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 P0Ds7xh]h
function z = zernfun(n,m,r,theta,nflag) X8ev uN
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. k*-_CO-h
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N >f^&^28
% and angular frequency M, evaluated at positions (R,THETA) on the Y6`9:97
% unit circle. N is a vector of positive integers (including 0), and G#HbiVH9
% M is a vector with the same number of elements as N. Each element Sr)/
Mf
% k of M must be a positive integer, with possible values M(k) = -N(k) jm =E_86_
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, V3$!`T}g4
% and THETA is a vector of angles. R and THETA must have the same uw
L T$
% length. The output Z is a matrix with one column for every (N,M) .hg<\-:_
% pair, and one row for every (R,THETA) pair. "}\2zub9
% @I]uK[qd
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike O*z x{a6
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), %bt2^
% with delta(m,0) the Kronecker delta, is chosen so that the integral ;J2U5Y NO
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ;} gvBI2e
% and theta=0 to theta=2*pi) is unity. For the non-normalized C N"Vw
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Fw+JhIVP
% n #p6i
% The Zernike functions are an orthogonal basis on the unit circle. [{Fr{La`D'
% They are used in disciplines such as astronomy, optics, and ( iP,F]
% optometry to describe functions on a circular domain. 8doT`rI1
% 7t\kof
% The following table lists the first 15 Zernike functions. uz
` H
% ~1S7\e7{
% n m Zernike function Normalization 37ll8
% -------------------------------------------------- .'lc[iI9)d
% 0 0 1 1 ynw^nmM
% 1 1 r * cos(theta) 2 #"O9\X/B
% 1 -1 r * sin(theta) 2 UIL5K
% 2 -2 r^2 * cos(2*theta) sqrt(6) ]b'K
BAMy
% 2 0 (2*r^2 - 1) sqrt(3) &&ecq
% 2 2 r^2 * sin(2*theta) sqrt(6) %pc0a^iB
% 3 -3 r^3 * cos(3*theta) sqrt(8) <.l5>mgkCw
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 3a:(\:?z
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) wC{=o`v
% 3 3 r^3 * sin(3*theta) sqrt(8) L -b~#
% 4 -4 r^4 * cos(4*theta) sqrt(10) Q&MZ/Nnf
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Aw4Qm2Kf
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) z Rz#0
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) dDi 1{s
% 4 4 r^4 * sin(4*theta) sqrt(10) kX'1.<[
% -------------------------------------------------- j6/ 3p|E
% L0UAS'hf
% Example 1: KFAB
% }.NR+:0
% % Display the Zernike function Z(n=5,m=1) 3Nr8H.u&q
% x = -1:0.01:1; Kw(/#C:$
% [X,Y] = meshgrid(x,x); U>e@m?
% [theta,r] = cart2pol(X,Y); 8ji!FZf
% idx = r<=1; )Si`>o3T-.
% z = nan(size(X)); vD:.1,72
% z(idx) = zernfun(5,1,r(idx),theta(idx)); -hQ=0h~\B.
% figure E"#<I*b
% pcolor(x,x,z), shading interp J0@m
Ol
% axis square, colorbar >Eik>dQ a
% title('Zernike function Z_5^1(r,\theta)') ?TMo6SU
% PgB=<#9
% Example 2: I4m)5G?O2
% s<E_74q1
% % Display the first 10 Zernike functions )09_CC!a
% x = -1:0.01:1; [mw#a9
% [X,Y] = meshgrid(x,x); 5Lum$C
c}
% [theta,r] = cart2pol(X,Y); VY=~cVkzS
% idx = r<=1; p&Nw:S
% z = nan(size(X)); 4d!&.Qo9
% n = [0 1 1 2 2 2 3 3 3 3];
6C
r$R]5
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; M[<O]p6
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 49/1#^T"Q>
% y = zernfun(n,m,r(idx),theta(idx)); zLOmtZ(['
% figure('Units','normalized') LMsbTF@E
% for k = 1:10 Y
+HVn0~qz
% z(idx) = y(:,k); 0Yfk/}5
% subplot(4,7,Nplot(k)) N%y%)MI 8
% pcolor(x,x,z), shading interp w V;y]'
% set(gca,'XTick',[],'YTick',[]) r\_rnM)_xN
% axis square $N,9e
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) bTO$B2eh|
% end ~+l%}4RZ
% xS,):R
% See also ZERNPOL, ZERNFUN2. ynZ!
q?}G?n4
% Paul Fricker 11/13/2006 !RiPr(m@y
(ter+rTv
<Y~V!9(~{Q
% Check and prepare the inputs: rp=?4^(u
% ----------------------------- <@F4{*
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ? 1Z\=s
error('zernfun:NMvectors','N and M must be vectors.') m6lNZb]
end d[TcA2nF
KC }B\~ +
if length(n)~=length(m) cTRCQ+W6:
error('zernfun:NMlength','N and M must be the same length.') H#w?$?nIWu
end Kz$Ijj
[jAhw>
n = n(:); Q=uwmg86
m = m(:); F4bF&% R
if any(mod(n-m,2)) S'ikr
error('zernfun:NMmultiplesof2', ... '\_ic=&u
'All N and M must differ by multiples of 2 (including 0).') ~Ja>x`5
end HK2`.'D
`kekc.*-[@
if any(m>n) qn+m lduU
error('zernfun:MlessthanN', ... 61jDI^:
'Each M must be less than or equal to its corresponding N.') zoUW}O
end !p0FJ].g,
KVQZ
if any( r>1 | r<0 ) BOh&Db*
error('zernfun:Rlessthan1','All R must be between 0 and 1.')
9]AKNQq m
end !u7WCw.D m
/f0_mi,bD
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) jg%D
G2
error('zernfun:RTHvector','R and THETA must be vectors.') Ln`c DZSM
end z,2m7C
9F,jvCM63
r = r(:); }$$b6G
theta = theta(:); d^lA52X6P
length_r = length(r); K"g[%O<
if length_r~=length(theta) hR=4w$
error('zernfun:RTHlength', ... (MxLw:AV
'The number of R- and THETA-values must be equal.') J~c]9t
end 1ViDS
Gi{1u}-0
% Check normalization: yM\1n
% -------------------- Z.h`yRhO
if nargin==5 && ischar(nflag) =?FA9wm
isnorm = strcmpi(nflag,'norm'); #m8Oy|Y9`
if ~isnorm -nHc52,
error('zernfun:normalization','Unrecognized normalization flag.') F,lQj7
end $}HSU>,%
else g$]9xn#_[
isnorm = false; HX<5i>]0\u
end 7L]fCw
p[
DtZkrj)D/
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TF{
xFb)
% Compute the Zernike Polynomials d}WAP m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Zu+Z7@$}/
Ex<-<tY
% Determine the required powers of r: qbT].,?!U
% ----------------------------------- "`
9W"A=
m_abs = abs(m); RrRCT.+E
rpowers = []; o9Agx{'oV
for j = 1:length(n) D.\p7
NJ
rpowers = [rpowers m_abs(j):2:n(j)]; j~L{=ojz%
end 9D
0ujup
rpowers = unique(rpowers); T?% F
{v2Q7ZO-
% Pre-compute the values of r raised to the required powers, UQhfR}(
% and compile them in a matrix: l(<o,Uv[`
% ----------------------------- 'zpj_QM
if rpowers(1)==0 {@C+Js5
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ?MN?.O9-
rpowern = cat(2,rpowern{:}); "lUw{3
rpowern = [ones(length_r,1) rpowern]; ? ZN8Ku
else &&>OhH`
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); .Mm8\].
rpowern = cat(2,rpowern{:}); 67J=#%\
end B)Gm"bLCOZ
;AHa|35\
% Compute the values of the polynomials: o[8Y %3
% -------------------------------------- WE=`8`Li
y = zeros(length_r,length(n)); Ip<STz]-
for j = 1:length(n) ! .!qJ%
s = 0:(n(j)-m_abs(j))/2; ;O.U-s
pows = n(j):-2:m_abs(j); g*!2.P
for k = length(s):-1:1 Bz]64/
p = (1-2*mod(s(k),2))* ...
\1|T
prod(2:(n(j)-s(k)))/ ... A$%%;O
prod(2:s(k))/ ... b-~Gt]%>m
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... f<y""0L9
prod(2:((n(j)+m_abs(j))/2-s(k))); oIf-s[uH
idx = (pows(k)==rpowers); _H%ylAt1j
y(:,j) = y(:,j) + p*rpowern(:,idx); {?#g*QF|^
end "iOT14J!7
(R Ttz
if isnorm jJg
'Y:K9q
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); jcevpKkRG
end iPI6 _h
end m"jqHGFV
% END: Compute the Zernike Polynomials ~6{;3"^<
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% hPhN7E03
du`],/ 6
% Compute the Zernike functions: Xgop1
% ------------------------------ X}g!Lp
idx_pos = m>0; FFP>Y*v(
idx_neg = m<0; +&Sf$t 1
$t[`}I
}
z = y; E!jM&\Z j
if any(idx_pos) /sC$;l
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); =V%s^
end 2hu;N
if any(idx_neg) @cSz!E}
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); V,{ydxfB
end U%j=)VD])
qnruatA
% EOF zernfun