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 X;
\+<LE
function z = zernfun(n,m,r,theta,nflag) |}s*E_/[
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. NqazpB*
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N u^+7hkk
% and angular frequency M, evaluated at positions (R,THETA) on the 58tARL Dr
% unit circle. N is a vector of positive integers (including 0), and Ha0M)0Anv
% M is a vector with the same number of elements as N. Each element 9iIhte.
% k of M must be a positive integer, with possible values M(k) = -N(k) m<T%Rb4?@
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Db}j?ik/
% and THETA is a vector of angles. R and THETA must have the same n`B:;2X,
% length. The output Z is a matrix with one column for every (N,M) 17%,7P9pg
% pair, and one row for every (R,THETA) pair. Pe_W;q.
% lHY+}v0
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ,*TmIPNK
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), [[Ls_ZL!=
% with delta(m,0) the Kronecker delta, is chosen so that the integral TVtvuvQ2K
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, J@HtoTDO3
% and theta=0 to theta=2*pi) is unity. For the non-normalized hc(#{]].
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. b5dD/-Vj
% hP%M?MKC
% The Zernike functions are an orthogonal basis on the unit circle. r4b 6 c
% They are used in disciplines such as astronomy, optics, and O-0x8 O^B
% optometry to describe functions on a circular domain. #_ ;lf1x!
% .]Y$o^mf
% The following table lists the first 15 Zernike functions. B?gOHG*vd>
% x*\Y)9Vgy
% n m Zernike function Normalization k<nZ+! M
% -------------------------------------------------- ~|DUt
% 0 0 1 1 A7Cm5>Y_S
% 1 1 r * cos(theta) 2 `iFmrC<
% 1 -1 r * sin(theta) 2 Fh&G;aEq
% 2 -2 r^2 * cos(2*theta) sqrt(6) y4
#>X
% 2 0 (2*r^2 - 1) sqrt(3) 9rA0lqr]5
% 2 2 r^2 * sin(2*theta) sqrt(6) FJGlP&v<
% 3 -3 r^3 * cos(3*theta) sqrt(8) 1APe=tJ
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) $D~0~gn~
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) #'nr
Er <
% 3 3 r^3 * sin(3*theta) sqrt(8) DZ3wCLQtK
% 4 -4 r^4 * cos(4*theta) sqrt(10) 13$%,q)
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) hlvK5Z
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ^,lIK+#Elz
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Q",t3i4
% 4 4 r^4 * sin(4*theta) sqrt(10) T$)^gHS
% -------------------------------------------------- ,a{P4Bq
% RtkEGxw*^
% Example 1: DD+7V@
% ?um;s-x)
% % Display the Zernike function Z(n=5,m=1) rQ{7j!Im
% x = -1:0.01:1; .FP$m?
% [X,Y] = meshgrid(x,x); ^&9zw\x;z
% [theta,r] = cart2pol(X,Y); /e5O"@
% idx = r<=1; T8?Ghbn
% z = nan(size(X)); p;`>e>$
% z(idx) = zernfun(5,1,r(idx),theta(idx));
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% figure pTth}JM>
% pcolor(x,x,z), shading interp hIYNhZv
% axis square, colorbar v|)4ocFK
% title('Zernike function Z_5^1(r,\theta)') "=HA Y
% @(EAq<5{
% Example 2: ,i^9 |Oeq
% =g7x'
kN
% % Display the first 10 Zernike functions W]$w@.oW[
% x = -1:0.01:1; k>Is:P
% [X,Y] = meshgrid(x,x); ]\-A;}\e
% [theta,r] = cart2pol(X,Y); W 8<&gh+
% idx = r<=1; t5^{D>S1
% z = nan(size(X)); T=
8 0,
% n = [0 1 1 2 2 2 3 3 3 3]; @o].He@L<j
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ol\Utq,
% Nplot = [4 10 12 16 18 20 22 24 26 28]; W<h)HhyG
% y = zernfun(n,m,r(idx),theta(idx)); hk;5w{t}}
% figure('Units','normalized') M><yGaaX/
% for k = 1:10 Ye%~I`@?
% z(idx) = y(:,k); ^ox=HNV
% subplot(4,7,Nplot(k)) rET\n(AJ
% pcolor(x,x,z), shading interp aL\PGdgO
% set(gca,'XTick',[],'YTick',[]) &N$<e(K
% axis square lf`{zc r:
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) dohA0
% end u4cnE"
% >%_ \;svZG
% See also ZERNPOL, ZERNFUN2. \{_q.;}
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% Paul Fricker 11/13/2006 O0x,lq
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YsC>i`n9
% Check and prepare the inputs: TIqtF&@o4
% ----------------------------- df8k7D;~e
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) .fqN|[>
error('zernfun:NMvectors','N and M must be vectors.') OU\ ~::
end {f_={k
G{~J|{t\yz
if length(n)~=length(m) |w~nVRb
error('zernfun:NMlength','N and M must be the same length.') /obfw^
end oi7@s0@
|u% )gk
n = n(:); *gb*LhgO
m = m(:); b<[Or^X
]
if any(mod(n-m,2)) e-/&$Qq
error('zernfun:NMmultiplesof2', ... LtF,kAIt7v
'All N and M must differ by multiples of 2 (including 0).') 20h}
[Q(
end 4/~E4"8
AEI>\Y
if any(m>n) H064BM
error('zernfun:MlessthanN', ... _IHV7*u{;
'Each M must be less than or equal to its corresponding N.') aH(J,XY
end h]&GLb&<?
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if any( r>1 | r<0 ) bn&TF3b
error('zernfun:Rlessthan1','All R must be between 0 and 1.') #<"~~2?
end %bn jgy
PCee<W_%YE
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) dh\'<|\K
error('zernfun:RTHvector','R and THETA must be vectors.') edq4D53
end CT<7mi!
wc@X.Q[
r = r(:); V*;(kEqj
theta = theta(:); St9?RD{4;
length_r = length(r); #pow ub
if length_r~=length(theta) 9Q^r
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error('zernfun:RTHlength', ... B2vh-%63
'The number of R- and THETA-values must be equal.') |Pax =oJ\M
end \ A#41
WM$
MPs
% Check normalization: 2DDtu[}
% -------------------- T@B/xAq5!
if nargin==5 && ischar(nflag) OX0%C.K)hZ
isnorm = strcmpi(nflag,'norm'); vzAax k%
if ~isnorm oG?Xk%7&\
error('zernfun:normalization','Unrecognized normalization flag.') &vMb_;~B
end Y;M|D'y+
else !;v|' I
isnorm = false; hpX9[3
end ^ig' bw+WS
`UyG_;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x.6:<y
% Compute the Zernike Polynomials M#6W(|V/
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wH&!W~M
2 c{34:
% Determine the required powers of r: %3-y[f
% ----------------------------------- g}{aZ$sta
m_abs = abs(m); (NU
NHxi5B
rpowers = []; R4cM%l_#W
for j = 1:length(n) ]y'>=a|T
rpowers = [rpowers m_abs(j):2:n(j)]; ql{OETn#
end %)W2H^
rpowers = unique(rpowers); OX!tsARC@
D2eckLT
% Pre-compute the values of r raised to the required powers, D_*WYV
% and compile them in a matrix: _S1>j7RQo
% ----------------------------- 5coyr`7mP
if rpowers(1)==0 Y eo]]i{
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); dn+KH+v
rpowern = cat(2,rpowern{:}); \'D0'\:vz
rpowern = [ones(length_r,1) rpowern]; 5L%'@`mX
else t\,PB{P:J
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); =s2*H8]
rpowern = cat(2,rpowern{:}); ,!y$qVg'\f
end Y"aJur=`
S`0(*A[W*
% Compute the values of the polynomials: WPMSm<[
% -------------------------------------- oW*16>IN9l
y = zeros(length_r,length(n)); $|@@Qk/T
for j = 1:length(n) +gtbcF@rx
s = 0:(n(j)-m_abs(j))/2; Id .nu/
pows = n(j):-2:m_abs(j); zKJ#`OhT
for k = length(s):-1:1 ]Ie 0S~
p = (1-2*mod(s(k),2))* ... v MH
prod(2:(n(j)-s(k)))/ ... "7F?@D$e
prod(2:s(k))/ ... 7'V@+5
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 3$>1FoSk
prod(2:((n(j)+m_abs(j))/2-s(k))); U$ElV]N
idx = (pows(k)==rpowers); ;))+>%SGCt
y(:,j) = y(:,j) + p*rpowern(:,idx); h2]P]@nW;W
end u?(d gJ
Vaw+.sG`AP
if isnorm 9vc2VB$
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); k9 I%PH
end G@X% +$I
end K;H&n1
% END: Compute the Zernike Polynomials +.FEq*V
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% L48_96
D8?Vn"
% Compute the Zernike functions: !``,gExH
% ------------------------------ {Gk1vcq
idx_pos = m>0; {]@= ijjf
idx_neg = m<0; '4Bm;&6M
KBc1{adDx@
z = y; >jLY"
if any(idx_pos) /%1ON9o>
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Vv=. -&'
end sBg.u
if any(idx_neg) xdt-
;w|
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); :{l_FY436
end z,p~z*4
G<J?"oQbRT
% EOF zernfun