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 s-V5\Lip,
function z = zernfun(n,m,r,theta,nflag) >w,o|
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. i:9f#
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N '&?OhSeN
% and angular frequency M, evaluated at positions (R,THETA) on the m+y5Q&;f
% unit circle. N is a vector of positive integers (including 0), and K`|%-k+D
% M is a vector with the same number of elements as N. Each element tI2V)i!
% k of M must be a positive integer, with possible values M(k) = -N(k) {)E)&lL
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, zZrUS'8
% and THETA is a vector of angles. R and THETA must have the same `Sh#>
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% length. The output Z is a matrix with one column for every (N,M) 1SddZ5
% pair, and one row for every (R,THETA) pair. $a'n{EP
% X,m6#vLK2
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike G}!dm0s$
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), _wMc7`6F
% with delta(m,0) the Kronecker delta, is chosen so that the integral V6opV&
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, } 0su[gy[
% and theta=0 to theta=2*pi) is unity. For the non-normalized El3Y1g3+3
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. jNKu5"HB
% ~RlsgtX"
% The Zernike functions are an orthogonal basis on the unit circle. XH9Y|FX%#
% They are used in disciplines such as astronomy, optics, and b`?$;5
% optometry to describe functions on a circular domain. }$6;g-|HX
% e^;<T9Esr
% The following table lists the first 15 Zernike functions. y~,mIM$[@
% 60
D0z
% n m Zernike function Normalization P?- #d\qi
% -------------------------------------------------- G/l 28yt
% 0 0 1 1 Lt\Wz'6Y
% 1 1 r * cos(theta) 2 !Ee#jCXS
% 1 -1 r * sin(theta) 2 3em&7QM
% 2 -2 r^2 * cos(2*theta) sqrt(6) _!vxX]
% 2 0 (2*r^2 - 1) sqrt(3) )U6-&-07
% 2 2 r^2 * sin(2*theta) sqrt(6) l*~ ".q;S
% 3 -3 r^3 * cos(3*theta) sqrt(8) P0R8
f
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ,ALEfepo
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) m tPmVze
% 3 3 r^3 * sin(3*theta) sqrt(8) s8i@HO
% 4 -4 r^4 * cos(4*theta) sqrt(10) Fjq~^_8
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -&L(0?*qo
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) I_QWdxn
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0A]+9@W;
% 4 4 r^4 * sin(4*theta) sqrt(10) 5somoV B
% -------------------------------------------------- X\\c=[#8-
% N*Is_V\R
% Example 1: lCyBdY9n
% =f
FTi1]/h
% % Display the Zernike function Z(n=5,m=1) XsOz
{?G
% x = -1:0.01:1; &bh%>[
% [X,Y] = meshgrid(x,x); -SyQ`V)T7N
% [theta,r] = cart2pol(X,Y); ,{tz%\,%
% idx = r<=1; E5>y?N
% z = nan(size(X)); qFqK.u
% z(idx) = zernfun(5,1,r(idx),theta(idx)); puv/+!q
% figure W~EDLL Z
% pcolor(x,x,z), shading interp `$kKTc:f
% axis square, colorbar itH`
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% title('Zernike function Z_5^1(r,\theta)') G54,`uz2
% >gj%q$@
% Example 2: K<BS%~,I
% lWiC$
% % Display the first 10 Zernike functions @ V_@r@A
% x = -1:0.01:1; 0!Zp4>l\Z
% [X,Y] = meshgrid(x,x); U};~ff+
% [theta,r] = cart2pol(X,Y); 2 q4dCbJ!
% idx = r<=1; 71g\fGG\
% z = nan(size(X)); 8y9`xRy
% n = [0 1 1 2 2 2 3 3 3 3]; .>/Tc
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; *x0nAo_n
% Nplot = [4 10 12 16 18 20 22 24 26 28]; am+'j5`Ys
% y = zernfun(n,m,r(idx),theta(idx)); ")gd)_FOS
% figure('Units','normalized') ,McwPHEMB
% for k = 1:10 ZxvqLu
% z(idx) = y(:,k); E%+ aqA)f
% subplot(4,7,Nplot(k)) $e99[y@
% pcolor(x,x,z), shading interp JDa=+\_
% set(gca,'XTick',[],'YTick',[]) { \r1A
% axis square @bkZ< Gq
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) <'Ppu
% end -Hx._I$l
% vt(A?$j|A
% See also ZERNPOL, ZERNFUN2. $qvk9 B0E
Xp_3EQl
% Paul Fricker 11/13/2006 X+R?>xq{=h
:!fP~(R'm
2D?V0>/
% Check and prepare the inputs: $y2"Q,n+
% ----------------------------- Nt>wzPd)
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) JA")L0a_
error('zernfun:NMvectors','N and M must be vectors.') YtQsSU
end rM{3]v{~
P7X':
if length(n)~=length(m) )P)Zds@F
error('zernfun:NMlength','N and M must be the same length.') W-72&\7
end }3}{} w0Y
$@VQ{S
n = n(:); c:$W5j('Z
m = m(:); ]>:LHW
if any(mod(n-m,2)) {j0c)SETN
error('zernfun:NMmultiplesof2', ... `1 tD&te0
'All N and M must differ by multiples of 2 (including 0).') =P,h5J
end vWGjc2_
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if any(m>n) hwon^?
error('zernfun:MlessthanN', ... 2O*(F>>dT
'Each M must be less than or equal to its corresponding N.') {I]X-+D|_
end tB,1+I=
)|d]0/<
if any( r>1 | r<0 ) H&yK{0H
error('zernfun:Rlessthan1','All R must be between 0 and 1.')
Z>O2
end EYLqg`2A
=Nc}XFq
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) K~U5jpc
error('zernfun:RTHvector','R and THETA must be vectors.') 0/vmj,&B(
end b }^ylm
qMHI-h_A
r = r(:); IM^K]$q$47
theta = theta(:); xDJs0P4
length_r = length(r); cyQ&w>'
if length_r~=length(theta) <8'-azpJ6<
error('zernfun:RTHlength', ... u4W2{
'The number of R- and THETA-values must be equal.') ;q3"XLV(T[
end 2G(RQ\Ro*
KA"D2j9wn
% Check normalization: 03{pxI
% -------------------- +O2z&a;q
if nargin==5 && ischar(nflag) e*zt;SR
isnorm = strcmpi(nflag,'norm'); ,[Bv\4Ah
if ~isnorm I Ceb2R
error('zernfun:normalization','Unrecognized normalization flag.') V>Zw" #Q
end Hxw 7Q?F
else AJ:(NV1=
isnorm = false; {dbPMx
end 4"=(kC~~
=/|2f; Q
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9/@7NNKJ
% Compute the Zernike Polynomials Q&X#(3&'
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7M#irCX
w7;,+Jq
% Determine the required powers of r: u=U.+\f5
% ----------------------------------- ]W7e2:Hra
m_abs = abs(m); {e1akg.
rpowers = []; [q%Rx!L
for j = 1:length(n) &* Aems{-
rpowers = [rpowers m_abs(j):2:n(j)]; p1O[QQ|
end Ag6^>xb^
rpowers = unique(rpowers); ZbZCW:8>k
gaIN]9wLm
% Pre-compute the values of r raised to the required powers, tr<iFT}C
% and compile them in a matrix: Avc9W[4
% ----------------------------- JxV0y
if rpowers(1)==0 BbV @ziL
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Hl3%+f
rpowern = cat(2,rpowern{:}); Zdm7As]
rpowern = [ones(length_r,1) rpowern]; ?Tr]zxtd
else %#zqZ|q
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 1dl@2CVS
rpowern = cat(2,rpowern{:}); `F^~*FnR,B
end 4$wn8!x2|
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% Compute the values of the polynomials: BSMM3jXb
% -------------------------------------- 5g$]ou
y = zeros(length_r,length(n)); _!} L\E~
for j = 1:length(n) *?-,=%,z/
s = 0:(n(j)-m_abs(j))/2; 9S y |:J0
pows = n(j):-2:m_abs(j); |@+/R .l
for k = length(s):-1:1 DC-tBbQkk
p = (1-2*mod(s(k),2))* ... }C<<l5/ z
prod(2:(n(j)-s(k)))/ ... {?zBc E:
prod(2:s(k))/ ... o-f;$]yp>
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 8(b
C.
prod(2:((n(j)+m_abs(j))/2-s(k))); /ZeN\ybx
idx = (pows(k)==rpowers); He}uE0^
y(:,j) = y(:,j) + p*rpowern(:,idx); EJz?GM
end z
:q9~
b":3J)Y6.
if isnorm +IM:jrT(
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); YIc|0[ ]*|
end ]8c%)%Vi
end hbOyrjanx
% END: Compute the Zernike Polynomials .EXe3!J)!
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @uJ^k
>B
fGz++;b<S
% Compute the Zernike functions: Wt ,t5
% ------------------------------ 0|^/ e-^
idx_pos = m>0; #3h~Z)+y
idx_neg = m<0; \mIm}+!H
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z = y; +pefk+
if any(idx_pos) T0Kjnzs
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); *2(W`m
end Pcs62aE
if any(idx_neg) &l0-0T>
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Q~y) V
end l[P VWM
B'kV.3t
% EOF zernfun