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 nF54tR[
function z = zernfun(n,m,r,theta,nflag) 1Ce@*XBU
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. s`M9
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N N|8P)
% and angular frequency M, evaluated at positions (R,THETA) on the 9A/\h3HrJ
% unit circle. N is a vector of positive integers (including 0), and ;X8yFq
% M is a vector with the same number of elements as N. Each element #o=y?(
% k of M must be a positive integer, with possible values M(k) = -N(k) !^^?dRd*v
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, dT`D:)*:
% and THETA is a vector of angles. R and THETA must have the same }\z.)B4,
% length. The output Z is a matrix with one column for every (N,M) 8;d:-Cp
% pair, and one row for every (R,THETA) pair. 8ZM?)#`@{
% `n#H5Oyn
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ;\a
YlV-
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 5QW=&zI`=
% with delta(m,0) the Kronecker delta, is chosen so that the integral mPOGidxix
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ]9YJ,d@J
% and theta=0 to theta=2*pi) is unity. For the non-normalized )_+rU|We
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. @GBxL*e
% 3VsW@SG7N
% The Zernike functions are an orthogonal basis on the unit circle. M`. tf_x
% They are used in disciplines such as astronomy, optics, and O}+.U<V
% optometry to describe functions on a circular domain. 9i'jjN
% v/Py"hQ
% The following table lists the first 15 Zernike functions. VvvRRP^q
% *i\Qo
% n m Zernike function Normalization ?+_Gs;DGVE
% -------------------------------------------------- zO~8?jDN4|
% 0 0 1 1 gD,1 06%
% 1 1 r * cos(theta) 2 2"0es40;0
% 1 -1 r * sin(theta) 2 OglEt[ "
% 2 -2 r^2 * cos(2*theta) sqrt(6) I(]}XZq
% 2 0 (2*r^2 - 1) sqrt(3) DNOueU
% 2 2 r^2 * sin(2*theta) sqrt(6) kY&k-K\
% 3 -3 r^3 * cos(3*theta) sqrt(8) $}<PL}+
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) :V1W/c
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) {%<OD8>p
% 3 3 r^3 * sin(3*theta) sqrt(8) _a5d?Q9Z
% 4 -4 r^4 * cos(4*theta) sqrt(10) k&&2Tq
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) s:OFVlC%\
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) VYu~26Zr
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =q>'19^Jx
% 4 4 r^4 * sin(4*theta) sqrt(10) '= _/ 1F*q
% -------------------------------------------------- y-T| #
% %dRo^E1p
% Example 1: DQNnNsP:M-
% W0(_~
% % Display the Zernike function Z(n=5,m=1) fdxLAC
% x = -1:0.01:1; &)8:h+&Z
% [X,Y] = meshgrid(x,x); "JVkVp[5D+
% [theta,r] = cart2pol(X,Y); vGc,vjC3x
% idx = r<=1; g$7{-OpB
% z = nan(size(X)); 0)%YNaskj
% z(idx) = zernfun(5,1,r(idx),theta(idx)); k'gh
% figure ,`wXg
% pcolor(x,x,z), shading interp ~Fe${2
% axis square, colorbar m#8m] Y
% title('Zernike function Z_5^1(r,\theta)') :}yi-/_8!
% *meZ8DV2DH
% Example 2: `k=bL"T>\
% K\>tA)IPSV
% % Display the first 10 Zernike functions <:(6EKJAq}
% x = -1:0.01:1; Vx(B{5>Vu
% [X,Y] = meshgrid(x,x); G %N
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% [theta,r] = cart2pol(X,Y); X'wE7=29M
% idx = r<=1; )!Jc3%(B
% z = nan(size(X)); P::TO-C
% n = [0 1 1 2 2 2 3 3 3 3]; Mx6@$tQ%
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; -|kA)M[
% Nplot = [4 10 12 16 18 20 22 24 26 28]; mYxuA0/k
% y = zernfun(n,m,r(idx),theta(idx)); 5j:0Yt
% figure('Units','normalized') 4FEk5D
% for k = 1:10 IN4=YrM^
% z(idx) = y(:,k); 9!f/aI
% subplot(4,7,Nplot(k)) AcS|c:3MUy
% pcolor(x,x,z), shading interp ie;]/va
% set(gca,'XTick',[],'YTick',[]) 9)0D~oUi
% axis square x N=i]~
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Dakoqke
% end - d8TD*^
% {SwQ[$k=_
% See also ZERNPOL, ZERNFUN2. %dJX-sm@
6^%UU
o%
% Paul Fricker 11/13/2006 ,RE\$~`w
d1T,eJ}
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% Check and prepare the inputs: y@ ML/9X8q
% ----------------------------- j H19k}D
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) |w_7_J2
error('zernfun:NMvectors','N and M must be vectors.') =2[7
E
end C2@,BCR
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if length(n)~=length(m) (]b!{kS
error('zernfun:NMlength','N and M must be the same length.') "nZ*{uv
end
-%2[2p
"Weg7mc#
n = n(:); qi;f^9M%
m = m(:); z)'M k[
if any(mod(n-m,2)) aT_&x@x
error('zernfun:NMmultiplesof2', ... 9!T[Z/}T
'All N and M must differ by multiples of 2 (including 0).') NXwz$}}Pp
end 6^uq?
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if any(m>n) k9)jjR*XxG
error('zernfun:MlessthanN', ... fYp'&Btb]x
'Each M must be less than or equal to its corresponding N.') GMMp|WV|
end thV>j9'
wm]^3qI2
if any( r>1 | r<0 ) _8"O$w
error('zernfun:Rlessthan1','All R must be between 0 and 1.') O_$m!5ug
end y|CP;:f;
f-}[_Y%;
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) )A!>=2M`
error('zernfun:RTHvector','R and THETA must be vectors.') sW)Zi
end a-l;vDs
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r = r(:); O?U'!o=
theta = theta(:); bSsh^Z
length_r = length(r); j*F`"df
if length_r~=length(theta) XD |E=s
error('zernfun:RTHlength', ... XS`M-{f`
'The number of R- and THETA-values must be equal.') #Xhdn\7
end rrQQZ5fh b
kjEEuEv
% Check normalization: ]d,S749(s
% -------------------- (:._"jp]
if nargin==5 && ischar(nflag) io,M{Ib
isnorm = strcmpi(nflag,'norm'); T6H}/#*tK
if ~isnorm U"q/rcA
error('zernfun:normalization','Unrecognized normalization flag.') A:aE|v/T&
end S>.SSXlM
else V2$h8\a
isnorm = false; s4 6}s{6
end hQ]H
/+\
M%1}/!J3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% QA2borfy
% Compute the Zernike Polynomials ]?3un!o3o
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +|bmT
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% Determine the required powers of r: ?"zY"*>4
% ----------------------------------- t<~ $
m_abs = abs(m); 6fd+Q
/
rpowers = []; |AcRIq
for j = 1:length(n) NG
rpowers = [rpowers m_abs(j):2:n(j)]; hGd<<\
end 70f Klp
rpowers = unique(rpowers); r) $+
2kdC]|H2?
% Pre-compute the values of r raised to the required powers, 5m?8yT}
% and compile them in a matrix: A;/-u<f
% ----------------------------- <Ard7UT
if rpowers(1)==0 vz^<YZMu
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); vFE;D@bz:
rpowern = cat(2,rpowern{:}); 1QmH{jM
rpowern = [ones(length_r,1) rpowern]; Y2d;E.DH8
else p3]_}Y
D[#
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); >Y_*%QGH_
rpowern = cat(2,rpowern{:}); MS0Fl|YA
end 0KMctPT]p
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% Compute the values of the polynomials: PZ{Dv'C
% -------------------------------------- OH5>vV'i
y = zeros(length_r,length(n)); h3*Zfl<]
for j = 1:length(n) w=^`w:5X
s = 0:(n(j)-m_abs(j))/2; 3dht!7/
pows = n(j):-2:m_abs(j); @;<ht c
for k = length(s):-1:1 ms!r ef4`+
p = (1-2*mod(s(k),2))* ... d+X}cq=
prod(2:(n(j)-s(k)))/ ... z;A>9vQ_J
prod(2:s(k))/ ... \e!vj.PU
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... z "+Mrew
prod(2:((n(j)+m_abs(j))/2-s(k))); L]d-hs
idx = (pows(k)==rpowers); 0PU8#2pR
y(:,j) = y(:,j) + p*rpowern(:,idx); AtF3%Zv2
end ,z;ky5Ct
uL3Eq>~x
if isnorm ;]gP@ h/
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ~4s'0 w^
end OCZ[D{i9@
end $/=nU*pd
% END: Compute the Zernike Polynomials iC W*]U
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %^1cyk
O!Oumw,$
% Compute the Zernike functions: wk6NG/<
% ------------------------------ E<C&Cjz:H
idx_pos = m>0; E2cB U{x
idx_neg = m<0; U$
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z I+\Oll#Q
z = y; Qu}W/j|3
if any(idx_pos) abJ"
[
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 2gzou|Y
end O~59FuL
if any(idx_neg) ]da^xWK
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); } ~"hC3w
end {dL?rQ>5L
)(tM/r4`c&
% EOF zernfun