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 _unoDoB
function z = zernfun(n,m,r,theta,nflag) Pw]r&)I`y[
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. AY<L8
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 4VeT]`C^h
% and angular frequency M, evaluated at positions (R,THETA) on the )p;t
'*]
% unit circle. N is a vector of positive integers (including 0), and 6bjZW ~
% M is a vector with the same number of elements as N. Each element 4|5;nxkGm8
% k of M must be a positive integer, with possible values M(k) = -N(k) hWFOed4C
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1,
nVgvn2N/
% and THETA is a vector of angles. R and THETA must have the same q27q/q8
% length. The output Z is a matrix with one column for every (N,M) |Rx+2`6Dp
% pair, and one row for every (R,THETA) pair. 2^Im~p~ByE
% xhho{
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike \h s7>5O^K
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), B:UPSX)A
% with delta(m,0) the Kronecker delta, is chosen so that the integral :8}Qt^p
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, '~f*O0_
% and theta=0 to theta=2*pi) is unity. For the non-normalized }aa]1X(u
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?6ssSjR}
% HGh)d` 8
% The Zernike functions are an orthogonal basis on the unit circle. o%z^@Cq
% They are used in disciplines such as astronomy, optics, and BdU .;_K
% optometry to describe functions on a circular domain. b%"/8rK
% WL'!M&h
% The following table lists the first 15 Zernike functions. . uGne
% F g):>];<9
% n m Zernike function Normalization FqnD"]A
% -------------------------------------------------- | a
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% 0 0 1 1 e?07o!7[;
% 1 1 r * cos(theta) 2 b">"NvlB
% 1 -1 r * sin(theta) 2 HpUJ_pZ
% 2 -2 r^2 * cos(2*theta) sqrt(6) Ygg(qB1q
% 2 0 (2*r^2 - 1) sqrt(3) %t1Z!xv_
% 2 2 r^2 * sin(2*theta) sqrt(6) =x(k)RTDu
% 3 -3 r^3 * cos(3*theta) sqrt(8)
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% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) rAW7Zp~KK
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) %a0q|)Nrj
% 3 3 r^3 * sin(3*theta) sqrt(8) 3^q9ll7Op
% 4 -4 r^4 * cos(4*theta) sqrt(10) 'zMmJl}\vd
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) U Lq`!1{
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) [rAi9LSO"
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /J!hKK^k
% 4 4 r^4 * sin(4*theta) sqrt(10) 7OXRR)]V
% -------------------------------------------------- )NZ&m$I|-
% ~fcC+"7q/
% Example 1: ,1<6=vL
% m%"=sX7/9
% % Display the Zernike function Z(n=5,m=1) 9M|#X1r{%{
% x = -1:0.01:1; hmb=_W
% [X,Y] = meshgrid(x,x); 69uDc
% [theta,r] = cart2pol(X,Y); AtAu$"ue
% idx = r<=1; l#>A.-R*`
% z = nan(size(X)); Dp} $q`F[
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ^$oEM0h
% figure
~Z#\f5yv@
% pcolor(x,x,z), shading interp B*QLKO:)i
% axis square, colorbar -I#<?=0B
% title('Zernike function Z_5^1(r,\theta)') q="ymx~
% 'q8:1i9\[
% Example 2: pg<cvok
% w5Ucj*A\
% % Display the first 10 Zernike functions nvodP"iV
% x = -1:0.01:1; !g~u'r'1
% [X,Y] = meshgrid(x,x); jcxeXp|00
% [theta,r] = cart2pol(X,Y); 1x+w|h
% idx = r<=1; "Vwk&~B%
% z = nan(size(X)); .^rsVNG
% n = [0 1 1 2 2 2 3 3 3 3]; r6 pz(rCs}
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 8 }Maj
% Nplot = [4 10 12 16 18 20 22 24 26 28]; /9-kG
% y = zernfun(n,m,r(idx),theta(idx)); 7Uh/Gl
% figure('Units','normalized') uH)v\Js
% for k = 1:10 5VLC\QgK^
% z(idx) = y(:,k); <Lq.J`|+
% subplot(4,7,Nplot(k)) FJsg3D*@J
% pcolor(x,x,z), shading interp {=y~O
% set(gca,'XTick',[],'YTick',[]) Reg%ah|$/=
% axis square +C=^,B!,
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) m&6)Vt
% end iN+&7#x;/
% `g(r.`t^
% See also ZERNPOL, ZERNFUN2. 82=>I*0Q
thQ)J |1
% Paul Fricker 11/13/2006 4Mjcx.21
k'1iquc#u
9m2Yrj93
% Check and prepare the inputs: (d=knoo7A
% ----------------------------- V#d8fRm
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 4jz2x #T
error('zernfun:NMvectors','N and M must be vectors.') f}zv@6#&
end sfEy
UE%~SVi.#
if length(n)~=length(m) ':?MFkYC
error('zernfun:NMlength','N and M must be the same length.') f\M;m9{(
end :|E-Dx4F6H
t3FfPV!P"
n = n(:); )R9QJSe
m = m(:); =%G<S'2'
if any(mod(n-m,2)) H7R6Ljd?&S
error('zernfun:NMmultiplesof2', ... Bis'59?U_
'All N and M must differ by multiples of 2 (including 0).') g{$F;qbkO
end *lws7R
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if any(m>n) q.g0Oz@z
error('zernfun:MlessthanN', ... [(4s\c
'Each M must be less than or equal to its corresponding N.') aMycvYzH
end K\vyfYi
|dQ-l !
if any( r>1 | r<0 ) dw
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error('zernfun:Rlessthan1','All R must be between 0 and 1.') /qX=rlQ/ n
end 3Zeh$DZ
~^pV>>LX|
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) .q7|z3@,
error('zernfun:RTHvector','R and THETA must be vectors.') \z>L,U
end ?3{:[*
[N4#R
r = r(:); j)neVPf%v
theta = theta(:); h1'j1uI
length_r = length(r); 8LM 91
if length_r~=length(theta) 1:r 8p6
error('zernfun:RTHlength', ... +:&,Ts/
'The number of R- and THETA-values must be equal.') #.kDin~!
end
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>SR!*3$5
% Check normalization: vw/L|b7G
% -------------------- _:0
if nargin==5 && ischar(nflag) L]C|&KP
isnorm = strcmpi(nflag,'norm'); =5jng.
if ~isnorm k;bdzcMkQ
error('zernfun:normalization','Unrecognized normalization flag.') vdLBf+Zi
end U94Tp A6
else ,xeJf6es
isnorm = false; KDODUohC
end sD[G?X
gLyE,1Z}u
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `Z3p( G
% Compute the Zernike Polynomials "DniDA
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% l6wN&JHTh
?'dsiA[
% Determine the required powers of r: vl{G;[6
% ----------------------------------- 0A}'@N@G)
m_abs = abs(m); %xq/eC7
rpowers = []; drr n&y
for j = 1:length(n) !X5~!b^*
rpowers = [rpowers m_abs(j):2:n(j)]; (")IU{>c6
end t3dvHU&Z:
rpowers = unique(rpowers); E9]/sFA-]
aOj(=s
% Pre-compute the values of r raised to the required powers, rX-V0
% and compile them in a matrix: d1"%sI
% ----------------------------- oY#62&wk4
if rpowers(1)==0 L1'#wH
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); uTemAIp
$u
rpowern = cat(2,rpowern{:}); +EtL+Y(U
rpowern = [ones(length_r,1) rpowern]; phT|w
H
else LZ97nvK
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); w;RG*rv
rpowern = cat(2,rpowern{:}); XpFoSW#K
end Dd(#
7(q EHZEr
% Compute the values of the polynomials: Hq[vh7Lux
% -------------------------------------- eX;Tufe*(Q
y = zeros(length_r,length(n)); HW{si]~q
for j = 1:length(n) BRTM]tRZ
s = 0:(n(j)-m_abs(j))/2; dKOW5\H'
pows = n(j):-2:m_abs(j); _;;'/rs
j
for k = length(s):-1:1 *@XJ7G[
p = (1-2*mod(s(k),2))* ... )4[Yplo
prod(2:(n(j)-s(k)))/ ... 0X`sQNx
prod(2:s(k))/ ... R::0.*FF
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ),G?f {`!
prod(2:((n(j)+m_abs(j))/2-s(k))); rNdeD~\
idx = (pows(k)==rpowers); 6vx0F?>_
y(:,j) = y(:,j) + p*rpowern(:,idx); J?yNZK$WqN
end Z*Sa%yf
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if isnorm *u2pk>y)
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); q'07
end .pP{;:Avpn
end ?,_$;g
% END: Compute the Zernike Polynomials ;*3OkNxa3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dG6 G
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% Compute the Zernike functions: =~)J:x\F
% ------------------------------ .RI{\ i`
idx_pos = m>0; U]Iypl`l
idx_neg = m<0; [d(@lbV0
X*Ibk-PUM
z = y; RXMzwk
if any(idx_pos) :# 1d;jx
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); k`LoRqF
end ZnX]Q+w
if any(idx_neg) s Zan.Kc#
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 6x1!!X+)+
end Z9:erKT
v6gfyGCJ
% EOF zernfun