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 xBw ua;
function z = zernfun(n,m,r,theta,nflag) 8jLO-^X<<
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. '=(yh{W
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N `sRys oW
% and angular frequency M, evaluated at positions (R,THETA) on the -*?{/QmKb
% unit circle. N is a vector of positive integers (including 0), and [E}pU8.t6
% M is a vector with the same number of elements as N. Each element Pb@$RAU63
% k of M must be a positive integer, with possible values M(k) = -N(k) {gDoktC@M
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ZQ_~
L!ot
% and THETA is a vector of angles. R and THETA must have the same q'biTn]2
% length. The output Z is a matrix with one column for every (N,M) lx82:_
% pair, and one row for every (R,THETA) pair. |FFMQ"
% V0F1X s`
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 1py>[II@
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ty9(mtH+
% with delta(m,0) the Kronecker delta, is chosen so that the integral n0^3F1Z
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ^c sOXP=Yp
% and theta=0 to theta=2*pi) is unity. For the non-normalized C$v
!emu
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Mt121Q&"
% C\cZ
% The Zernike functions are an orthogonal basis on the unit circle. )L,Nh~
% They are used in disciplines such as astronomy, optics, and K*j1Fy:
% optometry to describe functions on a circular domain. /"1[qT\F
% e#tWQM3
% The following table lists the first 15 Zernike functions. #Z_f/@b
% p!K]c D
% n m Zernike function Normalization ~~WX#Od*$
% -------------------------------------------------- 7{=+Va5
% 0 0 1 1 6~8dMy;w
% 1 1 r * cos(theta) 2 :Ui'x8yt
% 1 -1 r * sin(theta) 2 Lez]{%+.`[
% 2 -2 r^2 * cos(2*theta) sqrt(6) `
B+Pl6l)F
% 2 0 (2*r^2 - 1) sqrt(3) \&Oc}]
% 2 2 r^2 * sin(2*theta) sqrt(6) E0Kt4%b
% 3 -3 r^3 * cos(3*theta) sqrt(8) Jqt|'G3
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ]5eZLXM
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) T\T>\&nY+|
% 3 3 r^3 * sin(3*theta) sqrt(8) qNbgN{4
% 4 -4 r^4 * cos(4*theta) sqrt(10) FOX0
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) L0xh?B
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) d1d:5b
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) LO,k'gg<
% 4 4 r^4 * sin(4*theta) sqrt(10) )7N$lY<
% -------------------------------------------------- Xm.["&
% [\ppK C
% Example 1: (_~Dyvo
% =Xb:.
% % Display the Zernike function Z(n=5,m=1) v;R+{K87
% x = -1:0.01:1; ,#80`&\%
% [X,Y] = meshgrid(x,x); brt`oR
% [theta,r] = cart2pol(X,Y); p!cNn7{;
% idx = r<=1; jX91=78d
% z = nan(size(X)); =xHzhh
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 4:XVu
% figure ;8<lgZ9H<
% pcolor(x,x,z), shading interp #K[6Ai=We}
% axis square, colorbar Kdb:Q0B
% title('Zernike function Z_5^1(r,\theta)') @LDu08lr
% ~2U5Wt
% Example 2: ltG|#(
% g6<D 1r
% % Display the first 10 Zernike functions n'Z5rXg
% x = -1:0.01:1; i.>d#S
% [X,Y] = meshgrid(x,x); >`.$Tyw
% [theta,r] = cart2pol(X,Y); EoHrXv
% idx = r<=1; IgtTYxI
% z = nan(size(X)); fhQ}Z%$
% n = [0 1 1 2 2 2 3 3 3 3]; G!m;J8#m(
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; *Y9' tHI
% Nplot = [4 10 12 16 18 20 22 24 26 28]; L)/^%/!
% y = zernfun(n,m,r(idx),theta(idx)); >WW5;7$
% figure('Units','normalized') P}bw Ej
% for k = 1:10 ;"DI)hdz
% z(idx) = y(:,k); *6P)HU@
% subplot(4,7,Nplot(k)) H}&4#CQ'!
% pcolor(x,x,z), shading interp RB/;qdqR
% set(gca,'XTick',[],'YTick',[]) a6.0$'
% axis square '9q:gFO
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) {,CvWL
% end 6I$:mHEhd
% GxcW^{;
% See also ZERNPOL, ZERNFUN2. ?$rHyI
m^ [VM&%
% Paul Fricker 11/13/2006 5NAB^&{Z<X
5QJFNE
#_[W*-|L
% Check and prepare the inputs: YXjWk),
% ----------------------------- Z?tw#n[T
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) d7Devs
k
error('zernfun:NMvectors','N and M must be vectors.') ^B7C8YP
end >qjV(_?F-
e`D? x1-
if length(n)~=length(m) j+>&~
error('zernfun:NMlength','N and M must be the same length.') AwO'%+Bv
end lC(g&(\{
K
yFR;.F-
n = n(:); (J/!9NS:
m = m(:); G .k\N(l
if any(mod(n-m,2)) Z:s:NvFX
error('zernfun:NMmultiplesof2', ... 9\D 0mjn=l
'All N and M must differ by multiples of 2 (including 0).') b_j8g{/9
end @je vY81)
2w? 5vSv
if any(m>n) \Z ms
error('zernfun:MlessthanN', ... Di8;Tq
'Each M must be less than or equal to its corresponding N.') ^5d9n<_xnQ
end _Zs]za.#)|
U/I+A|S[
if any( r>1 | r<0 ) sz+Uq]Mn
error('zernfun:Rlessthan1','All R must be between 0 and 1.') JqYt^,,Q:
end Ks-aJ+}
)!(etB=`y
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Q.Ljz
Z
error('zernfun:RTHvector','R and THETA must be vectors.') O:3DIT1#>
end 8cyC\Rs
o|0QstSCl
r = r(:); K~JXP5`(
theta = theta(:); N`%f+eT(
length_r = length(r); 0al8%z9e@
if length_r~=length(theta) [v$NxmRu
error('zernfun:RTHlength', ... +4%:q~C
'The number of R- and THETA-values must be equal.') Jf=$h20x
end eEG]JH
6C|]Fm
% Check normalization: *=ymK*
% -------------------- &k2nt
if nargin==5 && ischar(nflag) =q-HR+
isnorm = strcmpi(nflag,'norm'); k_<8SG+`
if ~isnorm hu+% X.F4
error('zernfun:normalization','Unrecognized normalization flag.') pe1 _E
KU
end oPA
[vY
else 19t'
isnorm = false; vz_ZXy9Z
end `F<[\@\d5
.xp|w^
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P7iU_CgyW
% Compute the Zernike Polynomials JKsdPW<?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;c_pa0L
"gPAxt
% Determine the required powers of r: op%?V:
% ----------------------------------- ]XH}G9X^
m_abs = abs(m); zUhJr$N$
rpowers = []; 1#3 Qa{i
for j = 1:length(n) S(f V ,;Z
rpowers = [rpowers m_abs(j):2:n(j)]; =
5E:C P
end 4{r_EV[(
rpowers = unique(rpowers); a~-^$Fzgy
I2wT]L UV
% Pre-compute the values of r raised to the required powers, f1RfNiW.
% and compile them in a matrix: xf.2Ig
% ----------------------------- wCb%{iowH
if rpowers(1)==0 fii\&p7z
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); +i[w& P
rpowern = cat(2,rpowern{:}); /B?hM&@z
rpowern = [ones(length_r,1) rpowern]; G
Riu]
else ymsqJ
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); [,|Z<
rpowern = cat(2,rpowern{:}); 92k}ON
end D?w-uR%Y
%TJF+;
% Compute the values of the polynomials: DjT ekn
% -------------------------------------- ;')T}wuq
y = zeros(length_r,length(n)); \JLiA>@@
for j = 1:length(n) LEJ7. 82
s = 0:(n(j)-m_abs(j))/2; ,Wp0,>!
pows = n(j):-2:m_abs(j); zq5_&AeW
for k = length(s):-1:1 Lz
VvUVk
p = (1-2*mod(s(k),2))* ... ,QpDz{8
prod(2:(n(j)-s(k)))/ ... sKX%<n$
prod(2:s(k))/ ... %V$ujun`
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... JAA P5ur
prod(2:((n(j)+m_abs(j))/2-s(k))); `f:5w^A
idx = (pows(k)==rpowers); C3 %, pDh
y(:,j) = y(:,j) + p*rpowern(:,idx); [^gSWU
end pr-{/6j6
JHf}LZu
if isnorm k*4?fr
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); (!';
end ?nFT51t/4
end pg~`NN
% END: Compute the Zernike Polynomials N[}XLhbt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #oYX0wvl
VmTk4?V4
% Compute the Zernike functions: \~
% ------------------------------ 'FBvAk6
idx_pos = m>0; )N-+,Ms
idx_neg = m<0; `.dTkL
,gU9ywg
z = y; n20H{TA
if any(idx_pos) e<^tY0rR&
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); <,0&Ox
end mId{f
if any(idx_neg) T^GdN_qF
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); "VWxHRVg4M
end e7L;{+XI
q9Y0Lk
% EOF zernfun