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 ``\i58K{e
function z = zernfun(n,m,r,theta,nflag) dS!:JO27
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. JJ2_hVU
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ]<rkxgMW>
% and angular frequency M, evaluated at positions (R,THETA) on the MWpQ^dL_
% unit circle. N is a vector of positive integers (including 0), and $ig0j`
% M is a vector with the same number of elements as N. Each element %Iv,@}kvT+
% k of M must be a positive integer, with possible values M(k) = -N(k) g<f <Ip=
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, $r8 ^0ZRr
% and THETA is a vector of angles. R and THETA must have the same dj7hx"BI
% length. The output Z is a matrix with one column for every (N,M) IIF]/Ek]
% pair, and one row for every (R,THETA) pair. Et/\xL
% h!.^?NF
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike q?DTMKx
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ^l=!JP=M=
% with delta(m,0) the Kronecker delta, is chosen so that the integral [] `&vWZ
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, b#toM';T
% and theta=0 to theta=2*pi) is unity. For the non-normalized C=)A6
;=se
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. e .2ib?8
% #_J@-f7^
% The Zernike functions are an orthogonal basis on the unit circle. ?DQsc9y
% They are used in disciplines such as astronomy, optics, and A1D^a,
% optometry to describe functions on a circular domain. nvJf/90$
% Ix!Iw[CNd
% The following table lists the first 15 Zernike functions. `c5"d
% s{S4J'VW
% n m Zernike function Normalization >eqxV|]i
% -------------------------------------------------- ^*8G8'k;$
% 0 0 1 1 n}_JB>i~
% 1 1 r * cos(theta) 2 2w_W Adi
% 1 -1 r * sin(theta) 2 dzsmIV+
% 2 -2 r^2 * cos(2*theta) sqrt(6) o+QE8H43
% 2 0 (2*r^2 - 1) sqrt(3) uK$9Ll{lk
% 2 2 r^2 * sin(2*theta) sqrt(6) ,)Ju [
% 3 -3 r^3 * cos(3*theta) sqrt(8) 1#*a:F&re
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) D2!X?"[P
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Y*>#T
% 3 3 r^3 * sin(3*theta) sqrt(8) =/Mq 5.
% 4 -4 r^4 * cos(4*theta) sqrt(10) s'a/j)^
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) t2"O
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) G_{&sa
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) C8e
!H
% 4 4 r^4 * sin(4*theta) sqrt(10) D 38$`j
% -------------------------------------------------- EB=-H#
% TI#''XCB5
% Example 1: "B4;,+4kR
% =Z+nz^'b
% % Display the Zernike function Z(n=5,m=1) V_RTI.3p
% x = -1:0.01:1; #Jn_c0
% [X,Y] = meshgrid(x,x); *-q"3D`
% [theta,r] = cart2pol(X,Y); i;jw\ed
% idx = r<=1; OK6]e3UO
% z = nan(size(X)); =aj/,Q]
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 8lb%eb]U
% figure W?aI|U1
% pcolor(x,x,z), shading interp pUq1|)g
% axis square, colorbar ,M6Sy]Aj
% title('Zernike function Z_5^1(r,\theta)') (
Qcp{q
% O<"}|nbmQ[
% Example 2: wsN?[=l{s
% Bck7\
% % Display the first 10 Zernike functions #u"k~La
% x = -1:0.01:1; m&\h4$[kql
% [X,Y] = meshgrid(x,x); f3{MvAy[
% [theta,r] = cart2pol(X,Y); =p?WBZT|:
% idx = r<=1; SWQ5fcPu
% z = nan(size(X)); 6s\Kt3=
% n = [0 1 1 2 2 2 3 3 3 3]; W#BM(I
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; @,u/w4
% Nplot = [4 10 12 16 18 20 22 24 26 28]; \B 0ywN?
% y = zernfun(n,m,r(idx),theta(idx)); PSVc+s[Q+V
% figure('Units','normalized') 1_
C]*p
% for k = 1:10 Y&_&s7z
% z(idx) = y(:,k); 6290ZNvr
% subplot(4,7,Nplot(k)) J-)
XQDD
% pcolor(x,x,z), shading interp A~+S1
% set(gca,'XTick',[],'YTick',[]) 2fS[J'-o
% axis square 1~ t{aLPz
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) >teOm?@U
% end IlE_@gS8
% @@rEs40
% See also ZERNPOL, ZERNFUN2. pT1[<X!<s
.YnFH$;$
% Paul Fricker 11/13/2006 psC
mbN
\eb|eN0i
MpqZH{:?G
% Check and prepare the inputs: S.Ma$KL~'^
% ----------------------------- :ORR_f`>
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) .G(llA}
error('zernfun:NMvectors','N and M must be vectors.') )+"'oY$]}
end Ru>uL@w
nJ"YIT1K]p
if length(n)~=length(m) HJ[/|NZU$
error('zernfun:NMlength','N and M must be the same length.') _uKZ Ml
end d,tU#N{Q6
!F4@KAv
n = n(:); n?ctLbg
m = m(:); {^rs#, W
if any(mod(n-m,2)) 7 aYn0_NKp
error('zernfun:NMmultiplesof2', ... a/U2xq{x
'All N and M must differ by multiples of 2 (including 0).') -,aeM~
end hf<^/@^tK
;?~$h-9)
if any(m>n) >'xGp7}y
error('zernfun:MlessthanN', ... ND,Kldji
'Each M must be less than or equal to its corresponding N.') 5"]~oPK
end 8kOKwEX
EVUq--)~
if any( r>1 | r<0 ) {
"xln/
error('zernfun:Rlessthan1','All R must be between 0 and 1.') X3:XTuV
end c8M2 ^{O,`
qdG~!h7j
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) L9ap(
error('zernfun:RTHvector','R and THETA must be vectors.') P^Q[-e{
end 2Nm>5l
m6yIR6H
r = r(:); OxtOd\0$
theta = theta(:); d:q +
length_r = length(r); s/e"'Hz
if length_r~=length(theta) xc:!cA{V
error('zernfun:RTHlength', ... 9F-
)r'
'The number of R- and THETA-values must be equal.') ai^4'{#zi
end Hb(B?!M)
_l],
"[d
% Check normalization: u=NSsTP&
% -------------------- /.eeO k
if nargin==5 && ischar(nflag) \[>9UC%
isnorm = strcmpi(nflag,'norm'); =GBI0&U
if ~isnorm `L5~mb;7*
error('zernfun:normalization','Unrecognized normalization flag.') H,<7G;FPT
end -/dEsgO
else xwZ8D<e-,
isnorm = false; vF/ =J
end ia{c
Btd Xv4V
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Q U
F$@)A
% Compute the Zernike Polynomials ]G}B 0u3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Xvok1NM,
\#x}q'BC4
% Determine the required powers of r: qxJQPz
% ----------------------------------- eL.7#SIr}
m_abs = abs(m); pA#}-S%
rpowers = []; Dli^2hD
for j = 1:length(n) O^I[
(8Y8
rpowers = [rpowers m_abs(j):2:n(j)]; "4j:[9vR\
end wVA|!>v
rpowers = unique(rpowers); a>B[5I5
qy!Ou3^
% Pre-compute the values of r raised to the required powers, >(tn "2
% and compile them in a matrix: '7B"(dA&C
% ----------------------------- zN_:nY>
if rpowers(1)==0 oXt,e
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @TA9V@?)
rpowern = cat(2,rpowern{:}); 7C?.L70ZY
rpowern = [ones(length_r,1) rpowern]; l??;3kh1
else ,rwuy[Q8
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )I@L+
rpowern = cat(2,rpowern{:}); I#FF*@oeM
end }98>5%Uv
VnJMmMM
% Compute the values of the polynomials: i"^<CR@e
% -------------------------------------- y466A]|
y = zeros(length_r,length(n)); A~{f/%8D
for j = 1:length(n) :H[\;Z1_
s = 0:(n(j)-m_abs(j))/2; <<|H=![
pows = n(j):-2:m_abs(j); )06iV
for k = length(s):-1:1 w.+Eyu_I\
p = (1-2*mod(s(k),2))* ... lZt(&^T
prod(2:(n(j)-s(k)))/ ... 6j8<Q 2
prod(2:s(k))/ ... ;ggy5?>Qu
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... tllBCuAe
prod(2:((n(j)+m_abs(j))/2-s(k))); bYh9sO/l
idx = (pows(k)==rpowers); g.#+z'l
y(:,j) = y(:,j) + p*rpowern(:,idx); -05U%l1e
end {lz G*4?
a%J6f$A#
if isnorm 9 K
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); vh>{_
#
end C@HD(..#
end NyI;v=
% END: Compute the Zernike Polynomials ZAg;q#z j
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% L]2<&%N2
KLt%[$CTi
% Compute the Zernike functions: WY)^1Gb$ux
% ------------------------------ N^elVu4 K
idx_pos = m>0; ~j,TVY
idx_neg = m<0; G\Q9IcJ0dY
K:qOoY
z = y; n*qN29sx
if any(idx_pos) mR":z|6
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); de-0?6
end 3BMS_,P
if any(idx_neg) DB&SOe
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); j+748QAhh
end n2;9geq+
`.k5v7!o
% EOF zernfun