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 W`&hp6Jq
function z = zernfun(n,m,r,theta,nflag) m3ff;,
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~v83pu1!2s
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N d1kJRJ
% and angular frequency M, evaluated at positions (R,THETA) on the f X)#=c|5
% unit circle. N is a vector of positive integers (including 0), and smLQS+UE
% M is a vector with the same number of elements as N. Each element /@Zrq#o
zx
% k of M must be a positive integer, with possible values M(k) = -N(k) &[SC|=U'M
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, QM]YJr3rE
% and THETA is a vector of angles. R and THETA must have the same `lPfb[b
% length. The output Z is a matrix with one column for every (N,M) 'RRE|L,
% pair, and one row for every (R,THETA) pair. JLi|Td"1%
% 'QIqBU'~
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike o]:9')5^
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), G9:l'\
% with delta(m,0) the Kronecker delta, is chosen so that the integral 7)k\{&+P
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, )ANmIwmC#
% and theta=0 to theta=2*pi) is unity. For the non-normalized BUR*n;V`
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ]q-Y }1di8
% `@
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% The Zernike functions are an orthogonal basis on the unit circle. "vslZ`RU
% They are used in disciplines such as astronomy, optics, and :c[L3rJl
% optometry to describe functions on a circular domain. U?=Dg1
% rD>f|kA?L
% The following table lists the first 15 Zernike functions. hzRYec(
% 7=DdrG<
% n m Zernike function Normalization IMfqiH)
% -------------------------------------------------- m_l[MG\
% 0 0 1 1 5Dl/aHb
% 1 1 r * cos(theta) 2 ;'Nd~:-]
% 1 -1 r * sin(theta) 2 H4JTGt1"
% 2 -2 r^2 * cos(2*theta) sqrt(6) pD74+/DD
% 2 0 (2*r^2 - 1) sqrt(3) 7!$^r$t
% 2 2 r^2 * sin(2*theta) sqrt(6) @]#1(9P
% 3 -3 r^3 * cos(3*theta) sqrt(8) t_suF$
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) e!r-+.i(
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) @<Yy{~L|
% 3 3 r^3 * sin(3*theta) sqrt(8) I9Fr5p-%O
% 4 -4 r^4 * cos(4*theta) sqrt(10) 2>H24F
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :\}(&
>
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 9$m|'$p3sG
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ~WN:DXn
% 4 4 r^4 * sin(4*theta) sqrt(10) 3Le{\}-$.
% -------------------------------------------------- orvp*F{7[H
% FkRo
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% Example 1: f4Rf?w*
% nJLFfXWx
% % Display the Zernike function Z(n=5,m=1) fg{n(TE"8
% x = -1:0.01:1; 4NIRmDEd
% [X,Y] = meshgrid(x,x); (@}!0[[^
% [theta,r] = cart2pol(X,Y); Ip]KPrwp
% idx = r<=1; &yol_%C
% z = nan(size(X)); v 6Vcjm
% z(idx) = zernfun(5,1,r(idx),theta(idx)); H$KTo/
% figure S/I /-Bp~
% pcolor(x,x,z), shading interp LYg-
.~<I
% axis square, colorbar 3<zp
% title('Zernike function Z_5^1(r,\theta)') ~| 6[j<ziL
% C{XmVc.
% Example 2: L z1ME(
%
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% % Display the first 10 Zernike functions
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% x = -1:0.01:1; 9RI-Lq`
% [X,Y] = meshgrid(x,x); o7LuKRl
% [theta,r] = cart2pol(X,Y); d&s9t;@=
% idx = r<=1; u=_mvN
% z = nan(size(X)); :$9tF>
% n = [0 1 1 2 2 2 3 3 3 3]; P_#bow
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; qWKAM@
% Nplot = [4 10 12 16 18 20 22 24 26 28]; y<bDTeoo
% y = zernfun(n,m,r(idx),theta(idx)); SG4%}wn%
% figure('Units','normalized') M[112%[+4
% for k = 1:10 dmN&+t
% z(idx) = y(:,k); ~<OSYb
% subplot(4,7,Nplot(k)) Ezv
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% pcolor(x,x,z), shading interp Q&|\r
% set(gca,'XTick',[],'YTick',[]) :TC@tM~Oy
% axis square x7x\Y(@
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) LAe6`foW/
% end H ?y,ie#u
% az|N-?u
% See also ZERNPOL, ZERNFUN2. !GEJIefx_
-{vKus
% Paul Fricker 11/13/2006 y%bF&
\A6B,|@
f!
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% Check and prepare the inputs: r!a3\ep
% ----------------------------- B i<Q=x'Z;
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) r s?R:+
error('zernfun:NMvectors','N and M must be vectors.') y[_Q-
end '1)$'
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if length(n)~=length(m) yw3$2EW
error('zernfun:NMlength','N and M must be the same length.') -n<pPau2
end g]yBA7/S"
A;|D:;x3G
n = n(:); qXtC^n@x
m = m(:); x6ARzH\
if any(mod(n-m,2)) cXOK)g#
error('zernfun:NMmultiplesof2', ... "E?2xf|.
'All N and M must differ by multiples of 2 (including 0).') tlp@?(u
end @w !PaP
"?I y (*^
if any(m>n) ce3YCflt
error('zernfun:MlessthanN', ... t; {F%9j{
'Each M must be less than or equal to its corresponding N.') Ev(>z-{F
end "s_lP&nq
zb<6
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if any( r>1 | r<0 ) 2eol
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error('zernfun:Rlessthan1','All R must be between 0 and 1.') BC<^a )D=
end .oUTqki
z}ddqZ27G$
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) J9iy
error('zernfun:RTHvector','R and THETA must be vectors.') K_ ~"}
end k<{{*
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r = r(:); J@Q7p}
theta = theta(:); 1sdLDw_)p
length_r = length(r); 28J^DMOW
if length_r~=length(theta) Y@ksQ_u
error('zernfun:RTHlength', ... 0C6-GKbZ
'The number of R- and THETA-values must be equal.') > eIP.,9
end 6WJ)by
Z>W g*sZy)
% Check normalization: *8_wYYH
% -------------------- Uu(SR/R}
if nargin==5 && ischar(nflag) 9g"2^^wD
isnorm = strcmpi(nflag,'norm');
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if ~isnorm M={V|H0
error('zernfun:normalization','Unrecognized normalization flag.') ],a 5)kV
end 1@1U/ss1
else Rt!FPoN,y
isnorm = false; (/j/>9iro
end 4 k _vdz
C$D-Pt"+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !F1N~6f
% Compute the Zernike Polynomials ,+xB$e
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #[~pD:qqM
9"A`sGZ
% Determine the required powers of r: CtAwBQO
% ----------------------------------- h+&OQ%e=8
m_abs = abs(m); j=aI9p
rpowers = []; 'JfdV%M
for j = 1:length(n) 8UyMVY
rpowers = [rpowers m_abs(j):2:n(j)]; IrhA+)pdse
end "4+WZR]
rpowers = unique(rpowers); ( _)jkI
\
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% Pre-compute the values of r raised to the required powers, ]d0tE?9
% and compile them in a matrix: kDN:ep{/
% ----------------------------- cm[&?
if rpowers(1)==0 _EMwm&!
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); W!&'pg
rpowern = cat(2,rpowern{:}); e`TH91@
rpowern = [ones(length_r,1) rpowern]; C:C}5<fkx
else )V6Hl@v
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); U4-g^S[
rpowern = cat(2,rpowern{:}); *HO}~A%Lx
end ps%q9}J
X+S9{X#Cm
% Compute the values of the polynomials: `-l6S
% -------------------------------------- DV-;4AxxRq
y = zeros(length_r,length(n)); lfz2~Si5A
for j = 1:length(n) -[!P!d=
s = 0:(n(j)-m_abs(j))/2; O8u j`G 9
pows = n(j):-2:m_abs(j); I@%t.%O Jp
for k = length(s):-1:1 L>%o[tS
p = (1-2*mod(s(k),2))* ... r{ef .^&:
prod(2:(n(j)-s(k)))/ ... %_L\z*+
prod(2:s(k))/ ... % !>I*H
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... "a"]o
prod(2:((n(j)+m_abs(j))/2-s(k))); pDcjwlA%
idx = (pows(k)==rpowers); 9Hu/u=vB<
y(:,j) = y(:,j) + p*rpowern(:,idx); *
%M3PTY\
end i2(1ki/|O
;YX4:OBqr
if isnorm ); dT_
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); %S nd\
end mkF"
end \":m!K;Z
% END: Compute the Zernike Polynomials f[~L?B;_L
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,7NZu0
V8-oYwOR
% Compute the Zernike functions: U1RpLkibQ
% ------------------------------ !@'6)/
idx_pos = m>0; T {Uc:Z
idx_neg = m<0; &PK\|\\2
7`8Ik`lY
z = y; dJ""XaHqf
if any(idx_pos) rT5Ycm@
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); %V{7DA&C
end Qj6/[mUr~
if any(idx_neg) $8[r9L!
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); e9[|!/./5
end y2vUthRwo
4NG?_D5&
% EOF zernfun