex areas, by splitting $\Omega $ into convex sub-areas and observing that the integrals along shared perimeter segments, as depicted in Fig. 2, cancel out due to the opposing integration directions. Therefore, only the counterclockwise integral along the perimeter is needed to calculate the average of the x-derivative over an arbitrary connected region with a continuous perimeter. Importantly, when calculating the average of the y-derivative, the integral in the numerator must be performed clockwise.

Fig. 2. Geometry used for calculating the average x-derivative of a function over a non-convex region with continuous perimeter by dividing it into convex regions.

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3. Average gradient over a polygonal area

The average x-derivative of W over a polygonal subaperture defined by N vertices sorted counterclockwise, can be calculated using Eq. (3) as

(4)$${\left\langle {\frac{{\partial W(x,y)}}{{\partial x}}} \right\rangle _\Omega } = {{\left[ {\sum\limits_{q = 1}^N {\int\limits_{{y_q}}^{{y_{q + 1}}} {W({a_{q,q + 1}^xy + b_{q,q + 1}^x,y} )\textrm{d}y} } } \right]} \mathord{\left/ {\vphantom {{\left[ {\sum\limits_{q = 1}^N {\int\limits_{{y_q}}^{{y_{q + 1}}} {W({a_{q,q + 1}^xy + b_{q,q + 1}^x,y} )\textrm{d}y} } } \right]} {\iint\limits_\Omega {\textrm{d}x\textrm{d}y} }}} \right. } {\iint\limits_\Omega {\textrm{d}x\textrm{d}y} }},$$

where q+1 cycles back to 1. Similarly, with the N vertices sorted clockwise, the y-derivatives can be calculated as

(5)$${\left\langle {\frac{{\partial W(x,y)}}{{\partial y}}} \right\rangle _\Omega } = {{\left[ {\sum\limits_{q = 1}^N {\int\limits_{{x_q}}^{{x_{q + 1}}} {W({x,a_{q,q + 1}^yx + b_{q,q + 1}^y} )\textrm{d}x} } } \right]} \mathord{\left/ {\vphantom {{\left[ {\sum\limits_{q = 1}^N {\int\limits_{{x_q}}^{{x_{q + 1}}} {W({x,a_{q,q + 1}^yx + b_{q,q + 1}^y} )\textrm{d}x} } } \right]} {\iint\limits_\Omega {\textrm{d}x\textrm{d}y} }}} \right. } {\iint\limits_\Omega {\textrm{d}x\textrm{d}y} }}.$$

In these formulae, $a_{q,q + 1}^x$, $b_{q,q + 1}^x$, $a_{q,q + 1}^y$ and $b_{q,q + 1}^y$ are the coefficients of the equation of a line passing through the adjacent polygon vertices q and q+1,

(6)$$x = a_{q,q + 1}^xy + b_{q,q + 1}^x,$$

with $a_{q,q + 1}^x = {{({{x_{q + 1}} - {x_q}} )} \mathord{\left/ {\vphantom {{({{x_{q + 1}} - {x_q}} )} {({{y_{q + 1}} - {y_q}} )}}} \right.} {({{y_{q + 1}} - {y_q}} )}}$ and $b_{q,q + 1}^x = {{{x_q} - {y_q}({{x_{q + 1}} - {x_q}} )} \mathord{\left/ {\vphantom {{{x_q} - {y_q}({{x_{q + 1}} - {x_q}} )} {({{y_{q + 1}} - {y_q}} )}}} \right.} {({{y_{q + 1}} - {y_q}} )}}$. Similarly, for the y-derivative of W, we have

(7)$$y = a_{q,q + 1}^yx + b_{q,q + 1}^y,$$

with $a_{q,q + 1}^y = {{({{y_{q + 1}} - {y_q}} )} \mathord{\left/ {\vphantom {{({{y_{q + 1}} - {y_q}} )} {({{x_{q + 1}} - {x_q}} )}}} \right.} {({{x_{q + 1}} - {x_q}} )}}$ and $b_{q,q + 1}^y = {{{y_q} - {x_q}({{y_{q + 1}} - {y_q}} )} \mathord{\left/ {\vphantom {{{y_q} - {x_q}({{y_{q + 1}} - {y_q}} )} {({{x_{q + 1}} - {x_q}} )}}} \right.} {({{x_{q + 1}} - {x_q}} )}}$. These, with an explicit formula for the area of a polygon in the denominator (see page 193 of Ref. [81]), allow us to write the average x-derivative of W over the polygonal subaperture in terms of the vertices sorted counterclockwise as,

(8)$${\left\langle {\frac{{\partial W(x,y)}}{{\partial x}}} \right\rangle _\Omega } = {{2\sum\limits_{q = 1}^N {\int\limits_{{y_q}}^{{y_{q + 1}}} {W({a_{q,q + 1}^xy + b_{q,q + 1}^x,y} )\textrm{d}y} } } \mathord{\left/ {\vphantom {{2\sum\limits_{q = 1}^N {\int\limits_{{y_q}}^{{y_{q + 1}}} {W({a_{q,q + 1}^xy + b_{q,q + 1}^x,y} )\textrm{d}y} } } {\sum\limits_{q = 1}^N {({{x_q}{y_{q + 1}} - {y_q}{x_{q + 1}}} )} }}} \right. } {\sum\limits_{q = 1}^N {({{x_q}{y_{q + 1}} - {y_q}{x_{q + 1}}} )} }},$$

noting that for horizontal polygon sides parallel to the x-axis, ${y_q} = {y_{q + 1}}$ and, thus, the corresponding terms in the numerator are identically zero. Similarly, for the average of the y-derivative we have,

(9)$${\left\langle {\frac{{\partial W(x,y)}}{{\partial y}}} \right\rangle _\Omega } = {{2\sum\limits_{q = 1}^N {\int\limits_{{x_q}}^{{x_{q + 1}}} {W({x,a_{q,q + 1}^yx + b_{q,q + 1}^y} )\textrm{d}x} } } \mathord{\left/ {\vphantom {{2\sum\limits_{q = 1}^N {\int\limits_{{x_q}}^{{x_{q + 1}}} {W({x,a_{q,q + 1}^yx + b_{q,q + 1}^y} )\textrm{d}x} } } {\sum\limits_{q = 1}^N {({{x_q}{y_{q + 1}} - {y_q}{x_{q + 1}}} )} }}} \right. } {\sum\limits_{q = 1}^N {({{x_q}{y_{q + 1}} - {y_q}{x_{q + 1}}} )} }},$$

with the vertical segments parallel to the y-axis not contributing to the numerator and remembering that here the vertices are sorted clockwise.

4. Zernike polynomials in Cartesian coordinates

The Zernike polynomials are defined as the product of a radial coordinate polynomial of order n, and either a sine or a cosine azimuthal function of order m,

(10)$$Z_n^m({\rho ,\theta } )= N_n^mR_n^{|m |}(\rho )\left\{ {\begin{array}{c} {\cos ({|m |\theta } ),\,\,\,\,\textrm{for }m \ge 0}\\ {\sin ({|m |\theta } ),\,\,\,\,\textrm{for }m < 0} \end{array}} \right.$$

with $|m |\le n$, $n - |m |$ an even number and

(11)$$R_n^{|m |}(\rho )= \sum\limits_{s = 0}^{{{({n - |m |} )} \mathord{\left/ {\vphantom {{({n - |m |} )} 2}} \right.} 2}} {\frac{{{{({ - 1} )}^s}({n - s} )!}}{{s![{{{({n + |m |} )} \mathord{\left/ {\vphantom {{({n + |m |} )} 2}} \right.} 2} - s} ]![{{{({n - |m |} )} \mathord{\left/ {\vphantom {{({n - |m |} )} 2}} \right.} 2} - s} ]!}}} {\rho ^{n - 2s}}. $$

When the normalization factor $N_n^m$ is ${[{2{{({n + 1} )} \mathord{\left/ {\vphantom {{({n + 1} )} {({1 + {{\delta }_{m0}}} )}}} \right.} {({1 + {{\delta }_{m0}}} )}}} ]^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}}$ with ${{\delta }_{m0}}$ being the Kronecker delta function, the polynomials have unit norm over the unit circle [68]. From this definition, closed-form expressions in Cartesian coordinates can be obtained, as demonstrated by Carpio and Malacara [79]. We now reproduce their derivation, using a right-handed Cartesian coordinate system with the x-axis having zero-azimuth and with increasing azimuth toward the positive y-axis.

Let us start by making changes of variables within the summation $n^{\prime} = {{({n - |m |} )} \mathord{\left/ {\vphantom {{({n - |m |} )} 2}} \right.} 2}$ and $k = n^{\prime} - s$, to rewrite Eq. (10) using combinations (or binomial coefficients) as,

(12)$$Z_n^m({\rho ,\theta } )= N_n^m\sum\limits_{k = 0}^{n^{\prime}} {{{({ - 1} )}^{n^{\prime} - k}}\left( {\begin{array}{c} {n^{\prime}}\\ k \end{array}} \right)\left( {\begin{array}{c} {n^{\prime} + |m |+ k}\\ {n^{\prime}} \end{array}} \right)} \,{\rho ^{2k + |m |}}\left\{ {\begin{array}{c} {\cos ({|m |\theta } ),\,\,\,\,\textrm{for }m \ge 0}\\ {\sin ({|m |\theta } ),\,\,\,\,\textrm{for }m < 0} \end{array}} \right.$$

Now, if we think of the polynomials as defined over the complex plane, with x and $y$ being the real and imaginary parts of a complex number $z$ (i.e., $z = x + iy = \rho \,{\textrm{e}^{i\theta }}$), and using de Moivre’s formula for calculating the powers of complex numbers, we have

(13)$$Z_n^m({x,y} )= N_n^m\sum\limits_{k = 0}^{n^{\prime}} {{{({ - 1} )}^{n^{\prime} - k}}\left( {\begin{array}{c} {n^{\prime}}\\ k \end{array}} \right)\left( {\begin{array}{c} {n^{\prime} + |m |+ k}\\ {n^{\prime}} \end{array}} \right)} {({{x^2} + {y^2}} )^k}\left\{ {\begin{array}{c} {{\mathop{\rm Re}\nolimits} \{{{{({x + iy} )}^{|m |}}} \},\,\,\,\,\textrm{for }m \ge 0}\\ {{\mathop{\rm Im}\nolimits} \{{{{({x + iy} )}^{|m |}}} \},\,\,\,\,\textrm{for }m < 0} \end{array}} \right.$$

If we now note that ${({{x^2} + {y^2}} )^k}{({x + iy} )^{|m |}} = {({x - iy} )^k}{({x + iy} )^{k + |m |}}$ and using the binomial theorem on the right-hand-side of this equation, we get,

(14)$${({{x^2} + {y^2}} )^k}{({x + iy} )^{|m |}} = \left[ {\sum\limits_{v = 0}^k {\left( {\begin{array}{c} k\\ v \end{array}} \right){x^{k - v}}{{({ - 1} )}^v}{i^v}{y^v}} } \right]\left[ {\sum\limits_{u = 0}^{k + |m |} {\left( {\begin{array}{c} {k + |m |}\\ u \end{array}} \right){x^{k + |m |- u}}{i^u}{y^u}} } \right],$$

which after consolidating the monomials becomes,

(15)$${({{x^2} + {y^2}} )^k}{({x + iy} )^{|m |}} = \sum\limits_{v = 0}^k {{{({ - 1} )}^v}\left( {\begin{array}{c} k\\ v \end{array}} \right)\sum\limits_{u = 0}^{k + |{m{\kern 1pt} } |} {\left( {\begin{array}{c} {k + |m |}\\ u \end{array}} \right){i^{u + v}}{x^{2k + |{m{\kern 1pt} } |- u - v}}{y^{u + v}}} }. $$

Let us now make the change of indices $p^{\prime} = u + v$ (or equivalently, $u = p^{\prime} - v \ge 0$), to facilitate the separation of the real and imaginary parts,

(16)$${({{x^2} + {y^2}} )^k}{({x + iy} )^{|m |}} = \sum\limits_{v = 0}^k {{{({ - 1} )}^v}\left( {\begin{array}{c} k\\ v \end{array}} \right)\sum\limits_{p^{\prime} = v}^{k + |{m{\kern 1pt} } |+ v} {\left( {\begin{array}{c} {k + |m |}\\ {p^{\prime} - v} \end{array}} \right){i^{p^{\prime}}}{x^{2k + |{m{\kern 1pt} } |- p^{\prime}}}{y^{p^{\prime}}}} }. $$

We can now separate the real and imaginary parts by retaining even and odd powers of the imaginary number, $i$ respectively,

(17)$${\mathop{\rm Re}\nolimits} \{{{{({{x^2} + {y^2}} )}^k}{{({x + iy} )}^{|m |}}} \}= \sum\limits_{v = 0}^k {{{({ - 1} )}^v}\left( {\begin{array}{c} k\\ v \end{array}} \right)\sum\limits_{\,\,\,\,\,\,\,\,\,p^{\prime}\,\,\textrm{even}\atop \scriptstyle v \le p^{\prime} \le k + |m |+ v}^{} {\left( {\begin{array}{c} {k + |m |}\\ {p^{\prime} - v} \end{array}} \right){i^{p^{\prime}}}{x^{2k + |m |- p^{\prime}}}{y^{p^{\prime}}}} }, $$

(18)$${\mathop{\rm Im}\nolimits} \{{{{({{x^2} + {y^2}} )}^k}{{({x + iy} )}^{|m |}}} \}= \sum\limits_{v = 0}^k {{{({ - 1} )}^v}\left( {\begin{array}{c} k\\ v \end{array}} \right)\sum\limits_{\,\,\,\,\,\,\,\,\,p^{\prime}\,\,\textrm{odd} \atop \scriptstyle v \le p^{\prime} \le k + |m |+ v}^{} {\left( {\begin{array}{c} {k + |m |}\\ {p^{\prime} - v} \end{array}} \right){i^{p^{\prime} - 1}}{x^{2k + |m |- p^{\prime}}}{y^{p^{\prime}}}}} . $$

Therefore, Zernike polynomials in Eq. (13) with non-negative angular order $({m \ge 0} )$ after making a final change of indices $p^{\prime} = 2p$ can be written as

(19)$$Z_n^{m \ge 0}(x,y) = N_n^m\sum\limits_{k = 0}^{n^{\prime}} {\left( {\begin{array}{c} {n^{\prime}}\\ k \end{array}} \right)\left( {\begin{array}{c} {n^{\prime} + |m |+ k}\\ {n^{\prime}} \end{array}} \right)} \sum\limits_{v = 0}^k {\left( {\begin{array}{c} k\\ v \end{array}} \right)\sum\limits_{p = \lceil{v/2} \rceil }^{\lfloor{({k + |m |+ v} )/2} \rfloor } {{{({ - 1} )}^{n^{\prime} - k + v + p}}} \left( {\begin{array}{c} {k + |m |}\\ {2p - v} \end{array}} \right){x^{2k + |m |- 2p}}{y^{2p}}}, $$

where $\lceil{} \rceil $ denotes the ceiling function, $\lfloor{} \rfloor $ the floor function, and $p$ is a non-negative integer. Similarly, the change of indices $p^{\prime} = 2p - 1$ allows us to write the polynomials with negative azimuthal order as,

(20)$$Z_n^{m < 0}(x,y) = N_n^m\sum\limits_{k = 0}^{n^{\prime}} {\left( {\begin{array}{c} {n^{\prime}}\\ k \end{array}} \right)\left( {\begin{array}{c} {n^{\prime} + |m |+ k}\\ {n^{\prime}} \end{array}} \right)} \sum\limits_{v = 0}^k {\left( {\begin{array}{c} k\\ v \end{array}} \right)\sum\limits_{p = \lceil{({v + 1} )/2} \rceil }^{\lfloor{({k + |m |+ v + 1} )/2} \rfloor } {{{({ - 1} )}^{n^{\prime} - k + v + p - 1}}} } \left( {\begin{array}{c} {k + |m |}\\ {2p - 1 - v} \end{array}} \right){x^{2k + |m |- 2p + 1}}{y^{2p - 1}}$$

Here, we did not use the associative property to facilitate efficient computational implementation.

5. Average Zernike polynomial gradients over a polygonal area

Now that we have explicit formulae for the Zernike polynomials in Cartesian coordinates, we can calculate their average slopes over polygonal areas by first substituting the two-point form of the line in Eq. (6) into Eqs. (19) and (20),

(21)$$\begin{array}{l} Z_n^{m \ge 0}(a_{q,q + 1}^xy + b_{q,q + 1}^x,y) = N_n^m\sum\limits_{k = 0}^{n'} {\left( {\begin{array}{c} {n'}\\ k \end{array}} \right)\left( {\begin{array}{c} {n' + \left| m \right| + k}\\ {n'} \end{array}} \right)} \,\sum\limits_{v = 0}^k {\left( {\begin{array}{c} k\\ v \end{array}} \right)} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sum\limits_{p = \left\lceil {v/2} \right\rceil }^{\left\lfloor {\left( {k + \left| m \right| + v} \right)/2} \right\rfloor } {{{\left( { - 1} \right)}^{n' - k + v + p}}} \left( {\begin{array}{c} {k + \left| m \right|}\\ {2p - v} \end{array}} \right){\left( {a_{q,q + 1}^xy + b_{q,q + 1}^x} \right)^{2k + \left| m \right| - 2p}}{y^{2p}}, \end{array}$$

and

(22)$$\begin{array}{l} Z_n^{m < 0}(a_{q,q + 1}^xy + b_{q,q + 1}^x,y) = N_n^m\sum\limits_{k = 0}^{n^{\prime}} {\left( {\begin{array}{c} {n^{\prime}}\\ k \end{array}} \right)\left( {\begin{array}{c} {n^{\prime} + |m |+ k}\\ {n^{\prime}} \end{array}} \right)} \sum\limits_{v = 0}^k {\left( {\begin{array}{c} k\\ v \end{array}} \right)} \sum\limits_{p = \lceil{({v + 1} )/2} \rceil }^{\lfloor{({k + |m |+ v + 1} )/2} \rfloor } {{{({ - 1} )}^{n^{\prime} - k + v + p - 1}}} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\begin{array}{c} {k + |m |}\\ {2p - 1 - v} \end{array}} \right){({a_{q,q + 1}^xy + b_{q,q + 1}^x} )^{2k + |m |- 2p + 1}}{y^{2p - 1}} \end{array}. $$

Using the binomial theorem, we get,

(23)$$\begin{array}{l} Z_n^{m \ge 0}(a_{q,q + 1}^{x}y + b_{q,q + 1}^x,y) = N_n^m\sum\limits_{k = 0}^{n'} {\left( {\begin{array}{c} {n'}\\ k \end{array}} \right)\left( {\begin{array}{c} {n' + \left| m \right| + k}\\ {n'} \end{array}} \right)} \,\sum\limits_{v = 0}^k {\left( {\begin{array}{c} k\\ v \end{array}} \right)} \,\sum\limits_{p = \left\lceil {v/2} \right\rceil }^{\left\lfloor {\left( {k + \left| m \right| + v} \right)/2} \right\rfloor } {{{\left( { - 1} \right)}^{n' - k + v + p}}} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\begin{array}{c} {k + \left| m \right|}\\ {2p - v} \end{array}} \right)\sum\limits_{t = 0}^{2k + \left| m \right| - 2p} {\left( {\begin{array}{c} {2k + \left| m \right| - 2p}\\ t \end{array}} \right)} a_{q,q + 1}^{x}{}^{2k + \left| m \right| - 2p - t}b_{q,q + 1}^{x}{}^{t}y^{2k + \left| m \right| - t} \end{array},$$

and

(24)$$\begin{array}{l} Z_n^{m < 0}(a_{q,q + 1}^xy + b_{q,q + 1}^x,y) = N_n^m\sum\limits_{k = 0}^{n'} {\left( {\begin{array}{c} {n'}\\ k \end{array}} \right)\left( {\begin{array}{c} {n' + \left| m \right| + k}\\ {n'} \end{array}} \right)} \,\sum\limits_{v = 0}^k {\left( {\begin{array}{c} k\\ v \end{array}} \right)} \,\sum\limits_{p = \left\lceil {\left( {v + 1} \right)/2} \right\rceil }^{\left\lfloor {\left( {k + \left| m \right| + v + 1} \right)/2} \right\rfloor } {{{\left( { - 1} \right)}^{n' - k + v + p - 1}}} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\begin{array}{c} {k + \left| m \right|}\\ {2p - 1 - v} \end{array}} \right)\sum\limits_{t = 0}^{2k + \left| m \right| - 2p + 1} {\left( {\begin{array}{c} {2k + \left| m \right| - 2p + 1}\\ t \end{array}} \right)} a_{q,q + 1}^{x}{}^{2k + \left| m \right| - 2p + 1 - t}b_{q,q + 1}^{x}{}^{t}y^{2k + \left| m \right| - t}. \end{array}.$$

We can now perform the y-integration along an edge of the polygon, getting,

(25)$$\begin{array}{l} \int\limits_{{y_q}}^{{y_{q + 1}}} {Z_n^{m \ge 0}(a_{q,q + 1}^xy + b_{q,q + 1}^x,y)\textrm{d}y} = \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,N_n^m\sum\limits_{k = 0}^{n'} {\left( {\begin{array}{c} {n'}\\ k \end{array}} \right)\left( {\begin{array}{c} {n' + \left| m \right| + k}\\ {n'} \end{array}} \right)} \,\sum\limits_{v = 0}^k {\left( {\begin{array}{c} k\\ v \end{array}} \right)} \,\sum\limits_{p = \left\lceil {v/2} \right\rceil }^{\left\lfloor {\left( {k + \left| m \right| + v} \right)/2} \right\rfloor } {{{\left( { - 1} \right)}^{n' - k + v + p}}} \left( {\begin{array}{c} {k + \left| m \right|}\\ {2p - v} \end{array}} \right) \times \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sum\limits_{t = 0}^{2k + \left| m \right| - 2p} {\left( {\begin{array}{c} {2k + \left| m \right| - 2p}\\ t \end{array}} \right)} a_{q,q + 1}^{x}{}^{2k + \left| m \right| - 2p - t}b_{q,q + 1}^{x}{}^{t}\left( {\frac{{{y_{q + 1}}^{2k + \left| m \right| - t + 1} - {y_q}^{2k + \left| m \right| - t + 1}}}{{2k + \left| m \right| - t + 1}}} \right) \end{array},$$

and

(26)$$\begin{array}{l} \int\limits_{{y_q}}^{{y_{q + 1}}} {Z_n^{m < 0}(a_{q,q + 1}^xy + b_{q,q + 1}^x,y)\textrm{d}y} = \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,N_n^m\sum\limits_{k = 0}^{n^{\prime}} {\left( {\begin{array}{c} {n^{\prime}}\\ k \end{array}} \right)\left( {\begin{array}{c} {n^{\prime} + |m |+ k}\\ {n^{\prime}} \end{array}} \right)\,} \sum\limits_{v = 0}^k {\left( {\begin{array}{c} k\\ v \end{array}} \right)} \,\sum\limits_{p = \lceil{({v + 1} )/2} \rceil }^{\lfloor{({k + |m |+ v + 1} )/2} \rfloor } {{{({ - 1} )}^{n^{\prime} - k + v + p - 1}}} \left( {\begin{array}{c} {k + |m |}\\ {2p - 1 - v} \end{array}} \right)[{ \cdot{\cdot} \cdot } \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left. {\sum\limits_{t = 0}^{2k + |m |- 2p + 1} {\left( {\begin{array}{c} {2k + |m |- 2p + 1}\\ t \end{array}} \right)} a{{_{q,q + 1}^{x}}^{2k + |m |- 2p + 1 - t}}b{{_{q,q + 1}^{x}}^t}\left( {\frac{{{y_{q + 1}}^{2k + |m |- t + 1} - {y_q}^{2k + |m |- t + 1}}}{{2k + |m |- t + 1}}} \right)} \right] \end{array}. $$

Similarly, for the average along the y-axis, the integral along an edge of the polygon is

(27)$$\begin{array}{l} \int\limits_{{x_q}}^{{x_{q + 1}}} {Z_n^{m \ge 0}(x,a_{q,q + 1}^yx + b_{q,q + 1}^y)\textrm{d}x} = N_n^m\sum\limits_{k = 0}^{n'} {\left( {\begin{array}{c} {n'}\\ k \end{array}} \right)\left( {\begin{array}{c} {n' + \left| m \right| + k}\\ {n'} \end{array}} \right)} \,\sum\limits_{v = 0}^k {\left( {\begin{array}{c} k\\ v \end{array}} \right)} \,\sum\limits_{p = \left\lceil {v/2} \right\rceil }^{\left\lfloor {\left( {k + \left| m \right| + v} \right)/2} \right\rfloor } {{{\left( { - 1} \right)}^{n' - k + v + p}}} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\begin{array}{c} {k + \left| m \right|}\\ {2p - v} \end{array}} \right)\sum\limits_{t = 0}^{2p} {\left( {\begin{array}{c} {2p}\\ t \end{array}} \right)} a_{q,q + 1}^{y}{}^{2p - t}b_{q,q + 1}^{y}{}^{t}\left( {\frac{{{x_{q + 1}}^{2k + \left| m \right| - t + 1} - {x_q}^{2k + \left| m \right| - t + 1}}}{{2k + \left| m \right| - t + 1}}} \right) \end{array},$$

and

(28)$$\begin{array}{l} \int\limits_{{x_q}}^{{x_{q + 1}}} {Z_n^{m < 0}(x,a_{q,q + 1}^yx + b_{q,q + 1}^y)\textrm{d}x} = N_n^m\sum\limits_{k = 0}^{n'} {\left( {\begin{array}{c} {n'}\\ k \end{array}} \right)\left( {\begin{array}{c} {n' + \left| m \right| + k}\\ {n'} \end{array}} \right)} \,\sum\limits_{v = 0}^k {\left( {\begin{array}{c} k\\ v \end{array}} \right)} \,\sum\limits_{p = \left\lceil {\left( {v + 1} \right)/2} \right\rceil }^{\left\lfloor {\left( {k + \left| m \right| + v + 1} \right)/2} \right\rfloor } {{{\left( { - 1} \right)}^{n' - k + v + p - 1}}} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\begin{array}{c} {k + \left| m \right|}\\ {2p - 1 - v} \end{array}} \right)\sum\limits_{t = 0}^{2p - 1} {\left( {\begin{array}{c} {2p - 1}\\ t \end{array}} \right)} a_{q,q + 1}^{y}{}^{2p - 1 - t}b_{q,q + 1}^{y}{}^{t}\left( {\frac{{{x_{q + 1}}^{2k + \left| m \right| - t + 1} - {x_q}^{2k + \left| m \right| - t + 1}}}{{2k + \left| m \right| - t + 1}}} \right). \end{array}$$

These expressions can then be used in Eqs. (4) and (5) to evaluate the average Zernike gradients.

6. Application example: The Shack-Hartmann wavefront sensor

In the SHWS, each uniformly illuminated lenslet [82] forms an image onto a pixelated sensor that shifts relative to a reference position, proportionally to the average wavefront slope over the lenslet area [64,65,73,83–86]. The wavefront can then be estimated by fitting the measured slopes to the calculated slopes due to a linear combination of basis functions such as the Zernike polynomials. Here we compare the formulae derived in the previous section for calculating such average gradients against alternative methods in two simulated SHWSs with 100% fill factor square lenslets, with 7 and 23 fully illuminated lenslets across the pupil. All calculations were performed using an I9-7920X central processing unit (Intel, Santa Clara, CA, USA), using Matlab release 2017b (Mathworks, Natick, MA, USA). Slopes were calculated for all lenslets and Zernike polynomials up to the 9^th radial order using the following methods:

• Center value: Zernike slope at the lenslet center.
• 2D sampling average: evaluating the Zernike slope over a square grid of 20×20 uniformly spaced samples per lenslet, and then calculating their average.
• 2D integration: numerically integrating the Zernike slope over the lenslet area using Matlab’s integral2 function with default tolerances.
• 1D integration: numerical Zernike polynomial integration along the lenslet perimeter using Matlab’s integral function with default tolerances.
• Symbolic calculation: symbolic Zernike polynomial integration in Cartesian coordinates (defined in Eqs. (19) and (20)) along the lenslet perimeter, using Matlab’s symbolic toolbox.
• Analytical expression: evaluating the gradient average formulae derived in the previous section, that is, using Eqs. (25–28).

The symbolic calculation method is used here as the true value, although both the analytical expression and the symbolic calculation methods are exact and should, therefore, provide the same results up to 16 digits (the default precision of Matlab) other than for rounding errors. The relative errors were calculated as the (Euclidean) norm of the polynomial gradients for each method divided by the norm of the same gradient calculated using the symbolic calculation. For timing purposes, and although not exhaustively optimized, all gradient calculations above were implemented using matrix operations when possible.

The results plotted in Fig. 3, show that, as expected, due to its direct evaluation of the lowest order polynomial and least samples across all methods, the evaluation of the gradient at the lenslet center is the fastest but also the least accurate approach. The averaging of 400 uniformly spaced discrete samples per lenslet (2D sampling average) is approximately two orders of magnitude slower and five times more accurate. Both numerical integration methods (2D and 1D integration) require comparable calculation times which are an order of magnitude higher than the 2D sample averaging but reduce errors by about ten orders of magnitude. Interestingly, and within each radial order, all methods show larger errors for the polynomials with lower azimuthal order. Tip and tilt relative errors for the simulated SHWS with 37 lenslets (Fig. 3, top row), do not appear on this plot because they are exactly zero. In the SHWS with 401 lenslets, however (Fig. 3, bottom row) these errors, when not exactly zero, the relative error is smaller than 10⁻¹²%, which is within the tolerance of the numerical integration algorithms used. Finally, the formulae derived in the previous section (labeled Analytical expression in the figure legend), require two orders of magnitude less time than the numerical integration methods and (on average) greater than two orders of magnitude lower error. The latter is, as expected, within Matlab’s numerical error.

Fig. 3. Comparison of numerical and analytical methods for estimating average Zernike gradient values over square areas within a circular pupil.

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7. Summary

We showed that the average wavefront gradient over a subaperture can be reduced to a one-dimensional integral of the wavefront along the perimeter. This was followed by a re-derivation of expressions for the Zernike polynomials in Cartesian coordinates to facilitate their direct integration along the perimeter of polygons. The resulting formulae only require the evaluation of polynomials at the vertices. Finally, when used in two simulated SHWSs with square lenslets, these formulae show a superior combination of calculation time and accuracy when compared to numerical [73–77], and symbolic integration. Application of these formulae to hexagonal lenslets, also commonly used [24,87,88], is straightforward.

Funding

National Eye Institute (P30-EY026877, R01-EY025231, R01-EY028287, U01-EY025477); Research to Prevent Blindness (Challenge Grant).

Disclosures

The authors declare no conflicts of interest. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

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Average gradient of Zernike polynomials over polygons

Abstract

1. Introduction

2. Average gradient of a two-dimensional function

3. Average gradient over a polygonal area

4. Zernike polynomials in Cartesian coordinates

5. Average Zernike polynomial gradients over a polygonal area

6. Application example: The Shack-Hartmann wavefront sensor

7. Summary

Funding

Disclosures

References

Cited By

Figures (3)

Equations (28)

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