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 L8`v
function z = zernfun(n,m,r,theta,nflag) QEr<(wM-y
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 4}H+hk8-
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N kvwnqaX
% and angular frequency M, evaluated at positions (R,THETA) on the #gC[L=01
% unit circle. N is a vector of positive integers (including 0), and J
p?XV<3Z
% M is a vector with the same number of elements as N. Each element m4/qxm"Dx:
% k of M must be a positive integer, with possible values M(k) = -N(k) ,6>3aD1w~q
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, gC1LQ!:;Oi
% and THETA is a vector of angles. R and THETA must have the same -pC'C%Q
% length. The output Z is a matrix with one column for every (N,M) s47R,K$
% pair, and one row for every (R,THETA) pair. aC,adNub
% 'zYS:W
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike /QQRy_Z1)
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), d,^O[9UWo
% with delta(m,0) the Kronecker delta, is chosen so that the integral WoTeIkM9
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, O(-p
md,
% and theta=0 to theta=2*pi) is unity. For the non-normalized a3yNd
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. B7f<XBU6>
% vD#U+
% The Zernike functions are an orthogonal basis on the unit circle. G0
)[(s
% They are used in disciplines such as astronomy, optics, and a`'>VCg
% optometry to describe functions on a circular domain. 1$0Kvvg[
% c e;7
% The following table lists the first 15 Zernike functions. GQbr}xX.#
% F!X0Wo=
% n m Zernike function Normalization cr!8Tp;2A
% -------------------------------------------------- NVMn7H}>
% 0 0 1 1 j.&dHtp
% 1 1 r * cos(theta) 2 nqy*>X`
% 1 -1 r * sin(theta) 2 Q4cCg7|0
% 2 -2 r^2 * cos(2*theta) sqrt(6) 6&$.E! z
% 2 0 (2*r^2 - 1) sqrt(3) 7fR5V
% 2 2 r^2 * sin(2*theta) sqrt(6) @AZNF+
\W$
% 3 -3 r^3 * cos(3*theta) sqrt(8) $)#orZtzr
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) $}&a*c>
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) uz!8=,DFw
% 3 3 r^3 * sin(3*theta) sqrt(8) lAN&d;NU6Z
% 4 -4 r^4 * cos(4*theta) sqrt(10) @[;'b$T$
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) LA Crg
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) pbx*Y`v
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +@rFbsyJ.
% 4 4 r^4 * sin(4*theta) sqrt(10) E*YmHJ:k
% -------------------------------------------------- lq9|tt6Z
% _mqU:?Q5
% Example 1: bYP8
% a}@b2Wc*
% % Display the Zernike function Z(n=5,m=1) 4!/QB6
% x = -1:0.01:1; p :xyy*I
% [X,Y] = meshgrid(x,x); d_`MS@2
% [theta,r] = cart2pol(X,Y); d ~M;
% idx = r<=1; )]fiyXA
% z = nan(size(X)); *ak0(yLn)
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ZD&F ,2v
% figure RnH?95n?{
% pcolor(x,x,z), shading interp L/u|90)L
% axis square, colorbar d#T5=5#
% title('Zernike function Z_5^1(r,\theta)') No7-fX1B
% R[m-jUL
% Example 2: ?$/::uo
% 7rdmj[vu
% % Display the first 10 Zernike functions %NkiY iA
% x = -1:0.01:1; )xcjQkb
% [X,Y] = meshgrid(x,x); ;T^s&/>E
% [theta,r] = cart2pol(X,Y); h ;uzbu
% idx = r<=1; 7]rIq\bM
% z = nan(size(X)); hrKeOwKHU
% n = [0 1 1 2 2 2 3 3 3 3]; Qf_N,Bq{a
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; lj]M 1zEz&
% Nplot = [4 10 12 16 18 20 22 24 26 28]; +t,b/K(?]
% y = zernfun(n,m,r(idx),theta(idx)); `/WxEu3
% figure('Units','normalized') yP]>eLTSd
% for k = 1:10 :P-H8*n""
% z(idx) = y(:,k); 1`?o#w
% subplot(4,7,Nplot(k)) X4o#kW
% pcolor(x,x,z), shading interp uf?;;wg
% set(gca,'XTick',[],'YTick',[]) ^KbR@Ah
% axis square ft/k-64
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) '
wl})
% end Qb^{`
% m@<,bZkl
% See also ZERNPOL, ZERNFUN2. f hK<P_}
2(d
% Paul Fricker 11/13/2006 H{=]94
f_ MK4
L$Z!
% Check and prepare the inputs: JcRxNH
)<"
% ----------------------------- ?J$k
5;
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) !Mw/j`*
error('zernfun:NMvectors','N and M must be vectors.') `n6cpX5
end /'8%=$2Kw
FqJd
if length(n)~=length(m) N]yT/8
error('zernfun:NMlength','N and M must be the same length.') %rB,Gl:)g
end -)%\$z
'R1C-U3w,
n = n(:); PoZ$3V$(Lz
m = m(:); $-DW+|p.?^
if any(mod(n-m,2)) Hva!6vwO%O
error('zernfun:NMmultiplesof2', ... ]+G\1SN~
'All N and M must differ by multiples of 2 (including 0).') .;31G0<w2
end Sy?^+JdM/
\ MuKS4
if any(m>n) 0qR#o/~I
error('zernfun:MlessthanN', ... +;!^aNJ,
'Each M must be less than or equal to its corresponding N.') +Q"s!\5
end 3B[tbU(
PUea`rE?R
if any( r>1 | r<0 ) ;xq;c\N
error('zernfun:Rlessthan1','All R must be between 0 and 1.') atZNX1LD[/
end N]8/l:@
>mQD/U
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) D7R;IA-w
error('zernfun:RTHvector','R and THETA must be vectors.') rG-x 3>b
end =Q;dYx%I5
_0 [s]
r = r(:); c&#Q`m
theta = theta(:); [)X( Qtk
length_r = length(r); |@rf#,hTDp
if length_r~=length(theta) b7'A5]X
error('zernfun:RTHlength', ... [C
ezz5
'The number of R- and THETA-values must be equal.') Cjt].XR@
end 3-y2i/4}$
*` -
% Check normalization: 5!i\S[:
% -------------------- v*y,PY1*
if nargin==5 && ischar(nflag) 1jK2*y
isnorm = strcmpi(nflag,'norm'); WYvcN8F
if ~isnorm yqb$,$
error('zernfun:normalization','Unrecognized normalization flag.') }!kvoV)]1
end GOCe&?
else J"eE9FLM
isnorm = false; m2%
end 9-Qu5L~
06af{FXsGb
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2j^8{Agz
% Compute the Zernike Polynomials skLr6Cs|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -<\hcV`&
W1}d6Sbg
% Determine the required powers of r: of_Om$
% ----------------------------------- (Kkqyrb
m_abs = abs(m); EzU=q
E
rpowers = []; R4f_Kio
for j = 1:length(n) mj& 4FQ#O*
rpowers = [rpowers m_abs(j):2:n(j)]; Gd2t^tc
end x*_'uP oS
rpowers = unique(rpowers); (ap,3$hS
I!p[:.t7
% Pre-compute the values of r raised to the required powers, y $>U[^G[
% and compile them in a matrix: r[&/*~xL
% ----------------------------- sc# q03
if rpowers(1)==0 csv;u'
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ?Hf8<C} 3
rpowern = cat(2,rpowern{:}); D14i]
rpowern = [ones(length_r,1) rpowern]; pTcN8E&Unz
else &Y8S! W@4
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 3B ;aoejHm
rpowern = cat(2,rpowern{:}); }ILg_>uq[
end Xa=oEG
pM_oIH'8:
% Compute the values of the polynomials: 8CGjI?j
% -------------------------------------- ":Ll.=!
y = zeros(length_r,length(n)); 05[k@f$n
for j = 1:length(n) {b]V
e/\
s = 0:(n(j)-m_abs(j))/2; :J;*]o:
pows = n(j):-2:m_abs(j); A}(Q^|6
for k = length(s):-1:1 %b3s|o3An
p = (1-2*mod(s(k),2))* ... -@G,Ry-\t
prod(2:(n(j)-s(k)))/ ... J4^aD;j
prod(2:s(k))/ ... U=\!`_f':
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ~BD 80s:f
prod(2:((n(j)+m_abs(j))/2-s(k))); DH{^9HK
idx = (pows(k)==rpowers); Yuqt=\? #
y(:,j) = y(:,j) + p*rpowern(:,idx); $]S*(K3U~
end `@Tl7I\
tU>?j1
if isnorm ~jn~M_}K
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); -jQMh
end pv ;ZR
end *Bm
_
% END: Compute the Zernike Polynomials 4n,>EA85
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tc<ly{ 1c
0GP\*Y8
% Compute the Zernike functions: gj-MkeI)
% ------------------------------ uQ'Izdm
idx_pos = m>0; b1ma(8{{{
idx_neg = m<0; 4vbtB2
k86j&
.m_
z = y; Z@{e\sZ)
if any(idx_pos) T?V!%AqY:
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); k9vzxZ%s:
end 78-D/WY/X
if any(idx_neg) ?kKr/f4N
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); @<,YUp,%S
end r\DA&b
yV/A%y-P
% EOF zernfun