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 nF:4}qy\
function z = zernfun(n,m,r,theta,nflag) U>SShpmZA
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. T<>,lQs(a
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ,THw"bm
% and angular frequency M, evaluated at positions (R,THETA) on the `[yKFa
I
% unit circle. N is a vector of positive integers (including 0), and =%O6:YM
% M is a vector with the same number of elements as N. Each element m7V/zne
% k of M must be a positive integer, with possible values M(k) = -N(k)
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% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, DfD&)tsMQ
% and THETA is a vector of angles. R and THETA must have the same B-Hrex]
% length. The output Z is a matrix with one column for every (N,M) hfB%`x#akQ
% pair, and one row for every (R,THETA) pair. {TROoX~H?
% MchA{p&Ol
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike YP<ms
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ^DLfY-F+j
% with delta(m,0) the Kronecker delta, is chosen so that the integral |-ALklXr
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, E]d.z6k
% and theta=0 to theta=2*pi) is unity. For the non-normalized =XQ%t
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% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ,qwuLBW
% C): 1?@
% The Zernike functions are an orthogonal basis on the unit circle. ]/6z;
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% They are used in disciplines such as astronomy, optics, and 3=[mP,pLh
% optometry to describe functions on a circular domain. {Xy5pfW
Q
% J)>c9w
% The following table lists the first 15 Zernike functions. >Tx?%nQ
% .Hm>i
% n m Zernike function Normalization v1JzP#
% -------------------------------------------------- t?gic9
q
% 0 0 1 1 BlO<PMmhT&
% 1 1 r * cos(theta) 2 s8Q 5ui]
% 1 -1 r * sin(theta) 2 re<{
>
% 2 -2 r^2 * cos(2*theta) sqrt(6) 2,F.$X
% 2 0 (2*r^2 - 1) sqrt(3) F(n$
% 2 2 r^2 * sin(2*theta) sqrt(6) P+sW[:
% 3 -3 r^3 * cos(3*theta) sqrt(8) I{2hfKUe`
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) C )
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% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ]W!0$'o
% 3 3 r^3 * sin(3*theta) sqrt(8) T-L||yE,h
% 4 -4 r^4 * cos(4*theta) sqrt(10) Zi
i
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) & .j&0WE
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) _[3D
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :Yl-w-oe
% 4 4 r^4 * sin(4*theta) sqrt(10) JQI: sj
% -------------------------------------------------- 6 "sSo j
% *fxG?}YT
% Example 1: J@'wf8Ub
% ITBE|b
% % Display the Zernike function Z(n=5,m=1)
(ZizuHC
% x = -1:0.01:1; 'H!Uh]!
% [X,Y] = meshgrid(x,x); EVSX.'&f
% [theta,r] = cart2pol(X,Y); T^KKy0ZGM
% idx = r<=1; p6@)-2^
% z = nan(size(X)); dn3y\
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 7}>E J
% figure Il'fL'3
% pcolor(x,x,z), shading interp ~
7s!VR
% axis square, colorbar kevrsV]/$
% title('Zernike function Z_5^1(r,\theta)') 4VSU8tK|N]
% \b x$i*
% Example 2: ?`ZUR&
20
% q} >%8;nm
% % Display the first 10 Zernike functions X"Swi&4
% x = -1:0.01:1; >bW#Zs,6
% [X,Y] = meshgrid(x,x); oPM96
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% [theta,r] = cart2pol(X,Y); PZ9I`P!C
% idx = r<=1; R 9\*#c
% z = nan(size(X)); /x$ nje,.
% n = [0 1 1 2 2 2 3 3 3 3]; H{wl% G
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ?tbrbkx
% Nplot = [4 10 12 16 18 20 22 24 26 28]; c@7rqHU-0
% y = zernfun(n,m,r(idx),theta(idx)); lo+A%\1
% figure('Units','normalized') .q>iXE_c
% for k = 1:10 } Kgy
% z(idx) = y(:,k); ga +dt
% subplot(4,7,Nplot(k)) VPo".BvG6
% pcolor(x,x,z), shading interp S1_RjMbYM
% set(gca,'XTick',[],'YTick',[]) K|,
.C[
% axis square f:}
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% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ]=BB#
% end z}
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% 0 H:X3y+
% See also ZERNPOL, ZERNFUN2. hgq;`_;1,
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% Paul Fricker 11/13/2006 1qA;/-Zr<o
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% Check and prepare the inputs: JZx[W&]zT
% ----------------------------- bt?5*ETA
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) h376Be{P
error('zernfun:NMvectors','N and M must be vectors.') z b3tIRH
end 75lA%|
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if length(n)~=length(m) ^cWnF0)j.
error('zernfun:NMlength','N and M must be the same length.') ob]w;"
end R|(a@sL
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n = n(:); ih3n<gXF
m = m(:); ?r4>" [
if any(mod(n-m,2)) UN#S;x*
error('zernfun:NMmultiplesof2', ... nw<uyaU-t
'All N and M must differ by multiples of 2 (including 0).') }SZd
end i%?* @uj
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if any(m>n) "AqB$^S9t
error('zernfun:MlessthanN', ... DEgXQ[
'Each M must be less than or equal to its corresponding N.') h(DTa
end H PVEnVn
n@3>6_^rwT
if any( r>1 | r<0 ) ;'1d1\wiDQ
error('zernfun:Rlessthan1','All R must be between 0 and 1.') o8MZiU1Xf
end c71y'hnT
"[N!m1i:{
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) {!`6zBsP
error('zernfun:RTHvector','R and THETA must be vectors.') x+]"
end 2~V*5~fb
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r = r(:); ['D]>Ot68
theta = theta(:); Lw,h+@0
length_r = length(r); zt%Mx>V@
if length_r~=length(theta) >\8+:oS^
error('zernfun:RTHlength', ... LzL
So"n
'The number of R- and THETA-values must be equal.') 8P`"M#fI
end *
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!,PWb3S
% Check normalization: ~TtiO#,t
% -------------------- {;oPLr+Z
if nargin==5 && ischar(nflag) W,u:gzmhw
isnorm = strcmpi(nflag,'norm'); 7zc^!LrW<
if ~isnorm <UCl@5g&
error('zernfun:normalization','Unrecognized normalization flag.') K sCyFp
end L/[K"
else :T~ [
isnorm = false; HaYo!.(Fv
end 2mU.7!g)
:Dp0?&_
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Bbc^FHip
% Compute the Zernike Polynomials wIgS3K
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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unxqkU/<Z
% Determine the required powers of r: FI.\%x
% ----------------------------------- AZ<=o
m_abs = abs(m); xz]~ jL@-]
rpowers = []; 6u%&<")4HP
for j = 1:length(n) pCG}ZKa
rpowers = [rpowers m_abs(j):2:n(j)]; /wv0i3_e
end '"Nr, vQo
rpowers = unique(rpowers); m {}Lm)M
jiGTA:v
% Pre-compute the values of r raised to the required powers, y7<|_:00
% and compile them in a matrix: Wn6Sn{8W{
% ----------------------------- k:%%/
if rpowers(1)==0 Q{/Ef[(a@
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); xD 7]C|8o
rpowern = cat(2,rpowern{:}); g)B]FH1
rpowern = [ones(length_r,1) rpowern]; OTv)
else JGZBL{8
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); r_d!ikOT(
rpowern = cat(2,rpowern{:}); iow"n$/
end 9H~n_
[>9is=>o.
% Compute the values of the polynomials: ->jDb/a{C
% -------------------------------------- A}^mdw9
y = zeros(length_r,length(n)); A}w/OA97RO
for j = 1:length(n) %2h>-.tY
s = 0:(n(j)-m_abs(j))/2; fV~~J2IK
pows = n(j):-2:m_abs(j); dWW.Y*339
for k = length(s):-1:1 C,zohlpC
p = (1-2*mod(s(k),2))* ... 'fW-Y!k%
prod(2:(n(j)-s(k)))/ ... ^f@=:eWI
prod(2:s(k))/ ... +ai<
q>+
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... tX[WH\(xI
prod(2:((n(j)+m_abs(j))/2-s(k)));
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idx = (pows(k)==rpowers); MOC/KNb
y(:,j) = y(:,j) + p*rpowern(:,idx); R-14=|7a-
end u:b=\T L
4z)]@:`}z
if isnorm k{0o9,
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 4!$"ayGv;D
end <naz+QK'
end yQrD9*t&g
% END: Compute the Zernike Polynomials .]Z"C&"N]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k=^xVQuI
('~LMu_
% Compute the Zernike functions: `_h&glMJ,q
% ------------------------------ Hp?/a?\Xm
idx_pos = m>0; $Q0n
idx_neg = m<0; =u;MCQ[
JS77M-Ac
z = y; t,'<gI
if any(idx_pos) >sbu<|]a
7
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)');
8Y?;x}
end !'Kjx
if any(idx_neg) ]^]wP]R_
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 9u:Q,0\
end >3bCTE
V.Mry`9-
% EOF zernfun