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 rH&G<o&,
function z = zernfun(n,m,r,theta,nflag) 79ckLd9
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. e,HMwD
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 7^$)VBQ/
% and angular frequency M, evaluated at positions (R,THETA) on the ?i~g,P]NK
% unit circle. N is a vector of positive integers (including 0), and 5IW8=$k~.)
% M is a vector with the same number of elements as N. Each element 0DNU,u
% k of M must be a positive integer, with possible values M(k) = -N(k) f~=r*&U
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, <l$P&jSF3
% and THETA is a vector of angles. R and THETA must have the same yGTziv!
% length. The output Z is a matrix with one column for every (N,M) GWsd| kxU
% pair, and one row for every (R,THETA) pair. rK1-Mu
% u$%A#L[
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike fc@'9-pt
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), a2`%ghW3
% with delta(m,0) the Kronecker delta, is chosen so that the integral B8T\s)fxnX
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, XphE loL
% and theta=0 to theta=2*pi) is unity. For the non-normalized /.R<,/gj
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. %r%So_^
% n9%]-s\Hn
% The Zernike functions are an orthogonal basis on the unit circle. u-JpI-8h
% They are used in disciplines such as astronomy, optics, and 3JO]f5
% optometry to describe functions on a circular domain. 2*[QZ9U[@
% 2Il8f
% The following table lists the first 15 Zernike functions. 03=5Nof1
% TVaA>]Fv
% n m Zernike function Normalization ?cKZ_c
% -------------------------------------------------- 9sSN<7
% 0 0 1 1 +r]zs^'
% 1 1 r * cos(theta) 2 .2W"w)$nuq
% 1 -1 r * sin(theta) 2 wpXgPVZT
% 2 -2 r^2 * cos(2*theta) sqrt(6) fRB5U'
% 2 0 (2*r^2 - 1) sqrt(3) 4zjs!AK%
% 2 2 r^2 * sin(2*theta) sqrt(6) p[9s<lEh
% 3 -3 r^3 * cos(3*theta) sqrt(8) dRW$T5dac
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Z^yNLF *&V
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) {u"8[@@./
% 3 3 r^3 * sin(3*theta) sqrt(8)
UMU2^$\iS
% 4 -4 r^4 * cos(4*theta) sqrt(10) X|}2_B
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) N\NyXh$
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) _c`K+o"3
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) }rq9I"/L
% 4 4 r^4 * sin(4*theta) sqrt(10) :z&7W<
% -------------------------------------------------- ;f1qLI
% zF`3gl.
% Example 1: r^0F"9eOL
% Ag9?C*
% % Display the Zernike function Z(n=5,m=1) >Lft9e
% x = -1:0.01:1; s?2$ue&-f
% [X,Y] = meshgrid(x,x); V`kMCE;?l
% [theta,r] = cart2pol(X,Y); (W[V?!1
% idx = r<=1; `JB?c
% z = nan(size(X)); Z
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% z(idx) = zernfun(5,1,r(idx),theta(idx)); NX[4PKJ0C
% figure B07v^!Z>
% pcolor(x,x,z), shading interp cl@g
% axis square, colorbar @WMA }\Cc
% title('Zernike function Z_5^1(r,\theta)') s58C2
% t `kui.
% Example 2: Qm4o7x{q
% */^QH@ P
% % Display the first 10 Zernike functions OsqNB'X
% x = -1:0.01:1; 0[Ht_qxb
% [X,Y] = meshgrid(x,x); ^uBxgWIC
% [theta,r] = cart2pol(X,Y); iK5_u2]Q
% idx = r<=1; x/!5K|c
% z = nan(size(X)); -
e"jw#B
% n = [0 1 1 2 2 2 3 3 3 3]; nKoiG*PI
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Hc^W%t~
% Nplot = [4 10 12 16 18 20 22 24 26 28]; *`_{
% y = zernfun(n,m,r(idx),theta(idx)); Hnk:K9u.B:
% figure('Units','normalized') X5LBEOG
% for k = 1:10 lf(`SYQnOY
% z(idx) = y(:,k); 6eFp8bANN#
% subplot(4,7,Nplot(k)) (o5j'2:.
% pcolor(x,x,z), shading interp qpIC{'A.
% set(gca,'XTick',[],'YTick',[]) }e2VY
% axis square Ep9W- n?}
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) zc rY>t#l
% end ":a\z(*t
% 3cdTed-MIh
% See also ZERNPOL, ZERNFUN2. LbEM^D
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% Paul Fricker 11/13/2006 5"XC$?I<}
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% Check and prepare the inputs: =X\^J
% ----------------------------- ,R%q}IH#
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) I/Jb!R ~
error('zernfun:NMvectors','N and M must be vectors.') Ar*^;/
end Od f[*
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if length(n)~=length(m) MZ+^-@X
error('zernfun:NMlength','N and M must be the same length.') Xtt?]
end Bn@(zHG+5&
#(An6itl
n = n(:); svxw^0~a
m = m(:); YIw1
if any(mod(n-m,2)) x
}Ad_#q
error('zernfun:NMmultiplesof2', ... PB;eHy
'All N and M must differ by multiples of 2 (including 0).') 1-lu\"H`
end (x/k.&
VD_$$Gn*q
if any(m>n) 2hzsKkrA
{
error('zernfun:MlessthanN', ... _ODbY;M
'Each M must be less than or equal to its corresponding N.') _S>JKz
end QQWadVQo
}zhGS!fO
if any( r>1 | r<0 ) 'Ut7{rZ5
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 0lhVqy}:}o
end !1e6Ss
^#-nE7
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) <HbcNE~
error('zernfun:RTHvector','R and THETA must be vectors.') |*}4 m'c
end bv&;R
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r = r(:); tH-C8Qxy
theta = theta(:); X5j1`t,
length_r = length(r); yUpgoX(6
if length_r~=length(theta) "'D=,*
error('zernfun:RTHlength', ... +E{|63~q
'The number of R- and THETA-values must be equal.') I:mr}mv=i
end Hy^N!rBxfO
17`1SGZ
% Check normalization: 9I4K}R
% -------------------- ]*AR,0N&
if nargin==5 && ischar(nflag) V#iPj'*
isnorm = strcmpi(nflag,'norm'); J:Qa5MTWp
if ~isnorm K*~0"F>"0
error('zernfun:normalization','Unrecognized normalization flag.') r,h%[JKM
end /Njd[=B
else [PDNwh0g5
isnorm = false; )c)vTZy
end 9b9$GyI
XCBL}pNkR
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "g1)f"pL
% Compute the Zernike Polynomials O6LS(5j2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7eAX*Kgt<_
Fvbh\m
~
% Determine the required powers of r: @0/+_2MH-
% ----------------------------------- )r
jiY%F$
m_abs = abs(m); _no*k?o*
rpowers = []; ^zQ/mo,Z
for j = 1:length(n) oC0qG[yp9S
rpowers = [rpowers m_abs(j):2:n(j)]; V6@o]*
end fTK3,s1=
rpowers = unique(rpowers); UWd=!h^dt
uC(V
% Pre-compute the values of r raised to the required powers, =`H@%
% and compile them in a matrix: 7t0er'VC
% ----------------------------- oU.R2\Q
if rpowers(1)==0 toBHkiuD
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); E?
;0)'h
rpowern = cat(2,rpowern{:}); 2QyV%wz
rpowern = [ones(length_r,1) rpowern]; %`1q-,>v
else #Up86(Z
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); V<T9&8l+:
rpowern = cat(2,rpowern{:}); hYG6 pTCb
end `T5W}p[6
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% Compute the values of the polynomials: <^5Z:n!q
% -------------------------------------- lww!-(<ww
y = zeros(length_r,length(n));
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for j = 1:length(n) l*d(;AR
s = 0:(n(j)-m_abs(j))/2; ~d|A!S`
pows = n(j):-2:m_abs(j); Nh_Mz;ITuu
for k = length(s):-1:1 1SCR.@k<
p = (1-2*mod(s(k),2))* ... EVsC >rz
prod(2:(n(j)-s(k)))/ ... vunHNHltW0
prod(2:s(k))/ ... of%Ktm5Qi
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... CVL3VT1j0
prod(2:((n(j)+m_abs(j))/2-s(k))); #W4dkCd(pF
idx = (pows(k)==rpowers); \o*5
y(:,j) = y(:,j) + p*rpowern(:,idx); BBwy,\o#
end U`, 6 * MS
K8GP@yD]M
if isnorm +M\`#i\g>
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); eg;~zv
end \/zq7j
end su{poQ}K
% END: Compute the Zernike Polynomials aBNc(?ri
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @tNz Q8
"n(hfz0y%
% Compute the Zernike functions: S2sQOM@
% ------------------------------ hK L4cpK4
idx_pos = m>0; Jh,]r?Bd
idx_neg = m<0; 96( v
.WA-&b_
z = y; K*K,}W&}
if any(idx_pos) G\2CR*
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); m
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end SJb&m-
if any(idx_neg) fI ?>+I5
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); H<i]V9r
end n8~N$tDU
riY~%9iV'
% EOF zernfun