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 GmcxN<
function z = zernfun(n,m,r,theta,nflag) F
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. @|j`I1r.A
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N
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% and angular frequency M, evaluated at positions (R,THETA) on the B@63=a*kG
% unit circle. N is a vector of positive integers (including 0), and nv2Y6e}dG
% M is a vector with the same number of elements as N. Each element |rq~.cA
% k of M must be a positive integer, with possible values M(k) = -N(k)
u> %r(
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, +wY3E*hU
% and THETA is a vector of angles. R and THETA must have the same a+{YTR>0m
% length. The output Z is a matrix with one column for every (N,M) ;KbnaUAS8
% pair, and one row for every (R,THETA) pair. qWy{{A+
% ~lzV=c$t
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Ra;e#)7X
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 2Qc&6-;`
% with delta(m,0) the Kronecker delta, is chosen so that the integral /ZvNgaH5M
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, #OJsu
% and theta=0 to theta=2*pi) is unity. For the non-normalized M#=woj&[
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Bb}JyT
% 7Aq4YjbX
% The Zernike functions are an orthogonal basis on the unit circle. 6K[s),rdv
% They are used in disciplines such as astronomy, optics, and X:j&+d2g0/
% optometry to describe functions on a circular domain. 9 /t}S6b{
% H)
m!)=\'
% The following table lists the first 15 Zernike functions. n'ZlIh
% U:J~Oy_Z
% n m Zernike function Normalization @>ONp|}@qI
% -------------------------------------------------- U@BVVH?,o
% 0 0 1 1 VS%8f.7ep
% 1 1 r * cos(theta) 2 D4c}z#}*0
% 1 -1 r * sin(theta) 2 MP w@O0QS
% 2 -2 r^2 * cos(2*theta) sqrt(6) q~;P^i<Y
% 2 0 (2*r^2 - 1) sqrt(3) 8T&m{s
% 2 2 r^2 * sin(2*theta) sqrt(6) ~*L H[l>K
% 3 -3 r^3 * cos(3*theta) sqrt(8) r&o%n5B
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) S;Lqx5Cd
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 1&i!92:E
% 3 3 r^3 * sin(3*theta) sqrt(8) tCI8\~
% 4 -4 r^4 * cos(4*theta) sqrt(10) shYcfLJ
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?N,a {#w
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) RVXRF_I
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) {SqY77
% 4 4 r^4 * sin(4*theta) sqrt(10) Lyt6DvAp"
% -------------------------------------------------- ,HUs MCXQ
% S]K^wj[
% Example 1: B5=L</Aj
% |jEKUTv,G
% % Display the Zernike function Z(n=5,m=1) r\'3q'7p
% x = -1:0.01:1; M\enjB7k
% [X,Y] = meshgrid(x,x); ;}.jRmnJ
% [theta,r] = cart2pol(X,Y); R+]Fh4t
% idx = r<=1; pZlBpGQf
% z = nan(size(X)); f$*M;|c1c/
% z(idx) = zernfun(5,1,r(idx),theta(idx)); f*NtnD=rJ
% figure a_x$I?,
% pcolor(x,x,z), shading interp K{x<zv&,
% axis square, colorbar NV36Q^Am[
% title('Zernike function Z_5^1(r,\theta)') "h2;65@
% zp% MK+x
% Example 2: rZKv:x}{6
% I@pnZ-5
% % Display the first 10 Zernike functions 7M3q|7?
% x = -1:0.01:1; jdXkU
% [X,Y] = meshgrid(x,x); j MW|B
% [theta,r] = cart2pol(X,Y); !+U#^2Gz
% idx = r<=1; :2 QA#
% z = nan(size(X)); ##}a0\x|
% n = [0 1 1 2 2 2 3 3 3 3]; Af5In9WB5
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; uLe+1`Y5Ux
% Nplot = [4 10 12 16 18 20 22 24 26 28]; %/1`"M5ko
% y = zernfun(n,m,r(idx),theta(idx)); HR['y9U
% figure('Units','normalized') h&h]z[r R
% for k = 1:10 u'yePJTE
% z(idx) = y(:,k); Pkc4=i,`A
% subplot(4,7,Nplot(k)) qW?^_
% pcolor(x,x,z), shading interp ~AjbF(Ad
% set(gca,'XTick',[],'YTick',[]) jM2gu~
% axis square B&-;w_K
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) v@Otp
% end qW;nWfkYC
% 0EPF;
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% See also ZERNPOL, ZERNFUN2. _L%/NXu,
q'C'S#qqn
% Paul Fricker 11/13/2006 .zBSjh_=H
Da?0B9'
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% Check and prepare the inputs: OWtN=Gk
% ----------------------------- ~qFi0<-M
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) gAv?\9=a)W
error('zernfun:NMvectors','N and M must be vectors.') ~uzu*7U
end @^k$`W;
"%,zB_ng\<
if length(n)~=length(m) @zsr.d6Q
error('zernfun:NMlength','N and M must be the same length.') _.?$~;7
end h8pc<t\6
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n = n(:); 3zs~Y3M?i
m = m(:);
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if any(mod(n-m,2)) {BJ[h
error('zernfun:NMmultiplesof2', ... KXicy_@DC`
'All N and M must differ by multiples of 2 (including 0).') BCsW03sQ
end SV6Np?U
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if any(m>n) A' dt
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error('zernfun:MlessthanN', ... 5OpK~f5
'Each M must be less than or equal to its corresponding N.') { F.Ihw
end \-V
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if any( r>1 | r<0 ) Xa9G;J$
error('zernfun:Rlessthan1','All R must be between 0 and 1.') jUW{Z@{U
end zcIZJVYA
5#QB&A>
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) -bZ^A~<O,
error('zernfun:RTHvector','R and THETA must be vectors.') 42Kzdo|}
end -qid.
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r = r(:); 9R'rFI
theta = theta(:); pZjyzH{~
length_r = length(r); z~z.J]
if length_r~=length(theta) xV<NeU
error('zernfun:RTHlength', ... Rqvm%sAi
'The number of R- and THETA-values must be equal.') xU67ztS'E'
end ec"L*l"
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% Check normalization: Bz/NFNi[p
% -------------------- XK(<N<Z@|e
if nargin==5 && ischar(nflag) &W".fRH_O
isnorm = strcmpi(nflag,'norm'); mgH4)!Z*56
if ~isnorm KY2xKco
error('zernfun:normalization','Unrecognized normalization flag.')
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end j&Z:|WniK
else h r* KDT^!
isnorm = false;
LL kAA?P
end NrS1y"#d9
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% * k=Pk
% Compute the Zernike Polynomials L7a+ #mGE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Vj~R6
i FS?nZ~.
% Determine the required powers of r: |iO2,99i
% ----------------------------------- tao3Xr^?
m_abs = abs(m); ph^qQDA
rpowers = []; @}aK\
for j = 1:length(n) dIIsO{Zqv
rpowers = [rpowers m_abs(j):2:n(j)]; 3ywBq9FGhp
end bLaD1rnGi
rpowers = unique(rpowers); 0D$+WX
U/0NN>V
% Pre-compute the values of r raised to the required powers, ]2K>#sn-]
% and compile them in a matrix: mxP{"6
% ----------------------------- 9I^_n+E
if rpowers(1)==0 2{@:
:JZ
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); - DL/Hk_r
rpowern = cat(2,rpowern{:}); ]7'Q2OU7
rpowern = [ones(length_r,1) rpowern]; r(i<H%"Z
else .o.@cLdU
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);
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rpowern = cat(2,rpowern{:}); 1*;?uC\
end F}i rCi47c
pwU]r
% Compute the values of the polynomials: {l_R0
% -------------------------------------- D[;6xJ
y = zeros(length_r,length(n)); ]'2p"A0U
for j = 1:length(n) IxgnZX4N
s = 0:(n(j)-m_abs(j))/2; _%Mu{Ni&
pows = n(j):-2:m_abs(j); UmInAH4
for k = length(s):-1:1 y(6&90cr
p = (1-2*mod(s(k),2))* ... *A
c~
prod(2:(n(j)-s(k)))/ ... v |QFUa`
prod(2:s(k))/ ... AB}Qd\
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... a]
>|2JN<&
prod(2:((n(j)+m_abs(j))/2-s(k))); v_)cp9d]
idx = (pows(k)==rpowers); 6q6&N'We
y(:,j) = y(:,j) + p*rpowern(:,idx); ]<W1edr
end !>9*$E
|
V,|9$A;
if isnorm ^ /:]HG
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); K& 2p<\2
end &<.Z4GxS
end P%B1dRa
% END: Compute the Zernike Polynomials 6t/})Xv
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |WubIj*\{
(WN 'wp
% Compute the Zernike functions: |w /txn8G|
% ------------------------------ /KlA7MH 6
idx_pos = m>0; ,7/un8:%c
idx_neg = m<0; r/3!~??x
x1mxM#ql
z = y; +zz9u?2C`
if any(idx_pos) 98o;_tU'
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Ldt7?Y(V(
end &v3r#$Hj[
if any(idx_neg) #; }IHAR
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 7{az %I$h
end YfF&: "-NU
gEU)UIJ
% EOF zernfun