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 $`'^&o;&f
function z = zernfun(n,m,r,theta,nflag) tS2lex%
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. gzDb~UEoF
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N D0QXvrf
% and angular frequency M, evaluated at positions (R,THETA) on the >?e*;f$VdJ
% unit circle. N is a vector of positive integers (including 0), and _>5BFQ_
% M is a vector with the same number of elements as N. Each element ej<z]{`05
% k of M must be a positive integer, with possible values M(k) = -N(k) sZ'3PNpCP
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, $^7&bQ
% and THETA is a vector of angles. R and THETA must have the same d*3R0Q|#{
% length. The output Z is a matrix with one column for every (N,M) i=2+1;K
% pair, and one row for every (R,THETA) pair. q0y?$XS
% O!f* @
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Ro:-u7q
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), wCvD4C.WH
% with delta(m,0) the Kronecker delta, is chosen so that the integral rI]:| k
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, l}AB):<Z
% and theta=0 to theta=2*pi) is unity. For the non-normalized =GR
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% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. d(a6vEL4
% <R{\pz2w
% The Zernike functions are an orthogonal basis on the unit circle. Mdwh-Cis/
% They are used in disciplines such as astronomy, optics, and z|P& 8#txM
% optometry to describe functions on a circular domain. 0l_-
% *rEW@06^\
% The following table lists the first 15 Zernike functions. F"23>3
% dbZPt~S'$
% n m Zernike function Normalization jv0e&rt
% -------------------------------------------------- 1<R
\V
% 0 0 1 1 ;pB?8Z
% 1 1 r * cos(theta) 2 FpRK^MEkG
% 1 -1 r * sin(theta) 2 2N,*S
% 2 -2 r^2 * cos(2*theta) sqrt(6) t%dPj8~
% 2 0 (2*r^2 - 1) sqrt(3) OC\C^Yh*U
% 2 2 r^2 * sin(2*theta) sqrt(6) :,VyOmf
% 3 -3 r^3 * cos(3*theta) sqrt(8) kD;BwU[
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Ra*9d]N@
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) \aW5V: ?
% 3 3 r^3 * sin(3*theta) sqrt(8) qbAoab53
% 4 -4 r^4 * cos(4*theta) sqrt(10) Tf0#+6 1>
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Y2$%%@
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) E_y h9lk
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) @/7Rp8Fr
% 4 4 r^4 * sin(4*theta) sqrt(10) \|&5eeE@
% -------------------------------------------------- Q'=!1^&
% 4*dT|NU
% Example 1: =q"3a9pb7
% pI^n("|
% % Display the Zernike function Z(n=5,m=1) 7I.[1V`
% x = -1:0.01:1; 6% ,Q
% [X,Y] = meshgrid(x,x); (Pu*[STTT
% [theta,r] = cart2pol(X,Y); l/I W"A
% idx = r<=1; (?3(=+t
% z = nan(size(X)); |F=^Cu,
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ?QMs<
% figure l;;:3:
% pcolor(x,x,z), shading interp &`%C'KZ
% axis square, colorbar =Tj0dfO|"
% title('Zernike function Z_5^1(r,\theta)')
w\QpQ~OX
% 4v?S`w:6
% Example 2: eX$Biv1N
% F%|(pHk
% % Display the first 10 Zernike functions 7:;V[/
% x = -1:0.01:1; O,;SA
% [X,Y] = meshgrid(x,x); {M$8V~8D
% [theta,r] = cart2pol(X,Y); 6Rt pB\hq
% idx = r<=1; /;>EyWW
% z = nan(size(X)); GS^4tmc
% n = [0 1 1 2 2 2 3 3 3 3]; d8K^`k+x
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; NSkI2>+P
% Nplot = [4 10 12 16 18 20 22 24 26 28]; >]`x~cE.5
% y = zernfun(n,m,r(idx),theta(idx)); /za,&7sf
% figure('Units','normalized') #*`|}_6L
% for k = 1:10 K4tX4U[Z
% z(idx) = y(:,k); r9U1 O@c
% subplot(4,7,Nplot(k)) @GV^B'}*
% pcolor(x,x,z), shading interp SW=p5@Hy{
% set(gca,'XTick',[],'YTick',[]) [+1
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% axis square s0h)~z
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 8;5/_BwMu
% end Yl f4q/-
% =os j}(
% See also ZERNPOL, ZERNFUN2. +(<f(]bG
BKTsc/v2>:
% Paul Fricker 11/13/2006 ^q6~xC,/
|
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t&oNJq{
% Check and prepare the inputs: @PI\.y_w
% ----------------------------- v$c D!`+k
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) :z:Blp>nK/
error('zernfun:NMvectors','N and M must be vectors.') wVVe L$28
end ~:@H6Ke[
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if length(n)~=length(m) )<|T Ep4r-
error('zernfun:NMlength','N and M must be the same length.') :s5g6TR
end Z*)<E)
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n = n(:); BAhC-;B#R
m = m(:); t&xx-4
if any(mod(n-m,2)) $1v5*E
error('zernfun:NMmultiplesof2', ... ZUu^==a
'All N and M must differ by multiples of 2 (including 0).') x\%egw
end =bDG|:+
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if any(m>n) (gZ!o_
error('zernfun:MlessthanN', ... >g[W@FhT'k
'Each M must be less than or equal to its corresponding N.') jz)H?UuDY
end sa` Yan
s:ruCS
if any( r>1 | r<0 ) (TE2t7ab|M
error('zernfun:Rlessthan1','All R must be between 0 and 1.') B'Wky>5)
end _x!pMj(A
5-OvPTY`M
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) cC4T3]4l'
error('zernfun:RTHvector','R and THETA must be vectors.') |(SW
end R+K[/AA
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r = r(:); ueR42J%s
theta = theta(:); @I&"P:E0F;
length_r = length(r); +[ItkfSod!
if length_r~=length(theta) ;i9CQ0e?
error('zernfun:RTHlength', ... wLtTC4D
'The number of R- and THETA-values must be equal.') qo@dFKy
end MjpJAV/84
}]I?vyQ#V
% Check normalization: $ZS9CkN
% -------------------- v\ Ljm,+
if nargin==5 && ischar(nflag) (5> ibe
isnorm = strcmpi(nflag,'norm'); %\l,X{X
if ~isnorm qC )VT3
error('zernfun:normalization','Unrecognized normalization flag.') 'T[zh#v>S
end mw[4<vfB0a
else +!vRU`
isnorm = false; 0R&
U18)y
end Bt,Xe~$z-
O[!o1.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
`xUPML-
% Compute the Zernike Polynomials K_QCYS.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |Z ,G
6<Z:Xw
% Determine the required powers of r: WM"^#=+$
% ----------------------------------- 5F"?]'*/
m_abs = abs(m); O@iW?9C+
rpowers = []; tWnm{mF
for j = 1:length(n) W[Bu&?h$
rpowers = [rpowers m_abs(j):2:n(j)]; oui!fTy
end u7?juI#Cl
rpowers = unique(rpowers); !9,
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>|)0Amt
% Pre-compute the values of r raised to the required powers, %z~U@Mka
% and compile them in a matrix: ozC!q)j
% ----------------------------- =[JN'|Q+
if rpowers(1)==0 pGY]VwY
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @@IA35'tc
rpowern = cat(2,rpowern{:}); 2HXKz7da
rpowern = [ones(length_r,1) rpowern]; 4Umsc>yfK
else Net)l@IB]
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); [+g@@\X4
rpowern = cat(2,rpowern{:}); ;YDF*~9u
end t1jlxK
6#M0AG
% Compute the values of the polynomials: %i8>w:@NW
% -------------------------------------- "<x~{BN?
y = zeros(length_r,length(n)); N?;o_^C
for j = 1:length(n) T-C#xmY(
s = 0:(n(j)-m_abs(j))/2; X5Y
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pows = n(j):-2:m_abs(j); <zuE=0P~%
for k = length(s):-1:1 Rt^<xXX$
p = (1-2*mod(s(k),2))* ... JGcD{RU|
prod(2:(n(j)-s(k)))/ ... WEtA4zCO
prod(2:s(k))/ ... W@,p9=425
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 1G%PXrEj8
prod(2:((n(j)+m_abs(j))/2-s(k))); P0e ""9JOo
idx = (pows(k)==rpowers); JA(fam~{
y(:,j) = y(:,j) + p*rpowern(:,idx); t3t0vWE<,
end [fi'=Cb
2BDan^:-Av
if isnorm $-Pqs
^g
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); P4j 8`}&/
end MJ,ZXJXs
end BD7@Mj*|
% END: Compute the Zernike Polynomials _]xt65TL
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% QZ3(u<f
+'/}[1q1/T
% Compute the Zernike functions: d:hL
)x
% ------------------------------ 8i;)|z7
idx_pos = m>0; ] 5v4^mk
idx_neg = m<0; 2l@"p!ar=
ZQ~myqx,+L
z = y; &
8'(
if any(idx_pos) ncattp
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); RP,:[}mPl
end 5!F\h'E
if any(idx_neg) j-YJ."
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ~sIGI?5f
end z8/xGQn
eR-=<0Iw;
% EOF zernfun