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 }R`}Ey|{
function z = zernfun(n,m,r,theta,nflag) _#w5hXcu
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. L>!MEMqm
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N \oO&c
% and angular frequency M, evaluated at positions (R,THETA) on the mWuhXY^Q
% unit circle. N is a vector of positive integers (including 0), and 'h 7n}
% M is a vector with the same number of elements as N. Each element f0g&=k{OD
% k of M must be a positive integer, with possible values M(k) = -N(k) n;k
B_i*l
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, X
iM{YZ`B
% and THETA is a vector of angles. R and THETA must have the same +'UxO'v3]
% length. The output Z is a matrix with one column for every (N,M) $'b b)@_
% pair, and one row for every (R,THETA) pair. BA_l*h%=Cc
% aWb5w
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike i>;6Z s>S
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), @@|H8mP}H
% with delta(m,0) the Kronecker delta, is chosen so that the integral `(8RK
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 5S4`.'
% and theta=0 to theta=2*pi) is unity. For the non-normalized D$SO 6X~
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. #}xPOz7:
% >IHf5})R
% The Zernike functions are an orthogonal basis on the unit circle. #DcK{|ty
% They are used in disciplines such as astronomy, optics, and ~PC S_
% optometry to describe functions on a circular domain. i(kr#XsU
% DkBVk+
% The following table lists the first 15 Zernike functions. }j,G)\g#
% ,tuZ_"?M
% n m Zernike function Normalization 'Y5=A!*@tf
% -------------------------------------------------- _x?S0R1
% 0 0 1 1 dZ\T@9+j+
% 1 1 r * cos(theta) 2 IFWP&20
% 1 -1 r * sin(theta) 2 34~[dY
% 2 -2 r^2 * cos(2*theta) sqrt(6) .T}S[`Yx5
% 2 0 (2*r^2 - 1) sqrt(3) 66cPoG
% 2 2 r^2 * sin(2*theta) sqrt(6) r-o6I:y
% 3 -3 r^3 * cos(3*theta) sqrt(8) [Kd"M[1[<
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) .vXe}%
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) BO;LK-V
% 3 3 r^3 * sin(3*theta) sqrt(8) 'w}/o+x@
% 4 -4 r^4 * cos(4*theta) sqrt(10) 6}PoBhgSg-
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) JP 8v2)
p
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) [RHji47
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 2graLJ?9Z
% 4 4 r^4 * sin(4*theta) sqrt(10) jI807g+
% -------------------------------------------------- }C&kzJBEF
% If(IG]>`D
% Example 1: b=Y3O
% ^v@&
q
% % Display the Zernike function Z(n=5,m=1) oOK&+r7
% x = -1:0.01:1; WG3 .qLH%
% [X,Y] = meshgrid(x,x); PWs=0.Wj
% [theta,r] = cart2pol(X,Y); u/L\e.4
% idx = r<=1; GZ/vUe
% z = nan(size(X)); +)TOcxF%
% z(idx) = zernfun(5,1,r(idx),theta(idx)); I`EgR?5 `
% figure XJi^gT N
% pcolor(x,x,z), shading interp O.+X,CQG*
% axis square, colorbar gNzamorv[
% title('Zernike function Z_5^1(r,\theta)') 6o]X.plr
% `oo(\O7t=
% Example 2: G7H'OB
&
% ~UV$(5&-
% % Display the first 10 Zernike functions -AU!c^-o
% x = -1:0.01:1; STgYXA(
% [X,Y] = meshgrid(x,x); GFtE0IQ
% [theta,r] = cart2pol(X,Y); 8p~G)J3U
% idx = r<=1; ?TVR{e:
% z = nan(size(X)); -pm^k-%v
% n = [0 1 1 2 2 2 3 3 3 3]; 4f>
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% m = [0 -1 1 -2 0 2 -3 -1 1 3]; d/`Q,Vl
% Nplot = [4 10 12 16 18 20 22 24 26 28]; _ ?Z :m
% y = zernfun(n,m,r(idx),theta(idx)); I%31MU9
% figure('Units','normalized') 4
g^oy^~
% for k = 1:10 ?]u=5gqUU
% z(idx) = y(:,k); %1VfTr5
% subplot(4,7,Nplot(k)) zAdZXa[MRY
% pcolor(x,x,z), shading interp S4BU !
% set(gca,'XTick',[],'YTick',[]) >pn5nn1a
% axis square 6)~J5Fb
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ;p 'Ej'E
% end h ?%]uFJC
% . 'rC'FT
% See also ZERNPOL, ZERNFUN2. F%>`?NG+c
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% Paul Fricker 11/13/2006 Vtr5<:eEx
p8Wik<'^
\@HsMV2+zN
% Check and prepare the inputs: wsLfp82
% ----------------------------- YX:[],FP
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) LdM9k(
error('zernfun:NMvectors','N and M must be vectors.') w*"h#^1z
end JgY#W1>
L@HWm;aN
if length(n)~=length(m)
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error('zernfun:NMlength','N and M must be the same length.') csay\Q{
end 11>K\"K}
h\i>4^]X.
n = n(:); N/&t)7
m = m(:); x#_0
6
if any(mod(n-m,2)) i'bUX=JK
error('zernfun:NMmultiplesof2', ... |SF5'\d'
'All N and M must differ by multiples of 2 (including 0).') q!P{a^Fnc
end N^{+1u7
V,CVMbn/%N
if any(m>n) R59'KR2?
error('zernfun:MlessthanN', ... |}>;wZ[7
'Each M must be less than or equal to its corresponding N.') oCftI':@
end wO
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if any( r>1 | r<0 ) `<:D.9vO "
error('zernfun:Rlessthan1','All R must be between 0 and 1.') *N#{~
end x:O;Z~ |.
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) +(|6Wv
error('zernfun:RTHvector','R and THETA must be vectors.') `vFYeN;
end L'?0*t
\.%GgTF
r = r(:); wJQ"|
theta = theta(:); V]$Tbxg
length_r = length(r); 8Ekk"h6
if length_r~=length(theta) EecV%E
error('zernfun:RTHlength', ... fudIUG.
'The number of R- and THETA-values must be equal.') +/E
yX=
end KLn.vA.
xiW;Y{kZ
% Check normalization: a!US:^}lu
% -------------------- `:I<Jp
if nargin==5 && ischar(nflag) ZRd,V~iz
isnorm = strcmpi(nflag,'norm'); Y@Zv52,
if ~isnorm jw"]U jub
error('zernfun:normalization','Unrecognized normalization flag.') VTt{0 ~
end ,{br6*E
else WTcrfs)T
isnorm = false; GrB+Y!{{
end *uq}jlD`!
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }H4Z726
% Compute the Zernike Polynomials viJK%^U=-
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /T_ G9zc
[*k25N
% Determine the required powers of r: '!%Zf;Fjr
% ----------------------------------- x(Us
O}
m_abs = abs(m); 2/c^3[ccR
rpowers = []; W_E0+
for j = 1:length(n) : 6*FnKD
rpowers = [rpowers m_abs(j):2:n(j)]; [;F%6MPK^
end z[I3k
rpowers = unique(rpowers); kq
SpZoV0'
AMhHq/Dw
% Pre-compute the values of r raised to the required powers, nKzS2u=:Y
% and compile them in a matrix: DhAQ|SdCf
% ----------------------------- w)hH8jx{
if rpowers(1)==0 GuV.7&!x
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); x@ZxV*T^
rpowern = cat(2,rpowern{:}); i@C1}o-/
rpowern = [ones(length_r,1) rpowern]; : ;nvqb d
else xSQ:#o=8G
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); "0(H! }D
rpowern = cat(2,rpowern{:}); QyGTm"9l
end s5DEuu>g
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% Compute the values of the polynomials: /2!Wy6p
% -------------------------------------- k-$5H~(PZ
y = zeros(length_r,length(n)); \!erP!$x.
for j = 1:length(n) QD6in>+B@
s = 0:(n(j)-m_abs(j))/2; tR,&|?0
pows = n(j):-2:m_abs(j); )e$}sw{t
for k = length(s):-1:1 J m5).
p = (1-2*mod(s(k),2))* ... NEpomE(>x
prod(2:(n(j)-s(k)))/ ... ya<nD '%9
prod(2:s(k))/ ... ` n*e8T
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... W_%p'8,
prod(2:((n(j)+m_abs(j))/2-s(k))); }W:Rg}v
idx = (pows(k)==rpowers); =peodj^
y(:,j) = y(:,j) + p*rpowern(:,idx); O]>FNsh !
end UkE fuH
w$X"E*~>8
if isnorm Y~P1r]piB
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); w&vZ$n-|
end <}@*i
end 4pin\ZS:C
% END: Compute the Zernike Polynomials [IF5Iv\b
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s\)0f_I
s+@+<QE
% Compute the Zernike functions: ScgaWJ
% ------------------------------ 0%Y8M` ~s7
idx_pos = m>0; i;u#<y{E
idx_neg = m<0; 8QYP\7}o
m44"qp
z = y; &%@b;)]J
if any(idx_pos) ^/0c`JG!x
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); B1x# 7>K
end r9nyEzk
if any(idx_neg) Q"6:W2#v
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Z%Kkh2-uh
end b9F:X
A5UZUU^
% EOF zernfun