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 Cz` !j
function z = zernfun(n,m,r,theta,nflag) 2r4owB?
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. u_shC"X:
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N .5Sw
% and angular frequency M, evaluated at positions (R,THETA) on the R7pdwKD
% unit circle. N is a vector of positive integers (including 0), and MOi.bHCQJP
% M is a vector with the same number of elements as N. Each element xM"k qRZ
% k of M must be a positive integer, with possible values M(k) = -N(k) -^yb[b,
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, MEf`&<t
% and THETA is a vector of angles. R and THETA must have the same `f?v_Ui-$
% length. The output Z is a matrix with one column for every (N,M) ;/l$&:
% pair, and one row for every (R,THETA) pair.
[uqe|< :
% ]?tC+UKb
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike PT4Wox9U
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 2:3-mWE
% with delta(m,0) the Kronecker delta, is chosen so that the integral %&w 8E[
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, LX;w~fRr.
% and theta=0 to theta=2*pi) is unity. For the non-normalized ]zK'aod
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 2O;Lw@W
% (xxNQ]
l-(
% The Zernike functions are an orthogonal basis on the unit circle. RvrZtg5
% They are used in disciplines such as astronomy, optics, and O|wu;1pQ
% optometry to describe functions on a circular domain. Ad$CHx-
%
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% The following table lists the first 15 Zernike functions. P&I%!'<
% jd ]$U_U(
% n m Zernike function Normalization trlZ ^K
% -------------------------------------------------- %c:v70*h=
% 0 0 1 1 A8tzIh8
% 1 1 r * cos(theta) 2 "pUqYMB2i
% 1 -1 r * sin(theta) 2 f"i(+:la
% 2 -2 r^2 * cos(2*theta) sqrt(6) \A
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% 2 0 (2*r^2 - 1) sqrt(3) z 3((L
% 2 2 r^2 * sin(2*theta) sqrt(6) WRIOj Q:
% 3 -3 r^3 * cos(3*theta) sqrt(8) P5;n(E(19
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) V}=%/OY?
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 2yB)2n#ut
% 3 3 r^3 * sin(3*theta) sqrt(8) !'m
MGxkEb
% 4 -4 r^4 * cos(4*theta) sqrt(10) 9NzK1V0X
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) w(/#isC
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) "MS}@NLUW
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 5@ c/,6l
% 4 4 r^4 * sin(4*theta) sqrt(10) }7Lo}}
% -------------------------------------------------- 3X|7 R
% ct o+W}k
% Example 1: kD"BsL*6!
% I'sq0^
% % Display the Zernike function Z(n=5,m=1) '?$N.lj$d
% x = -1:0.01:1; !W\Zq+^^J3
% [X,Y] = meshgrid(x,x); lSW6\jX
% [theta,r] = cart2pol(X,Y); R{6~7<m.
% idx = r<=1; 7
k:w3M
% z = nan(size(X)); R k'5L
% z(idx) = zernfun(5,1,r(idx),theta(idx)); "p Rr>F a
% figure "Sx}7?8AB
% pcolor(x,x,z), shading interp (g(.gN]
% axis square, colorbar EuH[G_5e0
% title('Zernike function Z_5^1(r,\theta)') g<b(q|
% SK][UxoHm
% Example 2: ko7*9`
% FR57F(31
% % Display the first 10 Zernike functions mHj3ItXUu
% x = -1:0.01:1; 0;J#".(KQ
% [X,Y] = meshgrid(x,x); :6h$1
+6
% [theta,r] = cart2pol(X,Y); (v/mKG yg
% idx = r<=1; S2APqRg*
% z = nan(size(X)); H]I^?+)9
% n = [0 1 1 2 2 2 3 3 3 3]; O\~/J/u
<
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; NI<;L m
% Nplot = [4 10 12 16 18 20 22 24 26 28]; KCDbE6
% y = zernfun(n,m,r(idx),theta(idx)); ng0tNifZ;
% figure('Units','normalized') WSi`KNX
% for k = 1:10 U-]Rm}X\M
% z(idx) = y(:,k); (B/od# nU
% subplot(4,7,Nplot(k)) YZ0y_it)
% pcolor(x,x,z), shading interp DA9-F
% set(gca,'XTick',[],'YTick',[]) T> < Vw
% axis square \N|ma P
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) = n>aJ(=Pd
% end BdMmeM2h
% 'gD,HX
% See also ZERNPOL, ZERNFUN2. +KcD Y1[
31cC*
% Paul Fricker 11/13/2006 %B#(d)T*-
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isF
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% Check and prepare the inputs: tJ qd
% ----------------------------- Uo<iZ3J
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) U@'F9UB`
error('zernfun:NMvectors','N and M must be vectors.') )NjxKSiU@
end Y-ZTv(<
+t8{aaV
if length(n)~=length(m) dv7IHUFf
error('zernfun:NMlength','N and M must be the same length.') QIb4ghm,
end .dE2,9{Z
L_~vPp
n = n(:); ^.Xom~
m = m(:); 9im<J'
if any(mod(n-m,2)) o6b\
w
error('zernfun:NMmultiplesof2', ... $D%[}[2
'All N and M must differ by multiples of 2 (including 0).') {y\5 9
end WVMkLMg8d
Nn:>c<[
if any(m>n) l2.Lh<G
error('zernfun:MlessthanN', ... Shag4-*@hi
'Each M must be less than or equal to its corresponding N.') I_aSC 4
end <\6<-x(H5
tqMOh R
if any( r>1 | r<0 ) "TQ3{=j{
error('zernfun:Rlessthan1','All R must be between 0 and 1.') _Pe,84Ro
end VNggDKS~K
QRw/d}8l
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) FCp\w1+
error('zernfun:RTHvector','R and THETA must be vectors.') '*d);{D8
end ohW
qp2~
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r = r(:); jn:9Cr,o;g
theta = theta(:); ;Q{~jT
length_r = length(r); F,)\\$=,
if length_r~=length(theta) oBpoZ @[Z
error('zernfun:RTHlength', ... `9>1 w d
'The number of R- and THETA-values must be equal.') \~4IOu
end Z{p)rscX
M#'j7EMu
% Check normalization: /l.ox.4z#
% -------------------- c&]nAn(
if nargin==5 && ischar(nflag) up^D9(y\
isnorm = strcmpi(nflag,'norm'); }iBFo\vU
if ~isnorm !J/fJW>m6
error('zernfun:normalization','Unrecognized normalization flag.') 3{/Y&/\"'^
end JsY|Fv
else ,JVWn>s
isnorm = false; s<hl>vY_'
end &?wNL@n
KhFw%Z0s<
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wusj;v4C4M
% Compute the Zernike Polynomials KJQW ))%e
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ji] H|
8
mFy9{M
% Determine the required powers of r: nS$_VJ]~
% ----------------------------------- l+!eC
lM%
m_abs = abs(m); =TcT` ](o
rpowers = []; p@Va`:RDW
for j = 1:length(n) N#!**Q 0
rpowers = [rpowers m_abs(j):2:n(j)]; lq[o2\
end Jp#Onl+d6
rpowers = unique(rpowers); 8gK
<xp
WA1h|:Z
% Pre-compute the values of r raised to the required powers, [.[|rnil
% and compile them in a matrix: w /l\p3n
% ----------------------------- O% }EpIP_
if rpowers(1)==0 U1,f$McZs
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); u.~`/O
rpowern = cat(2,rpowern{:});
,fR /C
rpowern = [ones(length_r,1) rpowern]; UU;U,q
else t_\;G~O9-M
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 6&qT1nF1
rpowern = cat(2,rpowern{:}); <GRplkf`
end 5+yT{,(5
.`;
bQh'!
% Compute the values of the polynomials: qbZY[Q+F
% -------------------------------------- Mb
+
y = zeros(length_r,length(n)); E|`JmfLQu
for j = 1:length(n) T^F9A55y
s = 0:(n(j)-m_abs(j))/2; R'e>YDC
pows = n(j):-2:m_abs(j); G0^,@jF?b
for k = length(s):-1:1 wrJ:jTh
p = (1-2*mod(s(k),2))* ... 8RE" xJMff
prod(2:(n(j)-s(k)))/ ... %'vLkjI.
prod(2:s(k))/ ... 2n3g!M6~
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... .CY;-
prod(2:((n(j)+m_abs(j))/2-s(k))); 5<=ktA48[
idx = (pows(k)==rpowers); bayDdR4T
y(:,j) = y(:,j) + p*rpowern(:,idx); ?]In@h-
end 23_\UTM}1
fk!P#
if isnorm WPXLN'w+
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); *v6 j7<H
end *s[bq;$
end =T3O; i
% END: Compute the Zernike Polynomials ?x-:JME0
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *$/!.e
n `Ry!
% Compute the Zernike functions: iLR^ V!
% ------------------------------ (w/)u
idx_pos = m>0; ckCb)r_
idx_neg = m<0; S#g=;hD
UP?]5x>
z = y; XkE'k;AEx
if any(idx_pos) f;Uf=.#F
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); E6njmdu
end ;c;5O@R}3
if any(idx_neg) =xX)2h
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); =(%+S<}
end }+3v5Nz;
s?-J`k~q
% EOF zernfun