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 Fs&r^ [/b
function z = zernfun(n,m,r,theta,nflag) FaQc@4%o
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. @7K(_Wd
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N '
r/xBj[Z
% and angular frequency M, evaluated at positions (R,THETA) on the n50WHlMtt
% unit circle. N is a vector of positive integers (including 0), and N5. B"l
% M is a vector with the same number of elements as N. Each element uR6 `@F
% k of M must be a positive integer, with possible values M(k) = -N(k) ~3Y4_b5E
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, {A'_5 X9
% and THETA is a vector of angles. R and THETA must have the same ?z&5g-/b
% length. The output Z is a matrix with one column for every (N,M) w|c200Is}e
% pair, and one row for every (R,THETA) pair. S? #6{rx
% qKTzigjj
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 8kT`5`}lB
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), b_^y
Ke^W
% with delta(m,0) the Kronecker delta, is chosen so that the integral UCJx{7
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, oI-,6G}
% and theta=0 to theta=2*pi) is unity. For the non-normalized 33g$mUB
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. &O#,"u/q`
% 9e Fj+
% The Zernike functions are an orthogonal basis on the unit circle. ~z)JO'Z$
% They are used in disciplines such as astronomy, optics, and yxAy1P;dX
% optometry to describe functions on a circular domain. nF$HWp>
% 0+e
% The following table lists the first 15 Zernike functions. sE&1ZJ]7
% H$.K
% n m Zernike function Normalization e~7FK_y#0
% -------------------------------------------------- et?FX K"y
% 0 0 1 1 3S"
/l
% 1 1 r * cos(theta) 2 (eSsx/
% 1 -1 r * sin(theta) 2 8N \<o7t%
% 2 -2 r^2 * cos(2*theta) sqrt(6) ,Oe:SZJ>
% 2 0 (2*r^2 - 1) sqrt(3) inh
J|pe"
% 2 2 r^2 * sin(2*theta) sqrt(6) +lxjuEiae
% 3 -3 r^3 * cos(3*theta) sqrt(8) tAsap}(
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Jj?HOtaM
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) AEkjy h\
% 3 3 r^3 * sin(3*theta) sqrt(8) "6
~5RCZ
% 4 -4 r^4 * cos(4*theta) sqrt(10) W4UK?#S+
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 'q?Y5@s
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) S=\cF,Zs
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) <cU%yA710
% 4 4 r^4 * sin(4*theta) sqrt(10) zwz_K!229
% -------------------------------------------------- w!'y,yb%
% QiK-|hFj
% Example 1: -E~r?\;X
% tQas_K5
% % Display the Zernike function Z(n=5,m=1) @JGFG+J}
% x = -1:0.01:1; 5RAhm0Op~.
% [X,Y] = meshgrid(x,x); -K3d u&j
% [theta,r] = cart2pol(X,Y); YmOj.Q&
% idx = r<=1; fv k(eWB
% z = nan(size(X)); k||dX(gl
% z(idx) = zernfun(5,1,r(idx),theta(idx)); S`$%C=a.
% figure `mA;1S
% pcolor(x,x,z), shading interp i&?\Pp;5-j
% axis square, colorbar t<ZBp0
% title('Zernike function Z_5^1(r,\theta)') Lq;T\m_de
% fp*6Dv_
% Example 2: NGJst_
% b3F KDm[
% % Display the first 10 Zernike functions >]8(3&zd
% x = -1:0.01:1; +3J<vM}dy
% [X,Y] = meshgrid(x,x); tDRo)z
% [theta,r] = cart2pol(X,Y); 9!FU,4 X
% idx = r<=1; <bb!BS&w
% z = nan(size(X)); c@Br_-
% n = [0 1 1 2 2 2 3 3 3 3]; (~o"*1fk>
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; /QWXEL/M=
% Nplot = [4 10 12 16 18 20 22 24 26 28]; _VVq&t}
% y = zernfun(n,m,r(idx),theta(idx)); qS9<_if2
% figure('Units','normalized') `hdff0
% for k = 1:10 ;S
\s&. u
% z(idx) = y(:,k); :P/VBX h
% subplot(4,7,Nplot(k)) v?
OUd^
% pcolor(x,x,z), shading interp /Ry%K4$
% set(gca,'XTick',[],'YTick',[]) (qvH=VTwP
% axis square L9N}lH
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) i1XRBC9
% end tH4q*\U
% w^Yo)"6
% See also ZERNPOL, ZERNFUN2. 1ANFhl(l
URs]S~tk
% Paul Fricker 11/13/2006 }I-nT!D'y
&a=78Z
8lzoiA_9
% Check and prepare the inputs: 9TQVgkW
% ----------------------------- #-@Uq6Y
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) '(rD8 pc
error('zernfun:NMvectors','N and M must be vectors.') 1Acs0`3
end rhcax%Cd
VnVBA-#r|
if length(n)~=length(m) ]XbMqHGS
error('zernfun:NMlength','N and M must be the same length.') 3qn_9f ]
end l)*(UZ"
%~x?C4L8
n = n(:); } 6!/Nb
m = m(:); >mX6;6FF
if any(mod(n-m,2)) icIn>i<m
error('zernfun:NMmultiplesof2', ... ,}&TZkN{-
'All N and M must differ by multiples of 2 (including 0).') ?tL' X
end !u@P\8M}
pB\:.?.pd
if any(m>n) '/NpmNY:L
error('zernfun:MlessthanN', ... bj}Lxc ],
'Each M must be less than or equal to its corresponding N.') X!K> .r_Dg
end ""jW'%wR
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if any( r>1 | r<0 ) N|$9v{ j_
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ]t~.?)Ad+2
end S'8+jY
cI'n[G
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) \Q(a`6U
error('zernfun:RTHvector','R and THETA must be vectors.') P O 5Wi
end vRe X7
!5(DU~S*@S
r = r(:); hdCd:6
theta = theta(:); ]sqLGmUL
length_r = length(r); p|.5;)%|
if length_r~=length(theta) 4qp|g'uXT
error('zernfun:RTHlength', ... /uX*FZ
'The number of R- and THETA-values must be equal.') Y4HN1
end
j!>P7 8
E&zf<Y
% Check normalization: CTW\Dt5
% -------------------- Qgj# k
if nargin==5 && ischar(nflag) Ajm!;LA[jO
isnorm = strcmpi(nflag,'norm'); O&BvWik
if ~isnorm '0+~]4&}q
error('zernfun:normalization','Unrecognized normalization flag.') +4_, , I
end m..ajYSQ
else sdZ$3oE.
isnorm = false; K~vJ/9"|R
end DOJydYds
zplv.cf#q
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 88v8lt;R
% Compute the Zernike Polynomials 9GH5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s{Qae=$Q
[oVM9Q
% Determine the required powers of r: H5x7)1Ir|
% ----------------------------------- __'4Qt
m_abs = abs(m); ]"Uzn
rpowers = []; qIQ=OY=6
for j = 1:length(n) ih".y3
rpowers = [rpowers m_abs(j):2:n(j)]; xyL)'C
end B4RrUA32
rpowers = unique(rpowers); ]}!@'+=
G-T^1?
% Pre-compute the values of r raised to the required powers, ;7z6B|8
% and compile them in a matrix: ]nUr E6
% ----------------------------- C7ivAh
if rpowers(1)==0 {IJ;)<>&VE
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); %US&`BT!
rpowern = cat(2,rpowern{:}); ESRj<p%W
rpowern = [ones(length_r,1) rpowern]; aYaEy(m
else 9)1Ye
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); NSz}
rpowern = cat(2,rpowern{:}); VQHB}Y@^
end C*b[J
9Vm1q!lE
% Compute the values of the polynomials: sWo`dZ\6WB
% -------------------------------------- 5q0L<GOrj
y = zeros(length_r,length(n)); +_7a/3kh
for j = 1:length(n) _J!^iJ
s = 0:(n(j)-m_abs(j))/2; <3{MS],<<
pows = n(j):-2:m_abs(j); ~gd#cL%
for k = length(s):-1:1 Lmte ~oBi
p = (1-2*mod(s(k),2))* ... losqc *|
prod(2:(n(j)-s(k)))/ ... I@KM2KMN
prod(2:s(k))/ ... _eiqs
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 2/*u$~
prod(2:((n(j)+m_abs(j))/2-s(k))); w li cuY?
idx = (pows(k)==rpowers); 6h>#;M
y(:,j) = y(:,j) + p*rpowern(:,idx); B[@q.n
end SUUNC06V
+-@n}xb@
if isnorm RhE~Rwbx
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); |X8?B=
end 6]?%1HSi
end 1 jidBzu<
% END: Compute the Zernike Polynomials "sN%S's
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G{} 2"/
jjV'`Vy)
% Compute the Zernike functions: 754MQK|g
% ------------------------------ D!o[Sm}JO[
idx_pos = m>0; \ZLi Y
idx_neg = m<0; U*r54AyP
" !EnQB=
z = y; w[-)c6J yE
if any(idx_pos) <t"T'\3
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); LIcc0w3
end 5I2,za&e
if any(idx_neg) Gw<D'b)!
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); <==uK>pET
end TWp w/osW
n?@zp<
% EOF zernfun