非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 {kY`X[fvZ
function z = zernfun(n,m,r,theta,nflag) 1*eWvYo1
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. s525`Q;
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N //4p1^%
% and angular frequency M, evaluated at positions (R,THETA) on the 0X}w[^f
% unit circle. N is a vector of positive integers (including 0), and l")o!N?
% M is a vector with the same number of elements as N. Each element Bt`r6v;\
% k of M must be a positive integer, with possible values M(k) = -N(k) ;r2b@x:<_
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, s]V{}bY`
% and THETA is a vector of angles. R and THETA must have the same l#J>It\
% length. The output Z is a matrix with one column for every (N,M) OM.(g%2
% pair, and one row for every (R,THETA) pair. plz=G}Y
% QKL]O*
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike pqNoL*
H
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ua. 6?W)
% with delta(m,0) the Kronecker delta, is chosen so that the integral .,iw2:
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, #!F8n` C-
% and theta=0 to theta=2*pi) is unity. For the non-normalized [))2u:tbS\
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. *<
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% )9;kzp/
% The Zernike functions are an orthogonal basis on the unit circle. im^I9G
% They are used in disciplines such as astronomy, optics, and `b,g2XA
% optometry to describe functions on a circular domain. 07HX5 Hd
% ]T28q/B;k
% The following table lists the first 15 Zernike functions. 6b1 Uj<
% Q=9VuTE
% n m Zernike function Normalization cR@}
% -------------------------------------------------- KcMzZ!d7m
% 0 0 1 1 ;tIIEc
% 1 1 r * cos(theta) 2 `:^)"#z)
% 1 -1 r * sin(theta) 2 _|2";.1E
% 2 -2 r^2 * cos(2*theta) sqrt(6) XQ?)
% 2 0 (2*r^2 - 1) sqrt(3) H6+st`{
% 2 2 r^2 * sin(2*theta) sqrt(6) %%-Tjw o
% 3 -3 r^3 * cos(3*theta) sqrt(8) Bg
8t'dw?K
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) i~M.F=I5
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) wE=8jl*
% 3 3 r^3 * sin(3*theta) sqrt(8) ~m"M#1,ln3
% 4 -4 r^4 * cos(4*theta) sqrt(10) ZBh@%A
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +\]S<T*;
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) QH56tQq
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) }Q,C;!'"
% 4 4 r^4 * sin(4*theta) sqrt(10) ?2
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% -------------------------------------------------- v+,
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% bgx5{!A
% Example 1: Y{\2wU!Isn
% -ZMl[;OM
% % Display the Zernike function Z(n=5,m=1) )Z`viT
% x = -1:0.01:1; Z_TbM^N
% [X,Y] = meshgrid(x,x); [+5SEr}
% [theta,r] = cart2pol(X,Y); 6-E4)0\
% idx = r<=1; 8CHf. SXh
% z = nan(size(X)); eXtF[0f
% z(idx) = zernfun(5,1,r(idx),theta(idx)); )I1V2k$n
% figure :Y&W)V-
% pcolor(x,x,z), shading interp Zi '8~iEH
% axis square, colorbar 75cr!+
% title('Zernike function Z_5^1(r,\theta)') enO=-#
% 7B> cmi
% Example 2: jZgnt{
% r_2VExk
% % Display the first 10 Zernike functions 7.=s1~p
% x = -1:0.01:1; 0DjBqh$
% [X,Y] = meshgrid(x,x); (%^TTe
% [theta,r] = cart2pol(X,Y); KLM^O$=
% idx = r<=1; 4rCqN.J
% z = nan(size(X)); X\:(8C;+
% n = [0 1 1 2 2 2 3 3 3 3]; A-NC,3
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Kh_>V m/
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ?@7|Q/
% y = zernfun(n,m,r(idx),theta(idx)); qQ\hUii
% figure('Units','normalized') ZtZV:re=
% for k = 1:10 >WG91b<Xq
% z(idx) = y(:,k); VHkrPJ[
% subplot(4,7,Nplot(k)) i_9/!D
% pcolor(x,x,z), shading interp F;l<>|vG
% set(gca,'XTick',[],'YTick',[]) UEb'E;
% axis square eh#
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% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) q+=@kXs>+
% end I.0Usa"z
% w\[*_wQp
% See also ZERNPOL, ZERNFUN2. ^C#bW<T
Bc`A]U
% Paul Fricker 11/13/2006 g{.@|;d<p
nWg)zj:
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% Check and prepare the inputs: ~.=HN}E
% ----------------------------- IOsDVIXL\
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) g0U\AN
error('zernfun:NMvectors','N and M must be vectors.') G\+MT(&5
end <cd%n-
Cl){sP=8W
if length(n)~=length(m) &^<T/PiR
error('zernfun:NMlength','N and M must be the same length.') @g` ,'r
end 00
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n = n(:); JHOBg{Wg
m = m(:); Nv#, s_hG
if any(mod(n-m,2)) {dH<Un(4Z
error('zernfun:NMmultiplesof2', ... ]qTr4`.
'All N and M must differ by multiples of 2 (including 0).') L{ ^@O0S
end YZ+g<HXB
sX?7`n1U
if any(m>n) Rs`Y'_B
error('zernfun:MlessthanN', ... g#&##f
'Each M must be less than or equal to its corresponding N.') nf^k3QS\
end ooxzM `
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if any( r>1 | r<0 ) pNsLoNZ3w
error('zernfun:Rlessthan1','All R must be between 0 and 1.')
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end tOVm~C,R
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) xJ.!Q)[
error('zernfun:RTHvector','R and THETA must be vectors.') [l{eJ/W
end b,sc
T`G"2|ISS
r = r(:); SuuS!U+i>
theta = theta(:); hS/'b$#
length_r = length(r); 73<yrBxp
if length_r~=length(theta) ~n\ea:.
error('zernfun:RTHlength', ... n#,l&Bx
'The number of R- and THETA-values must be equal.') |a\TUzq
end H2KY$;X[
pZn%g]nRD
% Check normalization: HK.J/Zr
% -------------------- w#b2iE+Bw
if nargin==5 && ischar(nflag) \mGM#E
isnorm = strcmpi(nflag,'norm'); {bEEQCweNJ
if ~isnorm ApBThW*E
error('zernfun:normalization','Unrecognized normalization flag.') J8'zvH&I
end +.uk#K0o
else k"c_x*f
isnorm = false; e8v=n@0
end s]>%_(5
vRs5-T
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% hgltD8,
% Compute the Zernike Polynomials U0T N8O}Z
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }aIfIJ
'kK%sE
% Determine the required powers of r: z5/O8}Gz@
% ----------------------------------- >c eU!=>
m_abs = abs(m); gV;GC{pY
rpowers = []; &o.SmkJI
for j = 1:length(n) 'h=2_%l@Y
rpowers = [rpowers m_abs(j):2:n(j)]; 8m0sEV>
end !}7m^
rpowers = unique(rpowers); s9>!^MzBK
VV0$L=mo
% Pre-compute the values of r raised to the required powers, :Yqa[._AF
% and compile them in a matrix: U @|_5[nl
% ----------------------------- eW%jDsC
if rpowers(1)==0 vS#]RW&j
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 5K<C
rpowern = cat(2,rpowern{:}); 7m:, -xp
rpowern = [ones(length_r,1) rpowern]; GAKJc\o
else i2E7$[
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); - %|I
rpowern = cat(2,rpowern{:}); \9^@,kfP
end b.&YUg[#
<Z;BB)I&C`
% Compute the values of the polynomials: jEI L(0_H
% -------------------------------------- .VT,,0
y = zeros(length_r,length(n)); `314.a6S
for j = 1:length(n) Y`uCDfcQ
s = 0:(n(j)-m_abs(j))/2; {{\HU0g>&
pows = n(j):-2:m_abs(j); aT #|mk=\
for k = length(s):-1:1 iqeGy&F-
p = (1-2*mod(s(k),2))* ... W!*vO>^1W
prod(2:(n(j)-s(k)))/ ... %+~0+ev7r
prod(2:s(k))/ ... ^da-R;o]
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... v]~[~\|a
prod(2:((n(j)+m_abs(j))/2-s(k))); ix;8S=eP~{
idx = (pows(k)==rpowers); ?%(*bRV -
y(:,j) = y(:,j) + p*rpowern(:,idx); /_\4(vvf
end g:yK/1@Hk}
z?xd\x
if isnorm Z/x~:u_
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 0'uj*Y{L
end FceT'
end &0raa
% END: Compute the Zernike Polynomials q`hg@uwA{`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]Ea-?IhD
$ }53f'QjW
% Compute the Zernike functions: yyc&'J
% ------------------------------ U' Cp3>
idx_pos = m>0; 2ip~qZNw><
idx_neg = m<0; r+Y1m\
v]v f(]""
z = y; "'Ik{wGc
if any(idx_pos) xlAaIo)T
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); {8Ll\j@ "
end /_P`xm+=AC
if any(idx_neg) W2RS G~|
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); P1<;:!8'
end sp%7iNs
<Vim\
% EOF zernfun