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 NP#6'eH\
function z = zernfun(n,m,r,theta,nflag) #OMFv.
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. /BN_K8nb`
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Ez)hArxns
% and angular frequency M, evaluated at positions (R,THETA) on the XK+"
x!
% unit circle. N is a vector of positive integers (including 0), and _A/q bm
% M is a vector with the same number of elements as N. Each element VY1&YR}Y
% k of M must be a positive integer, with possible values M(k) = -N(k) on^m2pQ
*p
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, *r90IS}A$2
% and THETA is a vector of angles. R and THETA must have the same <6rc8jYz
% length. The output Z is a matrix with one column for every (N,M) :MPfCiAv
% pair, and one row for every (R,THETA) pair. .91@T.
% rGDx9KR4K!
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike L^E#"f
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), rWMG6+Scb
% with delta(m,0) the Kronecker delta, is chosen so that the integral 5Q$.q&,
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 1fOH$33
% and theta=0 to theta=2*pi) is unity. For the non-normalized zBjtPtiiI8
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. iVSN>APe
% je#OV,uHM
% The Zernike functions are an orthogonal basis on the unit circle. Kg;u.4.-M
% They are used in disciplines such as astronomy, optics, and WeiDg,]e$b
% optometry to describe functions on a circular domain. &02I-lD4+
% b0|;v-v
% The following table lists the first 15 Zernike functions. ^0tO2$
% 6"djX47j
% n m Zernike function Normalization Abc%VRsT
% -------------------------------------------------- @,^c?v
% 0 0 1 1 (~IoRhp^
% 1 1 r * cos(theta) 2 3 BQZ[%0@
% 1 -1 r * sin(theta) 2 V] 0T P#
% 2 -2 r^2 * cos(2*theta) sqrt(6) oniVC',
% 2 0 (2*r^2 - 1) sqrt(3) VFI\2n`
% 2 2 r^2 * sin(2*theta) sqrt(6) k}&7!G@T
% 3 -3 r^3 * cos(3*theta) sqrt(8) )45#lE3TH
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) $a#-d;
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) X/BcS[a
% 3 3 r^3 * sin(3*theta) sqrt(8) t9eEcqMg
% 4 -4 r^4 * cos(4*theta) sqrt(10) /$'|`jKsB
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 259R5X<V
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 2
r';)8:
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) oAprM Z7Y
% 4 4 r^4 * sin(4*theta) sqrt(10) Y
a/+|mv
% -------------------------------------------------- 0&$,?CL?
% vrq5 +K&||
% Example 1: IQ\5!e
% i9+qU
% % Display the Zernike function Z(n=5,m=1) csjCXT=Ve
% x = -1:0.01:1; 3j7Na#<tL3
% [X,Y] = meshgrid(x,x); Z{}+7P
% [theta,r] = cart2pol(X,Y); 5q,ZH6\
{
% idx = r<=1; $)#?4v<
% z = nan(size(X)); %'w?fqk
% z(idx) = zernfun(5,1,r(idx),theta(idx)); H8!)zZ
% figure ?1.WF}X'
% pcolor(x,x,z), shading interp nKnQ%R
% axis square, colorbar 5V*R
Dh
% title('Zernike function Z_5^1(r,\theta)') q|ZzGEj:OV
% (yK@(euG
% Example 2: U
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% ~J![Nx/
% % Display the first 10 Zernike functions 83!{?EPE
% x = -1:0.01:1; ('z:XW96
% [X,Y] = meshgrid(x,x); f=hT
o!i
% [theta,r] = cart2pol(X,Y); 7e:eL5f>~
% idx = r<=1; rrP_7D
% z = nan(size(X)); F*-+5nJ&@
% n = [0 1 1 2 2 2 3 3 3 3]; {YK7';_E*
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 25Uw\rKeO
% Nplot = [4 10 12 16 18 20 22 24 26 28]; j8)rz
% y = zernfun(n,m,r(idx),theta(idx)); G{74o8
% figure('Units','normalized') {,B.OM)J
% for k = 1:10 B:96E&
% z(idx) = y(:,k); kB9@
&t+
% subplot(4,7,Nplot(k)) `-w, 6
% pcolor(x,x,z), shading interp t{-*@8Ke
% set(gca,'XTick',[],'YTick',[]) 0uu)0:
% axis square 1*f*}M
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @TJ2
|_s6]
% end \SN>Yy
% Q9Vj8JO"{
% See also ZERNPOL, ZERNFUN2. s`en8%
H=*lj.x
% Paul Fricker 11/13/2006 $It3}?>C'
H X{K5 +
l5nm.i<M
% Check and prepare the inputs: ?c<uN~fC=
% ----------------------------- xW|8-q
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) .*B@1q
error('zernfun:NMvectors','N and M must be vectors.') w>e+UW25Y
end LP'~7FG
O7oq1JI]Y
if length(n)~=length(m) mwutv8?
error('zernfun:NMlength','N and M must be the same length.') vNHvuwK
end hmB`+?,z*
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n = n(:); v7/k0D .
m = m(:); uO>pl37@
if any(mod(n-m,2)) 7+;.Q
error('zernfun:NMmultiplesof2', ... qpjiQ,\:b
'All N and M must differ by multiples of 2 (including 0).') Y;"jsK{$
end 2UG>(R:
d;nk>6<|
if any(m>n) 3^iVDbAW{
error('zernfun:MlessthanN', ... ?pWda<&
'Each M must be less than or equal to its corresponding N.') 6_&S
?yA
end pfR~?jYzm
`!xI!Y\
if any( r>1 | r<0 ) y eam-8
error('zernfun:Rlessthan1','All R must be between 0 and 1.') L}7 TM:%
end L2c\i
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ;9<?~S
error('zernfun:RTHvector','R and THETA must be vectors.') {55f{5y3
c
end m%nRHT0KAf
6~l+wu<$
r = r(:); TR%8O;
theta = theta(:); gnYo/q=K
length_r = length(r); @;tM R|p
if length_r~=length(theta) N8DouDq
error('zernfun:RTHlength', ... +6x}yc:yd
'The number of R- and THETA-values must be equal.') G#~U\QlG-
end $b[Ha{9(v
] &SmeTe
% Check normalization: A-, hm=?
% -------------------- hj\A-Yf
if nargin==5 && ischar(nflag) 4aKppj
isnorm = strcmpi(nflag,'norm'); X3][C
if ~isnorm +-T|ov<
error('zernfun:normalization','Unrecognized normalization flag.') 4];>O
end p(cnSvg
else I%b5a`7
isnorm = false; 2.^CIJc
end 96S$Y~G#&
WM%w_,Z
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Dk&(QajL
% Compute the Zernike Polynomials l;$FR4}d
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #guK&?Fye
F&lH5
% Determine the required powers of r: Uw| -d[!
% ----------------------------------- h(^c5#.
m_abs = abs(m); ArScJ\/Nwv
rpowers = []; ^Nu j/
for j = 1:length(n) T`,G57-5
rpowers = [rpowers m_abs(j):2:n(j)]; RR|X4h0.
end Z|fi$2k0!
rpowers = unique(rpowers); Hy -)yR
"Pu917_P
% Pre-compute the values of r raised to the required powers, 4`ZoAr-5|
% and compile them in a matrix: n]7rHV}G
% ----------------------------- 76]Z~^Y
if rpowers(1)==0 !/ dH"h
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); l5]R*mR
rpowern = cat(2,rpowern{:}); hL&7D@
rpowern = [ones(length_r,1) rpowern]; S(^YTb7
else /q8B | (U
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); stMxlG"d
rpowern = cat(2,rpowern{:}); R+!oPWfb
end 'n^?DPvD
{NcJL< ;tS
% Compute the values of the polynomials: Aar]eY\
% -------------------------------------- TU;AO%5
y = zeros(length_r,length(n)); #DARZh U)
for j = 1:length(n) !T2{xmHKv$
s = 0:(n(j)-m_abs(j))/2; }x&XvI
pows = n(j):-2:m_abs(j); lHPnAaue@
for k = length(s):-1:1 rP,|
p = (1-2*mod(s(k),2))* ... @' %XdH
prod(2:(n(j)-s(k)))/ ... K4H27SH
prod(2:s(k))/ ... BG)zkn$
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 2Nx:Y+[
prod(2:((n(j)+m_abs(j))/2-s(k))); a8v\H8@X
idx = (pows(k)==rpowers); X-Ev>3H
y(:,j) = y(:,j) + p*rpowern(:,idx); +t&+f7
end ,izp^,`
^uphpABpD
if isnorm >o%X;U
3
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 1r*yYm'
end (kyRx+gA
end K>5bb
% END: Compute the Zernike Polynomials Yakrsi/jV}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S6\E
I5S
X\w["!B
% Compute the Zernike functions: u~VXe
% ------------------------------ *3OlWnZ?
idx_pos = m>0; vl6|i)D
idx_neg = m<0; eu8a<
j^v<rCzc(
z = y; $FDGHFM
if any(idx_pos) `:R9M+
OX
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); pAqPHD=
end Nf2lw]-G4
if any(idx_neg) 2yD ?f8P4
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Z-pZyDz
end N})vrB;1
N)a5~<fBG
% EOF zernfun