Zernike like functions on spherical cap: principle and applications in optical surface fitting and graphics rendering

Yi Zheng; Yi Zheng; Kai Wei; Kai Wei; Bin Liang; Bin Liang; Ying Li; Ying Li; Xinhui Chu; Xinhui Chu

doi:10.1364/OE.27.037180

1. Introduction

Complete orthogonal function on spherical cap is an essential tool in many attractive researches: (a) in high aperture system like microscopy, lithography and data storage, it serves as an appropriate expansion base for aberration study [1–3]; (b) it plays an important role in illumination design where lenses are routinely created to capture an entire hemisphere of light emitted from a source [4–5]; (c) for real-time high-quality rendering, bidirectional reflectance distribution function defined on hemisphere is applied to characterize the incident and outgoing radiances [6–8]. Accurately and efficiently representing bidirectional reflectance distribution function by a set of functions becomes particularly important in game and virtual reality; (d) human corneal is measured by video keratography, providing several thousand samples. Accurate modelling these data to a surface will serve to evaluation the overall image quality of eye, corneal refractive surgery and contact lens design [9,10]; (e) functions on spherical cap are also extensively used in geomagnetic field study over a portion of the earth [11,12] and cosmic microwave background research after removal galactic emission [13,14].

Function on spherical cap has been of interest for a long time. One expedient method is applying traditional planar functions like Zernike circular polynomials (ZCP) [15], Legendre polynomials, rectangular harmonics [16]. Yet it involves a flat surface approximation that becomes less accurate as the coverage analyzed becomes curve. Another prevalent approach is spherical harmonics (SH) which form a complete and orthogonal base on whole sphere. And an implementation of SH on spherical cap was described by Gorski [17], based on the Cholesky decomposition of the covariance matrix of the spherical harmonics on the spherical segment. However, the covariance matrix becomes ill-conditioned for function degree larger than 50, and so the method cannot be used to high degree requirement [13]. Other development widely accepted is spherical cap harmonic analysis (SCHA) for regional magnetic modelling presented by Haines [18]. SCHA consists of associated Legendre polynomials with non-integral degree and trigonometric functions with integral order. The advantage of SCHA is that it is derived from Laplacian equation and thus satisfies the necessary constraints that the curl and divergence be zero in source free domain. Yet SCHA employs two set of base functions which are mutually orthogonal in each set, but a function of a set is not orthogonal with respect to one of the other set [19]. It is a trade-off between fitting the potential field or its gradients. In addition, the instability in calculating non-integral degree from boundary condition and noise sensitive feature poses further difficulty for practitioners [20]. As function on spherical cap is widely used in a broad range of research fields, a further study leading to desirable performance is critical.

Purpose of the paper is to derive Zernike like functions on spherical cap. We initiate the study from analyzing geometrical character of spherical cap in Section 2. It is found to be homeomorphic to unit circle, where ZCP form a complete and orthogonal basis. Rules to derive new function set are introduced in Section 3. Calculation method of coefficients and normalization factors is presented in a matrix form. The architecture not only can be applied to spherical cap with arbitrary aperture but is also compatible with different mapping functions and weight factors. Three analytical functions are acquired by proposed approach and their properties are discussed in Section 4. Application of new functions in data fitting and graphic rendering is demonstrated in Section 5 to verify their special advantages.

2. Geometric analysis of spherical cap

Spherical cap is a portion of spherical surface as shown in Fig. 1(a).

Fig. 1. Spherical cap homeomorphic mapping to unit circle. (a) Schematic diagram of Spherical cap of semi-aperture angle ${\theta _b}$. (b) Spherical cap mapping to unit circle in $\Theta - \varphi$ coordinates. (c) Mapping functions in spherical cap, meaning of points A, B and C can be referenced to Table 1.

Download Full Size | PDF

Table 1. Mapping function and physical meaning

View Table | View all tables in this article

In spherical coordinates system, we use polar distance $r$, polar angle $\theta$ and azimuthal angle $\varphi$ to describe spherical cap.

(1)$$\left\{ \begin{array}{c} r = 1\\ 0 \le ({{\theta \mathord{\left/ {\vphantom {\theta {{\theta_b}}}} \right.} {{\theta_b}}}} )\le 1\\ 0 \le \varphi \le 2\pi \end{array} \right. $$

where ${\theta _b}$ is semi-angular aperture of the spherical cap considered, ranging from 0 to $\pi$, to fit with different spherical segment (${\pi \mathord{\left/ {\vphantom {\pi 2}} \right.} 2}$ for hemisphere for instance). Equation characterizing unit circle is written in Eq. (2) for comparison. $\rho$ represents the radial distance in cylindrical coordinates system.

(2)$$\left\{ \begin{array}{c} z = 0\\ 0 \le \rho \le 1\\ 0 \le \varphi \le 2\pi \end{array} \right. $$

The resemblances between spherical cap and unit circle are obvious. First, they are both two dimensional domain whether it is curved ($r = 1$) or flat ($z = 0$). Secondly, spherical cap and unit circle are rotational symmetric around Z axis. The azimuthal angle $\varphi$ ranges from 0 to $2\pi$. Thirdly, $\rho$ and $\theta$ describes the radial position of the concerned point on surface. Moreover, spherical cap can be homeomorphic mapped to a unit circle. A straightforward map function can be constructed by normalizing $\theta$ with boundary angle ${\theta _b}$, shown in Eq. (1). Method of projecting a spherical cap to a unit circle is not unique. Function $\Theta (\theta )$ is utilized to describe the mapping process. It should be a continuous increasing function satisfying boundary condition in Eq. (3). Then discussion of spherical cap function can be carried out on unit circle as shown in Fig. 1(b).

(3)$$\left\{ \begin{array}{l} \Theta (\theta ) = 0,\quad \theta = 0\\ \Theta (\theta ) = 1,\quad \theta = {\theta_b} \end{array} \right. $$

There are numerous candidates, satisfying boundary condition, for mapping function. Yet select one which leads to physical meaning and concise expression is a challenge. Several applicants are listed in Table 1 and shown in Fig. 1(c).

3. Zernike like functions derivation on spherical cap

3.1 Zernike circle polynomial (ZCP) overview and derive rules

Zernike circle polynomial was first presented by F. Zernike in his study of phase-contrast test [15] and now is broadly used in optical fabrication, testing, adaptive optics [21–22]. ZCP has several good properties. First, it forms a complete set of orthogonal polynomials over the unit circle. Advantage of using orthogonal functions is that coefficients are uniquely specified regardless of truncation. Secondly, they are invariant in form with respect to rotations of optical axes. Thirdly, their low order terms are analogous to Seidel aberrations. We want these good characters to be inherited. One effective method is to follow the rules that leads to ZCP. Fortunately, they are presented by Bhatia and Wolf [23].

(1) Functions should be invariant with respect to the rotation of axes about the origin of coordinates $x = y = 0$. This property distinguishes ZCP from other two-dimension orthogonal sets on unit circle and endows them with usefulness in optical application. The function $U$ will be invariant of rotation, if and only if, it is of the form: $(4)$$U_n^m(\Theta ,\varphi ) = R_n^m(\Theta ) \cdot \exp (im\varphi ) $$$ Where m is an integer (positive, negative or zero), called azimuthal order. And $R_n^m(\Theta )$ is a polynomial in $\Theta$ of degree n, containing no power of $\Theta$ lower than |m|. Moreover, $R_n^m(\Theta )$ is an even or an odd polynomial depending on m being even or odd. $R_n^m(\Theta )$ is denoted as radial polynomial in this paper.
(2) Functions should compose a complete set on two-dimension domain $\{ 0 \le \Theta \le 1, 0 \le \varphi \le 2\pi \}$. In the context completeness is crucial as it means that the system can represent all possible surface shapes. For degree $\le n$, the functions set $U_n^m(\Theta ,\varphi )$ contains $(n + 1)(n + 2)/2$ linearly independent polynomials. Hence every monomial ${x^i}{y^j}$ ($i \ge 0, j \ge 0\;$ integer) and consequently every polynomial may be expressed as a linear combination of the finite number of Zernike type functions. By Weierstrass approximation theorem [24], polynomials of x and y form a complete set on two-dimension area. It then follows that Zernike type functions is complete.
(3) They should be orthogonal to each other. Orthogonality, which implies linear independence between base functions, implies also good numerical behavior, avoids redundancy and ensures uniqueness of the representation. This property is expressed as an inner product of two functions in spherical coordinates. $(5)$$\left\{ \begin{array}{c} \left\langle {U_n^m,U_{n^{\prime}}^{m^{\prime}}} \right\rangle \equiv \left( {\int_0^{2\pi } {\exp (im\varphi ) \cdot \exp (im^{\prime}\varphi ) \cdot d\varphi } } \right) \cdot \left( {\int_0^{{\theta_b}} {W(\theta ) \cdot R_n^m(\Theta )R_{n^{\prime}}^{m^{\prime}}(\Theta ) \cdot \sin \theta \cdot d\theta } } \right)\\ \left\langle {U_n^m,U_{n^{\prime}}^{m^{\prime}}} \right\rangle = A \cdot {\delta_{m{\kern 1pt} m^{\prime}}} \cdot {\delta_{n{\kern 1pt} n^{\prime}}} \end{array} \right. $$$
Annotations to Eq. (5):
- i. $\delta$ is the Kronecker symbol and $A$ is normalization constant, usually the area of spherical segment.
- ii. $\sin \theta \cdot d\theta d\varphi$ is the differential area for spherical cap on unit sphere.
- iii. Those polynomials with different upper suffix ($m \ne m^{\prime}$) are orthogonal on the account of the orthognonality of the exponential factor over azimuthal domain and those with same upper suffix are orthogonal on the account of the radial polynomials over elevation domain.
- iv. $W(\theta )$ is weight factor depending on application circumstances. Please reference to weight factor discussion in appendix.
(4) The set should contain a function for each permissible pair of values of n (degree) and m (angular dependence), i.e. for integer values such that $n \ge 0$, $n \ge |m |$ and $n - |m |$ is even. By the rule, orthogonal sequence is set to ${\Theta ^{|m |}},{\Theta ^{|m |+ 2}},{\Theta ^{|m |+ 4}}, \cdots ,{\Theta ^{|m |+ 2k}}, \cdots$ with weight $W(\theta ) \cdot \sin (\theta )$ over the range of $0 \le \theta \le {\theta _b}$.

By mapping spherical cap to unit circle, rules in derivation ZCP can be applied to develop new functions on spherical segment. New functions could be acquired by Gram-Schmit process.

3.2 Zernike spherical cap functions derivation

The expression of new function can be written as follow: (according to the derive rules above)

(6)$$U_n^m(\Theta ,\varphi ) = N_n^m \cdot R_n^m(\Theta ) \cdot {\Phi _m}(m\varphi ) $$

where $N_n^m$ is normalization factor and ${\Phi _m}(m\varphi )$ is frequency function defined as:

(7)$${\Phi _m}(\varphi ) = \left\{ \begin{array}{l} \cos (m\varphi ),\\ {1 \mathord{\left/ {\vphantom {1 {\sqrt 2 }}} \right.} {\sqrt 2 }},\\ \sin (m\varphi ), \end{array} \right.\quad \begin{array}{c} {m > 0}\\ {m = 0}\\ {m < 0} \end{array} $$

$R_n^m(\Theta )$ is the radial polynomial as:

(8)$$R_n^m(\Theta ) = \sum\limits_{k = 0}^{{{(n - m)} \mathord{\left/ {\vphantom {{(n - m)} 2}} \right.} 2}} {c_{n,m}^{n - 2k}} {\Theta ^{n - 2k}},\quad c_{n,m}^n = 1 $$

where $c_{n,m}^{n - 2k}$ is the coefficient of radial polynomial. The expression of $U_n^m(\Theta ,\varphi )$ satisfies the first and second rule naturally. Coefficients can be calculated in required sequence to meet the orthogonal requirement. A special function is introduced to simplify the process.

(9)$$\textrm{s}(n )= \int_0^{{\theta _b}} {W(\theta ) \cdot {\Theta ^{2n}} \cdot \sin \theta d\theta } $$

Following Gram-Schmidt process, equations to calculate the coefficients are established in Eq. (10). The detailed derivation is presented in appendix.

(10)$${\textrm{P}_{n,m}} \cdot {\textrm{C}_{n,m}} ={-} {\textrm{B}_{n,m}} $$

where we have

(11)$$\begin{array}{l} k = (n - m)/2,\quad n > m\\ {\textrm{P}_{n,m}} = {\left[ {\begin{array}{cccc} {\textrm{s}(m)} &{\textrm{s}(m + 1)} &{\ldots } &{\textrm{s}(m + k - 1)}\\ {\textrm{s}(m + 1)} &{\textrm{s}(m + 2)} &{\ldots } &{\textrm{s}(m + k)}\\ \vdots & \vdots & \ddots & \vdots \\ {\textrm{s}(m + k - 1)} &{\textrm{s}(m + k)} &{\ldots } &{\textrm{s}(m + 2k - 2)} \end{array}} \right]_{k \times k}}\\ {\textrm{C}_{n,m}} = \left[ {\begin{array}{cccc} {c_{n,m}^m} &{c_{n,m}^{m + 2}} &{\ldots } &{c_{n,m}^{n - 2}} \end{array}} \right]_{k \times 1}^T\\ {\textrm{B}_{n,m}} = \left[ {\begin{array}{cccc} {\textrm{s}(m + k)} &{\textrm{s}(m + k + 1)} &{\ldots } &{\textrm{s}(m + 2k - 1)} \end{array}} \right]_{k \times 1}^T \end{array} $$

Then, coefficients are calculated by Eq. (12).

(12)$${\textrm{C}_{n,m}} ={-} {({\textrm{P}_{n,m}})^{ - 1}}{\textrm{B}_{n,m}} $$

Normalization factor can be calculated by the requirement of inner product of two functions to spherical cap area in Eq. (5).

(13)$$N_n^m({\theta _b}) = \sqrt {\frac{{2\pi ({1 - \cos {\theta_b}} )}}{{\int_0^{2\pi } {\int_0^{{\theta _b}} {W(\theta ) \cdot {{({U_n^m(\Theta ,\varphi )} )}^2}} } \cdot \sin \theta \cdot d\theta \cdot d\varphi }}} $$

4. Typical ZSCFs and their analytical expressions

4.1 Hemi-Spherical Harmonics (HSH)

Choose mapping function to $\sin (\theta )$, weight function to uniform and boundary angle to π/2. Orthogonal process is executed upon hemisphere and results are listed in Table 2. Considering their close connection to spherical harmonics, they are called as Hemi-spherical harmonics (HSH) in this paper.

Table 2. HSH expression (degree no more than 4)

View Table | View all tables in this article

The explicit hemispherical harmonics expression is

(14)$$\begin{aligned}HSH_n^m(s,\varphi ) &= {( - 1)^{|m|}}\sqrt {2({2n + 1} )\frac{{({n - |m |} )!}}{{({n + |m |} )!}}} \cdot P_n^{|m|}\{{\cos (\theta )} \}\cdot {\Phi _m}(\varphi ),\\ &\qquad \left( \begin{array}{l} n = 0,2,4, \cdots ;m = 0, \pm 2, \cdots , \pm n\\ n = 1,3,5, \cdots ;m ={\pm} 1, \pm 3, \cdots , \pm n \end{array} \right)\end{aligned} $$

where $P_n^{|m|}$ is the well-known associated Legendre function.

(15)$$P_n^{|m |}(x) = \frac{1}{{{2^n}n!}}{(1 - {x^2})^{\frac{{|m|}}{2}}}{\left( {\frac{d}{{dx}}} \right)^{n + |m|}}{({{x^2} - 1} )^n} $$

Isometric plots of the first 15 hemispherical harmonics are illustrated in Fig. 2(a).

Fig. 2. Hemispherical harmonics demonstration and its relationship with spherical harmonics. (a) Isometric plot for the first 15 hemispherical harmonics. (b) Filling lattice of HSH in SH diagram. Yellow blocks occupied by hemispherical harmonics and grey by complementary hemispherical harmonics.

Download Full Size | PDF

Comments on hemispherical harmonics.

(1) Orthogonality of HSH. It can be readily derived from orthogonality of SH on sphere and parity of radial polynomial.
(2) Completeness of HSH. HSH is found to be a subset of spherical harmonics for every integral pair $(n,m)$ meeting Zernike requirements. It is shown with yellow blocks in SH lattice in Fig. 2(b). Residual grey blocks are called as complementary hemispherical harmonics (CHSH) here. SH is the union of HSH and CHSH.CHSH and HSH are sets with value equal to zero and differential equals to zero respectively at boundary in Eq. (16). $(16)$$\left\{ \begin{array}{l} {\left. {\frac{{\partial HSH_n^m}}{{\partial \theta }}} \right|_{\theta = \pi /2}} = 0\\ { {CHSH_n^m} |_{\theta = \pi /2}} = 0 \end{array} \right. $$$
Moreover, every item in CHSH can be expressed as linear combinations of HSH on hemisphere. A simple proof is following. Take $CHSH_n^m$ for example. As a part of SH, it could first be written as $P_n^m(\cos (\theta ))$. Then replace all ${\cos ^2}(\theta )$ by $[1 - {\sin ^2}(\theta )]$. After organizing, we will obtain expression of following.
$(17)$$CHSH_n^m = ({{\alpha_1}{s^m} + {\alpha_2}{s^{m + 2}} + \cdots + {\alpha_{(n - m + 1)/2}}{s^{n - 1}}} )c \cdot {\Phi _m}(\varphi ) $$$ where $s = \sin(\theta )$, $c = \cos (\theta )$ and $\{{{\alpha_i}} \}$ are coefficients determined by associated Legendre polynomial correspondingly. Then substitute following Taylor expansion into Eq. (17) $(18)$$\cos (\theta ) = \sqrt {1 - {{\sin }^2}(\theta )} = ({1 - ({1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}){s^2} - ({1 \mathord{\left/ {\vphantom {1 8}} \right.} 8}){s^4} - ({1 \mathord{\left/ {\vphantom {1 {16}}} \right.} {16}}){s^6} - \cdots } ) $$$
The updating expression of $CHSH_n^m$ then becomes:
$(19)$$CHSH_n^m = {\beta _1} \cdot [{{s^m}{\Phi _m}(\varphi )} ]+ {\beta _2} \cdot [{{s^{m + 2}}{\Phi _m}(\varphi )} ]+ {\beta _3} \cdot [{{s^{m + 4}}{\Phi _m}(\varphi )} ] + \cdots $$$
$\{{{\beta_i}} \}$ are organized coefficients. Write the $HSH_n^m$, degree lower than (m + 2k), in matrix form:
$(20)$$\left[ {\begin{array}{c} {\begin{array}{c} {HSH_m^m}\\ {HSH_{m + 2}^m} \end{array}}\\ {\begin{array}{c} \vdots \\ {HSH_{m + 2k}^m} \end{array}} \end{array}} \right] = \left[ {\begin{array}{cccc} {c_{m,m}^m} &0 &0 &0\\ {c_{m + 2,m}^m} &{c_{m + 2,m}^{m + 2}} &0 &0\\ \vdots &\vdots &\ddots &0\\ {c_{m + 2k,m}^m} &{c_{m + 2k,m}^{m + 2}} & \cdots &{c_{m + 2k,m}^{m + 2k}} \end{array}} \right]\left[ {\begin{array}{c} {\begin{array}{c} {{s^m}{\Phi _m}(\varphi )}\\ {{s^{m + 2}}{\Phi _m}(\varphi )} \end{array}}\\ {\begin{array}{c} \vdots \\ {{s^{m + 2k}}{\Phi _m}(\varphi )} \end{array}} \end{array}} \right] $$$
Invert matrix and obtain ${s^{m + 2l}} \cdot {\Phi _m}(\varphi ),l = 0,1, \cdots k$ expressed by $HSH_m^m,HSH_{m + 2}^m, \cdots$, $HSH_{m + 2k}^m$. Put them into Eq. (19) and organize, we will get the following:
$(21)$$CHSH_n^m = \gamma _{n,m}^{m,m} \cdot HSH_m^m + \gamma _{n,m}^{m + 2,m} \cdot HSH_{m + 2}^m + \gamma _{n,m}^{m + 4,m} \cdot HSH_{m + 4}^m + \cdots $$$
Therefore, every item of $CHSH_n^m$ can be linear combination by $HSH$ of the same order m. And $HSH$ forms a complete set on hemisphere. The correlation coefficient can be calculated by Eq. (22).
$(22)$$\gamma _{n,m}^{m + k,m} = {{\left\langle {CHSH_n^m,HSH_{m + k}^m} \right\rangle } \mathord{\left/ {\vphantom {{\left\langle {CHSH_n^m,HSH_{m + k}^m} \right\rangle } {(2\pi )}}} \right.} {(2\pi )}}\quad k = 0,2,4, \cdots $$$
(3) Relation between HSH and ZCP. If circular radius in Eq. (2) is reconsidered, it will be found to be the projection of spherical radius r on X-Y plane or, in other word, $\rho = \sin (\theta ){|_{r = 1}}$. Thus ZCP comforms a complete, non-orthogonal set over hemispherical surface. And HSH can also be acquried by orthogonizing ZCP on hemisphere.
(4) For spherical cap with semi-aperture no more than ${\pi \mathord{\left/ {\vphantom {\pi 2}} \right.} 2}$, HSH can serve as a base to orthogonalize.
(5) Many systems in microscopy or lithography satisfy Abbe sine condition where $\sin (\theta )$ is the normalized radius in the pupil. Then the tip and tilt of spherical wavefront will be identical with the second and third term of HSH just as ZCP for planar wavefront. HSH composes a good base for describing aberration in these high numerical aperture systems just like ZCP does for traditional Gaussian systems.
(6) HSH is part of SH that naturally satisfies Laplacian equation in source free domain. The curl and divergence of HSH are both zero. With its completeness, orthogonality and analytical form, it can play active role in regional geomagnetic field modeling.

4.2 Zernike spherical functions (ZSF)

Mapping function is set to $\sin (\theta /2)$ and boundary angle to π. The new acquired results are listed in Table 3. They are nominated as Zernike spherical functions (ZSF) by resemblance to ZCP in expression.

Table 3. ZSF expression (degree no more than 4)

View Table | View all tables in this article

The analytical ZSF expression is

(23)$$ZSF_n^m(t,\varphi ) = \sqrt {2(n + 1)} \cdot R_n^m(t) \cdot {\Phi _m}(\varphi ),\quad \left( \begin{array}{l} n = 0,2,4, \cdots ;m = 0, \pm 2, \cdots , \pm n\\ n = 1,3,5, \cdots ;m ={\pm} 1, \pm 3, \cdots , \pm n \end{array} \right) $$

where $R_n^m(t)$ is well known Zernike polynomials expressed as

(24)$$R_n^m(t) = \sum\limits_{k = 0}^{(n - |m|)/2} {\frac{{{{( - 1)}^k}(n - k)!}}{{k!(\frac{{n + |m|}}{2} - k)!(\frac{{n - |m|}}{2} - k)!}}} {t^{n - 2k}} $$

Comments on ZSF:

(1) The orthogonality of ZSF is aperture invariant. It means its orthogonal property and expression in Eq. (23) will sustain, independent of aperture angle change. Just normalize variable t in Table 3 to ${{\sin ({\theta \mathord{\left/ {\vphantom {\theta 2}} \right.} 2})} \mathord{\left/ {\vphantom {{\sin ({\theta \mathord{\left/ {\vphantom {\theta 2}} \right.} 2})} {\sin ({{{\theta_b}} \mathord{\left/ {\vphantom {{{\theta_b}} 2}} \right.} 2})}}} \right.} {\sin ({{{\theta _b}} \mathord{\left/ {\vphantom {{{\theta_b}} 2}} \right.} 2})}}$. The character simplifies ZSP’s practice on arbitrary aperture with desirable performance.
(2) ZSF cannot replace SH role on full sphere. When polar angle approaches π, boundary of spherical cap degenerates to a point at the nadir. ZSF will become multivalued at the point. After all, full sphere is a closed surface, topologically different from spherical segment. To emphasize the difference, ZSF is said to be able to apply on full spherical cap instead of full sphere.
(3) For an optical system satisfying Herschel condition, the fourth term of ZSFs is $\cos (\theta )$, identical to wave front aberration for defocus. This leads ZSF to be a good basis describing aberration in such condition.

4.3 Longitudinal spherical functions (LSF)

Now the mapping function is written as $1 - \cos(\theta )$ and boundary angle as ${\pi \mathord{\left/ {\vphantom {\pi 2}} \right.} 2}$. Fortunately, another analytical expression is obtained (reference to Table 4). It is called Longitudinal Spherical Function (LSF) by the variable is along longitudinal (Z) axis.

(25)$$\begin{aligned}LSF_n^m(w,\varphi ) &= \sqrt {\frac{{{2^{|m| + 2.5}}}}{{\gamma _v^{(0,|m|- 0.5)}}}} {w^{|m|}}J\left( {\frac{{n - |m|}}{2},0,|m| - \frac{1}{2},2{w^2} - 1} \right) \cdot {\Phi _m}(\varphi ),\\ &\qquad\left( \begin{array}{l} n = 0,2, \cdots ;m = 0, \pm 2, \cdots , \pm n\\ n = 1,3, \cdots ;m ={\pm} 1, \pm 3, \cdots , \pm n \end{array} \right) \end{aligned}$$

where $J$ is well established Jacobi polynomials.

(26)$$J(v,a,b,x) = \frac{1}{{v!}}\sum\limits_{k = 0}^v {\frac{{{{( - v)}_k}{{(a + b + v + 1)}_k}{{(a + k + 1)}_{v - k}}}}{{k!}}} {\left( {\frac{{1 - x}}{2}} \right)^k} $$

Table 4. LSF expression (degree no more than 4)

View Table | View all tables in this article

The orthogonal property of Jacobi polynomials is shown in Eqs. (26) and (27).

(27)$$\left\{ \begin{array}{l} \int_{ - 1}^1 {{{(1 - x)}^a}{{(1 + x)}^b}} \cdot J(v,a,b,x) \cdot J(v^{\prime},a,b,x)dx = \gamma_v^{(a,b)} \cdot {\delta_{v,v^{\prime}}}\\ \gamma_v^{(a,b)} = \frac{{{2^{a + b + 1}}\Gamma (v + a + 1)\Gamma (v + b + 1)}}{{(2v + a + b + 1)\Gamma (v + 1)\Gamma (v + a + b + 1)}} \end{array} \right. $$

Comments on LSF:

(1) LSF can be utilized on arbitrary spherical cap including hyper spherical surface. Variable cos(θ) represents the vertical distance from the center (or observer) to the point concerned (or source). In applications containing hyper-spherical geometry and source-observer relation, LSF provides essential service. Head related transfer functions is such a case. It represents the acoustic transfer function from a sound source at a given location to human ear drums. A common inconvenience is the fact that practical constrains can prevent the measurement of head related transfer function on certain portions of the sphere, typically underneath the subject [25]. This leaves hundreds of measurements on hyper spherical cap to fit, where traditional SH confronts the difficulty of inversing an ill conditioned matrix.
(2) The orthogonality of LSF is aperture invariant. Normalizing variable w to ${{[{1 - \cos (\theta )} ]} \mathord{\left/ {\vphantom {{[{1 - \cos (\theta )} ]} {[{1 - \cos ({\theta_b})} ]}}} \right.} {[{1 - \cos ({\theta_b})} ]}}$ in Eq. (25) will serve the purpose.
(3) As reasons in previous part, LSF cannot either take the place of SH on full sphere. The maximum domain it can function is full spherical cap.

The performance of HSH, ZSF and LSF is compared in Table 5.

Table 5. Comparison of HSH, ZSF and LSF

View Table | View all tables in this article

5. Applications

5.1 Data fitting on spherical cap

In many applications a set of discrete data is given on a spherical cap. They can be freeform optical surface expressed by a series of points and directions, human cornea height data measured by keratoscope, observation of telescope for pointing error correction, regional geomagnetic measurements by satellite, etc. Fit data collected to a continuous surface is strongly required for interpolation, compression and analyzation. In this section, an example on hemisphere is used to verify the effectiveness of spherical cap functions, in comparison to ZCP and spherical harmonics. The fitting process is described generally by Eq. (28)

(28)$$\vec{c} = {({{F^T}F} )^{ - 1}}{F^T}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over m} $$

where $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over m}$ is the measurement vector, $\vec{c}$ is the fitting coefficients wanted. $F$ is the fit matrix generated by base functions. And ${F^T}F$ is called covariance matrix. Fitting ability of a function set is substantially depending on covariance matrix. Appearances of covariance matrix are showed in Figs. 3(a)–3(c) for SH, ZCP and HSH respectively (total function number equals to 66). Diagonal element of covariance matrix is self-correlation of a function and off-diagonal element represents cross-correlation of a function to the other. Lots of non-zero elements are observed on off-diagonal positions in Figs. 3(a) and 3(b). It means functions in SH and ZCP are strongly coupled. By contrast, covariance matrix of HSH delivers very clear no cross-correlation look in Fig. 3(c), verifying its orthogonal property. Then condition number of covariance matrix is studied. It measures how sensitive the output answer is to perturbation in the input data and to the roundoff errors made during solution process. Condition number is calculated for SH, ZCP and HSH with different fit degree and results are showed in Fig. 3(d). Condition number for HSH remains unchanged at 1.0 regardless of fit degree. And ZCP’s condition number increases linearly accordingly. Condition number for SH is very large, so a log scale is applied on Fig. 3(d). According to the result, fitting data on an incomplete sphere by SH confronts with difficulties as very sensitive to noises, numerical instability in calculation. HSH performs better than ZCP not only in smaller condition number, but also in its orthogonality: fitting results are uniquely specified regardless of truncation degree.

Fig. 3. Covariance matrix on hemi-sphere and condition number analysis. First 66 functions are taken out for demonstration. X and Y represent function sequence and Z axis is inner product value of two assigned function on hemisphere. (a) Covariance matrix for SH. (b) Covariance matrix for ZCP. (c) Covariance matrix for HSH. (d) Condition number with function degree. The magnitude of for HSH and ZCP is referenced to left Y axis and SH to the right log 10 axis.

Download Full Size | PDF

The convergence of fitting is verified by step function on hemisphere which is evenly divided into 36000 patches. The definition of step function is

(29)$$STEP(\theta - {\theta _0}) = \left\{ \begin{array}{l} 1,\\ 0, \end{array} \right.\;\begin{array}{c} {0 \le \theta \le {\theta _0}}\\ {{\theta _0} < \theta \le {\pi \mathord{\left/ {\vphantom {\pi 2}} \right.} 2}} \end{array} $$

where ${\theta _0}$ is the step position and is set to π/4, middle of the hemisphere. Besides the advantage succinct form of expression, step function contains abundant frequency information. Moreover, any symmetrical surface can be expressed as a linear combination of step functions modulated in step position and magnitude. It is widely used for performance evaluation of linear systems. The fitting results are shown in Fig. 4. From the meridian section, the resemblance of fit curve to step function increases with degree and residual error RMS decreases gradually accordingly.

Fig. 4. Fitting demonstration of HSH for step function on hemisphere. (a) Meridian section of fitting curves for 4, 20, 140-degree HSH with overhead view respectively on the right. (b) RMS residual fit error as function of the fit degree.

Download Full Size | PDF

5.2 Application in computer graphic rendering

Functions as HSH, ZSF and LSF can be regard as tool to carry frequency analysis on spherical cap, just like SH does on sphere. Some problem can get benefit from operation in frequency domain. Take computer graphic rendering for instance. Calculation of surface reflection radiance is the main problem in realistic illumination. It encounters three difficulties: the complexity of spatially varying bidirectional reflectance distribution function of real materials, the diversity of incident light due to intervening matter along light paths from sources to receiver, and the integration over the hemisphere of lighting directions at each point [26]. Equation (30) is the well accepted reflectance equation, where ${L_o}$ is the reflect radiance along $({\theta _o},{\varphi _o})$ direction. ${L_i}$ is the incident radiance over the hemisphere. And ${f_r}$ represents material reflectance distribution function.

(30)$${L_o}({\theta _o},{\varphi _o}) = \int_{{\varphi _i} = 0}^{2\pi } {\int_{{\theta _i} = 0}^{{\pi \mathord{\left/ {\vphantom {\pi 2}} \right.} 2}} {{f_r}({\theta _i},{\varphi _i};{\theta _o},{\varphi _o})} } \cdot {L_i}({\theta _i},{\varphi _i}) \cdot \cos {\theta _i} \cdot \sin {\theta _i}d{\theta _i}d{\varphi _i} $$

One method to circumvent the cumbersome integration in Eq. (23) is to expand ${f_r} \cdot \cos \theta$ and incident radiance ${L_i}$ with hemispherical cap functions. Then the two-dimension convolution reduces to a simple dot product. Traditionally the expansion is carried by SH [27]. Yet considering the hemispherical geometry of reflectance, hemispherical cap functions like HSH are more competent with the following reasons. Firstly, fitting bidirectional reflectance distribution function and local incident radiation by HSH is more stable and noise resistant than by SH. Secondly, it exempts the procedure of padding made-up data to lower hemisphere that SH requires. Moreover, total function number of HSH, degree no more than n, is ${{({n + 1} )({n + 2} )} \mathord{\left/ {\vphantom {{({n + 1} )({n + 2} )} 2}} \right.} 2}$. SH turns out to be $({n + 1} )({n + 1} )$. The nearly 50 percent reduction combined with orthogonality property will certainly improve the efficiency in both computation and memory storage.

6. Summary

We conceptualize and derive three sets of Zernike like functions on spherical cap. The orthogonal property ensures the uniqueness of representation without redundancy and the ability to fully characterize information on spherical cap is guaranteed by their completeness. With unitary condition number of covariance matrix, they also provide noise insensitive performance in fitting. Moreover, they can be readily applied in analyzation and computation with concise analytical form. Function obtained can be serviceable tool for a wide range of applications, including free form large-angle lens design in illumination, aberration description in high aperture optical system, regional geomagnetic field expression, graphics rendering etc.

Appendix

7.1 Gramm-Schmidt process for Zernike like functions

The Gramm-Schmidt algorithm is carried on radial polynomials with the same upper suffix. Take the calculation of $R_0^0(\Theta )$, $R_2^0(\Theta )$ and $R_4^0(\Theta )$ for example. They are written in Eq. (31).

(31)$$\left\{ \begin{array}{l} R_0^0(\Theta ) = 1\\ R_2^0(\Theta ) = {\Theta ^2} + c_{2,0}^0\\ R_4^0(\Theta ) = {\Theta ^4} + c_{4,0}^2{\Theta ^2} + c_{4,0}^0 \end{array} \right. $$

Coefficients in radial polynomials are calculated by orthogonal requirement. For $R_2^0(\Theta )$ the process is following.

(32.a)$$\int_0^{{\theta _b}} {R_0^0(\Theta ) \cdot R_2^0(\Theta ) \cdot W(\theta )\sin \theta \cdot d\theta } = 0 $$

(32.b)$$\int_0^{{\theta _b}} {{\Theta ^2} \cdot W(\theta )\sin \theta \cdot d\theta } + c_{2,0}^0\int_0^{{\theta _b}} {W(\theta )\sin \theta \cdot d\theta } = 0 $$

(32.c)$$s(1) + c_{2,0}^0s(0) = 0 $$

(32.d)$$c_{2,0}^0s(0) ={-} s(1) $$

For $R_4^0(\Theta )$ the process is following.

(33.a)$$\left\{ \begin{array}{l} \int_0^{{\theta_b}} {R_0^0(\Theta ) \cdot R_4^0(\Theta ) \cdot W(\theta )\sin \theta \cdot d\theta } = 0\\ \int_0^{{\theta_b}} {R_2^0(\Theta ) \cdot R_4^0(\Theta ) \cdot W(\theta )\sin \theta \cdot d\theta } = 0 \end{array} \right. $$

(33.b)$$\left\{ \begin{array}{l} s(2) + c_{4,0}^2 \cdot s(1) + C_{4,0}^0 \cdot s(0) = 0\\ s(3) + (c_{2,0}^0 + c_{4,0}^2) \cdot s(2) + (c_{2,0}^0c_{4,0}^2 + c_{4,0}^0) \cdot s(1) + c_{2,0}^0 \cdot c_{4,0}^0 \cdot s(0) = 0 \end{array} \right. $$

(33.c)$$\left[ {\begin{array}{cc} {s(0)} &{s(1)}\\ {s(1) + c_{2,0}^0 \cdot s(0)} &{s(2) + c_{2,0}^0 \cdot s(1)} \end{array}} \right]\left[ {\begin{array}{c} {c_{4,0}^0}\\ {c_{4,0}^2} \end{array}} \right] = \left[ {\begin{array}{c} { - s(2)}\\ { - s(3) - c_{2,0}^0 \cdot s(2)} \end{array}} \right] $$

(33.d)$$\left[ {\begin{array}{cc} 1 &0\\ {c_{2,0}^0} &1 \end{array}} \right]\left[ {\begin{array}{cc} {s(0)} &{s(1)}\\ {s(1)} &{s(2)} \end{array}} \right]\left[ {\begin{array}{c} {c_{4,0}^0}\\ {c_{4,0}^2} \end{array}} \right] ={-} \left[ {\begin{array}{cc} 1 &0\\ {c_{2,0}^0} &1 \end{array}} \right]\left[ {\begin{array}{c} {s(2)}\\ {s(3)} \end{array}} \right] $$

(33.e)$$\left[ {\begin{array}{cc} {s(0)} &{s(1)}\\ {s(1)} &{s(2)} \end{array}} \right]\left[ {\begin{array}{c} {c_{4,0}^0}\\ {c_{4,0}^2} \end{array}} \right] ={-} \left[ {\begin{array}{c} {s(2)}\\ {s(3)} \end{array}} \right] $$

The process to calculate coefficients in radial polynomials will be just the similar.

7.2 Discussion on weight factor

Weight factor is determined by application circumstances. Here following conditions are introduced. The first is uniform weight factor. For optical test, interferogram is formed by optical path difference between reference spherical wavefront and test wavefront. Illumination intensity distribution on wavefront is not so concerned. Therefore, the weight function can set to be uniform.

(34)$${W_{(1)}}(\theta ) = 1.0 $$

The second is for or electric field at focal point of an optical system of high aperture illuminated with a plane-polarized wave can be written [28,29] in the form of Eq. (35).

(35)$$E = \int_0^{2\pi } {\int_0^{{\theta _b}} {a(\theta )} } \cdot \exp (2\pi i \cdot \Phi ) \cdot (1 + \cos \theta ) \cdot \sin \theta \cdot d\theta d\varphi $$

where $a(\theta )$ is the illumination function (apodization factor), $\Phi$ is the wave-front aberration measured in wavelengths. If the system satisfies the sine condition, $a(\theta )$ equals to $\sqrt {\cos \theta }$. Then the weight function can be written as Eq. (36)

(36)$${W_{(2)}}(\theta ) = \sqrt {\cos \theta } \cdot (1 + \cos \theta ) $$

If a parabolic mirror is used to concentration electromagnetic radiation, the weight function will be

(37)$${W_{(3)}}(\theta ) = \textrm{se}{\textrm{c}^2}(\theta /2) \cdot (1 + \cos \theta ) $$

The apodization factor is discussed by Colin Sheppard in different conditions [29]. However, the orthogonal processes are same for different weight functions.

Funding

National Natural Science Foundation of China (11190011, 11973008, U1731134); Natural Science Foundation of Jiangsu Province (BK20141516).

Acknowledgments

The research is benefitted from discussion with Prof. Colin Sheppard, University of Wollongong and Dr. Hua Bai, NIAOT on orthogonal functions in high aperture systems. Writing assistance was provided by Prof. Feng-chuan Liu, Thirty-meter telescope international observatory.

Disclosures

For illustration, authors Yi Zheng, Kai Wei, Bin Liang, Ying Li and Xinhui Chu are represented below as YZ, KW, BL, YL and XHC. YZ: (P), KW: (P), BL: (P), YL: (P). The authors declare no conflicts of interest.

References

1. C. J. R. Sheppard, “Balanced diffraction aberrations, independent of the observation point: application to a tilted dielectric plate,” J. Opt. Soc. Am. A 30(10), 2150–2161 (2013). [CrossRef]

2. C. J. R. Sheppard, “Orthogonal aberration functions for high-aperture optical systems,” J. Opt. Soc. Am. A 21(5), 832–838 (2004). [CrossRef]

3. C. J. R. Sheppard, “Aberrations in high aperture conventional and confocal imaging system,” Appl. Opt. 27(22), 4782–4786 (1988). [CrossRef]

4. C. Gannon and R. Liang, “Using spherical harmonics to describe large-angle freeform lenses,” Appl. Opt. 57(28), 8143–8147 (2018). [CrossRef]

5. D. Ma, Z. Feng, and R. Liang, “Tailoring Freeform illumination optics in a double-pole coordinate system,” Appl. Opt. 54(9), 2395–2399 (2015). [CrossRef]

6. S. Rusinkiewicz, “A survey of BRDF representation for computer graphics. (1997),” http://www.cs.princeton.edu/∼smr/cs348c-97/surveypaper.html

7. P. M. Lam, C. S. Leung, and T. T. Wong, “Noise-resistant fitting for spherical harmonics,” IEEE Trans. Visual. Comput. Graphics 12(2), 254–265 (2006). [CrossRef]

8. H. Xu and Y. Sun, “Compact Representation of Spectral BRDFs Using Fourier Transform and Spherical Harmonic Expansion,” Comput. Graph. Forum 25(4), 759–775 (2006). [CrossRef]

9. C. M. Oliveira, A. Ferreira, and S. Franco, “Wavefront analysis and Zernike polynomial decomposition for evaluation of corneal optical quality,” J. Cataract Refractive Surg. 38(2), 343–356 (2012). [CrossRef]

10. D. Robert Iskander, “Modeling video-keratoscopic height data with spherical harmonics,” Optom. Vis. Sci. 86(5), 542–547 (2009). [CrossRef]

11. M. Mandea and M. Korte, Geomagnetic Observations and Models (Springer, 2011).

12. F. Vervelidou and E. Thebault, “Global maps of the magnetic thickness and magnetization of the Earth’s lithosphere,” Earth Planet Sp. 67(1), 173–191 (2015). [CrossRef]

13. D. J. Mortlock, A. D. Challinor, and M. P. Hobson, “Analysis of cosmic microwave background data on an incomplete sky,” Mon. Not. R. Astron. Soc. 330(2), 405–420 (2002). [CrossRef]

14. P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, and M. Ashdown, “Plank 2013 result. XII. Diffuse component separation,” Astron. Astrophys. 571, A12 (2014). [CrossRef]

15. F. Zernike, “Diffraction theory of knife-edge test and its improved form, the phase contrast method,” Mon. Not. R. Astron. Soc. 94(5), 377–384 (1934). [CrossRef]

16. L. R. Alldredge, “Rectangular harmonic analysis applied to the geomagnetic field,” J. Geophys. Res. 86(B4), 3021–3026 (1981). [CrossRef]

17. K. M. Gorski, “On determining the spectrum of primordial inhomogeneity from the COBE DMR sky maps: I. Method,” Astrophys. J. 430, L85 (1994). [CrossRef]

18. G. V. Haines, “Spherical cap harmonic analysis,” J. Geophys. Res. 90(B3), 2583–2591 (1985). [CrossRef]

19. E. Thebault, J. J. Schott, and M. Mandea, “Revised spherical cap harmonic analysis (R-SCHA): Validation and properties,” J. Geophys. Res. 111(B1), B01102 (2006). [CrossRef]

20. W. Jianqiang, C. Haoyuan, and C. Yinfu, “The analysis of the associated Legendre functions with non-integral degree,” Appl. Mech. Mater. 130-134, 3001–3005 (2011). [CrossRef]

21. V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58(7), 545–561 (2011). [CrossRef]

22. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29(10), 1174–1180 (1990). [CrossRef]

23. A. B. Bhatia and E. Wolf, “On the circle polynomials of Zernike and related orthogonal sets,” Math. Proc. Cambridge Philos. Soc. 50(1), 40–48 (1954). [CrossRef]

24. B. Courant and D. Hilbert, Methods of Mathematical Physics 1^st Ed. (Interscience Publishers, 1953), Vol. I, chap. 6.

25. Jen Ahrens, Mark R. P. Thomas, and Ivan Tashev, “HRTF Magnitude Modeling Using a Non-Regularized Least-Squares Fit of Spherical Harmonics Coefficients on Incomplete Data,” in Signal & Information Processing Association Summit & Conference (IEEE, 2013).

26. P.-P. Sloan, J. Kautz, and J. Snyder, “Precomputed radiance transfer for real-time rendering in dynamic, low-frequency lighting environments,” ACM Trans. Graphic. 21(3), 527–536 (2002). [CrossRef]

27. Jan Kautz, Peter-Pike Sloan, and John Snyder, “Fast, arbitrary BRDF shading for low-frequency lighting using spherical harmonics,” Proceedings of Eruographics Workshop on Rendering (2002).

28. C. J. R. Sheppard and P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44(4), 803–818 (1997). [CrossRef]

29. C. J. R. Sheppard and K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41(7), 1495–1505 (1994). [CrossRef]

	Function expression	Boundary	Physical meaning
$Θ_{1} (θ)$	$θ / θ_{b}$	$[0, π]$	Arc length from zenith to the point concerned	$\overset{⌢}{AB}$
$Θ_{2} (θ)$	$\sin (θ) / \sin (θ_{b})$	$[0, π / 2]$	Radial distance from Z axis to the point concerned	$\| BC \|$
$Θ_{3} (θ)$	$\sin (θ / 2) / \sin (θ_{b} / 2)$	$[0, π]$	Chord length from zenith to the point interested	$\| AB \|$
$Θ_{4} (θ)$	$(1 - \cos (θ)) / (1 - \cos (θ_{b}))$	$[0, π]$	Vertical distance from zenith to the point concerned	$\| AC \|$

		$H S H_{n}^{m} (s, φ)$ , $s = \sin (θ)$
n	m	Normalization facor $N_{n}^{m}$	Radial function $R_{n}^{m} (θ)$	Frequency function $Φ_{m} (φ)$
0	0	$\sqrt{2}$	1	$1 / \sqrt{2}$
1	±1	$\sqrt{3}$	$- s$	$\cos (φ), \sin (φ)$
2	0	$\sqrt{10}$	$(1 / 2) (- 3 s^{2} + 2)$	$1 / \sqrt{2}$
2	±2	$\sqrt{5 / 12}$	$3 s^{2}$	$\cos (2 φ), \sin (2 φ)$
3	±1	$\sqrt{7 / 6}$	$(3 / 2) (5 s^{3} - 4 s)$	$\cos (φ), \sin (φ)$
3	±3	$\sqrt{7 / 360}$	$- 15 s^{3}$	$\cos (3 φ), \sin (3 φ)$
4	0	$\sqrt{18}$	$(1 / 8) (35 s^{4} - 40 s^{2} + 8)$	$1 / \sqrt{2}$
4	±2	$\sqrt{1 / 20}$	$(15 / 2) (- 7 s^{4} + 6 s^{2})$	$\cos (2 φ), \sin (2 φ)$
4	±4	$\sqrt{1 / 2240}$	$105 s^{4}$	$\cos (4 φ), \sin (4 φ)$

		$Z S F_{n}^{m} (t, φ)$ , $t = \sin (θ / 2)$
n	m	Normalization factor	Radial function	Frequency function
0	0	$\sqrt{2}$	1	$1 / \sqrt{2}$
1	±1	2	$t$	$\cos (φ), \sin (φ)$
2	0	$\sqrt{6}$	$2 t^{2} - 1$	$1 / \sqrt{2}$
2	±2	$\sqrt{6}$	$t^{2}$	$\cos (2 φ), \sin (2 φ)$
3	±1	$\sqrt{8}$	$3 t^{3} - 2 t$	$\cos (φ), \sin (φ)$
3	±3	$\sqrt{8}$	$t^{3}$	$\cos (3 φ), \sin (3 φ)$
4	0	$\sqrt{10}$	$6 t^{4} - 6 t^{2} + 1$	$1 / \sqrt{2}$
4	±2	$\sqrt{10}$	$4 t^{4} - 3 t^{2}$	$\cos (2 φ), \sin (2 φ)$
4	±4	$\sqrt{10}$	$t^{4}$	$\cos (4 φ), \sin (4 φ)$

		$C S F_{n}^{m} (t, φ)$ , $w = 1 - \cos (θ)$
n	m	Normalization factor	Radial function	Frequency function
0	0	$\sqrt{2}$	1	$1 / \sqrt{2}$
1	±1	$\sqrt{6}$	$w$	$\cos (φ), \sin (φ)$
2	0	$\sqrt{5 / 2}$	$3 w^{2} - 1$	$1 / \sqrt{2}$
2	±2	$\sqrt{10}$	$w^{2}$	$\cos (2 φ), \sin (2 φ)$
3	±1	$\sqrt{7 / 2}$	$5 w^{3} - 3 w$	$\cos (φ), \sin (φ)$
3	±3	$\sqrt{14}$	$w^{3}$	$\cos (3 φ), \sin (3 φ)$
4	0	$\sqrt{18} / 8$	$35 w^{4} - 30 w^{2} + 3$	$1 / \sqrt{2}$
4	±2	$\sqrt{9 / 2}$	$7 w^{4} - 5 w^{2}$	$\cos (2 φ), \sin (2 φ)$
4	±4	$3 \sqrt{2}$	$w^{4}$	$\cos (4 φ), \sin (4 φ)$

	HSH	ZSF	LSF
Curl and divergence	Zero	Nonzero	Nonzero
Special item meaning	$H S H_{1}^{1}$ and $H S H_{1}^{- 1}$ are identical to tip and tilt of spherical wavefront respectively.	$Z S F_{2}^{0}$ is identical to defocus of spherical wavefront	None
Fit range	No larger than hemisphere	All kind of spherical cap	All kind of spherical cap
Orthogonal invariance of aperture	No	Yes	Yes

Zernike like functions on spherical cap: principle and applications in optical surface fitting and graphics rendering

Abstract

1. Introduction

2. Geometric analysis of spherical cap

3. Zernike like functions derivation on spherical cap

3.1 Zernike circle polynomial (ZCP) overview and derive rules

3.2 Zernike spherical cap functions derivation

4. Typical ZSCFs and their analytical expressions

4.1 Hemi-Spherical Harmonics (HSH)

4.2 Zernike spherical functions (ZSF)

4.3 Longitudinal spherical functions (LSF)

5. Applications

5.1 Data fitting on spherical cap

5.2 Application in computer graphic rendering

6. Summary

Appendix

7.1 Gramm-Schmidt process for Zernike like functions

7.2 Discussion on weight factor

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (4)

Tables (5)

Equations (44)

Optics Express