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 a=&{B'^G
function z = zernfun(n,m,r,theta,nflag) lSK<LytB
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. (>M?
iB
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N w6<zPrA
% and angular frequency M, evaluated at positions (R,THETA) on the F|!
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% unit circle. N is a vector of positive integers (including 0), and ;!Q}g19C
% M is a vector with the same number of elements as N. Each element "Kc1@EX=
% k of M must be a positive integer, with possible values M(k) = -N(k) 3#Qek2
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, rGUu K0L&
% and THETA is a vector of angles. R and THETA must have the same -W'T3_
% length. The output Z is a matrix with one column for every (N,M) ,]H2F']4Z
% pair, and one row for every (R,THETA) pair. MCO`\"`l
% ukwO%JAr
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike +LB2V3UZ
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), zn2Qp
% with delta(m,0) the Kronecker delta, is chosen so that the integral 3u@=]0ZN
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, cTu"Tu\Qw
% and theta=0 to theta=2*pi) is unity. For the non-normalized \?~cJMN
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. (Y:?qy
% U C..)9
% The Zernike functions are an orthogonal basis on the unit circle. `FHKQS5
% They are used in disciplines such as astronomy, optics, and /M5R<rl
% optometry to describe functions on a circular domain. ck\TTNA
% BVe c
% The following table lists the first 15 Zernike functions. .
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% A|
s\5"??
% n m Zernike function Normalization |$G|M=*LN
% -------------------------------------------------- 4"d'iY
% 0 0 1 1 "fOxS\er
% 1 1 r * cos(theta) 2 [Nv)37|W
% 1 -1 r * sin(theta) 2 ..;ep2jSs
% 2 -2 r^2 * cos(2*theta) sqrt(6) $9rQ w1#e
% 2 0 (2*r^2 - 1) sqrt(3) ~jDf,a2
% 2 2 r^2 * sin(2*theta) sqrt(6) |?<^4U8
% 3 -3 r^3 * cos(3*theta) sqrt(8) aU?HIIA
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) cllnYvr3
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ~fY\;
% 3 3 r^3 * sin(3*theta) sqrt(8) ,HECHA_"
% 4 -4 r^4 * cos(4*theta) sqrt(10) u`Abko<D
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) N-YCOSUu
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) -W.bOr
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) h)pYV>!d
% 4 4 r^4 * sin(4*theta) sqrt(10) e!oL!Zg
% -------------------------------------------------- ~=k?ea/>
% M+GtUE~"
% Example 1: nNpXkI:
% `L7Cf&W\l8
% % Display the Zernike function Z(n=5,m=1) O*udV E>
% x = -1:0.01:1; 5# B M
% [X,Y] = meshgrid(x,x); 4gh`
>
% [theta,r] = cart2pol(X,Y); |H&&80I
% idx = r<=1; @BoZZ
% z = nan(size(X)); s7"5NU-
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ?L+|b5RS
% figure sj8lvIY5
% pcolor(x,x,z), shading interp \%Lj !\
% axis square, colorbar
PaZd^0'!Z
% title('Zernike function Z_5^1(r,\theta)') bBgyLyg
% `9mc+
% Example 2: *^i"q\n5(
% Z7J4rTA
% % Display the first 10 Zernike functions pIl[)%F
% x = -1:0.01:1;
, )PpE&
% [X,Y] = meshgrid(x,x); Zy=DY
% [theta,r] = cart2pol(X,Y); X]c>clk,
% idx = r<=1; ()(^B}VK
% z = nan(size(X)); v(~EO(n.
% n = [0 1 1 2 2 2 3 3 3 3]; sfzDE&>'
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; w-P;E!gTt
% Nplot = [4 10 12 16 18 20 22 24 26 28]; XVzsqi*Z
% y = zernfun(n,m,r(idx),theta(idx)); LX{mr{
% figure('Units','normalized') Nn-EtM0w
% for k = 1:10 _3zJ.%
% z(idx) = y(:,k); 9{CajtN
% subplot(4,7,Nplot(k)) oq[r+E-]$@
% pcolor(x,x,z), shading interp {Lugdf'
% set(gca,'XTick',[],'YTick',[]) BE)&.}l
% axis square *X8Pa;x
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) cQrXrij;!
% end &?Z<"+B8S
% Yd]
% See also ZERNPOL, ZERNFUN2. m*vz
dZuPR
% Paul Fricker 11/13/2006 `Ln1g@
(je`sV
OXS.CFZM
% Check and prepare the inputs: kJpr:4;@_
% ----------------------------- lY[\eQ
1:
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Wn&9R
j
error('zernfun:NMvectors','N and M must be vectors.') hCob^o
end FZtT2Z4&i
D*t[5,~j
if length(n)~=length(m) iHeu<3O
error('zernfun:NMlength','N and M must be the same length.') )WsR
8tk
end =55V<VI
@T] G5|\ok
n = n(:); uTNy{RBD+
m = m(:); dpcU`$kt
if any(mod(n-m,2)) RmJ|g<
error('zernfun:NMmultiplesof2', ... Uj^Y\w-@Z
'All N and M must differ by multiples of 2 (including 0).') 7ea%mg\
end #6mr'e1
i4lB]k
if any(m>n) A u"BDP
error('zernfun:MlessthanN', ... !im%t9
'Each M must be less than or equal to its corresponding N.') W4"1H0s`l
end $ZlzS`XF7
s:ojlmPb
if any( r>1 | r<0 ) jJAr #|
error('zernfun:Rlessthan1','All R must be between 0 and 1.') y=zs6HaS
end FTu<$`!1L
`l
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) o.wXaS8
error('zernfun:RTHvector','R and THETA must be vectors.') ?dmwz4k0
end )3^#CD
&/?OP)N,}
r = r(:); )kIjZ
theta = theta(:); MbeK{8~E%l
length_r = length(r); oxLO[js
if length_r~=length(theta) _ygdv\^Tet
error('zernfun:RTHlength', ... 4iY
<7l8
'The number of R- and THETA-values must be equal.') ]L?WC
end Awe'MG p%
-qG7, t
% Check normalization: 2 ]}e4@{
% -------------------- )h1 `?q:5
if nargin==5 && ischar(nflag) H[N~)3x
isnorm = strcmpi(nflag,'norm'); vj"['6Xa
if ~isnorm S2?)Sb`
error('zernfun:normalization','Unrecognized normalization flag.') B-V
end W?0u_F
else +/rh8?
isnorm = false; 2[Xe:)d
end o<rbC <
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4F+G;'JV
% Compute the Zernike Polynomials pIY3ft\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - XB[2h
Ni#y=cb
% Determine the required powers of r: :@S=0|:j
% ----------------------------------- ~>$z1o&}.
m_abs = abs(m); R^rA.7T
rpowers = []; n +dRAIqB
for j = 1:length(n) *}Rd%'
rpowers = [rpowers m_abs(j):2:n(j)]; :AyZe7:(D
end rLcXo%w
rpowers = unique(rpowers); \b?O+;5Cj
a KIS%M#Y
% Pre-compute the values of r raised to the required powers, >Sm#-4B-
% and compile them in a matrix: $it>*%
% ----------------------------- ,&jjpeZP
if rpowers(1)==0 Y^gIvX
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ;V^ I>-fnm
rpowern = cat(2,rpowern{:}); ^?T,>ZI
rpowern = [ones(length_r,1) rpowern]; \>+BvF
else `!.c_%m2
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); \ $
:)Ka
rpowern = cat(2,rpowern{:}); t}gK)"g
end 4}Hf"L[ l
EI@ep~
% Compute the values of the polynomials: RMa#z [{0
% -------------------------------------- hcQv!!Q"k$
y = zeros(length_r,length(n)); SpZmwa #\
for j = 1:length(n) &sGLm~m#
s = 0:(n(j)-m_abs(j))/2; "~T06!F45
pows = n(j):-2:m_abs(j); fw0Z- 9*
for k = length(s):-1:1 EiWd =jDm
p = (1-2*mod(s(k),2))* ... s_76)7
prod(2:(n(j)-s(k)))/ ... uQkQ#'e|
prod(2:s(k))/ ... E /V`NqC
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Y4*?QBYA
prod(2:((n(j)+m_abs(j))/2-s(k))); > u=nGeO
idx = (pows(k)==rpowers); -3C$br
y(:,j) = y(:,j) + p*rpowern(:,idx); (Jk:Qz5
end yJw4!A 1!
cQ/T:E7$`
if isnorm ^7C,GaDsn
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 2^&5D,}0
end yj9Ad*.
end 1JN/oq;
% END: Compute the Zernike Polynomials =4Wjb
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \>4x7mF!
zxvowM
% Compute the Zernike functions: iPrAB*
% ------------------------------ PSa"u5 O
idx_pos = m>0; |R (rb-v
idx_neg = m<0; *1_A$14l
`Dv&.
z = y; y#5;wb<1
if any(idx_pos) RQ[6svfP
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 8wA'a'V.
end 1iE*-K%Q
if any(idx_neg) ,y/N^^\
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); +6x:+9S
end CB?,[#r5f
tNCKL.yU
% EOF zernfun