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 _^ q\XPS
function z = zernfun(n,m,r,theta,nflag) 8VP"ydg-U
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. =9pw uH
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N l@N;sI<O-
% and angular frequency M, evaluated at positions (R,THETA) on the % Cu.u)/+
% unit circle. N is a vector of positive integers (including 0), and JAlU%n?R
% M is a vector with the same number of elements as N. Each element !8Z2X!$m{<
% k of M must be a positive integer, with possible values M(k) = -N(k) 6X7s 4
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, g@@&sB-A"
% and THETA is a vector of angles. R and THETA must have the same <Zp^lDxa
% length. The output Z is a matrix with one column for every (N,M) L6:W'u^
% pair, and one row for every (R,THETA) pair. i s L{9^
% [dj5$l|
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 4l&"]9D
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), E
&7@#'l
% with delta(m,0) the Kronecker delta, is chosen so that the integral {J~(#i
k
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, g4:VR:o
% and theta=0 to theta=2*pi) is unity. For the non-normalized M[aT2A
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 2wx!Lpr<i_
% xfq]9<
% The Zernike functions are an orthogonal basis on the unit circle. )Fqy%uR8
% They are used in disciplines such as astronomy, optics, and 5M%,N-P^
% optometry to describe functions on a circular domain. tu\mFHvlg
% iOT)0@f'
% The following table lists the first 15 Zernike functions. r^$\t0h(U8
% [kbC'Eh*
% n m Zernike function Normalization E'5Ajtw;
% -------------------------------------------------- 2Co@+I[,4&
% 0 0 1 1 3{N\A5~
% 1 1 r * cos(theta) 2 aje^Z=]
% 1 -1 r * sin(theta) 2 ?ork^4 $s
% 2 -2 r^2 * cos(2*theta) sqrt(6) [6D>f?z
% 2 0 (2*r^2 - 1) sqrt(3) J &!B|TS
% 2 2 r^2 * sin(2*theta) sqrt(6) u8Y~_)\MA
% 3 -3 r^3 * cos(3*theta) sqrt(8) dQ: ?<zZ
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Bvz62?
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) l8z%\p5cR
% 3 3 r^3 * sin(3*theta) sqrt(8) GDF{Lf)/v
% 4 -4 r^4 * cos(4*theta) sqrt(10) NQ?x8h3
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) NuU'0_")/
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) (NX)oP
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) R0%?:!
F
% 4 4 r^4 * sin(4*theta) sqrt(10) ]Ap`
% -------------------------------------------------- >DL/..
% 81Z4>F:
% Example 1: U.: sK*
% Fse['O~
% % Display the Zernike function Z(n=5,m=1) >):m-I
% x = -1:0.01:1; MDk*j,5V
% [X,Y] = meshgrid(x,x); Hk,lX r
% [theta,r] = cart2pol(X,Y); /Zc#j^_
% idx = r<=1; kLJlS,nh\r
% z = nan(size(X)); v"rl5x
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 3[VWTq)D=
% figure J' W}7r
% pcolor(x,x,z), shading interp @7-=zt+f
% axis square, colorbar $,TGP+vH
% title('Zernike function Z_5^1(r,\theta)') [FGgkd}
% O@s{uZ|A6
% Example 2: Yv^p=-E
% c4\C[$
% % Display the first 10 Zernike functions Jy9bY
% x = -1:0.01:1; R*087X7
N|
% [X,Y] = meshgrid(x,x); U
IfH*6X
% [theta,r] = cart2pol(X,Y); 2}w#3K
% idx = r<=1; <
kz[:n:
% z = nan(size(X)); q/$GE,"
% n = [0 1 1 2 2 2 3 3 3 3]; be7L="vZw
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; IV0[!D
% Nplot = [4 10 12 16 18 20 22 24 26 28]; X(]Zr
% y = zernfun(n,m,r(idx),theta(idx)); (#$$nQj
% figure('Units','normalized') Ox^:)ii
% for k = 1:10 ibXe"X/_
% z(idx) = y(:,k); =+<d1W`>0
% subplot(4,7,Nplot(k)) [ByQ;s5tY
% pcolor(x,x,z), shading interp [(|^O>k8c
% set(gca,'XTick',[],'YTick',[]) \^&
% axis square 34ha26\np
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ~Q?!W0ZBE
% end A[`G^$
% ZHCr2^w6
% See also ZERNPOL, ZERNFUN2. .5.8;/
/
~].ggcl`w
% Paul Fricker 11/13/2006 g` [` P@
>Q=Ukn;k
!2$ z *C2;
% Check and prepare the inputs: dx@QWTNE
% ----------------------------- Cp^g'&
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) P?
(vW&B
error('zernfun:NMvectors','N and M must be vectors.') H8f]}
end 'z5h3J
L,?/'!xV
if length(n)~=length(m) $w)~xE5;
error('zernfun:NMlength','N and M must be the same length.') .%'Z~|K4
end {oUAP1V^
R-
n = n(:); X\\7$
m = m(:); %v{1#~u
if any(mod(n-m,2)) rQJ"&CapT
error('zernfun:NMmultiplesof2', ... T 6Ctf#
'All N and M must differ by multiples of 2 (including 0).') R{?vQsLk
end >.<ooWw
\~#WY5
if any(m>n) +}aC-&
error('zernfun:MlessthanN', ... B[F-gq-
'Each M must be less than or equal to its corresponding N.') X3wX`V}
end {U"^UuU]
__I/F6{ 9V
if any( r>1 | r<0 ) nNaXp*J
error('zernfun:Rlessthan1','All R must be between 0 and 1.') HI`q1m.
end C!&y
\4{2eU
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) jQ=~g-y
error('zernfun:RTHvector','R and THETA must be vectors.') inAAgW#s}
end !tXZ%BP.u
~e">_;k6
r = r(:); d-B7["z,
theta = theta(:); q'G,!];qL
length_r = length(r); xx)-d,S
if length_r~=length(theta) \.#p_U5In
error('zernfun:RTHlength', ... +}@8p[`)
'The number of R- and THETA-values must be equal.') h2w}wsb0l
end {v` 2sB
hoQ7).>
% Check normalization: S1J<9xqSQ8
% -------------------- @hif$
if nargin==5 && ischar(nflag) 4woO;Gm
isnorm = strcmpi(nflag,'norm'); lA^+Flh
if ~isnorm 1J}8sG2`
error('zernfun:normalization','Unrecognized normalization flag.') `f9gC3Hk
end 2p!"p`b~
else wX,F`e3"/
isnorm = false; %gd(wzco
end vq!uD!lr
&:5\"b
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% u~1o(Zn
=
% Compute the Zernike Polynomials 7&B$HZ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% z@Hp,|Vy[
|Au ]1}
% Determine the required powers of r: E9}{1A
% ----------------------------------- "TS
m_abs = abs(m); '+Xlw
rpowers = []; a9U_ug58
for j = 1:length(n) 'ZP)cI:+X
rpowers = [rpowers m_abs(j):2:n(j)]; ;V5yXNQ
end Vj?DA5W`'
rpowers = unique(rpowers); 5x8+xw3Eh
#1.YKo
% Pre-compute the values of r raised to the required powers, {ZsdLF#
% and compile them in a matrix: T=Z.TG|lIx
% ----------------------------- k`{7}zxS
if rpowers(1)==0 Wu1{[a|
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); MJ{%4S{K,p
rpowern = cat(2,rpowern{:}); XORk!m|
rpowern = [ones(length_r,1) rpowern]; ^U[D4UM
else ut2~rRiK
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); !j:`7PT\
rpowern = cat(2,rpowern{:}); As&vFt P
end OJAIaC\
o@bNpflb`
% Compute the values of the polynomials: 1|r,dE2k9
% -------------------------------------- LiQgR
6j
y = zeros(length_r,length(n)); xiblPF_n3
for j = 1:length(n) I=DxRgt
s = 0:(n(j)-m_abs(j))/2; zj{r^D$
pows = n(j):-2:m_abs(j); XT>.`, sv
for k = length(s):-1:1 g\SrO {*
p = (1-2*mod(s(k),2))* ... _<c$)1
prod(2:(n(j)-s(k)))/ ... Cq)IayD@
prod(2:s(k))/ ... 4qi[r)G
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 6NWn(pZ]p
prod(2:((n(j)+m_abs(j))/2-s(k))); rQ`i8GF
idx = (pows(k)==rpowers); 5Por "&%
y(:,j) = y(:,j) + p*rpowern(:,idx); a>O9pX
end N_pUv
Ev"|FTI/
if isnorm nC1zzFFJ
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); <^?1uzxH8A
end yp.[HMRD
end mEyK1h1G@
% END: Compute the Zernike Polynomials LUX*P7*B
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% y
!$alE
*Lqg=9kzr
% Compute the Zernike functions: KJ2Pb"s
% ------------------------------ $Fkaa<9;P
idx_pos = m>0; (6l+lru[
idx_neg = m<0; nrm+z"7
NEt1[2X%
z = y; Fs_]RfG
if any(idx_pos) %UUH"
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); z!;1i[|x
end L|-98]8>
if any(idx_neg) c9qR'2
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); mm[2wfTE
end G;NF5`*4mc
b$%0.s
% EOF zernfun