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
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function z = zernfun(n,m,r,theta,nflag) ~O|0.)71]
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 97&6i TYA
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N DV.MvFV
% and angular frequency M, evaluated at positions (R,THETA) on the !nYAyjf
% unit circle. N is a vector of positive integers (including 0), and >l7
o/*4
% M is a vector with the same number of elements as N. Each element WW_X:N~~e\
% k of M must be a positive integer, with possible values M(k) = -N(k) N CsUC
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, lA ,%'+-
% and THETA is a vector of angles. R and THETA must have the same oC?b]tzj
% length. The output Z is a matrix with one column for every (N,M) +0a',`yc
% pair, and one row for every (R,THETA) pair. xFvSQ`sp
% =kCpCpET
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike mee-Qq:}
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), n/8fv~zU
% with delta(m,0) the Kronecker delta, is chosen so that the integral [+%*s3`c#
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ~/.&Z`ls
% and theta=0 to theta=2*pi) is unity. For the non-normalized +HcH]D;
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Fb}9cpz{
% fklMYu4:n
% The Zernike functions are an orthogonal basis on the unit circle. C[Fh^
% They are used in disciplines such as astronomy, optics, and w5|"cD#8A
% optometry to describe functions on a circular domain. 8<G@s`*
% LnL<WI*Pq
% The following table lists the first 15 Zernike functions. Ay_<?F+&
% +u
Lu.-N
% n m Zernike function Normalization lg=[cC2
% -------------------------------------------------- 5eU/ [F9
% 0 0 1 1 duqu}*Jw
% 1 1 r * cos(theta) 2 N ;hq
% 1 -1 r * sin(theta) 2 E }yxF.
% 2 -2 r^2 * cos(2*theta) sqrt(6) Rza\n8
% 2 0 (2*r^2 - 1) sqrt(3) {P3,jY^
% 2 2 r^2 * sin(2*theta) sqrt(6) f9rToH
% 3 -3 r^3 * cos(3*theta) sqrt(8) xpnnWHdaq
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) HW d,1
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) b9v Kux
% 3 3 r^3 * sin(3*theta) sqrt(8) xv ja
% 4 -4 r^4 * cos(4*theta) sqrt(10) |~/{lE=I
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *83+!DV|
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) M aEh8*
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) jgYiuM3c\
% 4 4 r^4 * sin(4*theta) sqrt(10) 5_O.p3$tV
% -------------------------------------------------- AsLAm#zq
% 'X?`+2wK
% Example 1: '=ZE*nGC
% $M8'm1R9
% % Display the Zernike function Z(n=5,m=1) 3!
+5MsR+
% x = -1:0.01:1; oT_,k}L IX
% [X,Y] = meshgrid(x,x); l5MxJ>?4%B
% [theta,r] = cart2pol(X,Y); JDs<1@ \
% idx = r<=1; }Yt0VtLt
% z = nan(size(X)); x O)nS _I
% z(idx) = zernfun(5,1,r(idx),theta(idx)); t (1z+
% figure 5M(?_qj
% pcolor(x,x,z), shading interp qB&*"gf
% axis square, colorbar #"Zr#P{P
% title('Zernike function Z_5^1(r,\theta)') JrQN-e!
% s 2$R2,
% Example 2: 7OZs~6(
% w_-{$8|
% % Display the first 10 Zernike functions bZi>
% x = -1:0.01:1; k-89(
% [X,Y] = meshgrid(x,x); QVP
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% [theta,r] = cart2pol(X,Y); I?PKc'b
% idx = r<=1; *7R3EUUk
% z = nan(size(X)); 5GY%ZRHh
% n = [0 1 1 2 2 2 3 3 3 3]; G ;z2}Ei
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ecFI"g
% Nplot = [4 10 12 16 18 20 22 24 26 28]; h8h4)>:
% y = zernfun(n,m,r(idx),theta(idx)); ]EK"AuEz`
% figure('Units','normalized') @#V{@@3$
% for k = 1:10 o1Xk\R{
% z(idx) = y(:,k); +F/ '+
% subplot(4,7,Nplot(k)) -0kwS4Hx2
% pcolor(x,x,z), shading interp V^0*S=N
% set(gca,'XTick',[],'YTick',[]) `KL`^UqR
% axis square
V`%m~#Me
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) /Ly%-py-$
% end "qF&%r'
% eL(T
% See also ZERNPOL, ZERNFUN2. [qy@g5`
%0]&o,
w{
% Paul Fricker 11/13/2006 *s!8BwiE
&
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% Check and prepare the inputs: lf Giw^
% ----------------------------- 'UB<;6wy
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) j{HxX
error('zernfun:NMvectors','N and M must be vectors.') `$i`i 'S
end +(<CE#bb[
A$
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`
if length(n)~=length(m) &
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error('zernfun:NMlength','N and M must be the same length.') rdC(+2+Ay
end B@F 1!8l
jem$R/4"
n = n(:); 9<Bf5d
m = m(:); weu'<C
if any(mod(n-m,2)) 0zEn`rq&
error('zernfun:NMmultiplesof2', ... @^P=jXi<
'All N and M must differ by multiples of 2 (including 0).') b\^.5SEw
end 9M7{.XR,
9]S}m[8k
if any(m>n) h)YqC$A-s
error('zernfun:MlessthanN', ... ! g}9xIL
'Each M must be less than or equal to its corresponding N.') 0h; -Yg
end Q0r_+0[7j
aMHIOA%Kh
if any( r>1 | r<0 ) VRxBi!d
error('zernfun:Rlessthan1','All R must be between 0 and 1.') C ]#R7G
end H8\N~>
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) NfN#q:w1
error('zernfun:RTHvector','R and THETA must be vectors.') B4{A(-Tc
end Ck[Z(=b$$:
xi.;`Q^#
r = r(:); !|`YNsR
theta = theta(:); E-Mp|y /V
length_r = length(r); +ivz
if length_r~=length(theta) ,{.&xJ$
error('zernfun:RTHlength', ... +)V6"XY-(
'The number of R- and THETA-values must be equal.') Gd'^vqo<
end (K2 p3M^
sd=i!r)ya
% Check normalization: Pajr`gU
% -------------------- 1ltoLd\{
if nargin==5 && ischar(nflag) ;/YSQt)rc>
isnorm = strcmpi(nflag,'norm'); lxxK6;r~>
if ~isnorm -nU_eDy
error('zernfun:normalization','Unrecognized normalization flag.') $D45X<
end #}A!Bk
else on(W^ocnD
isnorm = false; VR_1cwKBM
end hup]Jk
&'(:xjN
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TM"i9a? ;
% Compute the Zernike Polynomials EKDv3aFQZ#
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xxedezNko
L=VuEF
% Determine the required powers of r: 9t)t-t#P;
% ----------------------------------- $y`|zK|G-
m_abs = abs(m); QALMF rWH
rpowers = []; s~TYzfA
for j = 1:length(n) NcPzmW{#;g
rpowers = [rpowers m_abs(j):2:n(j)]; V#Wd
end 3"<{YEj8U
rpowers = unique(rpowers); N-5lILuJJ
qC]D9
A
% Pre-compute the values of r raised to the required powers, mT~:k}u~W
% and compile them in a matrix: m2 OP=z@)
% ----------------------------- (apAUIE
if rpowers(1)==0 |"ck;.)
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 2Gx&ECa,
rpowern = cat(2,rpowern{:}); <iTaJa$0m
rpowern = [ones(length_r,1) rpowern]; 578Dl(I#)
else w%L0mH2]ng
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ()P?f ed
rpowern = cat(2,rpowern{:}); T$k) ^'
end Ib!`ChZ
[.0R"|$sy+
% Compute the values of the polynomials: fPspJug
% -------------------------------------- 8XTVpf4
y = zeros(length_r,length(n)); !WrUr]0IP
for j = 1:length(n) 56L>tP
s = 0:(n(j)-m_abs(j))/2; EI+.Q
pows = n(j):-2:m_abs(j); 4cs`R+]o
for k = length(s):-1:1 /TpM#hkq/2
p = (1-2*mod(s(k),2))* ... }z[O_S,X
prod(2:(n(j)-s(k)))/ ... rYc?y
prod(2:s(k))/ ... (z"Cwa@e
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 8)sqj=
prod(2:((n(j)+m_abs(j))/2-s(k))); g*8sh
idx = (pows(k)==rpowers); `33+OW
y(:,j) = y(:,j) + p*rpowern(:,idx); RMsr7M4<91
end 3"q%-M|+Q
0xH$!?{b
if isnorm _a c_8m
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); %*LdacjZ
end "IB)=Hc
end kigc+R
% END: Compute the Zernike Polynomials =<FFFoF*C_
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% iT
IW;Cv
lK}F>6^\
% Compute the Zernike functions: d~YDg{H
% ------------------------------ ^@jOS{f l
idx_pos = m>0; _Z2VS"yH
idx_neg = m<0; 2\m+
B< 6*Ktc
z = y; Is-Kz}4L
if any(idx_pos) W"z!sf5U
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Px)VDs=k
end T|oz_c\e
if any(idx_neg) R1?g6. Mq
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); E-&=I> B5
end F4d L{0;j
-rU *)0PR
% EOF zernfun