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 9coN >y
function z = zernfun(n,m,r,theta,nflag) bVgmjt2&>
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ]r&dWF
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N bnZ`Wc*5b
% and angular frequency M, evaluated at positions (R,THETA) on the 8+|7*Ud
% unit circle. N is a vector of positive integers (including 0), and ^J-"8%
% M is a vector with the same number of elements as N. Each element (@(rz/H
% k of M must be a positive integer, with possible values M(k) = -N(k) 'Dx_n7&=
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, PrQs_tNi
% and THETA is a vector of angles. R and THETA must have the same CqAv^n7 }
% length. The output Z is a matrix with one column for every (N,M) o0&pSCK
% pair, and one row for every (R,THETA) pair. {G i:W/jJ
% 8GKqPS+
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike +)<H,?/
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), k62KZ5| D
% with delta(m,0) the Kronecker delta, is chosen so that the integral 5^0K5R6GQf
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, A5q%ytI
% and theta=0 to theta=2*pi) is unity. For the non-normalized `21$e
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. _/pdZM,V
% 6Gj69Lr
% The Zernike functions are an orthogonal basis on the unit circle. J_@`:l0,z
% They are used in disciplines such as astronomy, optics, and kf-/rC)>
% optometry to describe functions on a circular domain. wK*b2r}0/
% ;n2b$MB?nM
% The following table lists the first 15 Zernike functions. z$]HZ#aRE
% }'c@E0"
% n m Zernike function Normalization 6$'0^Ftm'
% -------------------------------------------------- =JDa[_lpN
% 0 0 1 1 Op
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% 1 1 r * cos(theta) 2 EG
oe<.
% 1 -1 r * sin(theta) 2 4+2hj*I
% 2 -2 r^2 * cos(2*theta) sqrt(6) xA#'%|"
% 2 0 (2*r^2 - 1) sqrt(3) tLc~]G*\`s
% 2 2 r^2 * sin(2*theta) sqrt(6) }DzN-g<K
% 3 -3 r^3 * cos(3*theta) sqrt(8) X)^&5;\`
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) R1/87eB
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) s]@k,%
% 3 3 r^3 * sin(3*theta) sqrt(8) -)o0P\cTEt
% 4 -4 r^4 * cos(4*theta) sqrt(10) #fkOm
Y7X
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) lKA2~ o
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) d_!lRQ^N
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) nv-_\M
% 4 4 r^4 * sin(4*theta) sqrt(10) KX $Q`lM
% -------------------------------------------------- =2tl149m/z
% `mo>~c7
% Example 1: "PtOe[Xk
% f^D4aEU
% % Display the Zernike function Z(n=5,m=1) $/XR/
% x = -1:0.01:1; Yv7`5b{N.
% [X,Y] = meshgrid(x,x); r<XlIi
% [theta,r] = cart2pol(X,Y); AOVoOd+6
% idx = r<=1; {WYmO1
% z = nan(size(X)); |vf /M|
% z(idx) = zernfun(5,1,r(idx),theta(idx)); BdYl
sYp
% figure d*(wU>J '
% pcolor(x,x,z), shading interp z ;KUIWg
% axis square, colorbar }RPeAcbU_
% title('Zernike function Z_5^1(r,\theta)') oEuo@\U05v
% 8C4=f
% Example 2: 4}>1I}!k
% Da WzQe=
% % Display the first 10 Zernike functions AYLCdCoK.
% x = -1:0.01:1; D-!#TN`Y
% [X,Y] = meshgrid(x,x); AcCM
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% [theta,r] = cart2pol(X,Y); cc|"^-j-7
% idx = r<=1; $v*0\O
% z = nan(size(X)); ~hxo_&
% n = [0 1 1 2 2 2 3 3 3 3]; v*.#LJEm
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 76M`{m
% Nplot = [4 10 12 16 18 20 22 24 26 28]; q=|0lZ$`V_
% y = zernfun(n,m,r(idx),theta(idx)); Me|+)}'p5h
% figure('Units','normalized') DHO+JtO
% for k = 1:10 h1uD >heGl
% z(idx) = y(:,k); ko<iG]Dv'
% subplot(4,7,Nplot(k)) ?=lnYD j
% pcolor(x,x,z), shading interp lS:R##
% set(gca,'XTick',[],'YTick',[]) Vy:MK9U2
% axis square \mt>R[
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 5NECb4FG
% end <0hJo=6a8
% GP/Gv
% See also ZERNPOL, ZERNFUN2. 9X2l H~C
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% Paul Fricker 11/13/2006 *;I F^u1
WP-'gC6K=
}:5>1FfX=
% Check and prepare the inputs: D@yuldx'/
% ----------------------------- b2vc
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) :%hxg
error('zernfun:NMvectors','N and M must be vectors.') ^MZdht
end nPj/C7j
Mi5"XQ>/
if length(n)~=length(m) 9ywPWT[^
error('zernfun:NMlength','N and M must be the same length.') ,UD,)ZPf[
end 9u\&kQxqD
@Fl&@ $
n = n(:); G2x5% `
m = m(:); \I4*|6kA
if any(mod(n-m,2)) 8'kA",P
error('zernfun:NMmultiplesof2', ... 3C8W]yw/s
'All N and M must differ by multiples of 2 (including 0).') Jc#()4
end XU}sbbwu
$*Q_3]AY]
if any(m>n) ,6%{9oW9Z:
error('zernfun:MlessthanN', ... vKX
$Nf
'Each M must be less than or equal to its corresponding N.') 3GS oHsNk
end 9N[vNg<n
y/}>)o4Q
if any( r>1 | r<0 ) Hkv4t5F
error('zernfun:Rlessthan1','All R must be between 0 and 1.') }0({c~z\
end ?=]*r>a3
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) :K3nJ1G&
error('zernfun:RTHvector','R and THETA must be vectors.') 3-Q*umh
end h69: Tj!
fQ&:1ec
r = r(:); rX%qWhiEJ
theta = theta(:); 1MV\
^l_
length_r = length(r); SRN:!-
if length_r~=length(theta) 042sjt
error('zernfun:RTHlength', ... ezt_ct/Z
'The number of R- and THETA-values must be equal.') J]f\=;z;<a
end C_[V[k0(
GLe(?\Ug=
% Check normalization: Z:#-4CiP
% -------------------- #_+T@|r
if nargin==5 && ischar(nflag) fNi&1J-/
isnorm = strcmpi(nflag,'norm'); !P, 9Sg&5)
if ~isnorm UC^Bn1
error('zernfun:normalization','Unrecognized normalization flag.') -o+_PL
$\
end sBuVm<H
else F*QD\sG:
isnorm = false; sX3Vr&r
end n
?%3=~9
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4[]R?lL
% Compute the Zernike Polynomials C61KY7iyR
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $J#}3;a
qVFz-!6b
% Determine the required powers of r: _c|>m4+X
% ----------------------------------- _9Kdcoh
m_abs = abs(m); o_gpBaWD
rpowers = []; y@AKb
for j = 1:length(n) -/aDq?<<
rpowers = [rpowers m_abs(j):2:n(j)]; VwoCRq*
end v&U'%1|
rpowers = unique(rpowers); H{x}gBQ
/|y3M/;F
% Pre-compute the values of r raised to the required powers, 2I:vie
% and compile them in a matrix: 0+O)~>v
% ----------------------------- VG'oy
if rpowers(1)==0 V9"Kro
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); o(~>a
rpowern = cat(2,rpowern{:}); }0uSm%,"
rpowern = [ones(length_r,1) rpowern]; :H<u@%
else {"e/3
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);
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rpowern = cat(2,rpowern{:}); 3Fh<%<=
end {!B0&x
pM\)f
% Compute the values of the polynomials: )F<<M+q=
% -------------------------------------- @6i^wC
y = zeros(length_r,length(n)); C9Fc(Y?_
for j = 1:length(n) 2s
EdN$O
s = 0:(n(j)-m_abs(j))/2; K4xZT+Qb
pows = n(j):-2:m_abs(j); L5cNCWpo
for k = length(s):-1:1 &I?1(t~hT
p = (1-2*mod(s(k),2))* ... w"-bO ~5h
prod(2:(n(j)-s(k)))/ ... ZzI^*Nyg
prod(2:s(k))/ ... ;4F[*VF!w
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 7%8,*T
prod(2:((n(j)+m_abs(j))/2-s(k))); QA.B.U7!
idx = (pows(k)==rpowers); &}w,bG$
y(:,j) = y(:,j) + p*rpowern(:,idx); F& H~JJ
end ,^|+n()O
Yq/|zTe{
if isnorm uGLVY%N
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 5cyl:1Ln
end .'"+CKD.N
end u!nt0hS
% END: Compute the Zernike Polynomials -H.;73Kb[
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )sB`!:~HjP
+ 7E6U*
% Compute the Zernike functions: *D;B%j^;
% ------------------------------ c.&vWmLSGE
idx_pos = m>0; 8c__ U<
idx_neg = m<0; zv#i\8h^p
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z = y; fylW)W4C
if any(idx_pos) ,i*^fpF`F"
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Z#>k:v
end \s<iM2]Kl
if any(idx_neg) =q[3/'2V$?
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); H7#RL1qM&
end %
C6 H(
15U=2j*.b
% EOF zernfun