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 Ya(3Z_f+VZ
function z = zernfun(n,m,r,theta,nflag) H )CoByaj
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 7|jy:F,w%
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N <j/wK]d*/
% and angular frequency M, evaluated at positions (R,THETA) on the e)m6xiZ
% unit circle. N is a vector of positive integers (including 0), and 3Tp8t6*nL
% M is a vector with the same number of elements as N. Each element *`LrvE@t
% k of M must be a positive integer, with possible values M(k) = -N(k) Mpco8b-b
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, DLD9
% and THETA is a vector of angles. R and THETA must have the same +KWO`WR
% length. The output Z is a matrix with one column for every (N,M) C6h[L
% pair, and one row for every (R,THETA) pair. oOaLD{g>
% D7muf
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike @(+\*]?^&
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), d_ x
jW
% with delta(m,0) the Kronecker delta, is chosen so that the integral 3to!C"~\K-
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ,X;$-.
% and theta=0 to theta=2*pi) is unity. For the non-normalized |_QpB?b
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. *'tGi_2?(
% U#Iwe=
% The Zernike functions are an orthogonal basis on the unit circle. ]&9=f#k%
% They are used in disciplines such as astronomy, optics, and }E[vW
% optometry to describe functions on a circular domain. G9GHBwT
% f6nuh&!-
% The following table lists the first 15 Zernike functions. hpYv*WH:
% 4mtO"'|
% n m Zernike function Normalization TBky+]p@
% -------------------------------------------------- `Q#)N0
% 0 0 1 1 R(,m!
% 1 1 r * cos(theta) 2 p=#/H,2
% 1 -1 r * sin(theta) 2 j}`ku9S~
% 2 -2 r^2 * cos(2*theta) sqrt(6) WFhppi
% 2 0 (2*r^2 - 1) sqrt(3) :ln?PT
% 2 2 r^2 * sin(2*theta) sqrt(6) "5'eiYms
% 3 -3 r^3 * cos(3*theta) sqrt(8) BUV4L5(
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) {d]B+'
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 2oOos%0
% 3 3 r^3 * sin(3*theta) sqrt(8) X.FoX
% 4 -4 r^4 * cos(4*theta) sqrt(10) 3x7fa^umR
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 8~~ k?
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 33wVP}e5
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) fY?:SPR+
% 4 4 r^4 * sin(4*theta) sqrt(10) -B!
a
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% -------------------------------------------------- &4#%xg
% 9_.pLLx
% Example 1: Xwjm T
% G2 V$8lh
% % Display the Zernike function Z(n=5,m=1) EwgNd Gcj
% x = -1:0.01:1; P}( c0/
% [X,Y] = meshgrid(x,x); s{{8!Q
% [theta,r] = cart2pol(X,Y); )EQI>1_
% idx = r<=1; VUP.
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% z = nan(size(X));
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% z(idx) = zernfun(5,1,r(idx),theta(idx));
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% figure Qh6vH9(D
% pcolor(x,x,z), shading interp -N5h` Ii7
% axis square, colorbar >Z<ZT
% title('Zernike function Z_5^1(r,\theta)') o?~27
% X+<9-]=
% Example 2: {7MY*&P$,
% Y,EF'Ot
% % Display the first 10 Zernike functions %cDDu$9;
% x = -1:0.01:1; +2}Ar<elP
% [X,Y] = meshgrid(x,x); L<XX?I\p
% [theta,r] = cart2pol(X,Y); ^,?>6O
% idx = r<=1; Pgq(yPC
% z = nan(size(X)); l@u
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% n = [0 1 1 2 2 2 3 3 3 3]; O8N1gf;t
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; i_+e&Bjd4j
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Z=;=9<vA
% y = zernfun(n,m,r(idx),theta(idx)); qW|h"9sr
% figure('Units','normalized') 4>fj@X(3
% for k = 1:10 (~! @Uz5
% z(idx) = y(:,k); 6 b?K-)kL
% subplot(4,7,Nplot(k)) T+rym8.p
% pcolor(x,x,z), shading interp nD>X?yz2
% set(gca,'XTick',[],'YTick',[]) k`]76C7
% axis square zlTLp-^Y
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ZtP/|P5@
% end w^n&S=E E~
% AW9%E/{
% See also ZERNPOL, ZERNFUN2. !vc5NKv#n
/R?*i@rvf
% Paul Fricker 11/13/2006 45iO2W uur
h.Sbds
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% Check and prepare the inputs: +W8#] u|
% ----------------------------- 4`cf FowK~
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) uC#]F@
error('zernfun:NMvectors','N and M must be vectors.') S$R=!3* "V
end 0"+QWh
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if length(n)~=length(m) jGJf[:M&Pm
error('zernfun:NMlength','N and M must be the same length.') ^L
Xr4
end R`@7f$;wG
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n = n(:); &9.3-E47*
m = m(:); #q9BU:
if any(mod(n-m,2)) 5H 1x-b
error('zernfun:NMmultiplesof2', ... @T.F/Pjhc
'All N and M must differ by multiples of 2 (including 0).') gu'+kw
end m}: X\G(6Q
\,:7=
if any(m>n) Gz8JOl
error('zernfun:MlessthanN', ... #.Ly
'Each M must be less than or equal to its corresponding N.') ANj%q9e!Yi
end Bxj4rC[
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if any( r>1 | r<0 )
!y!s/i&P%
error('zernfun:Rlessthan1','All R must be between 0 and 1.') -~lrv#5Q
end _n4`mL8>kH
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) M$f_I +
error('zernfun:RTHvector','R and THETA must be vectors.') I>-}ys`[
end |BGzdBm^x:
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r = r(:); AP=SCq;
theta = theta(:); \S~<C[P
length_r = length(r); &qa16bz
if length_r~=length(theta) &;GoCU Le
error('zernfun:RTHlength', ... y4!fu<[i
'The number of R- and THETA-values must be equal.') Y!|};
end P5"B7>L:
soKR*gJ,
% Check normalization: mcQ\"9 ;pY
% -------------------- +OB&PE
if nargin==5 && ischar(nflag) nRX<$OzTV
isnorm = strcmpi(nflag,'norm'); D6e<1W
if ~isnorm {z'Gg
error('zernfun:normalization','Unrecognized normalization flag.') WCp[6g&%O
end $.B}zY{
else W$Aypy
isnorm = false; &N%-.&t'
end !yV)EJ:$
~$Z_#,|i?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% y G>sBc
% Compute the Zernike Polynomials X<1ymb3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0nlh0u8#
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% Determine the required powers of r: icK U)
% ----------------------------------- rj5)b:c}
m_abs = abs(m); [Kbna>`
rpowers = []; SC2g5i`
for j = 1:length(n) Ew9MWlk
rpowers = [rpowers m_abs(j):2:n(j)]; \nQEvcH
end mj y+_
rpowers = unique(rpowers); *I9G"R8
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% Pre-compute the values of r raised to the required powers, _g"su#
% and compile them in a matrix: 6|%HCxWO
% ----------------------------- YeF'r.Y
if rpowers(1)==0 HlX7A1i/
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); hDEZq>&
rpowern = cat(2,rpowern{:}); $5>x)jr:w+
rpowern = [ones(length_r,1) rpowern]; \z2d=E
else #mO.[IuD
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); %5(v'/dQ
rpowern = cat(2,rpowern{:}); '9|R7
end Gs}lw'pK
[{_K[5i
% Compute the values of the polynomials: D/WzYc2h]
% -------------------------------------- 9Mv4=k^7|4
y = zeros(length_r,length(n)); nON"+c*
for j = 1:length(n) Q $>SYvW
s = 0:(n(j)-m_abs(j))/2; <^8OYnp
pows = n(j):-2:m_abs(j); An
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for k = length(s):-1:1 lHPhZ(Z
p = (1-2*mod(s(k),2))* ... ;!>>C0s"
prod(2:(n(j)-s(k)))/ ... }HZ'i;~r|9
prod(2:s(k))/ ... /p@0Q[E
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ]}AyDy6C
prod(2:((n(j)+m_abs(j))/2-s(k))); k${F7I(Tb
idx = (pows(k)==rpowers); %M05& <
y(:,j) = y(:,j) + p*rpowern(:,idx); Wy$Q!R=i
end 2l4`h)_q
5cl%>U
if isnorm 5wMEp" YHE
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); m^,3jssdA
end ;V1e>?3
end _n<
@Jk~
% END: Compute the Zernike Polynomials rHgrCMW
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% gH/k}M7tA#
UIw6~a3E
% Compute the Zernike functions: ,Onu%
% ------------------------------ V{kgDpB
idx_pos = m>0; rYr.mX
idx_neg = m<0; *|:]("i
g/soop\:
z = y; oI%.oP}G
if any(idx_pos) h'G8@j;
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); q0
8
end GD .>u
if any(idx_neg) rx;zd ?
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); OAz-w
end #Y<b'7yJ
~#
|p=Y
% EOF zernfun