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 S'8+jY
function z = zernfun(n,m,r,theta,nflag) 9Y'pT.Gyb
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Fz';H
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 3 a.!9R>
% and angular frequency M, evaluated at positions (R,THETA) on the
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% unit circle. N is a vector of positive integers (including 0), and o+&Om~W
% M is a vector with the same number of elements as N. Each element Gmi?xGn
% k of M must be a positive integer, with possible values M(k) = -N(k) Y&j`HO8f
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, <O
0Q]`i
% and THETA is a vector of angles. R and THETA must have the same $QwpoVp`~
% length. The output Z is a matrix with one column for every (N,M) Mq)]2>"v
% pair, and one row for every (R,THETA) pair. +1YEOOfVY
% OQ hQ!6
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike <+g77NL
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 05R"/r*
% with delta(m,0) the Kronecker delta, is chosen so that the integral k:Y\i]#yP
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, =~h b&
% and theta=0 to theta=2*pi) is unity. For the non-normalized 38p"lT
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. HzGwO^tbK
% =Q40]>bpx
% The Zernike functions are an orthogonal basis on the unit circle. &{.IUg
% They are used in disciplines such as astronomy, optics, and BP@tI|
% optometry to describe functions on a circular domain. e' o2PW
% 9>w~B|/
% The following table lists the first 15 Zernike functions. RB+Jp
% Au'y(KB
% n m Zernike function Normalization o& FOp'
% -------------------------------------------------- "H[K3
% 0 0 1 1 yiQ ?p:DM
% 1 1 r * cos(theta) 2 wpM2{NTP
% 1 -1 r * sin(theta) 2 zp;!HP;/=
% 2 -2 r^2 * cos(2*theta) sqrt(6) UgGa]b[9A
% 2 0 (2*r^2 - 1) sqrt(3) xj;:B( i
% 2 2 r^2 * sin(2*theta) sqrt(6) O*l,&5
% 3 -3 r^3 * cos(3*theta) sqrt(8) I U"
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) "ktuq\a@
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) XQ}J4J~Vm
% 3 3 r^3 * sin(3*theta) sqrt(8) bh1$
A
% 4 -4 r^4 * cos(4*theta) sqrt(10) z1Bi#/i
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) AE}cHBwZE
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ]6^<VC`5D
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?I6rW JcQ6
% 4 4 r^4 * sin(4*theta) sqrt(10) BA:x*(%~
% -------------------------------------------------- 1 ;$XX#7o
% s6 g"uF>k
% Example 1: }8x+F2i
% sh_;98^
% % Display the Zernike function Z(n=5,m=1) ]##aAh-P4&
% x = -1:0.01:1; F)hj\aHm k
% [X,Y] = meshgrid(x,x); q k^FyZ<
% [theta,r] = cart2pol(X,Y); ]qT&6:;-]
% idx = r<=1; "iK=
8
% z = nan(size(X)); HXa[0VOx
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ]@Zv94Z(
% figure E)$>t}$
% pcolor(x,x,z), shading interp gUru=p
% axis square, colorbar D8wf`RUt
% title('Zernike function Z_5^1(r,\theta)') - j3Lgm
% 6/8K2_UeoW
% Example 2: G^W0!u,@
% '%rT]u3U
% % Display the first 10 Zernike functions =NtHV4=b
% x = -1:0.01:1; gPKf8{#%e
% [X,Y] = meshgrid(x,x); 8<C*D".T$
% [theta,r] = cart2pol(X,Y); |&= -Nm
% idx = r<=1; [j0[c9.p[
% z = nan(size(X));
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% n = [0 1 1 2 2 2 3 3 3 3]; T%eBgseS
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 8D )nM|
% Nplot = [4 10 12 16 18 20 22 24 26 28]; *,$5EN
% y = zernfun(n,m,r(idx),theta(idx)); zkRAul32|
% figure('Units','normalized') GM%OO)dO}
% for k = 1:10 WY!\^| ,
% z(idx) = y(:,k); ~9+01UU^
% subplot(4,7,Nplot(k)) $K^l=X
% pcolor(x,x,z), shading interp }pMVl
% set(gca,'XTick',[],'YTick',[]) Dds-;9
% axis square wN!\$i@E:
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) V6][*.i!9
% end [LnPV2@e
% src9EeiV
% See also ZERNPOL, ZERNFUN2. !l
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% Paul Fricker 11/13/2006 U.<j2Kum
s=n4'`y1
lr>NG,N
% Check and prepare the inputs: ]
]U )wg
% ----------------------------- C(XV
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Q ]CMm2L^f
error('zernfun:NMvectors','N and M must be vectors.') !XtG6ON=
end S $p>sItO
#BLHHK/[
if length(n)~=length(m) ;_bRq:!j;
error('zernfun:NMlength','N and M must be the same length.') 0~ho/ _
end J 4gtm"2)
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ubq{'
n = n(:); l}uZxKuYx
m = m(:); S&!(h
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if any(mod(n-m,2)) i&:SWH=
error('zernfun:NMmultiplesof2', ... NuQ!huh
'All N and M must differ by multiples of 2 (including 0).') 7
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end k77IXT_7u
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if any(m>n) MR1I"gqE}I
error('zernfun:MlessthanN', ... sGu.G
'Each M must be less than or equal to its corresponding N.') %P0
end 0 %~~IT}U
K
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if any( r>1 | r<0 ) *K|~]r(F?
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 3*h"B$g!
end s:tX3X
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ,->K)Rs ;
error('zernfun:RTHvector','R and THETA must be vectors.') R0RxcBtG
end 7% D 4
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r = r(:); QR&e~rks
theta = theta(:); "UTW(~D'
length_r = length(r); V5K/)\#
if length_r~=length(theta) 'SFAJ
error('zernfun:RTHlength', ... -hXKCb4YU
'The number of R- and THETA-values must be equal.') H'k}/<%Q
end T<B}Z11R
C<D$Y,[w
% Check normalization: $+Ze"E
% -------------------- =%m{|HQ`
if nargin==5 && ischar(nflag) 2f[;U"
isnorm = strcmpi(nflag,'norm'); I}_}VSG(
if ~isnorm A08kwYxiW
error('zernfun:normalization','Unrecognized normalization flag.') wtYgHC}X
end 2=_$&oT**
else $P{`-Y }a
isnorm = false; lI?P_2AaS
end $2a"Ec!7
v'i'I/
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F^.A~{&L
% Compute the Zernike Polynomials i#la'ICwJ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% { U;yW)
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% Determine the required powers of r: CGi;M=xr
% ----------------------------------- !i"zM}
m_abs = abs(m); M.Yp'Av
rpowers = []; P PJ^;s
for j = 1:length(n) OyO]; Yk
rpowers = [rpowers m_abs(j):2:n(j)]; i47LX;}
end <MbhBIejr
rpowers = unique(rpowers); "Wj{+|f
E]'
f&0s
% Pre-compute the values of r raised to the required powers, _f^6F<!
% and compile them in a matrix: %6 *c40
% ----------------------------- UH MJ(.Wa-
if rpowers(1)==0 ?0; 2ct
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); v.|#^A?Qx
rpowern = cat(2,rpowern{:}); )k Wxp
rpowern = [ones(length_r,1) rpowern]; w1tM !4r
else /wLBmh1"
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7W)W9=&BT
rpowern = cat(2,rpowern{:}); ;].X;Ky<
end blQ&QQL
G=zNZ
% Compute the values of the polynomials: Eiu/p&ct
% -------------------------------------- tu}!:5xi
y = zeros(length_r,length(n)); bny5e:= d
for j = 1:length(n) _Q1p_sdg
s = 0:(n(j)-m_abs(j))/2; k;^$Pd?t
pows = n(j):-2:m_abs(j); f]r*;YEc4
for k = length(s):-1:1 GNJ/|9
p = (1-2*mod(s(k),2))* ... Q$U5[TZm
prod(2:(n(j)-s(k)))/ ...
!Vyf2xS"
prod(2:s(k))/ ... iE'' >Z
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 9qftMDLZJ\
prod(2:((n(j)+m_abs(j))/2-s(k))); M=raKb?F
idx = (pows(k)==rpowers); -zFJ)!/?
y(:,j) = y(:,j) + p*rpowern(:,idx); tpGT~Y(
end 2p&$bft
v^JzbO~|gj
if isnorm BzfR8mD
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); fn,n'E]
end :GIBB=D9
end _z#"BN
% END: Compute the Zernike Polynomials A;L
]=J
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A&M_ J
2 q4p-
% Compute the Zernike functions: t)&U'^
% ------------------------------ a>OYJe
idx_pos = m>0; Br!;Ac&N
idx_neg = m<0; <mFDC?j
YD[HBF)~j
z = y; +E</A:|}S
if any(idx_pos) ;}SGJ7
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); v/ N[)<
end
e;`(*
if any(idx_neg) PRU&y/zZmG
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); !CU-5bpu
end yn\c;Z
&?R/6"J
% EOF zernfun