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 )tJL@Qo
function z = zernfun(n,m,r,theta,nflag) 3xc:Y>
*`
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Vx0MG{vG1
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N :k Rv
% and angular frequency M, evaluated at positions (R,THETA) on the Z`e$~n(Bh
% unit circle. N is a vector of positive integers (including 0), and E>o&GYc
% M is a vector with the same number of elements as N. Each element L2:oZ&:u`J
% k of M must be a positive integer, with possible values M(k) = -N(k) [I#Q
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, NHst7$Y<
% and THETA is a vector of angles. R and THETA must have the same wI|bBfd(
% length. The output Z is a matrix with one column for every (N,M) c`Lpqs`
% pair, and one row for every (R,THETA) pair. Q&\ZC?y4
% 89 _&X[X
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ?14X8Mb8W_
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ,Gf+U7'K
% with delta(m,0) the Kronecker delta, is chosen so that the integral ":5~L9&G
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, &e5^v
% and theta=0 to theta=2*pi) is unity. For the non-normalized K*hf(w9="%
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. H{p[Ghp
% vLVSZX
% The Zernike functions are an orthogonal basis on the unit circle. p]atH<^;K
% They are used in disciplines such as astronomy, optics, and s2,`eV
% optometry to describe functions on a circular domain. #l8K8GLuf
% i[V,IP +
% The following table lists the first 15 Zernike functions. lk5_s@V
l
% 0~LnnDN
% n m Zernike function Normalization 'eTpcrS3
% -------------------------------------------------- *}50q9)/
% 0 0 1 1 NpjsZcA
% 1 1 r * cos(theta) 2 /
r`Y'rm
% 1 -1 r * sin(theta) 2 &k {t0>
% 2 -2 r^2 * cos(2*theta) sqrt(6) ]}*G[[
^p
% 2 0 (2*r^2 - 1) sqrt(3) ^ ^U)WB
% 2 2 r^2 * sin(2*theta) sqrt(6) pJ<)intcbE
% 3 -3 r^3 * cos(3*theta) sqrt(8) qCv}+d)
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) zXA= se0U
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 2l;ge>DJ
% 3 3 r^3 * sin(3*theta) sqrt(8) QZeb+r
% 4 -4 r^4 * cos(4*theta) sqrt(10) &QHA_+88W
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) IrVM|8vT3
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) vErbX3RY2
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) _ ;v_L
% 4 4 r^4 * sin(4*theta) sqrt(10) -F~9f>
% -------------------------------------------------- mAtG&my)
% 0.3[=a43
% Example 1: **
"s~
% 60SenHKles
% % Display the Zernike function Z(n=5,m=1) -bG#h)yj
% x = -1:0.01:1; 0o\=0bH&s
% [X,Y] = meshgrid(x,x); y[Fw>g1`q
% [theta,r] = cart2pol(X,Y); v:!7n
% idx = r<=1; iz$v8;w
% z = nan(size(X));
Q}`2Y^.
% z(idx) = zernfun(5,1,r(idx),theta(idx)); pRa oR
% figure Amq8q
% pcolor(x,x,z), shading interp bC>yIjCTn
% axis square, colorbar UBpM8 /U
% title('Zernike function Z_5^1(r,\theta)') Z2Y583D
% +=lcN~U2
% Example 2: Ix l"'Q_z
% LP-KD
% % Display the first 10 Zernike functions uc{Qhw!;:
% x = -1:0.01:1; m/"=5*pA
% [X,Y] = meshgrid(x,x); [~&:`I1
% [theta,r] = cart2pol(X,Y); pu
m9x)y1
% idx = r<=1; 7{6cLYl
% z = nan(size(X)); ~P.-3
% n = [0 1 1 2 2 2 3 3 3 3]; pR^Y|NG!
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; jmwQc&
% Nplot = [4 10 12 16 18 20 22 24 26 28]; =iQ`F$M
% y = zernfun(n,m,r(idx),theta(idx)); Toa#>Z*+Rb
% figure('Units','normalized') DdA}A>47
% for k = 1:10 QI^8b\36
% z(idx) = y(:,k); d}A2I
% subplot(4,7,Nplot(k)) Tef3
Z6
% pcolor(x,x,z), shading interp jL[Is2<@
% set(gca,'XTick',[],'YTick',[]) %.Q2r ?j
% axis square lyc{Z%!3
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) R,b O{2O
% end 8;dbU*
% z ]4g`K+
% See also ZERNPOL, ZERNFUN2. A0 w `o
!n?*vN=S
% Paul Fricker 11/13/2006 .]d
tRH<
26klW:2*
u\& [@v
% Check and prepare the inputs: F7PZV+\
% ----------------------------- 3Tte8]0
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) <38@b
]+
error('zernfun:NMvectors','N and M must be vectors.') .TrQ +k>
end "oGM>@q=B
h[v3G<C ~r
if length(n)~=length(m) I3y4O^?
error('zernfun:NMlength','N and M must be the same length.') {UVm0AeUq
end 7)5$1
.}u(&
n = n(:); 9/qS*Zdh)
m = m(:); W1,L>Az^Ts
if any(mod(n-m,2)) i1H80m s
error('zernfun:NMmultiplesof2', ... IgnY*2FT
'All N and M must differ by multiples of 2 (including 0).') ^T
J
end V5^b6$R@
&_x/Dzu!z
if any(m>n) y5t Ap
error('zernfun:MlessthanN', ... CjukD%>sde
'Each M must be less than or equal to its corresponding N.') af5`ktx
end ,f""|X5
2dlV'U_g
if any( r>1 | r<0 ) Kgio}y
error('zernfun:Rlessthan1','All R must be between 0 and 1.') HC`3AQ12!&
end \EfwS%
P
4 ~|TKd{
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ~0$F
V
error('zernfun:RTHvector','R and THETA must be vectors.') ~;4k UJD
end wk7_(gT`0
1&L){ hg
r = r(:); Y{:/vOj
theta = theta(:); P!&CH4+
length_r = length(r); :[rKSA]@
if length_r~=length(theta) uTloj.
error('zernfun:RTHlength', ... 8qLgB
'The number of R- and THETA-values must be equal.') u! FSXX<
end .7^-*HT}
s
!vROJ
% Check normalization: YxqQg
% -------------------- L/+J|_J)
if nargin==5 && ischar(nflag) <UL|%9=~
isnorm = strcmpi(nflag,'norm'); 4E(5Ccb
if ~isnorm -"tgEC\tD
error('zernfun:normalization','Unrecognized normalization flag.') NB#*`|qt
end hd
BC ^n
else aw~EK0yU
isnorm = false; :pu{3-n.
end ;l4\^E1
"4AQpD
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ._nKM5.
% Compute the Zernike Polynomials IbaL.t\>
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% R}26 "+~
,DOmh<b
% Determine the required powers of r: oVsazYJ|?
% ----------------------------------- #E@i @'T
m_abs = abs(m); R51!j>[fqM
rpowers = []; Cb-E<W&2D
for j = 1:length(n) 1}M.}G2u/
rpowers = [rpowers m_abs(j):2:n(j)]; [1MEA;
end WYzaD}
rpowers = unique(rpowers); *g6o ;c
3pH`]m2
% Pre-compute the values of r raised to the required powers, <~*Ol+/
% and compile them in a matrix: OkUpgXU
% ----------------------------- /j@r~mt/pA
if rpowers(1)==0 6+z]MT
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); GK%ovK
rpowern = cat(2,rpowern{:}); gQDK?aQX
rpowern = [ones(length_r,1) rpowern]; \ \}/2#1=c
else <BA&S
_=4
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ,LO-!\L
rpowern = cat(2,rpowern{:}); D.!7jA#
end y]%,Y=%X
%"^XxVJ*
% Compute the values of the polynomials: ~l6Y<-!
% -------------------------------------- 3:#rFb
y = zeros(length_r,length(n)); ,e_#
for j = 1:length(n) wO%:WL$5
s = 0:(n(j)-m_abs(j))/2; /CE d14.
pows = n(j):-2:m_abs(j); = lD]sk
for k = length(s):-1:1 O3:
dOL/C
p = (1-2*mod(s(k),2))* ... <]^D({`
prod(2:(n(j)-s(k)))/ ... BAHx7x#(
prod(2:s(k))/ ... S$WM&9U
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... c10).zZ
prod(2:((n(j)+m_abs(j))/2-s(k))); nHrCSfK
idx = (pows(k)==rpowers); mh]$g<*m
y(:,j) = y(:,j) + p*rpowern(:,idx); LTct0Gh
end W10fjMC}^
1z:N$O_v
if isnorm Zx 1z
hc
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); b15qy? `y
end :/qO*&i,N
end (=/;rJ`q
% END: Compute the Zernike Polynomials =fo/+m5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6Qu*'
&Z!2xfQy>
% Compute the Zernike functions: uJ[Vv4N%9
% ------------------------------ w\*/(E<:
idx_pos = m>0; E%B Gf}h
idx_neg = m<0; ]S geZ07
AoeW<}MO
z = y; efR$s{n!
if any(idx_pos) /)TEx}wk
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); $(=1A>40
end q:^Cw8
if any(idx_neg)
%'z3es0
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); _6
`4_<c=
end jRAL(r|
2A+,. S_!x
% EOF zernfun