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 _b1w<T
`
function z = zernfun(n,m,r,theta,nflag) $l!+SLK
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 9t^Q_ [hG
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Q)b*;
@
% and angular frequency M, evaluated at positions (R,THETA) on the ~i)IY1m"
% unit circle. N is a vector of positive integers (including 0), and qOd*9AS'|M
% M is a vector with the same number of elements as N. Each element PgF7ug%,@C
% k of M must be a positive integer, with possible values M(k) = -N(k) om'DaG`A
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 0(~,U!g[=
% and THETA is a vector of angles. R and THETA must have the same 2V 9vS
% length. The output Z is a matrix with one column for every (N,M) 7L\kna<
% pair, and one row for every (R,THETA) pair. tZn=[X~Vw@
% %knPeo&
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike W2\Q-4D
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), qC?\i['`
% with delta(m,0) the Kronecker delta, is chosen so that the integral ]$gBX=
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, `:fc*n,*
% and theta=0 to theta=2*pi) is unity. For the non-normalized _laLTP*
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. .|g67PH=
% +8etCx
% The Zernike functions are an orthogonal basis on the unit circle. 56R)631]p
% They are used in disciplines such as astronomy, optics, and ;'x\L<b/)
% optometry to describe functions on a circular domain. j,c8_;X!
% dJ0qg_ U&
% The following table lists the first 15 Zernike functions. j*aYh^
% A&~<qgBTp
% n m Zernike function Normalization ~J:"sUR
% -------------------------------------------------- Ie%twc
% 0 0 1 1 Lp?JSMe
% 1 1 r * cos(theta) 2 "|:I]ZB
% 1 -1 r * sin(theta) 2
0^PI&7A?y
% 2 -2 r^2 * cos(2*theta) sqrt(6) 47c` ) *Hc
% 2 0 (2*r^2 - 1) sqrt(3) rZBOWT
% 2 2 r^2 * sin(2*theta) sqrt(6) x>yeF,q1
% 3 -3 r^3 * cos(3*theta) sqrt(8) ]8i2'x
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) uBe1{Z
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) mVBF2F<4
% 3 3 r^3 * sin(3*theta) sqrt(8) Rr'^l]
% 4 -4 r^4 * cos(4*theta) sqrt(10) _(<D*V[
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) C/!c? $J
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) :RnFRAcr
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) '"=Mw;p
% 4 4 r^4 * sin(4*theta) sqrt(10) 0bQm:J[(#
% -------------------------------------------------- %hu] =
% \dL#PI3
% Example 1: j`9+pI
% Z=vzF0
% % Display the Zernike function Z(n=5,m=1) gTp){
% x = -1:0.01:1; u,6 'yB'u
% [X,Y] = meshgrid(x,x); 8'(|1
% [theta,r] = cart2pol(X,Y); '5mzlR
% idx = r<=1; ;S FmbZ%~
% z = nan(size(X)); D*oJz3[
% z(idx) = zernfun(5,1,r(idx),theta(idx)); W_zAAIY_Y
% figure vh~:{akR
% pcolor(x,x,z), shading interp >qSaF
% axis square, colorbar {bUd"Tu
% title('Zernike function Z_5^1(r,\theta)') wb>>bV+U
% o9:GKc
% Example 2: xCd9b:jG
% +C{ %pF
% % Display the first 10 Zernike functions l|[8'*]r!
% x = -1:0.01:1; OudD1( )W
% [X,Y] = meshgrid(x,x); cN> z`xl
% [theta,r] = cart2pol(X,Y); 7b2N'^z}
% idx = r<=1; J@{yWgLg
% z = nan(size(X)); q1nGj
% n = [0 1 1 2 2 2 3 3 3 3]; ,'CDKzY
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; bm{L6D E
% Nplot = [4 10 12 16 18 20 22 24 26 28]; {GS7J
% y = zernfun(n,m,r(idx),theta(idx)); `3$S^|v
% figure('Units','normalized') HgwL~vG
% for k = 1:10 ?^F#}>C
% z(idx) = y(:,k); ~lR"3z_Z}
% subplot(4,7,Nplot(k)) /#PEEN
% pcolor(x,x,z), shading interp ] Qp0|45=
% set(gca,'XTick',[],'YTick',[]) x0])&':!
% axis square P^%.7C
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 5@+8*Fdk
% end 5Dy800.B2
% /:a~;i
% See also ZERNPOL, ZERNFUN2. sa~.qmqu
>sE5zj|V
% Paul Fricker 11/13/2006 Aa5IccR
/hue]ZaQq
vXnTPjbE
% Check and prepare the inputs: Ml)Xq-&wc
% ----------------------------- saH +C@_,
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) %aX<p{EY
error('zernfun:NMvectors','N and M must be vectors.') 7oPBe1P,K+
end T8.@}a
$cev,OW6]
if length(n)~=length(m) BZqb
o `9
error('zernfun:NMlength','N and M must be the same length.') 3<x_[0v`K1
end .cA[b
<3;/,>^ Pm
n = n(:); g]C+uj^
m = m(:); ?K7m:Dx
if any(mod(n-m,2)) c@{,&,vsj
error('zernfun:NMmultiplesof2', ... A+j~oR
'All N and M must differ by multiples of 2 (including 0).') SvH=P!`+
end (r,RwWYm
>RxZ-.,a
if any(m>n) :L9\`&}FS
error('zernfun:MlessthanN', ... u>(s.4]+
'Each M must be less than or equal to its corresponding N.') J#CF S G
end Mg95us
kTG}>I
if any( r>1 | r<0 ) EOV<|WF>
error('zernfun:Rlessthan1','All R must be between 0 and 1.') uH]n/Kv1,
end \O?#gW\tR
&l%#OI}OE
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 4qjY,QJ
error('zernfun:RTHvector','R and THETA must be vectors.') 6^['g-\2
end dL")E|\\k
8|7fd|6~
r = r(:); $cH'9W}3K
theta = theta(:); 4;|&}Ij
length_r = length(r); Y(/VW&K&:
if length_r~=length(theta) A0S6 4(
error('zernfun:RTHlength', ... lp?geav
'The number of R- and THETA-values must be equal.') NF0} eom
end Vm&fw".J
[HIg\N$I8C
% Check normalization: 33couAP#
% -------------------- 1O9V Ej5
if nargin==5 && ischar(nflag) a +*|P
isnorm = strcmpi(nflag,'norm'); =Ze~6vS,
if ~isnorm uZ Id.+Rk
error('zernfun:normalization','Unrecognized normalization flag.') O>w$
end @8@cpm
else ~v9\4O
isnorm = false; 9ZG.%+l
end bQ0m=BzF
p=9G)VO
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Old5E&
% Compute the Zernike Polynomials L<QqQ"`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LtH;#Q
34]f[jJ|
% Determine the required powers of r: [F+lVb
% ----------------------------------- G?=X!up(
m_abs = abs(m); 'fcJ]%-=
rpowers = []; |!I# T
for j = 1:length(n) :?jOts>uP
rpowers = [rpowers m_abs(j):2:n(j)]; X"8Jk4y
end u-j$4\'
rpowers = unique(rpowers); sh}=#eb
PWL Mux
% Pre-compute the values of r raised to the required powers, )F]E[sga
% and compile them in a matrix: D4n~2]
% ----------------------------- R$(,~~MH
if rpowers(1)==0 6P?
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); .'+Tnu(5q
rpowern = cat(2,rpowern{:}); )#Y*]
rpowern = [ones(length_r,1) rpowern]; 5@Ot@o
else 2}I1z_dq~
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); $>5|TG
0i
rpowern = cat(2,rpowern{:}); 49_b)K.tB
end yZ 6560(q
Y'bDEdeT
% Compute the values of the polynomials: K-k;`s#
% -------------------------------------- E n{vCN
y = zeros(length_r,length(n)); F7#
for j = 1:length(n) ~2V|]Y;s
s = 0:(n(j)-m_abs(j))/2; -`iZBC50
pows = n(j):-2:m_abs(j); (Pc:A!}
for k = length(s):-1:1 "-A@>*g
p = (1-2*mod(s(k),2))* ... uQ9P6w=Nt
prod(2:(n(j)-s(k)))/ ... :%xiH%C>
prod(2:s(k))/ ... v~ZdMQvwt
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... s+C&\$E
prod(2:((n(j)+m_abs(j))/2-s(k))); %{&yXi:mS
idx = (pows(k)==rpowers); 9dJARSUuF
y(:,j) = y(:,j) + p*rpowern(:,idx); ~naL1o_FZ
end 8>6+]]O
ga6M8eOI
if isnorm cm6cW(x6
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); V8`t7[r
end JQi)6A?J
end L!c7$M5xJ
% END: Compute the Zernike Polynomials t~Cul+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vUvIZa
ISa2|v;M
% Compute the Zernike functions: &JtK<g
% ------------------------------ ZnI_<iFR*
idx_pos = m>0; pDCQ?VW
idx_neg = m<0; ~H7m7
Z-*L[
z = y; w2YfFtgD,
if any(idx_pos) B;2os ^*
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); /b@8#px
end ~*- eL.
if any(idx_neg) u!
x9O8y
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); :
JD%=w_
end 2jOh~-LU
I|n<B"Q6^
% EOF zernfun