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 m#S ZI}
function z = zernfun(n,m,r,theta,nflag) my} P\r.
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. .9ROa#7U;n
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N MRC5c:(
% and angular frequency M, evaluated at positions (R,THETA) on the CjST*(,b
% unit circle. N is a vector of positive integers (including 0), and bZlAK)
% M is a vector with the same number of elements as N. Each element @=,J6
% k of M must be a positive integer, with possible values M(k) = -N(k) UG!&n@R
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, D=OU61AA
% and THETA is a vector of angles. R and THETA must have the same xp&I~YPH
% length. The output Z is a matrix with one column for every (N,M) xj~6,;83xR
% pair, and one row for every (R,THETA) pair. {Ise (>V
% ^{Vm,nAQqs
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike r;'!qwr
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), es6e-y@e
% with delta(m,0) the Kronecker delta, is chosen so that the integral rcbixOT
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, mb/3
#)
% and theta=0 to theta=2*pi) is unity. For the non-normalized gTq-\k(
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 4Cfwz-Qo
% r'!l`
gm,S
% The Zernike functions are an orthogonal basis on the unit circle. #2MwmIeA
% They are used in disciplines such as astronomy, optics, and dKMuo'H'%
% optometry to describe functions on a circular domain. bHMlh^{`%
% 6%'{Cq1DE
% The following table lists the first 15 Zernike functions. /#
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% rvG qUmSUs
% n m Zernike function Normalization XmnqZWB
% -------------------------------------------------- "s*{0'jo
% 0 0 1 1 q{@Wn]!k
% 1 1 r * cos(theta) 2 '|cuVxcE55
% 1 -1 r * sin(theta) 2 af_zZf!0
% 2 -2 r^2 * cos(2*theta) sqrt(6) F+6ZD5/
% 2 0 (2*r^2 - 1) sqrt(3) E`s_Dr}K
% 2 2 r^2 * sin(2*theta) sqrt(6) 6RF01z|~_
% 3 -3 r^3 * cos(3*theta) sqrt(8) PQ[TTLG\&
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) PY2`RZ/ @
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) y#MLxm
% 3 3 r^3 * sin(3*theta) sqrt(8) z_H2L"Z
% 4 -4 r^4 * cos(4*theta) sqrt(10) Q,4F=b
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4a 5n*6G!
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Kzm_AHA)
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;e{2?}#8&
% 4 4 r^4 * sin(4*theta) sqrt(10) h1Lp:@:|
% -------------------------------------------------- %MIu;u FR
% ]@f6O*&=
% Example 1: m<yA]
';s
% c`>\R<Z ]
% % Display the Zernike function Z(n=5,m=1) w iq{Jo#
% x = -1:0.01:1; P]TT
% [X,Y] = meshgrid(x,x); 0{,zE
% [theta,r] = cart2pol(X,Y); GGBe/X
% idx = r<=1; =UV?Pi*M>
% z = nan(size(X)); ,'9tR&S$_
% z(idx) = zernfun(5,1,r(idx),theta(idx)); VgdkCdWRm_
% figure .$yw;go3
% pcolor(x,x,z), shading interp 06`__$@h
% axis square, colorbar Z:*U/_G
% title('Zernike function Z_5^1(r,\theta)') {)[i\=,`{
% -3V~YhG
% Example 2: =.%ZF]Oe+#
% <r kW4
% % Display the first 10 Zernike functions </%H 'V@
% x = -1:0.01:1; X+3)DE\2
% [X,Y] = meshgrid(x,x); $i1A470C
% [theta,r] = cart2pol(X,Y); lVFX@I =pI
% idx = r<=1; y((_V%F}
% z = nan(size(X)); AWi87q
% n = [0 1 1 2 2 2 3 3 3 3]; MT5A%|H e
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; gv,T<A?Z2
% Nplot = [4 10 12 16 18 20 22 24 26 28]; =6dKC_Q
% y = zernfun(n,m,r(idx),theta(idx)); <Z;7=k
% figure('Units','normalized') G225Nz;Y*
% for k = 1:10 KH7]`CU
% z(idx) = y(:,k); |:?.-tq
% subplot(4,7,Nplot(k)) <7 rK
% pcolor(x,x,z), shading interp JA}'d7yEa
% set(gca,'XTick',[],'YTick',[]) =4D_-Q
% axis square +E:(-$"R
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Dmi;# WY
% end %(Ys-GeGr
% F:g{rm[
% See also ZERNPOL, ZERNFUN2. Z:hrrq9
c-T
^
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% Paul Fricker 11/13/2006 a(~YrA%~
J*Hn/m
V[M#qZS
% Check and prepare the inputs: L8zqLDi&
% ----------------------------- =s]{
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ]*@$%iCPE
error('zernfun:NMvectors','N and M must be vectors.') $.1'Ym
end Zz-;jkX)
c #!6
if length(n)~=length(m) Yel(}Ny
error('zernfun:NMlength','N and M must be the same length.') ?>8zU;Aj
end Bg
h$P
iq:[+
n = n(:); EAB+kY
m = m(:); lnWiE}F
if any(mod(n-m,2)) F"H!CJJu&
error('zernfun:NMmultiplesof2', ... w2+]C&B*
'All N and M must differ by multiples of 2 (including 0).') aTm.10{^
end j*u9+.
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if any(m>n) \,!q[nC
error('zernfun:MlessthanN', ... SU'9+=_$
'Each M must be less than or equal to its corresponding N.') ;QQ7vo
end
;"^9L
,rI
|+
if any( r>1 | r<0 ) $0SZlq>En
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ~k0)+D}
end E@6r{uZ#
/&:9VMMj
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) PJ@ ,01
error('zernfun:RTHvector','R and THETA must be vectors.') $jm<'
4
end a.IF%hP0xo
AV4HX\`{P0
r = r(:); +fQL~0tA
theta = theta(:); ^(JHRH~=h
length_r = length(r); #ljg2:I+
if length_r~=length(theta) !s*''v*
error('zernfun:RTHlength', ... mMAr8~A=
'The number of R- and THETA-values must be equal.') K=?F3tX^
end nj0AO0
}l?_Cfvu
% Check normalization: w00\1'-Kz
% -------------------- }!]x|zU.=
if nargin==5 && ischar(nflag) 25c!-.5D
isnorm = strcmpi(nflag,'norm'); o;>3z*9?3
if ~isnorm $A@3ogoS&
error('zernfun:normalization','Unrecognized normalization flag.') wLN2`ucC
end ,(27p6!
else :@`(}5F4
isnorm = false; >X,Ag
end KbdfSF$
nl9Cdi]o
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% eQVPxt2N
% Compute the Zernike Polynomials Rfc&OV
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `6N-MsP
e_k
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% Determine the required powers of r: $:E}Nj]{&
% ----------------------------------- if[o?6U4t
m_abs = abs(m); d<Q+D1
rpowers = []; "]s|D@^4#b
for j = 1:length(n) RvS q KW8
rpowers = [rpowers m_abs(j):2:n(j)]; Y-3[KH D
end U?F^D4CV\
rpowers = unique(rpowers); \_Kt6=
BZ;}ROmqk
% Pre-compute the values of r raised to the required powers, EcU'*
% and compile them in a matrix: /1W7<']>xV
% ----------------------------- ,J(5@8(>a
if rpowers(1)==0 NVc!g
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 7vpN6YP
rpowern = cat(2,rpowern{:}); u:uSsAn0$
rpowern = [ones(length_r,1) rpowern]; *Qg5Z
else y+";
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); i$JG^6,O
rpowern = cat(2,rpowern{:}); Q_kT}6#(J=
end Vo 6y8@\
-RH4y 2
% Compute the values of the polynomials: Cj !i)-
% -------------------------------------- =,d* {m~A
y = zeros(length_r,length(n)); h*#2bS~nl-
for j = 1:length(n) !0OD(XT
s = 0:(n(j)-m_abs(j))/2; 'lN*Ys iDi
pows = n(j):-2:m_abs(j); 1t[;` iZ
for k = length(s):-1:1 sUbz)BS#.
p = (1-2*mod(s(k),2))* ... C~KWH@
prod(2:(n(j)-s(k)))/ ... 6A$_&?
prod(2:s(k))/ ... P~\a)Szy
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... V%BJNJ
prod(2:((n(j)+m_abs(j))/2-s(k))); Sj0 ucnuHi
idx = (pows(k)==rpowers); ! 2Xr~u7a
y(:,j) = y(:,j) + p*rpowern(:,idx); (~G5t(+
end 2E3?0DL",
[W9e>Nsp0
if isnorm K$<`4#i
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Ld\LKwo
end qIDWl{b<
end s!@=rq
% END: Compute the Zernike Polynomials 1 ;\]D9i
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% E/~"j
(:?5 i`
% Compute the Zernike functions: +~w?Xw,
% ------------------------------ ]_ejDN\>{V
idx_pos = m>0; #QTfT&m+G}
idx_neg = m<0; rL%]S&M9
FDF3zzP0
z = y; g[EM]q,
if any(idx_pos) FJa[ToZ4+
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); R=vbUA
end bkr~13S{+
if any(idx_neg) `Di ^6UK(
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); S,*{q(
end !2zo]v4?
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% EOF zernfun