function z = zernpol(n,m,r,nflag) pm6#azQ
%ZERNPOL Radial Zernike polynomials of order N and frequency M. o$No@~%v
% Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of U~x]2{}
% order N and frequency M, evaluated at R. N is a vector of <\&9Odqc
% positive integers (including 0), and M is a vector with the \d@5*q
% same number of elements as N. Each element k of M must be a hq=;ZI
% positive integer, with possible values M(k) = 0,2,4,...,N(k) :21d
% for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd. R is +RooU?Aq
% a vector of numbers between 0 and 1. The output Z is a matrix b>uD-CSA
% with one column for every (N,M) pair, and one row for every 5CY%h
% element in R. ok:uTeJI
% 4IeCb?
% Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- E8PDIjp
% nomials. The normalization factor Nnm = sqrt(2*(n+1)) is mku@n;Hl_
% chosen so that the integral of (r * [Znm(r)]^2) from r=0 to xKUL}>8
% r=1 is unity. For the non-normalized polynomials, Znm(r=1)=1 5BVvT
`<
% for all [n,m]. P 45Irir
% K<>kT4
% The radial Zernike polynomials are the radial portion of the F3|^b{'zO
% Zernike functions, which are an orthogonal basis on the unit , PlH|
% circle. The series representation of the radial Zernike FNQ<k[#K'~
% polynomials is ,2M}qs"P7G
% Z8SwW<{ $
% (n-m)/2 d[a(uWEl
% __ nR_Zrm
% m \ s n-2s CHgip&(.F
% Z(r) = /__ (-1) [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r "hk#pQ
% n s=0 o[ 5dR<
% qf? "v;
% The following table shows the first 12 polynomials. p D<w@2K
% bZ`v1d
(r
% n m Zernike polynomial Normalization vIV|y>;g
% --------------------------------------------- CWdsOS=
% 0 0 1 sqrt(2) @DysM~I
% 1 1 r 2 BSm"]!D8*
% 2 0 2*r^2 - 1 sqrt(6) +z;*r8d<X
% 2 2 r^2 sqrt(6) H>TO8;5(
% 3 1 3*r^3 - 2*r sqrt(8) f[o~d`z
% 3 3 r^3 sqrt(8) UoT`/.
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(10) As,`($=
% 4 2 4*r^4 - 3*r^2 sqrt(10) Y1PR?c
Q
% 4 4 r^4 sqrt(10) y'2|E+*V
% 5 1 10*r^5 - 12*r^3 + 3*r sqrt(12) ^k6_j\5j
% 5 3 5*r^5 - 4*r^3 sqrt(12) & zDuh[j}
% 5 5 r^5 sqrt(12) xM jn=\}
% --------------------------------------------- /!5Wd(:
% )?rq8VO
% Example: h^3gYL7O6
% 5u$.!l8Nl
% % Display three example Zernike radial polynomials R"t#dG]1t
% r = 0:0.01:1; '/>Mr!H#
% n = [3 2 5]; s#X/
F
% m = [1 2 1]; )iN;1>
% z = zernpol(n,m,r); 3aqH!?rVU
% figure \YvG+7a
% plot(r,z) hr
vTFJ
% grid on '@{:FrG*U
% legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') e9LX0=
% @)Vpj\jM-C
% See also ZERNFUN, ZERNFUN2. |
eBwcC#^
"Z@P&jl
% A note on the algorithm. qDdO-fPev
% ------------------------ Y%
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% The radial Zernike polynomials are computed using the series m.<_WXH
% representation shown in the Help section above. For many special 5H3o?x
% functions, direct evaluation using the series representation can 65LtCQ}
% produce poor numerical results (floating point errors), because o#qdgZ
% the summation often involves computing small differences between j)J |'b|
% large successive terms in the series. (In such cases, the functions dseI~}
% are often evaluated using alternative methods such as recurrence j yHa}OT
% relations: see the Legendre functions, for example). For the Zernike f{9+,z
% polynomials, however, this problem does not arise, because the ^to*ET{0
% polynomials are evaluated over the finite domain r = (0,1), and o5Oig
% because the coefficients for a given polynomial are generally all owNwj
% of similar magnitude. #49l\>1z
% $?GggP d
% ZERNPOL has been written using a vectorized implementation: multiple tc~gn!"
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] vXKL<