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 Ijq1ns_tx8
function z = zernfun(n,m,r,theta,nflag) e`ti*1]q
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. DK 4 8
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N mY?^]3-_
% and angular frequency M, evaluated at positions (R,THETA) on the {Y-<#U~iH
% unit circle. N is a vector of positive integers (including 0), and 8<ZxE(v
% M is a vector with the same number of elements as N. Each element An cmSi
% k of M must be a positive integer, with possible values M(k) = -N(k) (3YCe {
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 6KPM4#61o
% and THETA is a vector of angles. R and THETA must have the same nPh5(&E
% length. The output Z is a matrix with one column for every (N,M) pMM,ox"
% pair, and one row for every (R,THETA) pair. rtf\{u9 }g
% n[ip'*2L
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 2='gC|&s6
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 3Z#k9c_b
% with delta(m,0) the Kronecker delta, is chosen so that the integral d;O16xcM/
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, DJ;il)^
% and theta=0 to theta=2*pi) is unity. For the non-normalized @~%R%Vu
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. aOHf#!/"sb
% 'PRsZ`x.
% The Zernike functions are an orthogonal basis on the unit circle. (@*[^@ipV
% They are used in disciplines such as astronomy, optics, and >2l1t}"\
% optometry to describe functions on a circular domain. (#GOXz
% -b+VzVJZ
% The following table lists the first 15 Zernike functions. ,K=\Y9l3
% ~pA_E!3W
% n m Zernike function Normalization U 2am1}
% -------------------------------------------------- 8enlF\I8g
% 0 0 1 1 (`PgvBL:
% 1 1 r * cos(theta) 2 nN1\
% 1 -1 r * sin(theta) 2 PjZsMHW%
% 2 -2 r^2 * cos(2*theta) sqrt(6) JVbR5"+.
% 2 0 (2*r^2 - 1) sqrt(3) ! iuDmL
% 2 2 r^2 * sin(2*theta) sqrt(6) h`n,:Y^++P
% 3 -3 r^3 * cos(3*theta) sqrt(8) 7/QQ&7+NkS
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) =W'a6)WE
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) *TQXE:vZ[
% 3 3 r^3 * sin(3*theta) sqrt(8) 1'DD9d{qN
% 4 -4 r^4 * cos(4*theta) sqrt(10) i[1K~yXq:
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 9TRS#iVL+*
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) l"^'uGB'
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) U@21N3_@_
% 4 4 r^4 * sin(4*theta) sqrt(10) o))z8n?b
% -------------------------------------------------- 8qGK"%{ ~
% 1!_$HA
% Example 1: %+gYZv-
% #DK@&Gv
% % Display the Zernike function Z(n=5,m=1) Xkcy~e
% x = -1:0.01:1; )90 Q
% [X,Y] = meshgrid(x,x); .CGPG,\2
% [theta,r] = cart2pol(X,Y); @9_H4V
% idx = r<=1; A+P9M \u.
% z = nan(size(X)); u6^cLQO+
% z(idx) = zernfun(5,1,r(idx),theta(idx)); _N!L?b83P
% figure J%ng8v5ex
% pcolor(x,x,z), shading interp -xs@rV`
% axis square, colorbar 91%QO?hz
% title('Zernike function Z_5^1(r,\theta)') ,aOi:aaZRT
% "ee:Z_Sz
% Example 2: zOJ4I^^
% dsck:e5agZ
% % Display the first 10 Zernike functions PyQt8Qlz
% x = -1:0.01:1; Xc"l')1H
% [X,Y] = meshgrid(x,x); k4:$LFw@
% [theta,r] = cart2pol(X,Y); jb;!"HC
% idx = r<=1; -]yM<dP
% z = nan(size(X)); IoO t n
% n = [0 1 1 2 2 2 3 3 3 3]; n
N.6?a
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; q.sErr[zc
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ]}7FTMGbY
% y = zernfun(n,m,r(idx),theta(idx)); eC`} oEz
% figure('Units','normalized') BG ,ln(Vz
% for k = 1:10 ;
kPx@C
% z(idx) = y(:,k); %N5gQXg
% subplot(4,7,Nplot(k)) 4<%(Y-_sF
% pcolor(x,x,z), shading interp C\"nlNKw
% set(gca,'XTick',[],'YTick',[]) iF]G$@rbU
% axis square Do1 Ip&X
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) sG-$d\
1d
% end <Y%km[Mh
% 9N1Uv,OtB
% See also ZERNPOL, ZERNFUN2. +/xmxh$ $
5cahbx1"
% Paul Fricker 11/13/2006 P5$d#Y(=
SURbH;[
S-x'nu$u
% Check and prepare the inputs: %f[0&)1!.v
% ----------------------------- 581Jp'cje
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) z[Sq7bbYO
error('zernfun:NMvectors','N and M must be vectors.') iCd$gwA>F
end &CP0T:h
o[=h=&@5p
if length(n)~=length(m) 0,a/t
jSr
error('zernfun:NMlength','N and M must be the same length.') scr`] tD
end W5 l)mAv
MU_8bK9m
n = n(:); 2ed4xhV
m = m(:); DX3xWdnr
if any(mod(n-m,2)) T}fH
error('zernfun:NMmultiplesof2', ... KD TG9KC
'All N and M must differ by multiples of 2 (including 0).') KWuc*!
end VtM:~|v
jLc"1+
if any(m>n) {7EnM1]
error('zernfun:MlessthanN', ... NT(gXEZ
'Each M must be less than or equal to its corresponding N.') }jL_/gvgy
end $a
/jfpV
-@*[
if any( r>1 | r<0 ) sd(Yr6~..
error('zernfun:Rlessthan1','All R must be between 0 and 1.') a4a/]q4T
end |[6jf!F
*\gS 2[S
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) k[_)5@2
error('zernfun:RTHvector','R and THETA must be vectors.') `vbd7i
end I`e$U
A(Tqf.,G
r = r(:); zY11.!2
theta = theta(:); YgiLfz iT
length_r = length(r); YJ6y]r
K2,
if length_r~=length(theta) m!'moumL;
error('zernfun:RTHlength', ... .~3s~y*s
'The number of R- and THETA-values must be equal.') f&=WgITa
end Kivr)cIG
dWR-}>
% Check normalization: `Zdeq.R]
% -------------------- adCTo
if nargin==5 && ischar(nflag) *8I+D>x
isnorm = strcmpi(nflag,'norm'); b\Wlpb=QZ
if ~isnorm )Z/L
error('zernfun:normalization','Unrecognized normalization flag.') &XvSAw+D@
end wP|Amn+;
else 0fOx&"UAB
isnorm = false; HQ187IwpTm
end s(2/]f$
1')_^]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8D
H~~by
% Compute the Zernike Polynomials BB$(0mM^
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #
dA-dN
Z91{*?
% Determine the required powers of r: uT
Z#85L`
% ----------------------------------- sY-
]
Q
m_abs = abs(m); >$/<~j]
rpowers = []; LMGo8%2I
for j = 1:length(n) +VSq [P
rpowers = [rpowers m_abs(j):2:n(j)]; YK{E=<:
end d*B^pDf
rpowers = unique(rpowers); =/#+,
g+RgDt9
% Pre-compute the values of r raised to the required powers, ',_E;(
% and compile them in a matrix: )qID<j#
% ----------------------------- 1WP(=7$.
if rpowers(1)==0 -J6G=+s/
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); -%G}T}"_
rpowern = cat(2,rpowern{:}); dvc=<!"'S
rpowern = [ones(length_r,1) rpowern]; Hxr)`i46
else )%zOq:{\5
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7u=R5
rpowern = cat(2,rpowern{:}); |T;]%<O3E
end 15l{gbCW
mVs<XnA47
% Compute the values of the polynomials: ,N1I\f
% -------------------------------------- W5SCm(QS5
y = zeros(length_r,length(n)); *+UgrsRk
for j = 1:length(n) ~+)sL1lx
s = 0:(n(j)-m_abs(j))/2; `;c{E%qeq
pows = n(j):-2:m_abs(j); nOQvBc
for k = length(s):-1:1 <E&8g[x6
p = (1-2*mod(s(k),2))* ... f(*ygI
prod(2:(n(j)-s(k)))/ ... yQ03&{#
prod(2:s(k))/ ... x
&
ZW
f?
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ;1MRBk,
prod(2:((n(j)+m_abs(j))/2-s(k))); K2o\+t
idx = (pows(k)==rpowers); 6rll0c~
y(:,j) = y(:,j) + p*rpowern(:,idx); }\?]uNH
end q}+Fm?B
-Pt']07E
if isnorm {/2
_"H3:
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); EpCT !e
end DkA@KS1Dq
end xm*6I
% END: Compute the Zernike Polynomials GF/!@N
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - %'ys
#wS/QrRE
% Compute the Zernike functions: (/[wM>q:r
% ------------------------------ O/ih9,
idx_pos = m>0; tj1M1s|a
idx_neg = m<0; gLzQM3{X9
N ]|P||fC
z = y; t,IQ|B&0
if any(idx_pos) ' 2:HBJ
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 50R&;+b
end Ls2g#+
if any(idx_neg) ]w5j?h"b
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); T$pBgS>
end p02E:?
,&ld:v?~
% EOF zernfun