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 qS7*.E~j|]
function z = zernfun(n,m,r,theta,nflag) #{]=>n)j
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. .f6_[cS;g
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N @!F9}n
AP
% and angular frequency M, evaluated at positions (R,THETA) on the 6qw_ |A&g
% unit circle. N is a vector of positive integers (including 0), and Gis'IX(
% M is a vector with the same number of elements as N. Each element @Xh4ZMyEx
% k of M must be a positive integer, with possible values M(k) = -N(k) 5}By2Tx
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ).&$pXj
% and THETA is a vector of angles. R and THETA must have the same YV2^eGr.
% length. The output Z is a matrix with one column for every (N,M) %+'&$
% pair, and one row for every (R,THETA) pair. CsE|pXVG
% J=Jw"? f
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike F:H76O` 8
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), |Rl|Th
% with delta(m,0) the Kronecker delta, is chosen so that the integral 7'<4'BGzl]
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, (*2"dd
% and theta=0 to theta=2*pi) is unity. For the non-normalized q2SkkY$_]y
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 5 Fd ]3
% lF#Kg!-l
% The Zernike functions are an orthogonal basis on the unit circle. ^yb_aC w
% They are used in disciplines such as astronomy, optics, and T^Z#x-Q
% optometry to describe functions on a circular domain. '}}DPoV
% &"CS1P|
% The following table lists the first 15 Zernike functions. 2R_k$kHl
% g VuN a)
% n m Zernike function Normalization a`{'u)@
% -------------------------------------------------- z,NHH):~
% 0 0 1 1 H;Bj\-Pa
% 1 1 r * cos(theta) 2 $iB(N ZV
% 1 -1 r * sin(theta) 2 BpKP]V
% 2 -2 r^2 * cos(2*theta) sqrt(6) Q/+a{m0f
% 2 0 (2*r^2 - 1) sqrt(3) !YoKKG~_0
% 2 2 r^2 * sin(2*theta) sqrt(6) |UBJu `%
% 3 -3 r^3 * cos(3*theta) sqrt(8)
d,H%
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) jrW7AT)\
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) %?cPqRHJ ~
% 3 3 r^3 * sin(3*theta) sqrt(8) bb<Vh2b>R
% 4 -4 r^4 * cos(4*theta) sqrt(10) F )tNA?p)
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Psv-y
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) (z.Vwl5
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :@p`E}1r{
% 4 4 r^4 * sin(4*theta) sqrt(10) a:8@:d1T K
% -------------------------------------------------- (g;Ff`P
Pc
% "y`?KY$[N
% Example 1: <6`,)(dj
% QO%LSRw
% % Display the Zernike function Z(n=5,m=1) fHgfI@{=j
% x = -1:0.01:1; d#W[<,
% [X,Y] = meshgrid(x,x); %? g]{
% [theta,r] = cart2pol(X,Y); K}zw%!ex
% idx = r<=1; `ybZE+S.
% z = nan(size(X)); 68d @By
% z(idx) = zernfun(5,1,r(idx),theta(idx)); O-|3k$'\z
% figure E>[~"~x"pV
% pcolor(x,x,z), shading interp oNdO@i%.q4
% axis square, colorbar ' ZB%McS
% title('Zernike function Z_5^1(r,\theta)') nQgn^z#
% 1|%$ie
% Example 2: ^.4<#Qs
% <&NR3^Eq
% % Display the first 10 Zernike functions [IYs4Y5
% x = -1:0.01:1; Xu
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% [X,Y] = meshgrid(x,x); E^qKkl
% [theta,r] = cart2pol(X,Y); +I')>6
% idx = r<=1; C/cyqxVl}
% z = nan(size(X)); O=mJ8W@
% n = [0 1 1 2 2 2 3 3 3 3]; 7j]@3D9[:p
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; :4A^~+J
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Xak~He
% y = zernfun(n,m,r(idx),theta(idx)); oDRNM^gz
% figure('Units','normalized') `j2z=5
% for k = 1:10 N$3F4b%+
% z(idx) = y(:,k); X$xqu\t7
% subplot(4,7,Nplot(k)) \gzNMI*
% pcolor(x,x,z), shading interp leiza?[
% set(gca,'XTick',[],'YTick',[]) Y8N&[L[z&
% axis square &oR&NKk
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) =]"PSY7p
% end fL@[B{XMM
% lyT~>.?{
% See also ZERNPOL, ZERNFUN2. 8Ej2JMc
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% Paul Fricker 11/13/2006 vbWX`skU
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f{vnZ|WD
% Check and prepare the inputs: d2(n3Xf
% ----------------------------- 4v{gc/g
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ' E@D
error('zernfun:NMvectors','N and M must be vectors.') y K{~
end N@) D,~
1\3n
if length(n)~=length(m) cBAA32wf
error('zernfun:NMlength','N and M must be the same length.') 4iw+3 Q|
end ?iq:Gf
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n = n(:); SAqX[c
m = m(:); N_T;&wibO
if any(mod(n-m,2))
mjw:Z,
error('zernfun:NMmultiplesof2', ... )D@
NX/}
'All N and M must differ by multiples of 2 (including 0).') +XQS
-=
end zi5;>Iv0}
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if any(m>n) m!PN1$9V
error('zernfun:MlessthanN', ... {:? -)Xq
'Each M must be less than or equal to its corresponding N.') S4\T (
end [3\}Ca1
d6Z;\f7[
if any( r>1 | r<0 ) '91Ak,cWB
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ]v+\v re
end -dza_{&+iZ
%II |;<
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) &<~`?-c
error('zernfun:RTHvector','R and THETA must be vectors.') _|#)tWy}
end 8J>s|MZ
0R; ;ou
r = r(:); e}Db-7B_~
theta = theta(:); f-3lJ?6
length_r = length(r);
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if length_r~=length(theta) No h*1u*
error('zernfun:RTHlength', ... :lcoS J
'The number of R- and THETA-values must be equal.') BK-{z).)
end JWEqy+,Fjw
/Jo*O=Lpo
% Check normalization: d#A.A<p*
% -------------------- Q.!D2RZc
if nargin==5 && ischar(nflag) AJj6@hi2P
isnorm = strcmpi(nflag,'norm'); j]jwQRe
if ~isnorm i5rAb<q`
error('zernfun:normalization','Unrecognized normalization flag.') V a<L[8
end k/*r2 C
else o8Tt|Lxb$8
isnorm = false; RU@`+6j+
end -[G+*3Y{7
/9i2@#J}W1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2r\f!m'
% Compute the Zernike Polynomials 4D0"Y#&G
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !x&/M*nBE
8*lVO2
% Determine the required powers of r: {Z$Aw4a"d
% ----------------------------------- }]/"auk
m_abs = abs(m); hX<0{pXM4
rpowers = []; {&m^*YN/
for j = 1:length(n) 0>>tdd7
rpowers = [rpowers m_abs(j):2:n(j)]; Z?dz@d%C
end JH5ckgdZ
rpowers = unique(rpowers); E QMn'>
)88z=5.
% Pre-compute the values of r raised to the required powers, Ij4oH
% and compile them in a matrix: iz&)FuOr
% ----------------------------- Fq9AO~z
if rpowers(1)==0 YGNO]Q~A
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); |&3[YZY
rpowern = cat(2,rpowern{:}); 6>X7JMRY
rpowern = [ones(length_r,1) rpowern]; bF<FX_}!s!
else RYy_Ppn96f
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); #T&''a
rpowern = cat(2,rpowern{:}); '
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end 3N[t2Y1r
$O?&!8);,
% Compute the values of the polynomials: iJT_*,P^
% -------------------------------------- ]haZ T\
y = zeros(length_r,length(n)); 4uwI=U UB
for j = 1:length(n) Jzo|$W
s = 0:(n(j)-m_abs(j))/2; lEh; MJ
pows = n(j):-2:m_abs(j); $@s&qi_&R
for k = length(s):-1:1 ;3'ta!.c
p = (1-2*mod(s(k),2))* ... !Qy%sY
prod(2:(n(j)-s(k)))/ ... wL\OAM6R
prod(2:s(k))/ ... -X3yCK?re
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... krFuEaO
prod(2:((n(j)+m_abs(j))/2-s(k))); M2l0x @|
idx = (pows(k)==rpowers); jZx.MBVy]
y(:,j) = y(:,j) + p*rpowern(:,idx); XShi[7
end ~+
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'pa[z5{k+
if isnorm J>y}kzCz
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 49W@?:b
end u$O`
\=
end V:s$V.{!
% END: Compute the Zernike Polynomials
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% m&a.i
B
9J+p.N
% Compute the Zernike functions: zk#"n&u0
% ------------------------------ 98'/yZ
idx_pos = m>0;
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idx_neg = m<0; <Nwqt[.
n@[_lNa4GD
z = y; >pdWR1ox
if any(idx_pos) ]{^'{ z$i
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ?71?Vd
end MVP|l_2!
if any(idx_neg) G9 v'a&
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 3dheT}XV?p
end X$BN&DD
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% EOF zernfun