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 wauM|/KG
function z = zernfun(n,m,r,theta,nflag) T[-Tqi NT
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Qnx?5R-}ZU
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 39x
4(
% and angular frequency M, evaluated at positions (R,THETA) on the '8LHX6FXK
% unit circle. N is a vector of positive integers (including 0), and d>0 j!+s
% M is a vector with the same number of elements as N. Each element @P">4xVX{
% k of M must be a positive integer, with possible values M(k) = -N(k) 55Xfu/hQ
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 8mC$p6Okd
% and THETA is a vector of angles. R and THETA must have the same Z ?ATWCa
% length. The output Z is a matrix with one column for every (N,M) (rQ)0g@
% pair, and one row for every (R,THETA) pair. >ktekO:H
% Icx)+Mq
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike (e32oP"
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), P!!:p2fo
% with delta(m,0) the Kronecker delta, is chosen so that the integral v?o("I[ C
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, M8VsU*aU
% and theta=0 to theta=2*pi) is unity. For the non-normalized !
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% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. !N/?b^y
% WV;[v g]
% The Zernike functions are an orthogonal basis on the unit circle. ]sqp^tQ`e
% They are used in disciplines such as astronomy, optics, and X=VaBy4#
% optometry to describe functions on a circular domain. %htbEKWR
% d 1 O+qS
% The following table lists the first 15 Zernike functions. _@Y17L.
% ^oEaE#I
% n m Zernike function Normalization ig'4DmNC
% -------------------------------------------------- w!R J8
% 0 0 1 1 5IP@_GV|
% 1 1 r * cos(theta) 2 .VkLF6
% 1 -1 r * sin(theta) 2 ^ lG^.
% 2 -2 r^2 * cos(2*theta) sqrt(6) YVO~0bX:
% 2 0 (2*r^2 - 1) sqrt(3) \r}*<CRr6
% 2 2 r^2 * sin(2*theta) sqrt(6) LufZ,
% 3 -3 r^3 * cos(3*theta) sqrt(8) KA."[dVa
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) RohD.`D
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) OKCX>'j:S
% 3 3 r^3 * sin(3*theta) sqrt(8) ROj=XM:+
% 4 -4 r^4 * cos(4*theta) sqrt(10) _2eL3xXha.
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )J&!>GP
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) P#2;1ki>
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) {\kDu#18Ld
% 4 4 r^4 * sin(4*theta) sqrt(10) y9Q"3LLic`
% -------------------------------------------------- `(L<Q%
% w&}UgtEm
% Example 1: !Op18hP$
% (z'!'?v;
% % Display the Zernike function Z(n=5,m=1) 5G#K)s(QC
% x = -1:0.01:1; 8;P_KRaE
% [X,Y] = meshgrid(x,x); p+R8Mo;I
% [theta,r] = cart2pol(X,Y); I`}x 9t
% idx = r<=1; dYhLk2
% z = nan(size(X)); LiD-su
D
% z(idx) = zernfun(5,1,r(idx),theta(idx)); hN_,Vyf
% figure yGPi9j{QXq
% pcolor(x,x,z), shading interp XXZ$^W&
% axis square, colorbar +isaqfy/
% title('Zernike function Z_5^1(r,\theta)') z(beT e
% 0"M0tA#
% Example 2: 'p(I!]"uo
% :=%`\\
% % Display the first 10 Zernike functions 3yIC@>&y(8
% x = -1:0.01:1; 0N3S@l#,\A
% [X,Y] = meshgrid(x,x); +luW=j0V
% [theta,r] = cart2pol(X,Y); bq`0$c%hN
% idx = r<=1; f%Bm x{Ttq
% z = nan(size(X)); (?zZvW8
% n = [0 1 1 2 2 2 3 3 3 3]; )IZ~!N|-w
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; x20sB
% Nplot = [4 10 12 16 18 20 22 24 26 28]; (`Q_^Bfyl
% y = zernfun(n,m,r(idx),theta(idx)); pi?U|&.1z
% figure('Units','normalized') L}%4YB
% for k = 1:10 K\>CXa
% z(idx) = y(:,k); Z= P=oldH
% subplot(4,7,Nplot(k)) NYZI;P1DA
% pcolor(x,x,z), shading interp 5VPP 2;J
% set(gca,'XTick',[],'YTick',[]) a0x/ ?)DO
% axis square cc$+"7/J^c
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ;u: }rA)
% end Fh$Xcz~i
% cX/["AM
% See also ZERNPOL, ZERNFUN2. ^aO\WKkA
a=3{UEi'o
% Paul Fricker 11/13/2006 (1b%);L7
FzGla} )
5%6r,?/7KM
% Check and prepare the inputs: !ZlNPPrq}
% ----------------------------- .% EEly
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) t Sf`
error('zernfun:NMvectors','N and M must be vectors.') B%Spmx8
end BpKgUwf;C
k"2xyzt*
if length(n)~=length(m) /.aDQ>
error('zernfun:NMlength','N and M must be the same length.') JMq00_
end O~AOZ^a:2
p#dpDjh
n = n(:); o$DJL11E
m = m(:); vMOit,{
if any(mod(n-m,2)) .v:K`y;f\(
error('zernfun:NMmultiplesof2', ... URD<KIN>
'All N and M must differ by multiples of 2 (including 0).') {?9s~{Dl
end pJE317 p'
\WVrn >%xu
if any(m>n) GlVD!0
error('zernfun:MlessthanN', ... <ctn_"p Z
'Each M must be less than or equal to its corresponding N.') glppb$oB\
end cHMS[.=;
m,U`hPJ
if any( r>1 | r<0 ) zk@KuBLL
error('zernfun:Rlessthan1','All R must be between 0 and 1.') {^#62Y
end <j.bG 7
3J{`]v5`
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) XK>/i}y
error('zernfun:RTHvector','R and THETA must be vectors.') t>T |\WAAL
end bG0t7~!{E
_KkLH\1g$
r = r(:); +`x8[A)-
theta = theta(:); O9k9hRE]z
length_r = length(r); 98os4}r
if length_r~=length(theta) r^k:$wJbRK
error('zernfun:RTHlength', ... ~o+HAc`=v
'The number of R- and THETA-values must be equal.') M"]~}*
end >]k'3|vV
'%`Wy@
% Check normalization: !#n lWX:~
% -------------------- rQbL86+
if nargin==5 && ischar(nflag) )-2o}KU]>
isnorm = strcmpi(nflag,'norm'); gHC -Y 0_
if ~isnorm wvm`JOP:A
error('zernfun:normalization','Unrecognized normalization flag.') $3sS&i<
end Q+[e)YO)
else tw]RH(g+#
isnorm = false; e1X*}OI
end "}]1OL S V
<m80e),~
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _1`*&k
JL~
% Compute the Zernike Polynomials DLkNL?a
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~3.1.
'A
*/n)_
% Determine the required powers of r: EW{z?/
% ----------------------------------- V$+xJ m
m_abs = abs(m); })|+tZ
rpowers = []; |Q^ZI
for j = 1:length(n) +'?p $@d
rpowers = [rpowers m_abs(j):2:n(j)]; XGEAcN
end H>[1DH#b
rpowers = unique(rpowers); dvk?A$
\c+)Y}:D
% Pre-compute the values of r raised to the required powers, *lg1iP{]
% and compile them in a matrix: qbkvwL9
% ----------------------------- l,*v/95h
if rpowers(1)==0 u7&r'rZ1_!
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); !Ljs9 =UF
rpowern = cat(2,rpowern{:}); y5.Z <Y
rpowern = [ones(length_r,1) rpowern]; 9/RbfV[)
else 5f7;pS<
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); SG8H~]CO)
rpowern = cat(2,rpowern{:}); 50(/LV1
end qu8i Jq
b1jh2pG(V
% Compute the values of the polynomials: #"6(Q2|
l
% -------------------------------------- LQ?J
r>4
y = zeros(length_r,length(n)); +}X?+Epm
for j = 1:length(n) }.7!@!q.
s = 0:(n(j)-m_abs(j))/2; Va06(Cq
pows = n(j):-2:m_abs(j); Gu<3*@Ng
for k = length(s):-1:1 cU5x8[2
p = (1-2*mod(s(k),2))* ... L*9^-,
prod(2:(n(j)-s(k)))/ ... _Q/D%7[pa
prod(2:s(k))/ ... @?{n`K7{`
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Ywt_h;:
prod(2:((n(j)+m_abs(j))/2-s(k))); |,5b[Y"Dt
idx = (pows(k)==rpowers); q,2]]K7y
y(:,j) = y(:,j) + p*rpowern(:,idx); BN@*CG
end >\8Bu#&s4
i)\`"&.j>N
if isnorm ;k/y[ x}
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); LS4c|Dv
end s'ntf
end $# @G!
% END: Compute the Zernike Polynomials g||{Qmr=1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% '@wYr|s4
=+97VO(w]G
% Compute the Zernike functions: KSuP'.l
% ------------------------------ ,m!j2H}8
idx_pos = m>0; bP6QF1L
idx_neg = m<0; `,aPK/
WYwsTsG{_
z = y; [Zl
if any(idx_pos) Qwk
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 18Pc4~>0
end *(s+u~, I
if any(idx_neg) 8=T;R&U^M
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); vAq`*]W+
end 6t
TLyI$+
+XJj:%yt
% EOF zernfun