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 gtU1'p"
function z = zernfun(n,m,r,theta,nflag) cf8-]G?tK
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. T{MC-j _T9
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Ueyw;Y
% and angular frequency M, evaluated at positions (R,THETA) on the =V $j6
% unit circle. N is a vector of positive integers (including 0), and <+E%E4
% M is a vector with the same number of elements as N. Each element m@qqVRn#)
% k of M must be a positive integer, with possible values M(k) = -N(k) (i`(>I.(/
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, @ RR\lZ
% and THETA is a vector of angles. R and THETA must have the same b](o]O{v
% length. The output Z is a matrix with one column for every (N,M) ym%slg
% pair, and one row for every (R,THETA) pair. TQ9'76INb
% bkQ3c-C<
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 7[o {9Yp&
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), (Pi-uL<[a
% with delta(m,0) the Kronecker delta, is chosen so that the integral *Z kss
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 2_pz3<,\
% and theta=0 to theta=2*pi) is unity. For the non-normalized L7q | ^`
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ZlR!s!vv
% ?ApRJm:T
% The Zernike functions are an orthogonal basis on the unit circle. D1"7s,Hmu
% They are used in disciplines such as astronomy, optics, and M []OHw
% optometry to describe functions on a circular domain. |O (G nsZ
% 0-xCp ~vE
% The following table lists the first 15 Zernike functions. d'zT:g
% m6n hC
% n m Zernike function Normalization U</+ .$b
% -------------------------------------------------- 960qvz!
% 0 0 1 1 !wh=dQgMe
% 1 1 r * cos(theta) 2
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#A
% 1 -1 r * sin(theta) 2 EF;,Gjh5p
% 2 -2 r^2 * cos(2*theta) sqrt(6) J+2R&3;_O
% 2 0 (2*r^2 - 1) sqrt(3) Pz473d
% 2 2 r^2 * sin(2*theta) sqrt(6) -<oZ)OfU
% 3 -3 r^3 * cos(3*theta) sqrt(8) b=LF%P
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) c^S&F9/U*
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ]h@{6N'oNS
% 3 3 r^3 * sin(3*theta) sqrt(8) 9*p G?3*I
% 4 -4 r^4 * cos(4*theta) sqrt(10) !<Z{@7oH
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `"Dy%&U
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) |=3 *;}
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?)cJZ>$!w
% 4 4 r^4 * sin(4*theta) sqrt(10) .cR*P<3O
% -------------------------------------------------- (!n-Age
% N$Hqa^!'T
% Example 1: `^%GN8d}nm
% 1g i}H)
% % Display the Zernike function Z(n=5,m=1) raQYn?[
% x = -1:0.01:1; >eo8
% [X,Y] = meshgrid(x,x); L?f qcW{
% [theta,r] = cart2pol(X,Y); 3wNN<R
% idx = r<=1; kPJ~X0Fr{t
% z = nan(size(X)); FOp_[rR
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 2u&c
&G
% figure ,k%8yK
% pcolor(x,x,z), shading interp \X opU"
% axis square, colorbar l$`G:%qHj
% title('Zernike function Z_5^1(r,\theta)') r5)f82pQ
% ,4Y sZ
% Example 2: Wf1-"Q
% 4~WlP,,M
% % Display the first 10 Zernike functions )/TVJAJ
% x = -1:0.01:1; }85#[~m'
% [X,Y] = meshgrid(x,x); +~:0Dxv W
% [theta,r] = cart2pol(X,Y); h.LSMU (O
% idx = r<=1; YPQCOG
% z = nan(size(X)); s=j O;K$
% n = [0 1 1 2 2 2 3 3 3 3]; j&}B<f _6J
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; +-k`x0v
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 2$Y3[$
% y = zernfun(n,m,r(idx),theta(idx)); Vx(;|/:
% figure('Units','normalized') :+A;TV
% for k = 1:10 j)@oRWL<
% z(idx) = y(:,k); EEg O
% subplot(4,7,Nplot(k)) *EE|?vn
% pcolor(x,x,z), shading interp (QhAGk&lu
% set(gca,'XTick',[],'YTick',[]) |vN$"mp^a
% axis square ^ N_`^m
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) G[B*TM6$
% end c=<v.J@K
% Q,TaJ]
% See also ZERNPOL, ZERNFUN2. &`5 :GLV
~pwY6Q
% Paul Fricker 11/13/2006 F13%)G(
[ 1D)$"
@%7/2k
% Check and prepare the inputs: 2X +7bM
% ----------------------------- EkV!hqs*
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7dx4~dF
error('zernfun:NMvectors','N and M must be vectors.') BC&^]M
end C890+(D~
QD6Z=>?S
if length(n)~=length(m) M,Po54u
error('zernfun:NMlength','N and M must be the same length.') |O^V)bZmx
end w7[0
.;}pU!S~R
n = n(:); 6UtG-WHHt
m = m(:);
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if any(mod(n-m,2)) r6/<&1[
error('zernfun:NMmultiplesof2', ... J]_)gb'1BR
'All N and M must differ by multiples of 2 (including 0).') Pyit87h{
end ol1AD: Ho
%hrsE5k^,
if any(m>n) gB'`I(q5.
error('zernfun:MlessthanN', ... A`
oa|k!U
'Each M must be less than or equal to its corresponding N.') pzYG?9cwz
end | eK,Td%
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if any( r>1 | r<0 ) .|tQ=l@I
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ZlUFJ*pk
end IrUpExJ
.jy)>"h0
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) HXz iDnj
error('zernfun:RTHvector','R and THETA must be vectors.') S:DcfR=a
end aj+zmk~-
i,^>uf
r = r(:); 6YB-}>?
theta = theta(:); 8VKb*
length_r = length(r); VN1#8{
if length_r~=length(theta) "1E?3PFJ
error('zernfun:RTHlength', ... ei(|5h
'The number of R- and THETA-values must be equal.') F12S(5Z0%
end GWVEIZ
sT@u3^>
% Check normalization: _q2`m
% -------------------- |2tSUOZ
if nargin==5 && ischar(nflag) * ,|)~$=>
isnorm = strcmpi(nflag,'norm'); qLU15cOM
if ~isnorm 8yNRxiW:
error('zernfun:normalization','Unrecognized normalization flag.') #p;4:IT
end wK/}E h\^
else GA}hp%
isnorm = false; )[F46?$vrk
end 8JFnB(3xU
w/)e2CH
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k|)^!BdO
% Compute the Zernike Polynomials w`w `q'
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ss*LgK_
b*9m2=6
% Determine the required powers of r: #h}IUR
% ----------------------------------- O p!
m_abs = abs(m); =+kvL2nx-
rpowers = []; hPNQGVv
for j = 1:length(n) T(t
<Ay?c
rpowers = [rpowers m_abs(j):2:n(j)]; {3_F fsg`
end 4'7
v!I9
rpowers = unique(rpowers); vUA)#z<
|sDG>Zq?
% Pre-compute the values of r raised to the required powers, |^>L`6uo
% and compile them in a matrix: 6Vu}kK)
% ----------------------------- mRix0XBI~
if rpowers(1)==0 =2GP^vh
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); t\E-6u
rpowern = cat(2,rpowern{:}); I3F6-gH
rpowern = [ones(length_r,1) rpowern]; QqT6P`0u
else 2xz%'X%
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 3tx0y
rpowern = cat(2,rpowern{:}); QZz{74]n
end o>QFdx
N23+1 h
% Compute the values of the polynomials: >"m@qkh
% -------------------------------------- UGezo3}
y = zeros(length_r,length(n)); 'IqK M
for j = 1:length(n) '/n%}=a=
s = 0:(n(j)-m_abs(j))/2; 9|?(GG
pows = n(j):-2:m_abs(j); vi()1LS/!
for k = length(s):-1:1 2!"\;/
p = (1-2*mod(s(k),2))* ... 6>P
prod(2:(n(j)-s(k)))/ ... ._F6- pl
prod(2:s(k))/ ... Oujlm|
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... HnYFE@Nl:U
prod(2:((n(j)+m_abs(j))/2-s(k))); {O3oUE+
idx = (pows(k)==rpowers); 6#lC(ko'
y(:,j) = y(:,j) + p*rpowern(:,idx); i32_ZB Z?y
end Ot8S'cB1,$
d
>wmg*J
if isnorm +X|m>9
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi);
EY[Q%
end p7ns(g@9
end Z ^9{Qq
% END: Compute the Zernike Polynomials AD4L`0D
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dp%pbn6w
O'yjB$j
% Compute the Zernike functions: wshp{ y
% ------------------------------ ]oWZ{#r2
idx_pos = m>0; <PuB3PEvV
idx_neg = m<0; spoWdRM2
9OO_Hp#|9
z = y; VTgbJ{?
if any(idx_pos) "3>*i!i
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); O%Gsk'mo
end R*TGn_J`
if any(idx_neg) R4rm>zisVX
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); e7)%=F/)
end Lw+1|
,mBKya)
% EOF zernfun