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 A5\ Hq
function z = zernfun(n,m,r,theta,nflag) MO| Dwuaf
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ?|Z~mE
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N g-ZXj4Ph!
% and angular frequency M, evaluated at positions (R,THETA) on the {,(iL8,^
% unit circle. N is a vector of positive integers (including 0), and q<^MC/]
% M is a vector with the same number of elements as N. Each element 6f
t6;*,
% k of M must be a positive integer, with possible values M(k) = -N(k) .!+7|us8l\
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, k}qCkm27
% and THETA is a vector of angles. R and THETA must have the same f<oU"WM
% length. The output Z is a matrix with one column for every (N,M) Brd9"M|d
% pair, and one row for every (R,THETA) pair. z TPNQ0=|
% 'R-g:X\{
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ^qVBg BPb
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), A@:U|)+4
% with delta(m,0) the Kronecker delta, is chosen so that the integral SjF(;0kC
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, |TQ4:P1T
% and theta=0 to theta=2*pi) is unity. For the non-normalized %<p/s;eu
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. YRv96|c,
% ^ rUq{
% The Zernike functions are an orthogonal basis on the unit circle. M0?%r`
% They are used in disciplines such as astronomy, optics, and CY*GCkH
% optometry to describe functions on a circular domain. [}l 90 lP
% s +qodb+
% The following table lists the first 15 Zernike functions. 8\C][ y
% +%WW8OX
% n m Zernike function Normalization (u='&ka
% -------------------------------------------------- ~4twI*f
% 0 0 1 1 .A_R6~::
% 1 1 r * cos(theta) 2 *XYp~b
% 1 -1 r * sin(theta) 2 9KJ}Ai
% 2 -2 r^2 * cos(2*theta) sqrt(6) =&Tuh}
% 2 0 (2*r^2 - 1) sqrt(3)
=}I=s@
% 2 2 r^2 * sin(2*theta) sqrt(6) 2 J3/Eu
% 3 -3 r^3 * cos(3*theta) sqrt(8) {Xr 9]g`
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) C(8!("tU
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 6hcK%0z
% 3 3 r^3 * sin(3*theta) sqrt(8) > sQ&5-i
% 4 -4 r^4 * cos(4*theta) sqrt(10) })?-)fFD
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) i\DU<lD5VN
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) >Y+m54EE
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,Jn` qvmi
% 4 4 r^4 * sin(4*theta) sqrt(10) qzO5p=}
% -------------------------------------------------- Y" rODk1
% JBZ1DZAWC
% Example 1: ~v:IgS
% ""_G4{
% % Display the Zernike function Z(n=5,m=1) @6aJh< c
% x = -1:0.01:1; \}Iq-Je
% [X,Y] = meshgrid(x,x); $A/?evJi8R
% [theta,r] = cart2pol(X,Y); OjG`s-91&
% idx = r<=1; F0r2=f(?
% z = nan(size(X)); R(8?9-w
% z(idx) = zernfun(5,1,r(idx),theta(idx)); m~P30)
% figure R9"}-A
% pcolor(x,x,z), shading interp I36%oA
% axis square, colorbar <%rm?;PBl
% title('Zernike function Z_5^1(r,\theta)') P&@,Z#\
% O,vC:av
% Example 2: yx*<c#Uf
% 0L ,!o[L*
% % Display the first 10 Zernike functions R7!v=X]i
% x = -1:0.01:1; nG{o$v_|
% [X,Y] = meshgrid(x,x); &N+`O)$
% [theta,r] = cart2pol(X,Y); CPeu="[
% idx = r<=1; oe3=QE
% z = nan(size(X));
]w$cqUhM
% n = [0 1 1 2 2 2 3 3 3 3]; 4sBvW
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; WiQVZ{
% Nplot = [4 10 12 16 18 20 22 24 26 28]; UWK|_RT6SA
% y = zernfun(n,m,r(idx),theta(idx)); 2+C:Em0yI
% figure('Units','normalized') L<B)BEE.
% for k = 1:10 z}Us+>z+jc
% z(idx) = y(:,k); gN73)uJ0
% subplot(4,7,Nplot(k)) P|p
X
F~
% pcolor(x,x,z), shading interp MA}}w&
% set(gca,'XTick',[],'YTick',[]) i3d2+N`
% axis square :O,r3O6
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 6X?:mn'%QF
% end G)M! ,
Q
% h+Yd
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% See also ZERNPOL, ZERNFUN2. ]>*VEe}hJ
v<<ATs%w
% Paul Fricker 11/13/2006 (\r^0>H
.jC5 y&
q@;1{
% Check and prepare the inputs: .}Ys+d1b9c
% ----------------------------- q4G$I?4
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) d<HO~+9
error('zernfun:NMvectors','N and M must be vectors.') V}7)>i$A
end qbCU&G|)
#a2Z.a<V
if length(n)~=length(m) >}2
,2
error('zernfun:NMlength','N and M must be the same length.') mO(Y>|mm
end j8PeO&n>
9}Z;(,6/.\
n = n(:); fE&s 6w&
m = m(:); mW+5I-~
if any(mod(n-m,2)) k'PvQl"I
error('zernfun:NMmultiplesof2', ... >H5t,FfQL
'All N and M must differ by multiples of 2 (including 0).') C]l)Pz$
end ;T8(byH ?
R#8cOmZ
if any(m>n) ) j&khHD
error('zernfun:MlessthanN', ... *QIYq
'Each M must be less than or equal to its corresponding N.') v6[VdWOx5
end \.p;
4V&
i_*.
if any( r>1 | r<0 ) @p}_"BHYWt
error('zernfun:Rlessthan1','All R must be between 0 and 1.')
B!8X?8D
end 1^V.L+0s]
[wiB1{/Ls.
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) "!7Hu7
error('zernfun:RTHvector','R and THETA must be vectors.') Ea'jAIFPpO
end GO@<?>K
55UPd#E'
r = r(:); BA@M>j6d
theta = theta(:); skTaIGRL
length_r = length(r); 5[r}'08b
if length_r~=length(theta) ~Cw7.NA{3
error('zernfun:RTHlength', ... 4,h)<(d{
'The number of R- and THETA-values must be equal.') )'e1@CR
end UJ%.KU%Q}
Ruq>+ }4
% Check normalization: +ZiYl[_|
% --------------------
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if nargin==5 && ischar(nflag) v6Y[_1
isnorm = strcmpi(nflag,'norm'); XeY[;}9
if ~isnorm `d4xX@
error('zernfun:normalization','Unrecognized normalization flag.') Q=vo5)t
end IR:{ { (
else 2@pEiq3
isnorm = false; P$N5j~*
end Mqk|H~l5c
* a1q M?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "lC>_A
% Compute the Zernike Polynomials F2_'U' a
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PVdN)tG5
9^N(s7s
% Determine the required powers of r: f}4A,%:1
% ----------------------------------- H.C*IL9
m_abs = abs(m); V?)V2>]
rpowers = []; w^ofH-R/
for j = 1:length(n) 4}cxSl]jf!
rpowers = [rpowers m_abs(j):2:n(j)]; !+z^VcV
end ips)-1
rpowers = unique(rpowers); f\q5{#"z
,L~aa?Nb-
% Pre-compute the values of r raised to the required powers, re#]zc<
% and compile them in a matrix: 5 $$Cav
% ----------------------------- 61&{I>~1
if rpowers(1)==0 Lc[TIX
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); G/fBeK$.
rpowern = cat(2,rpowern{:}); ;#IrHR*Bk
rpowern = [ones(length_r,1) rpowern]; K3h7gY| .
else G,^ ?qbHg
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); W?P4oKsql*
rpowern = cat(2,rpowern{:}); rUyGTe(@h
end k{b|w')
+%KkzdS'
% Compute the values of the polynomials: h)j#?\KYm9
% -------------------------------------- aK|
y = zeros(length_r,length(n)); tX1`/}``
for j = 1:length(n) V51kX{S
s = 0:(n(j)-m_abs(j))/2; 0`p"7!r
pows = n(j):-2:m_abs(j); )D'#>!Y
for k = length(s):-1:1 TvT>UBqj=
p = (1-2*mod(s(k),2))* ... Ex*{iJ;\
prod(2:(n(j)-s(k)))/ ... ;V?(j3b[
prod(2:s(k))/ ... 6@FhDj2X
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... }aXS MxCd
prod(2:((n(j)+m_abs(j))/2-s(k))); 4MW oGV9
idx = (pows(k)==rpowers); tQUKw@@Q
y(:,j) = y(:,j) + p*rpowern(:,idx); Otq1CD9
end KD+&5=Y
)1@%!fr
if isnorm (e!Yu#-
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Knb(MI6
end WS.g`%
end n<> ^cD
% END: Compute the Zernike Polynomials Fn4yx~0
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% T3"'`Sd9;
45<gO1
% Compute the Zernike functions: P0OMu/
% ------------------------------ t98S[Z(-%+
idx_pos = m>0; p W5D!z
idx_neg = m<0; ?Ov~\[) F
"zTy_0[;
z = y; hy%5LV<(
if any(idx_pos) &sBD0R(a
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); s?->2gxhx
end +|pYu<OY
if any(idx_neg) ,g*3u
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ~Jsu"kr
end l7VTuVGUJ
t>*(v#WeZ
% EOF zernfun