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 Syv[[Ek
function z = zernfun(n,m,r,theta,nflag) QP/%+[E.
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. };;\&#
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Bu|Uz0Y
% and angular frequency M, evaluated at positions (R,THETA) on the C_xOk'091
% unit circle. N is a vector of positive integers (including 0), and !lQGoXQ'4
% M is a vector with the same number of elements as N. Each element W[Kv
Qt3%
% k of M must be a positive integer, with possible values M(k) = -N(k) mG4$
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 7{F(NJUO1
% and THETA is a vector of angles. R and THETA must have the same b-4gHW
% length. The output Z is a matrix with one column for every (N,M) 0kC}qru'
% pair, and one row for every (R,THETA) pair. >Y,3EI\
% .x\fPjB
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike '](4g/%
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), !Rp
% with delta(m,0) the Kronecker delta, is chosen so that the integral N6K%Wkz
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, YDh6XD<Z
% and theta=0 to theta=2*pi) is unity. For the non-normalized /IQl
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 8/q6vk><
% oVi_X98R
% The Zernike functions are an orthogonal basis on the unit circle. OS|uZ<"Rq3
% They are used in disciplines such as astronomy, optics, and j{}-zQ]n
% optometry to describe functions on a circular domain. x~1.;dBF
% *;^!FBT
% The following table lists the first 15 Zernike functions. fDe4 [QQ8
% 5WhR|
% n m Zernike function Normalization Ce&nMgd~
% -------------------------------------------------- 5gP<+S#>T
% 0 0 1 1 @L?X}'0xI4
% 1 1 r * cos(theta) 2 (EZ34,k'S
% 1 -1 r * sin(theta) 2 W5'07N^
% 2 -2 r^2 * cos(2*theta) sqrt(6) O5}/OH|j
% 2 0 (2*r^2 - 1) sqrt(3) Q
I!c= :u
% 2 2 r^2 * sin(2*theta) sqrt(6) -^A=U7
% 3 -3 r^3 * cos(3*theta) sqrt(8) <(|No3jx
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) e| AA7
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) >R|*FYam
% 3 3 r^3 * sin(3*theta) sqrt(8) aJh=4j~.
% 4 -4 r^4 * cos(4*theta) sqrt(10) *Nfn6lVB
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) _PTo!aJL
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) [h"#Gwb=;
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) TTNgnP
% 4 4 r^4 * sin(4*theta) sqrt(10) -Vj'QqZ
% -------------------------------------------------- lo }[o0X
% !: e0cV
% Example 1: *7L*:g
% 44s
K2
% % Display the Zernike function Z(n=5,m=1) ,p(4OZz5,
% x = -1:0.01:1; w8~J5XS
% [X,Y] = meshgrid(x,x); $`nKq4Y
% [theta,r] = cart2pol(X,Y); y&y(<
% idx = r<=1; sy^k:y?
% z = nan(size(X)); XTIRY4{
d
% z(idx) = zernfun(5,1,r(idx),theta(idx)); W@S'mxk#*
% figure 84PD`A
% pcolor(x,x,z), shading interp 7Pt*V@DHS
% axis square, colorbar |=OO$z;q|
% title('Zernike function Z_5^1(r,\theta)') hl4@Y#n
% , N:'Z
% Example 2: ]mU,y$IQ
% DNgQ.lV
% % Display the first 10 Zernike functions A<6V$e$:2
% x = -1:0.01:1; )p.+39]{2
% [X,Y] = meshgrid(x,x); 2>{_O?UN
% [theta,r] = cart2pol(X,Y); ~;ink
% idx = r<=1; j/zD`ydj
% z = nan(size(X)); Kuh! b`9
% n = [0 1 1 2 2 2 3 3 3 3]; 47Y|1
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Z&mV1dxR
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Pn{yk`6E
% y = zernfun(n,m,r(idx),theta(idx)); lYd#pNN
% figure('Units','normalized') #unE>#DW
% for k = 1:10 b0a'Y"oef4
% z(idx) = y(:,k); Z$R2Z$f
% subplot(4,7,Nplot(k)) k&nhF9Y4
% pcolor(x,x,z), shading interp B3I\=
% set(gca,'XTick',[],'YTick',[]) vcB+h;x
% axis square =N,KVMxw
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 0/.#V*KM
% end >Kl78w:
% UQ|zSalv,
% See also ZERNPOL, ZERNFUN2. ;WIL?[;w
~qNpPIrGr
% Paul Fricker 11/13/2006 -X@;"0v
QN(f8t(
Q, E!Ew3
% Check and prepare the inputs: X.0/F6U
% ----------------------------- 1{ #Xa=
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) V mQ7M4j*
error('zernfun:NMvectors','N and M must be vectors.') -
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)O@ ;
end AK<ZP?0
~H0~5v F
if length(n)~=length(m) +DKrX
error('zernfun:NMlength','N and M must be the same length.') 'Rfvr7G/?
end <.3@-z>w2,
hoC}@8_
n = n(:); 1at$_\{.(
m = m(:); [Hdk=p
if any(mod(n-m,2)) xsRMF&8L
error('zernfun:NMmultiplesof2', ... o,) p *glO
'All N and M must differ by multiples of 2 (including 0).') -b@E@uAX/
end :Puv8[1i
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if any(m>n) 2h&pm
error('zernfun:MlessthanN', ... 9\)NFZ3Mz
'Each M must be less than or equal to its corresponding N.') {s8''+Q#(-
end qn@Qd9Sf
+2oZB]GPL
if any( r>1 | r<0 ) F dv&kK!
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ~E^EF{h
end NQfIY`lt'
HXU"]s2Z
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Ao96[2U6
error('zernfun:RTHvector','R and THETA must be vectors.') wri[#D {
end af{;4Cr
xSb/98;
r = r(:); uMsKF %m
theta = theta(:); v03~=(
length_r = length(r); B4R,[WE"
if length_r~=length(theta) },a|WL3^
error('zernfun:RTHlength', ... D .Cm&
'The number of R- and THETA-values must be equal.') !xo@i XL
end U7crbj;c)d
%o4d43uZ
% Check normalization: N5/TV%u
% -------------------- \g4\a?i
if nargin==5 && ischar(nflag) Q,\lS
isnorm = strcmpi(nflag,'norm'); -I=}SZ
if ~isnorm `?JrC3
error('zernfun:normalization','Unrecognized normalization flag.') ZuS+p0H"
end %n}.E304
else +G/~v`Bv
isnorm = false; {OAy@6
+
end Tjs-+$P+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4kXx(FE
% Compute the Zernike Polynomials *C\4%l
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [RpFC4W
U}A+jJ
% Determine the required powers of r: cjN4U [
% ----------------------------------- 3C,e>zE}
m_abs = abs(m); N_0&3PUSM
rpowers = []; #gN{8Yk>
for j = 1:length(n) XVv7W5/q]
rpowers = [rpowers m_abs(j):2:n(j)]; VDnAQ[T@d
end KktTR`W
rpowers = unique(rpowers); #-lk=>
wFqz.HoB
% Pre-compute the values of r raised to the required powers, *fd` .}
% and compile them in a matrix: M.OWw#?p:_
% ----------------------------- D 0n2r
if rpowers(1)==0 _)Qt,$
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 0_] aF8j
rpowern = cat(2,rpowern{:}); #kb(2Td
rpowern = [ones(length_r,1) rpowern]; Ne9
.wd
else p>}N9v;Bo
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); {Zseu$c
rpowern = cat(2,rpowern{:}); kT$4X0}
end =kc{ Q@Dk
8dZH&G@;
% Compute the values of the polynomials: 9hguC yr@h
% -------------------------------------- VR:b1XWX
y = zeros(length_r,length(n)); 1$Hf`h2
for j = 1:length(n) pP/o2
s = 0:(n(j)-m_abs(j))/2; 3p4bOT5
pows = n(j):-2:m_abs(j); j_H
T
for k = length(s):-1:1 }E1Eq
p = (1-2*mod(s(k),2))* ... v'@LuF'e8
prod(2:(n(j)-s(k)))/ ... 7I44BC*R~
prod(2:s(k))/ ... ah<f&2f
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... [c W
prod(2:((n(j)+m_abs(j))/2-s(k))); ^X;>?_Bk
idx = (pows(k)==rpowers); h=U 4
y(:,j) = y(:,j) + p*rpowern(:,idx); *xjIl<`pK
end JWdG?[$
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if isnorm AN1bfF:C
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); h n]6he
end U&/S
end O71rLk;
% END: Compute the Zernike Polynomials *oWzH_
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ce)Wvuh
v}mmY>M%
% Compute the Zernike functions: 'Hia6<m3
% ------------------------------ vSL{WT]m
idx_pos = m>0; a|53E<5X
idx_neg = m<0; VsMN i#?
ZT8j9zs
z = y; A3$b_i @P
if any(idx_pos) 1e+?O7/
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); lKwcT!Q4
end j w462h
if any(idx_neg) Y'~&%|9+T
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ;$/G T
end Smux&e
!Yf0y;e|:
% EOF zernfun