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 K/(LF}
function z = zernfun(n,m,r,theta,nflag) #~j $J
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. _x5-!gK
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N B#."cg4VR
% and angular frequency M, evaluated at positions (R,THETA) on the (a!E3y5,
% unit circle. N is a vector of positive integers (including 0), and F@/syX;bb5
% M is a vector with the same number of elements as N. Each element 8;=?F>]xn
% k of M must be a positive integer, with possible values M(k) = -N(k) &h[)nD
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Ew}GPJ
% and THETA is a vector of angles. R and THETA must have the same |QzJHP @
% length. The output Z is a matrix with one column for every (N,M) aJm5`az)
% pair, and one row for every (R,THETA) pair. sUF5Yq:9
% _BG`!3U+
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike _6FDuCVD-
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), dY?l
oFz
% with delta(m,0) the Kronecker delta, is chosen so that the integral &\?{%xj
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, w}}+8mk[
% and theta=0 to theta=2*pi) is unity. For the non-normalized 9F,XjPK=
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. IwFf8?
3
% Qvny$sr2
% The Zernike functions are an orthogonal basis on the unit circle. l$BKE{rg
% They are used in disciplines such as astronomy, optics, and /XRgsF
% optometry to describe functions on a circular domain. F`Vp
% s5 Fn("h]n
% The following table lists the first 15 Zernike functions. R U[
% -'W:P'BG
% n m Zernike function Normalization UL7%6v{'*
% -------------------------------------------------- TuMZHB7h;
% 0 0 1 1 XSZjuQ<[3
% 1 1 r * cos(theta) 2 uJ*|SSN~
% 1 -1 r * sin(theta) 2 w*SF Q_6YE
% 2 -2 r^2 * cos(2*theta) sqrt(6) r~;.8qs
% 2 0 (2*r^2 - 1) sqrt(3) Vfw +m1sS
% 2 2 r^2 * sin(2*theta) sqrt(6) [-[|4|CnOm
% 3 -3 r^3 * cos(3*theta) sqrt(8) `).;W
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 7Ph+Vs+h
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) e ]>{?Z
% 3 3 r^3 * sin(3*theta) sqrt(8) mR{%f?B
% 4 -4 r^4 * cos(4*theta) sqrt(10) {iq{<;)U?U
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) JvUHoc$sI
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) >|T?87
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1 _W5@)
% 4 4 r^4 * sin(4*theta) sqrt(10) OQX ek@~2
% -------------------------------------------------- G[yN*C
% Q!%CU8!`&
% Example 1: ;rta#pRn
%
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% % Display the Zernike function Z(n=5,m=1) I|[aa$G
% x = -1:0.01:1; }\ui}\
% [X,Y] = meshgrid(x,x); ;Wr,VU]
% [theta,r] = cart2pol(X,Y); Z42v@?R.!W
% idx = r<=1; }Lwj~{
% z = nan(size(X)); 13{"sY:PT#
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ;lWy?53=@
% figure T{K+1SPy4
% pcolor(x,x,z), shading interp -ap;Ul?
% axis square, colorbar eEe8T=mD
% title('Zernike function Z_5^1(r,\theta)') <Q-ufF85)
% S+OI?QS
% Example 2: m9>nvrQ
% g?o$:>c
% % Display the first 10 Zernike functions N<Q}4%^c
% x = -1:0.01:1; XKU=VOY
% [X,Y] = meshgrid(x,x); 7#|NQ=yd
% [theta,r] = cart2pol(X,Y); *&2#;mf3
% idx = r<=1; .y[K =p3
% z = nan(size(X)); E.% F/mM
% n = [0 1 1 2 2 2 3 3 3 3]; 1aMBCh<}JN
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; yZ)ScB^
% Nplot = [4 10 12 16 18 20 22 24 26 28]; RBgkC+2
% y = zernfun(n,m,r(idx),theta(idx)); 5BCaE)J
% figure('Units','normalized') $BBfsaJPT
% for k = 1:10 |)JoxqR
% z(idx) = y(:,k); @x J^JcE
% subplot(4,7,Nplot(k)) x}>tX
% pcolor(x,x,z), shading interp n_ez6{
% set(gca,'XTick',[],'YTick',[]) ujWHO$uz!
% axis square /7"1\s0 U
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) tw3d>H`
% end z=Vvb
% =L
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% See also ZERNPOL, ZERNFUN2. >`\*{]
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% Paul Fricker 11/13/2006 t@ JPnA7~
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Sf'5/9<DW+
% Check and prepare the inputs: dO//
% ----------------------------- 7ER 2h*
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) coFg69\^
error('zernfun:NMvectors','N and M must be vectors.') q@-qA]
end (Mm{"J3uv
#f~a\}$I
if length(n)~=length(m) Y-c~"#
error('zernfun:NMlength','N and M must be the same length.') ;VFr5.*x
end t5Mo'*j
=
W=\dsdnu*
n = n(:); ,"VQ0Z1
m = m(:); _~(Xd@c(
if any(mod(n-m,2)) .XB] X
error('zernfun:NMmultiplesof2', ... ZAH<!@qh
'All N and M must differ by multiples of 2 (including 0).') +?:V\niQI
end hw'2q9J|
MH Yf8HN
if any(m>n) 2*L/c-
error('zernfun:MlessthanN', ... bgK(l d`
'Each M must be less than or equal to its corresponding N.') RZtL<2.@
end nm-Y?!J
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if any( r>1 | r<0 ) ~&/Gx_KU
error('zernfun:Rlessthan1','All R must be between 0 and 1.') a*[\edcHU
end piFQ7B
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) LG|,g3&
error('zernfun:RTHvector','R and THETA must be vectors.') ibc/x v2
end `~]ReJ!X%
ZO1J";>u
r = r(:); p,8Z{mLn
theta = theta(:); dR+$7N$
length_r = length(r); v+sbRuo8
if length_r~=length(theta) A,e^bM
error('zernfun:RTHlength', ... _D4}[`
'The number of R- and THETA-values must be equal.') Wd5t,8*8
end 8 vw]u_e
T_Y }1n|7[
% Check normalization: x+e
_pb
% -------------------- UVJ(iNK"
if nargin==5 && ischar(nflag) 9p4U\hx
isnorm = strcmpi(nflag,'norm'); 8!SiTOzR?
if ~isnorm k#) .E X
error('zernfun:normalization','Unrecognized normalization flag.') @GtZK
end uP]o39b;V
else A%2}?Ds
isnorm = false; OP}p;(
end UYOn
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% " G6jUTt
% Compute the Zernike Polynomials %Ab_PAw
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% p ri{vveN@
q*nz4QTOE
% Determine the required powers of r: 8|NJ(D-$
% -----------------------------------
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m_abs = abs(m); -(}1o9e\7
rpowers = []; G9inNz*Cx
for j = 1:length(n) ji
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rpowers = [rpowers m_abs(j):2:n(j)]; # :w2Hf6Q
end =+S3S{\CK
rpowers = unique(rpowers); 9 lJj/
]/Qy1,
% Pre-compute the values of r raised to the required powers, xN8JrZE&
% and compile them in a matrix: )N6[rw<