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 "o6a{KY(
function z = zernfun(n,m,r,theta,nflag) F!pgec%]'
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. (yxHXO9N
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N A[6D40o
% and angular frequency M, evaluated at positions (R,THETA) on the 1$1[6
\3v
% unit circle. N is a vector of positive integers (including 0), and Z@d(0 z
% M is a vector with the same number of elements as N. Each element 9zs!rlzQ
% k of M must be a positive integer, with possible values M(k) = -N(k) w/UZ6fu
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 7v{s?h->$
% and THETA is a vector of angles. R and THETA must have the same c3]X#Qa#m$
% length. The output Z is a matrix with one column for every (N,M) Exu>%
% pair, and one row for every (R,THETA) pair. `iT{H]po
%
##_Jz 5P
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike lS!uL9t.
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), `Eq~W@';Q0
% with delta(m,0) the Kronecker delta, is chosen so that the integral NPY\ >pf
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, `+?g96
% and theta=0 to theta=2*pi) is unity. For the non-normalized RjW<
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% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. I2G:jMPy
% rwh,RI)
)g
% The Zernike functions are an orthogonal basis on the unit circle. KYN{Dh]-}
% They are used in disciplines such as astronomy, optics, and RP|/rd]-k
% optometry to describe functions on a circular domain. -H-:b7
%
roNRbA]
% The following table lists the first 15 Zernike functions. 8AgKK=C=
% jSc!"Trl]
% n m Zernike function Normalization JT(6Uf
% -------------------------------------------------- 'wm :Xa
% 0 0 1 1 <A+n[h
% 1 1 r * cos(theta) 2 7ea<2va,
% 1 -1 r * sin(theta) 2 "Di8MMGOY
% 2 -2 r^2 * cos(2*theta) sqrt(6) yuA+YZ
% 2 0 (2*r^2 - 1) sqrt(3) TVs#,
% 2 2 r^2 * sin(2*theta) sqrt(6) !${7 )=|=1
% 3 -3 r^3 * cos(3*theta) sqrt(8) 14Y<-OO:
k
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 9hn+eU
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) pB0p?D)n
% 3 3 r^3 * sin(3*theta) sqrt(8) mMSQW6~j
% 4 -4 r^4 * cos(4*theta) sqrt(10) bpp{Z1/4
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) % 8hjMds
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Z(c3GmY
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) F(}~~EtPHo
% 4 4 r^4 * sin(4*theta) sqrt(10) 43m@4Yb
% -------------------------------------------------- J,SP1-L
% )oAx t70
% Example 1: pEp`Z,p
% Ef~Ar@4fA
% % Display the Zernike function Z(n=5,m=1) -'%>Fon
% x = -1:0.01:1; Ql8s7 %
% [X,Y] = meshgrid(x,x); ky#5G-X
% [theta,r] = cart2pol(X,Y); 'JK"3m}nT
% idx = r<=1; X"Ca
% z = nan(size(X)); 8gn12._x
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ~H4wsa39
% figure Z`]r)z%f
% pcolor(x,x,z), shading interp W{W8\
% axis square, colorbar
bo|3sN+D
% title('Zernike function Z_5^1(r,\theta)') 1Xn:B_pP
% rHaj~s 4
% Example 2: c$P68$FB
% zN3b`K. i
% % Display the first 10 Zernike functions |w].*c}Z
% x = -1:0.01:1; 6Q*Zy[=
% [X,Y] = meshgrid(x,x); {3`cSm6c
% [theta,r] = cart2pol(X,Y); s~'"&0Gz
% idx = r<=1;
YG_|L[/#
% z = nan(size(X)); z; Jz^m-
% n = [0 1 1 2 2 2 3 3 3 3]; 9_-6Lwj6t
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; qd<-{
% Nplot = [4 10 12 16 18 20 22 24 26 28]; <q.Q,_cW
% y = zernfun(n,m,r(idx),theta(idx)); %EH{p@nM&-
% figure('Units','normalized') ?
FlQ\q
% for k = 1:10 rt0_[i
% z(idx) = y(:,k); Mj6
0?k
% subplot(4,7,Nplot(k)) c> 0R_
% pcolor(x,x,z), shading interp pf$gvL
% set(gca,'XTick',[],'YTick',[]) )_olJCdaP^
% axis square lG5KZ[/Or
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) =&I9d;7
% end oJ?,X^~_
% )uu(I5St
% See also ZERNPOL, ZERNFUN2. mg]t)+ PQ
ShesJj
% Paul Fricker 11/13/2006 ykYef
sBk|KG
/>n!2'!
% Check and prepare the inputs: o~7D=d?R
% ----------------------------- Z4oD6k5oc
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) " GY3sam
error('zernfun:NMvectors','N and M must be vectors.') `ZU]eAV
end H'+3<t>
$0R5 ]]db)
if length(n)~=length(m) Re+oCJ
error('zernfun:NMlength','N and M must be the same length.') C8W_f( i~
end K@%gvLa\
Eh^gR`I
n = n(:); NL,6<ZOon,
m = m(:); K~4bT=
if any(mod(n-m,2)) &NOCRabc
error('zernfun:NMmultiplesof2', ... eX1_=?$1P
'All N and M must differ by multiples of 2 (including 0).') Tm$8\c4V:*
end _Wq;bKG
KZO[>qC"R
if any(m>n) ,Wtgj=1!.
error('zernfun:MlessthanN', ... z%BX^b$Hj
'Each M must be less than or equal to its corresponding N.') jGoQXiX
end 9oIfSr,y
0"Euf41
if any( r>1 | r<0 ) n0G@BE1Y=
error('zernfun:Rlessthan1','All R must be between 0 and 1.') '14 86q@[$
end l[i1,4
wwv+s ~(0
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Nv|0Z'M
error('zernfun:RTHvector','R and THETA must be vectors.') nBLb1T
end "&{.g1i9
&O#1*y
Z
r = r(:); tt
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Q
theta = theta(:); &-w. rF@
length_r = length(r); *CbV/j"P?
if length_r~=length(theta) pFV~1W:
error('zernfun:RTHlength', ... 0|i|z!N>
'The number of R- and THETA-values must be equal.') #2lvRJB
end +~*e B
g[HuIn/
% Check normalization: .;S1HOHz4
% -------------------- fdHFSnQ g
if nargin==5 && ischar(nflag) -PU.Uw]
isnorm = strcmpi(nflag,'norm'); -%Ce
if ~isnorm 7z&$\qu2
error('zernfun:normalization','Unrecognized normalization flag.') =(Y0wZP|
end qq_ZkU@xg
else ; xQhq*
isnorm = false; ?>I
end 6__HqBQ
'1fyBU
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +JM@ kdE5b
% Compute the Zernike Polynomials HuKOb4g
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \rO!lvX
ksJ 1:_
% Determine the required powers of r: [wnaF|h
% ----------------------------------- Z6Z/Y()4Tl
m_abs = abs(m); O(9*VoD
rpowers = []; JoZzX{eu"
for j = 1:length(n) ^<uQ9p^B
rpowers = [rpowers m_abs(j):2:n(j)]; GXNkl?#
end ?Iij[CbU
rpowers = unique(rpowers); k7L4~W
feeHXKD|
% Pre-compute the values of r raised to the required powers, z]LVq k
% and compile them in a matrix: J83C]2~7
% ----------------------------- _34%St!lg
if rpowers(1)==0 .7:ecFKk
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); A:(qF.Tm
rpowern = cat(2,rpowern{:}); L/%{,7l<^?
rpowern = [ones(length_r,1) rpowern]; ]scr@e
else 1jAuW~
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); g[~J107%A
rpowern = cat(2,rpowern{:}); x{GFCy7
end gS`Z>+V5!c
v6E5#pse8
% Compute the values of the polynomials: l8_RA
% -------------------------------------- 4cJ/XgX
y = zeros(length_r,length(n)); b^()[4M;
for j = 1:length(n) +^J;ic
s = 0:(n(j)-m_abs(j))/2; N pQOLX/<?
pows = n(j):-2:m_abs(j); )nK+`{;@!
for k = length(s):-1:1 nPl,qcyY
p = (1-2*mod(s(k),2))* ... (Kg)cc[B`
prod(2:(n(j)-s(k)))/ ... cS@p`A7Tpo
prod(2:s(k))/ ... "T<7j.P?
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ...
fD8GAav
prod(2:((n(j)+m_abs(j))/2-s(k))); qLKL*m
idx = (pows(k)==rpowers); 1!E}A!;
y(:,j) = y(:,j) + p*rpowern(:,idx); }jFRuT;35
end 1|>bG#|
:Dtm+EQ
if isnorm |<y1<O>F
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); { ~(XO@;b
end qw)Ou]L=
end D4$"02"
% END: Compute the Zernike Polynomials m")p]B&i=
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% z U*Mk
%rEP.T\i
% Compute the Zernike functions: U^K8^an$
% ------------------------------ r?pFc3~N
idx_pos = m>0; 9\kEyb$F=
idx_neg = m<0; RmOkb~
tn(6T^u
z = y; ,zJ:a>v
if any(idx_pos) E5*pD*#
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 1$:O9{F
end xf|C{XV@H
if any(idx_neg) !RjC0,
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); hVJ}EF0
end YhN:t?
UjyrmQf
% EOF zernfun