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 A5Jadz~
function z = zernfun(n,m,r,theta,nflag) ;
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. kc2B_+Y1
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N H>/,Re
% and angular frequency M, evaluated at positions (R,THETA) on the 0BC@wV
% unit circle. N is a vector of positive integers (including 0), and UmVn: a
% M is a vector with the same number of elements as N. Each element j_rO_m <8
% k of M must be a positive integer, with possible values M(k) = -N(k) =cl#aS}e8
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, vb~%u;zrC@
% and THETA is a vector of angles. R and THETA must have the same @sn:%/x _
% length. The output Z is a matrix with one column for every (N,M) j>JBZ#g
% pair, and one row for every (R,THETA) pair. B1}i0pV,,
% >V(C>^%->
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 4xW~@meNB
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 66?`7j X
% with delta(m,0) the Kronecker delta, is chosen so that the integral T/|!^qLF
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, HMUx/M.j
% and theta=0 to theta=2*pi) is unity. For the non-normalized /1LN\Eu
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. !b`fykC
% D/=05E%[81
% The Zernike functions are an orthogonal basis on the unit circle. P[ o"%NZ'
% They are used in disciplines such as astronomy, optics, and !6|_`l>G,
% optometry to describe functions on a circular domain. 2*D2jw
% m%J?5rR3
% The following table lists the first 15 Zernike functions. [6VM4l"
% q,fp
DNo
% n m Zernike function Normalization `S((F|Ty=;
% -------------------------------------------------- .'M.yE~5J
% 0 0 1 1 2Di~}* 9&
% 1 1 r * cos(theta) 2 AIOGa<^
% 1 -1 r * sin(theta) 2 YTTy6*\,_
% 2 -2 r^2 * cos(2*theta) sqrt(6) Kc]cJ`P4.
% 2 0 (2*r^2 - 1) sqrt(3) w-WAgAch
% 2 2 r^2 * sin(2*theta) sqrt(6) R,D/:k'~k
% 3 -3 r^3 * cos(3*theta) sqrt(8) {($m LfC4
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Qf0P"s`
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8)
%t_'rv
% 3 3 r^3 * sin(3*theta) sqrt(8) qsp3G7\'=
% 4 -4 r^4 * cos(4*theta) sqrt(10) TgV-U
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) A&1EOQ=N
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) T+XcEI6w
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6'*6tS
% 4 4 r^4 * sin(4*theta) sqrt(10) fAStM:
% -------------------------------------------------- a'`i#U
% 60~*$`
% Example 1: 1N _"Mm{
% d
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% % Display the Zernike function Z(n=5,m=1) ,Z|O y|+'
% x = -1:0.01:1; 0*:n<T9
% [X,Y] = meshgrid(x,x); rs4:jS$)
% [theta,r] = cart2pol(X,Y); fX9b1x
% idx = r<=1; >;G_o="X
% z = nan(size(X)); wa[J\lW
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Onqapm0
% figure <8%+-[(
% pcolor(x,x,z), shading interp X ([^i;mr
% axis square, colorbar TH4f"h+B3"
% title('Zernike function Z_5^1(r,\theta)') q:up8-LAr
% 8Ie0L3d-
% Example 2: Y]R=z*i%
% LL:N/1ysG
% % Display the first 10 Zernike functions nS$4[!0
% x = -1:0.01:1; CNuE9|W(vI
% [X,Y] = meshgrid(x,x); dT1UYG}>j
% [theta,r] = cart2pol(X,Y); s7E %Et
% idx = r<=1; q@1A2L\Om
% z = nan(size(X)); zhE4:g9v
% n = [0 1 1 2 2 2 3 3 3 3]; "j`T'%EV
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; xg%{p``
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ZK{1z|
% y = zernfun(n,m,r(idx),theta(idx)); `o_i+?E
% figure('Units','normalized') ,f>^q"
% for k = 1:10 U#Kw+slM
% z(idx) = y(:,k); \Q`#E'?
% subplot(4,7,Nplot(k)) BB,-HhYT0
% pcolor(x,x,z), shading interp 78T;b7!-C
% set(gca,'XTick',[],'YTick',[]) aG"
% axis square MAqETjB
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) p^{yA"MQ
% end N<(rP1)`v
% %xx;C{g;a
% See also ZERNPOL, ZERNFUN2. oMn'{+(w
'#K~hep
% Paul Fricker 11/13/2006 ^l(,'>Cn
"
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% Check and prepare the inputs: ,iiI5FR
% ----------------------------- ?fU{?nI}>p
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ieEtC,U
error('zernfun:NMvectors','N and M must be vectors.') M(^IRI-
end qyE*?73W
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if length(n)~=length(m) _n*gj-
error('zernfun:NMlength','N and M must be the same length.') ('_S1?y
end _Axw$oYS
VF-[O
n = n(:); UA0R)BH'
m = m(:); y(Pv1=e
if any(mod(n-m,2)) T2T?)_f /
error('zernfun:NMmultiplesof2', ... <p_2&&?
'All N and M must differ by multiples of 2 (including 0).') ~8Ef`zL
end }q/[\3
sQzr+]+#9
if any(m>n) p{V(! v|
error('zernfun:MlessthanN', ... '~6l
6wi
'Each M must be less than or equal to its corresponding N.') /{ 8 .Jcx$
end ]_y;Igaj
Q!fk|D+j
if any( r>1 | r<0 ) )/v`k>E
error('zernfun:Rlessthan1','All R must be between 0 and 1.') d D^?%,a
end ]%5gPfv[T
+zFEx%3^
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) G|$n,X1O(
error('zernfun:RTHvector','R and THETA must be vectors.') MIv,$
end %+$!ctn
#
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r = r(:); ^F"eHUg
theta = theta(:); n{F&GE="
length_r = length(r); SMm$4h R
if length_r~=length(theta) G>^ _&(c@2
error('zernfun:RTHlength', ... T6rjtq
'The number of R- and THETA-values must be equal.') tUFXx\p
end Yceex}X*5
M<)Vtn
% Check normalization: `MMZR=LA
% -------------------- Gc!&I+kd
if nargin==5 && ischar(nflag) iEBxBsz_
isnorm = strcmpi(nflag,'norm'); "k7C
if ~isnorm %t-}dC&
error('zernfun:normalization','Unrecognized normalization flag.') "CT`]:GGK
end Z5>}
else 3DrW[\
isnorm = false; y{qKb:~wv
end ViG-tb
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FE?^}VH
% Compute the Zernike Polynomials xHwcP2 1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5NYYrA8,^
U| 1&=8l
% Determine the required powers of r: cNRe >
% ----------------------------------- q}7(w$&
m_abs = abs(m); 6~(iLtd#
rpowers = []; jowR!rqf
for j = 1:length(n) [IuF0$w=dj
rpowers = [rpowers m_abs(j):2:n(j)]; |Q~5TL>b
end 8J#TP7;
rpowers = unique(rpowers); T;JA.=I
ZGWZ2>k
% Pre-compute the values of r raised to the required powers, wo!;Bxo
N
% and compile them in a matrix: d[Rs
% ----------------------------- u*H
V
if rpowers(1)==0 c:z<8#A}
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); V_7QWIdiy>
rpowern = cat(2,rpowern{:}); 4EEXt<c.
rpowern = [ones(length_r,1) rpowern]; 0Z~G:$O/i
else q1o)l
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); |-k~Fa
rpowern = cat(2,rpowern{:}); bG9$ &,
end #kDJ>r |&-
syLpnNx=
% Compute the values of the polynomials: C")NNs=
% -------------------------------------- Q|J$R
y = zeros(length_r,length(n)); XB-l[4?
for j = 1:length(n) BnLE+X
s = 0:(n(j)-m_abs(j))/2; ~C2[5r{So
pows = n(j):-2:m_abs(j); 0(dXU\Y
for k = length(s):-1:1 t12 xPtN1
p = (1-2*mod(s(k),2))* ... *6%r2l'kZ
prod(2:(n(j)-s(k)))/ ... f)K1j{TZ
prod(2:s(k))/ ... 'gwh:8Xc
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... <swYo<?J#
prod(2:((n(j)+m_abs(j))/2-s(k))); 5%Q[X
idx = (pows(k)==rpowers); /WKp\r(Hp
y(:,j) = y(:,j) + p*rpowern(:,idx); QZ51}i
end 6*H F`@(
b:}+l;e52
if isnorm ' fm}&0
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); J~vK`+Zs
end kUG3_ *1
.
end ^iq$zHbc0u
% END: Compute the Zernike Polynomials WH^rM`9
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% HNtl>H
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% Compute the Zernike functions: D#,P-0+%
% ------------------------------ w_!]_6%{b
idx_pos = m>0; +b]+5!
idx_neg = m<0; *aF<#m v
(GdL(H#IL
z = y; 6-@n$5W0
if any(idx_pos) C7[CfcPA
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); )FrXD3p
end %v(\;&@
if any(idx_neg) &<sN(;%0R
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); \;G 97o
end #E(
n
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% EOF zernfun