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 .L7Yf+yFg
function z = zernfun(n,m,r,theta,nflag) sQ}%7BMK
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. j\'+wVyo
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N W 9Vz[
% and angular frequency M, evaluated at positions (R,THETA) on the LR3`=Z9
% unit circle. N is a vector of positive integers (including 0), and X#DL/#z k
% M is a vector with the same number of elements as N. Each element nFe` <Al$N
% k of M must be a positive integer, with possible values M(k) = -N(k) h zZ-$IX X
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ~J1;tZS
% and THETA is a vector of angles. R and THETA must have the same 8nIMZV
% length. The output Z is a matrix with one column for every (N,M) G7Z vfLR{:
% pair, and one row for every (R,THETA) pair. 1a&/Zlr
% .6#cDrK
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 6KEykw
j
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), y98JiNq
% with delta(m,0) the Kronecker delta, is chosen so that the integral [O7w =
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 2"leUur~rO
% and theta=0 to theta=2*pi) is unity. For the non-normalized 19F ;oFp
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ut4r~~Ar
% }A1|jY)x
% The Zernike functions are an orthogonal basis on the unit circle. Yz=h"Zr
% They are used in disciplines such as astronomy, optics, and u3Usq=Ij{
% optometry to describe functions on a circular domain. "mPSA Z
% w dGpt_
% The following table lists the first 15 Zernike functions. '7Mep
]
% 7deAr$?Wx
% n m Zernike function Normalization 7`IUMYl#~
% -------------------------------------------------- AozmO
% 0 0 1 1 1mHwYT+
% 1 1 r * cos(theta) 2 ;Y'8:ncDn
% 1 -1 r * sin(theta) 2 GS
;HtUQ
% 2 -2 r^2 * cos(2*theta) sqrt(6) 7~wFU*P1
% 2 0 (2*r^2 - 1) sqrt(3) s~=KhP~
% 2 2 r^2 * sin(2*theta) sqrt(6) R2}kz.
% 3 -3 r^3 * cos(3*theta) sqrt(8) ]<27Sw&yaG
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) EI1W
.V>@
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 5/B#) gm
% 3 3 r^3 * sin(3*theta) sqrt(8) K,f* SXM
% 4 -4 r^4 * cos(4*theta) sqrt(10) h@*lWi2K7
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) N[qA2+e$Z
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) VK2@2`$
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) { p1lae
% 4 4 r^4 * sin(4*theta) sqrt(10) nJFk4v4:2
% -------------------------------------------------- {y,nFxLq
% (U|)xA]y!
% Example 1: (M ]XNn
% Mv.Ciyc
% % Display the Zernike function Z(n=5,m=1) 6xH;:B)d
% x = -1:0.01:1; j4;Du>obQ
% [X,Y] = meshgrid(x,x); Ci~f#{
% [theta,r] = cart2pol(X,Y); }m6f^fs}
% idx = r<=1; O(VxMO
% z = nan(size(X)); (y1$MYZQ
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 9s!
2 wwh
% figure ]SFWt/<
% pcolor(x,x,z), shading interp HSNOL
% axis square, colorbar JOBz{;:R{
% title('Zernike function Z_5^1(r,\theta)') m8'@UzB
% WgE@8 9
% Example 2: 807al^s
x
% sffhPX\I
% % Display the first 10 Zernike functions jm+ V$YBP
% x = -1:0.01:1; ?4^};wDb2
% [X,Y] = meshgrid(x,x); N99[.mErU
% [theta,r] = cart2pol(X,Y); 0|g[o:;fl_
% idx = r<=1; :'Zx{F`
% z = nan(size(X)); {'NBp0i
% n = [0 1 1 2 2 2 3 3 3 3]; ?RHn @$g8M
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; WFouoXlG0
% Nplot = [4 10 12 16 18 20 22 24 26 28]; tKwn~T
% y = zernfun(n,m,r(idx),theta(idx)); rwy+~
% figure('Units','normalized') Qh*)pt]n
% for k = 1:10 Nepi|{
% z(idx) = y(:,k); ^f9>l;Lb
% subplot(4,7,Nplot(k)) 5J
ySFG3
% pcolor(x,x,z), shading interp ton1oq
% set(gca,'XTick',[],'YTick',[]) 4S tjj!ew
% axis square Z a!
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% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) sa*g
% end oz LH ]*
% E Zi &]
% See also ZERNPOL, ZERNFUN2. j !`B'{cH
t<Ot|Ex
% Paul Fricker 11/13/2006 GQb i$kl
vm8$:W2 }
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% Check and prepare the inputs:
JnY$fs*"
% ----------------------------- P>(&glr|
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Qlw>+y-i
error('zernfun:NMvectors','N and M must be vectors.') >z(wf>2J
end K4:
$=
,]ga[
if length(n)~=length(m) S#tY@h@XV
error('zernfun:NMlength','N and M must be the same length.') ;+a2\j+
end gljo;f:
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n = n(:); z`4c 4h]I
m = m(:); p}uncIod
if any(mod(n-m,2)) 6#U^<`
error('zernfun:NMmultiplesof2', ... e4DMO*6
'All N and M must differ by multiples of 2 (including 0).') #AShbl jm+
end V C-d0E0
5MR,UgT
if any(m>n) M%I@<~wl
error('zernfun:MlessthanN', ... b?8)7.{F{
'Each M must be less than or equal to its corresponding N.') R:M,tL-l
end U6<M/>RG$
X d3}Vn=
if any( r>1 | r<0 ) 49AW6H.JT
error('zernfun:Rlessthan1','All R must be between 0 and 1.') c+g@Z"es
end ##cnFQCB
1yMr~Fo
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 4jX3lq|
error('zernfun:RTHvector','R and THETA must be vectors.') {WQq}-(
end $5NKFJc
gv|"OlB
r = r(:); Od##U6e`
theta = theta(:); ! \sMR
length_r = length(r); zU&L.+
if length_r~=length(theta) "u492^
error('zernfun:RTHlength', ... |
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'The number of R- and THETA-values must be equal.') ; b*i3*!g
end :5b0np!
X:|8vS+0gU
% Check normalization: $=)gpPT
% -------------------- O6X"RsI}
if nargin==5 && ischar(nflag) B$XwTJ>
isnorm = strcmpi(nflag,'norm'); >P=Q #;v
if ~isnorm "g0(I8
error('zernfun:normalization','Unrecognized normalization flag.') T.ML$"f
end !Ms[eB
else pDl3!m
isnorm = false; F9a^ED0l\
end D d,2;#_
*2e!M^K<
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |ZiC`Nt
% Compute the Zernike Polynomials e#S0Fk)z
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% l63hLz
b?T
% Determine the required powers of r: H,y4`p 0
% ----------------------------------- }Wh6zT)
m_abs = abs(m); =r9r~SR#
rpowers = []; &%mXYj3y5
for j = 1:length(n) t e,[f
rpowers = [rpowers m_abs(j):2:n(j)]; gE])!GMM3
end k~.&j"K
rpowers = unique(rpowers); k|xtr&1N.!
Ba'LRz
% Pre-compute the values of r raised to the required powers, ~ G6"3"
% and compile them in a matrix: 2=iH$v
% ----------------------------- ._PzYE|m2
if rpowers(1)==0 ?LK 2g
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 1:M@&1LYp
rpowern = cat(2,rpowern{:}); U;q];e:,=}
rpowern = [ones(length_r,1) rpowern]; AUe# RP
else 5d\q-d
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ~Z'w)!h
rpowern = cat(2,rpowern{:}); 8|%^3O 0X
end ~j9O$s~)
j+-P :xvP
% Compute the values of the polynomials: .2)
=vf'd
% -------------------------------------- bm% $86
y = zeros(length_r,length(n)); /JkC+7H4
for j = 1:length(n) /Q{P3:k
s = 0:(n(j)-m_abs(j))/2; m P'^%TE
pows = n(j):-2:m_abs(j); !\Xm!I8
for k = length(s):-1:1 L+}n@B
p = (1-2*mod(s(k),2))* ... Pr ]Ka
prod(2:(n(j)-s(k)))/ ... -E"GX
prod(2:s(k))/ ... H1n1-!%d
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Bun><Y
@
prod(2:((n(j)+m_abs(j))/2-s(k))); /FP5`:PfL
idx = (pows(k)==rpowers); n\z,/'d"
y(:,j) = y(:,j) + p*rpowern(:,idx); @&|l^ 1
end :GpDg
d5 7i)=
if isnorm A][fLlpr
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); e[_m<e
end sZGj"_-Hzu
end <Z}SKR"U%
% END: Compute the Zernike Polynomials uvP2Wgt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% D,qu-k[jMI
3psU?8(
% Compute the Zernike functions: 29CINC
% ------------------------------ \^7C0R-hX
idx_pos = m>0; +l3=3
idx_neg = m<0; @c9^q>Uv
D^%^xq)E
z = y; *}k;L74|
if any(idx_pos) YW u cvw&
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); p~HW5\4
end JM1R ;i6
if any(idx_neg) t58e(dgi
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); b306&ZVEk
end HK|ynBAo
WOuEW w=
% EOF zernfun