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 `2l
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function z = zernfun(n,m,r,theta,nflag) J*nWCL
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. {[:]}m(c
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N RTQtXv6mD
% and angular frequency M, evaluated at positions (R,THETA) on the E=$li
% unit circle. N is a vector of positive integers (including 0), and DU|>zO%
% M is a vector with the same number of elements as N. Each element hRaX!QcG3
% k of M must be a positive integer, with possible values M(k) = -N(k) 4qvE2W}&
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, mO8E-D*3
% and THETA is a vector of angles. R and THETA must have the same ~/l5ys
% length. The output Z is a matrix with one column for every (N,M) p"tCMB
% pair, and one row for every (R,THETA) pair. S!6 ? b5
% ,9YgznQ
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ^_5t5>
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), O]VHX![Y$
% with delta(m,0) the Kronecker delta, is chosen so that the integral #dhce0m
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, OYLg-S
% and theta=0 to theta=2*pi) is unity. For the non-normalized A(}D76o_
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. M"!{Dx~
% w:HRzU>
% The Zernike functions are an orthogonal basis on the unit circle.
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% They are used in disciplines such as astronomy, optics, and F1GFn|OA
% optometry to describe functions on a circular domain. )l6(ss!J
% kK%@cIXS3
% The following table lists the first 15 Zernike functions. :D:Y-cG*n<
% K*9~g('
% n m Zernike function Normalization 6^NL>|?
% -------------------------------------------------- {'NXJ!I;t
% 0 0 1 1 )uRR!<"~
% 1 1 r * cos(theta) 2 mPJ@hr%3
% 1 -1 r * sin(theta) 2 lEXI<b'2
% 2 -2 r^2 * cos(2*theta) sqrt(6) tb/`*Yl@
% 2 0 (2*r^2 - 1) sqrt(3) *6/OLAkyF
% 2 2 r^2 * sin(2*theta) sqrt(6) :zp9L/eh
% 3 -3 r^3 * cos(3*theta) sqrt(8) rk8Cea
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) .Ge`)_e
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 9][A1+"
% 3 3 r^3 * sin(3*theta) sqrt(8) Vu5Djx'
% 4 -4 r^4 * cos(4*theta) sqrt(10) ,{Ga7rH*
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) %G/(7l[W
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) #&,~5
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 7 0Wy]8<P
% 4 4 r^4 * sin(4*theta) sqrt(10) p|n!R $_g\
% -------------------------------------------------- FM,o&0HSd
% ,buo&DT{L
% Example 1: <[A;i
% $J9/AFzO"
% % Display the Zernike function Z(n=5,m=1) Rg SB?
% x = -1:0.01:1; ~9Cw5rwH<;
% [X,Y] = meshgrid(x,x); fRK=y+gl@
% [theta,r] = cart2pol(X,Y); KMP[Ledr
% idx = r<=1; zn#lFPj12
% z = nan(size(X)); *hlinQKs
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 9S/X ,|i
% figure D!rD-e
% pcolor(x,x,z), shading interp \2[sUY<W
% axis square, colorbar S
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% title('Zernike function Z_5^1(r,\theta)') Nn{/_QG
% q854k+C
% Example 2: yC\!6pg
% L*zfZ&
% % Display the first 10 Zernike functions S.|%dz
% x = -1:0.01:1; TXbnK"XQ
% [X,Y] = meshgrid(x,x); 6F; |x
% [theta,r] = cart2pol(X,Y); tvOyT6 ]
% idx = r<=1; ?o`fX
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% z = nan(size(X)); ZO&F15$P
% n = [0 1 1 2 2 2 3 3 3 3]; 4XNkto
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; nVoP:FHH
% Nplot = [4 10 12 16 18 20 22 24 26 28]; %
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% y = zernfun(n,m,r(idx),theta(idx)); HY,VJxR[
% figure('Units','normalized') 7VW/v4n
% for k = 1:10 \me-#: Gu
% z(idx) = y(:,k); qF4=MQm\aE
% subplot(4,7,Nplot(k)) ,~>u<Wc!S
% pcolor(x,x,z), shading interp \OVw
% set(gca,'XTick',[],'YTick',[]) o?><(A|
% axis square b5?k)s2
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 5@EX,$h
% end Fiaeo0
% )NnkoCNeE
% See also ZERNPOL, ZERNFUN2. v-XB\|f
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% Paul Fricker 11/13/2006 cdTG ]n
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q(Zu;ecBN
% Check and prepare the inputs: 7l3Dxw/N
% -----------------------------
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) T#ehJq 5
error('zernfun:NMvectors','N and M must be vectors.') iCdq-r/r!6
end Kgb<uXk
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if length(n)~=length(m) %:Y'+!bX
error('zernfun:NMlength','N and M must be the same length.') ew1bb K>
end LEA^o"NW.
v2 }>/b)
n = n(:); BV
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m = m(:); hSXZu?/
if any(mod(n-m,2)) tx]!|x" F
error('zernfun:NMmultiplesof2', ... ZqfoO!Ta
'All N and M must differ by multiples of 2 (including 0).') $}.#0c8I
end w
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Y 016Xg5
if any(m>n) 7vEZb.~4z
error('zernfun:MlessthanN', ... YiC_,8A~
'Each M must be less than or equal to its corresponding N.') ~i=5NUE
end lQ| i
Ws
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if any( r>1 | r<0 ) E[tEW0ub
error('zernfun:Rlessthan1','All R must be between 0 and 1.') '/@i}
digf
end q@}tv=}
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 6]Ri$V&"
error('zernfun:RTHvector','R and THETA must be vectors.') 5 0<
end 0ae}!LO
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r = r(:); Q*]y=Za#:
theta = theta(:); Bu#\W
length_r = length(r); |1UJKJwX
if length_r~=length(theta) Rs53R$PIR
error('zernfun:RTHlength', ... g BV66L
'The number of R- and THETA-values must be equal.') }bYk#6KX
end CxJH)H$
RaAvPIJa |
% Check normalization: qrY]tb^K
% -------------------- $GX9-^og=T
if nargin==5 && ischar(nflag) W(jP??up
isnorm = strcmpi(nflag,'norm'); CChCxB
if ~isnorm ,dSP%?vV
error('zernfun:normalization','Unrecognized normalization flag.') dwmZ_m.
end ~jM!8]=
else 5
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isnorm = false; Tw!_=zy(Gw
end HsAKz]Mq
EALgBv>#ZL
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +t<'{KZ7;
% Compute the Zernike Polynomials u;=a=>05IR
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t"FB}%G
at5=Zo[bP
% Determine the required powers of r: uOQl;}Lk5
% ----------------------------------- NZt
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m_abs = abs(m); @1+({u#B
rpowers = []; .{66q#.
for j = 1:length(n) ,B$m8wlI|
rpowers = [rpowers m_abs(j):2:n(j)]; NEcE-7aT
end Un{ 9reX5
rpowers = unique(rpowers); {{Z3M>Q
btv.M
% Pre-compute the values of r raised to the required powers, ]B9Ut&mF;
% and compile them in a matrix: V.~C.x
% ----------------------------- KmaMS(A(3
if rpowers(1)==0 p|VgtQ/)%
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Hy#<fKz`!
rpowern = cat(2,rpowern{:}); WcKL=Z?(
rpowern = [ones(length_r,1) rpowern]; o 9{~F`{p
else \,yX3R3}.~
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); IjnO2X
rpowern = cat(2,rpowern{:}); w$z]Z-
end VVm8bl.q
_.K<#S
% Compute the values of the polynomials: nZ~J&QK-
% -------------------------------------- -aF\
u[b
y = zeros(length_r,length(n)); E:S (v
for j = 1:length(n) ky|Py
s = 0:(n(j)-m_abs(j))/2; VXIB9
/*i
pows = n(j):-2:m_abs(j); 1g bqHxWI
for k = length(s):-1:1 [Z{0|NR
p = (1-2*mod(s(k),2))* ... w[?E
oFI$Y
prod(2:(n(j)-s(k)))/ ... +oR wXO3W
prod(2:s(k))/ ... U+'h~P'4
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... _Sn7z?
prod(2:((n(j)+m_abs(j))/2-s(k))); ,5/zTLd
idx = (pows(k)==rpowers); o~={M7m
y(:,j) = y(:,j) + p*rpowern(:,idx); J#jx)K!
end [+z*&~'
Bd-@@d.H<
if isnorm !i*bb~
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); #ybtjsu'"U
end <R@w0b>
end kSH|+K\M4
% END: Compute the Zernike Polynomials "I)`gy&
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
9M!J7 W
;PF!=8dW
% Compute the Zernike functions: dsD!)$
% ------------------------------ pv){R;f
idx_pos = m>0; CJ#1j>
idx_neg = m<0; 4l`"P~=2<
b$G&i'd
z = y; cuW&X9\m,
if any(idx_pos) C6cEt5
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); '}.Z' %;
end 1*ui|fuK
if any(idx_neg) =}7[ypQM`]
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ew{(@p+$
end n*vzp?+Y
mq*Efb)!
% EOF zernfun