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 !]7Z),s
function z = zernfun(n,m,r,theta,nflag) d%NO_=I.
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Ly/"da
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N A#RA;Dt:
% and angular frequency M, evaluated at positions (R,THETA) on the y|Tb&XPD
% unit circle. N is a vector of positive integers (including 0), and Zm!T4pL
% M is a vector with the same number of elements as N. Each element l4u_Z:<w
% k of M must be a positive integer, with possible values M(k) = -N(k) kUUeyq
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, q3TAWNzI0
% and THETA is a vector of angles. R and THETA must have the same &z8@ rk|
% length. The output Z is a matrix with one column for every (N,M) .Ebg>j:\
% pair, and one row for every (R,THETA) pair. R2yiExw<
% 7RM$%'n\
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike PsMoH/+"
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), $3zs?Fd`
% with delta(m,0) the Kronecker delta, is chosen so that the integral v#{Sx>lO
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, qasbK:}
% and theta=0 to theta=2*pi) is unity. For the non-normalized Z0s}65BR
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. QI'ul e
% wZ6LiYiHl
% The Zernike functions are an orthogonal basis on the unit circle. URdCV{@42
% They are used in disciplines such as astronomy, optics, and =<MSM\Rb
% optometry to describe functions on a circular domain. FM$XMD0=
% ET;YAa*
% The following table lists the first 15 Zernike functions. O{SP4|0JV
% .(^KA{
% n m Zernike function Normalization 1p=^I'#
% -------------------------------------------------- .w/w]
Eq
% 0 0 1 1 3&:Us|}
% 1 1 r * cos(theta) 2 fmrd 7*MW
% 1 -1 r * sin(theta) 2 YAQ]2<H
% 2 -2 r^2 * cos(2*theta) sqrt(6) ZpvURp,I
% 2 0 (2*r^2 - 1) sqrt(3) cw|3W]
% 2 2 r^2 * sin(2*theta) sqrt(6) / E}L%OvE
% 3 -3 r^3 * cos(3*theta) sqrt(8) C?m2R(RF
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) s.`:9nj
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) T'B4 3Q
% 3 3 r^3 * sin(3*theta) sqrt(8) "c` $U]M%
% 4 -4 r^4 * cos(4*theta) sqrt(10) N^z4I,GV(
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) }5
^2g!M
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) i#]}k
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) j>Wb$p6S
% 4 4 r^4 * sin(4*theta) sqrt(10) jLo(Uf
% -------------------------------------------------- KM
li!.(b
% X)^eaw]Q0
% Example 1: S^(OjS
% CC&o pC
% % Display the Zernike function Z(n=5,m=1) 15dhr]8E
% x = -1:0.01:1; Ro3C(aRx
% [X,Y] = meshgrid(x,x); 9oBK(Sf@^
% [theta,r] = cart2pol(X,Y); ~A^E_
% idx = r<=1; 4o?_G[
% z = nan(size(X)); '0q.zzv|_
% z(idx) = zernfun(5,1,r(idx),theta(idx)); NU[{ANbl
% figure V&)Jvx}^
% pcolor(x,x,z), shading interp N$]B$vv
% axis square, colorbar VZuluV
% title('Zernike function Z_5^1(r,\theta)') PJ}d-
% 4A0
,N8ja}
% Example 2: y0s=yN_
% z5.Uv/n\1
% % Display the first 10 Zernike functions ov;1=M~RF
% x = -1:0.01:1; .5$"qb
?
% [X,Y] = meshgrid(x,x); cG!\P: re
% [theta,r] = cart2pol(X,Y); A1>fNilC9
% idx = r<=1; DR]=\HQ
% z = nan(size(X)); ZtHTl\z
% n = [0 1 1 2 2 2 3 3 3 3]; 7p1f*N[X
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; s1
mKz0q
% Nplot = [4 10 12 16 18 20 22 24 26 28]; u7||]|2
% y = zernfun(n,m,r(idx),theta(idx)); ,GOH8h
% figure('Units','normalized') :Kq]b@X
% for k = 1:10 FgwIOpqE*
% z(idx) = y(:,k); RfoEHN
% subplot(4,7,Nplot(k)) H!SFSgAu
% pcolor(x,x,z), shading interp m&S *S_c
% set(gca,'XTick',[],'YTick',[]) hK]mnA[Y
% axis square ,bTpD!
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) _43'W{%
% end P^'TI[\L9
% i?{)o]i
% See also ZERNPOL, ZERNFUN2. a4d7;~tZ
U80h0t%
% Paul Fricker 11/13/2006 *Aqd["q
KC+jHk
xP{)+$n
% Check and prepare the inputs: *jQ?(Tf
% ----------------------------- LX'z7fh
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) <n1panS
error('zernfun:NMvectors','N and M must be vectors.') '`s+e#rs4{
end -v %n@8p
]`eP"U{
if length(n)~=length(m) 52,[dP,g
error('zernfun:NMlength','N and M must be the same length.') 8
$qj&2 N
end }G/!9Zq
= Ed0vw
n = n(:); ;_X2E~i[
m = m(:); `!( IQ&
if any(mod(n-m,2)) 3xIelTf*
error('zernfun:NMmultiplesof2', ... %6.WGuO
'All N and M must differ by multiples of 2 (including 0).') 7Is:hx|:
end \s?8}k
/hN;\Z[@
if any(m>n) fI
v?HD:j
error('zernfun:MlessthanN', ... a%nf
)-}|
'Each M must be less than or equal to its corresponding N.') c_4K
end zq(4@S-TU
(b;Kl1Ql]
if any( r>1 | r<0 ) @}\i`H1s
error('zernfun:Rlessthan1','All R must be between 0 and 1.') xyD2<?dGUb
end 5>6:#.f%!e
G^|!'V
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) k{F]^VXQ
error('zernfun:RTHvector','R and THETA must be vectors.') a[_IG-l|i4
end KAJR.YNm
"&:H }Jd
r = r(:); F| jl=i
theta = theta(:); \483S]_-z{
length_r = length(r); bj6;>Ezp3(
if length_r~=length(theta) eo*l^7
error('zernfun:RTHlength', ... a]/KJn/B(
'The number of R- and THETA-values must be equal.') B:Y F|k}T
end e9R H[:
xtBu]I)%
% Check normalization: PI.Zd1r
% -------------------- ,j6R/sg
if nargin==5 && ischar(nflag) _\8jnpT:
isnorm = strcmpi(nflag,'norm'); P;`Awp?
if ~isnorm K491QXG
error('zernfun:normalization','Unrecognized normalization flag.') h,?%,GI
end 8_VGB0~3i
else $1$0M
isnorm = false; jddhX]>I
end aGd
wuD
~N%+ZXh&E
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% qSGM6kb
% Compute the Zernike Polynomials Pr:\zI
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% hVz] wKP
H:|.e)$i
% Determine the required powers of r: 0l3[?YtXc
% ----------------------------------- %AN,cE*
m_abs = abs(m); OwT _W)$
rpowers = []; NLr a"Z
for j = 1:length(n) q_6fr$-Qh
rpowers = [rpowers m_abs(j):2:n(j)]; TQu.jC
end 'ieTt_1.G
rpowers = unique(rpowers); \%&A? D
8_E(.]U
% Pre-compute the values of r raised to the required powers, EDz;6Z*4N
% and compile them in a matrix: }hsNsQ
% ----------------------------- t7xJ$^p[|K
if rpowers(1)==0 dl"=ZI
'^
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ttdY]+Fj
rpowern = cat(2,rpowern{:}); Zs]n0iwM'@
rpowern = [ones(length_r,1) rpowern]; _9]vlxgtG(
else :tbgX;tCs5
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); R `Fgne$4
rpowern = cat(2,rpowern{:}); ol41%q*
end MhR`
2{.g7bO
% Compute the values of the polynomials: Yn[>Y)
% -------------------------------------- Z;V(YK(WO.
y = zeros(length_r,length(n)); H[nco#
for j = 1:length(n) v)T#
iw[
s = 0:(n(j)-m_abs(j))/2; t
V(
WhP
pows = n(j):-2:m_abs(j); UWnF2,<s;
for k = length(s):-1:1 B$6KI
p = (1-2*mod(s(k),2))* ... 0zA;%oP
prod(2:(n(j)-s(k)))/ ... eAo+w*D(
prod(2:s(k))/ ... SswcO9JCX3
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ;<q2
prod(2:((n(j)+m_abs(j))/2-s(k))); Z1jxu;O(
idx = (pows(k)==rpowers); <{k`K[)
y(:,j) = y(:,j) + p*rpowern(:,idx); tT!'qL.*
end vQ*RrHG?c
8HFCmY#
if isnorm kc0MQ TJU
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); <$yA*
end q01 L{~>bz
end g<(!>:h
% END: Compute the Zernike Polynomials wgIm{;T[u
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {f\wIZ-K A
#2s}s<Sc;
% Compute the Zernike functions: ;-8.~Sm
% ------------------------------ JH{/0x#+
idx_pos = m>0; zt:
!hM/Vt
idx_neg = m<0; 1Xo0(*O
'5Yzo^R;
z = y; .SjJG67OyA
if any(idx_pos) D
h;5hu2"
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); _qR?5;v
end AwXzI;F^
if any(idx_neg) g=[OH
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); J{^md0l
end j&,Gv@
kBhjqI*
% EOF zernfun