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 4s/4z@3a
function z = zernfun(n,m,r,theta,nflag) u`Djle
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. )
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% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N TF 6_4t6
% and angular frequency M, evaluated at positions (R,THETA) on the x8%Q TTY
% unit circle. N is a vector of positive integers (including 0), and _F
xq
% M is a vector with the same number of elements as N. Each element "m +Eu|{
% k of M must be a positive integer, with possible values M(k) = -N(k) yA*~O$~Y
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, PETrMu<
% and THETA is a vector of angles. R and THETA must have the same E :*!an
% length. The output Z is a matrix with one column for every (N,M) 1\q(xka{
% pair, and one row for every (R,THETA) pair. XOzPi*V**
% 5sC{5LJzC
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike x!bFbi#!"
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 9)l-5o:D
% with delta(m,0) the Kronecker delta, is chosen so that the integral E^L
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, [P)'LY6F
% and theta=0 to theta=2*pi) is unity. For the non-normalized y
%Get
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. .$)'7
% {'Nvs_{6
% The Zernike functions are an orthogonal basis on the unit circle. # 'G/&&<
% They are used in disciplines such as astronomy, optics, and 6gwjrGje\
% optometry to describe functions on a circular domain. BZEY^G
% Woa5Ov!n0
% The following table lists the first 15 Zernike functions. {U(-cdU{e`
% _Hi;Y
% n m Zernike function Normalization ]]@jvU_?kS
% -------------------------------------------------- a*hOT_;#
% 0 0 1 1 i`7{q~d=
% 1 1 r * cos(theta) 2 6FG h=~{3,
% 1 -1 r * sin(theta) 2 )hK5_]"lmj
% 2 -2 r^2 * cos(2*theta) sqrt(6) A/RHb^N
% 2 0 (2*r^2 - 1) sqrt(3) kCxmC<34
% 2 2 r^2 * sin(2*theta) sqrt(6) L
q8}z-?
% 3 -3 r^3 * cos(3*theta) sqrt(8) 4q[C'
J
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) (:2:_FL
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 8lI#D)}
% 3 3 r^3 * sin(3*theta) sqrt(8) H,txbJ
% 4 -4 r^4 * cos(4*theta) sqrt(10) {YWj`K
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,WA7Kp9
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) t5N@z
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !y$Hr[v
% 4 4 r^4 * sin(4*theta) sqrt(10) 0F=UZf&
% -------------------------------------------------- CeS8I-,
% q7mqzMDk
% Example 1: #)q}Jw4]j
% 1;3oGuHj8
% % Display the Zernike function Z(n=5,m=1) f!ehq\K1k
% x = -1:0.01:1; 2G:)27Q-
% [X,Y] = meshgrid(x,x); wx -NUTRim
% [theta,r] = cart2pol(X,Y); )5gcLD/zI
% idx = r<=1; lxj_(Uo
% z = nan(size(X)); =$Sf]L
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Gnp,~F"
% figure i;lzFu)G
% pcolor(x,x,z), shading interp rmpJG|(
% axis square, colorbar ?l`DkUo*j
% title('Zernike function Z_5^1(r,\theta)') 5^cPG" 4@
% mfFC@~|g
% Example 2: 'VFxg,
% 5p"n g8nR
% % Display the first 10 Zernike functions QR2J;Oj_
% x = -1:0.01:1; ['R2$z
% [X,Y] = meshgrid(x,x); |;'V":yDs
% [theta,r] = cart2pol(X,Y); c.6u)"@$
% idx = r<=1; h'^7xDw
% z = nan(size(X)); pA3j@w
% n = [0 1 1 2 2 2 3 3 3 3]; Ttn=VX{
\
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; *ntq;]
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 1c~c_Cc4
% y = zernfun(n,m,r(idx),theta(idx)); /@R|*7K;9
% figure('Units','normalized') -en:81a#
% for k = 1:10 b')CGqbbmT
% z(idx) = y(:,k); ySP1WK
% subplot(4,7,Nplot(k)) ,&
=(DJ
% pcolor(x,x,z), shading interp 5fv eQI~!
% set(gca,'XTick',[],'YTick',[]) l-_voOP
% axis square VF!?B>
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) \hQ[5>
% end E}c(4RY
% <i^Bq=E<rJ
% See also ZERNPOL, ZERNFUN2. 6g8{;6x
zCL/^^#
% Paul Fricker 11/13/2006 ()JM161
C>$5<bx
Et(Q$/W
% Check and prepare the inputs: [0yKd?e
% ----------------------------- sI/Hcm
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7A8jnq7m/
error('zernfun:NMvectors','N and M must be vectors.') =#^%; 6 6z
end t9&)9,my
!EF~I8d\]
if length(n)~=length(m) +
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error('zernfun:NMlength','N and M must be the same length.') *6e`km
end oaHg6PT!
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n = n(:); TU$/3fp*
m = m(:); *rSMD_>
if any(mod(n-m,2)) d,iW#,
error('zernfun:NMmultiplesof2', ... ;TF(opW:
'All N and M must differ by multiples of 2 (including 0).') 24Z7;'
end ylLQKdcL
9bl&\Ykt.
if any(m>n) '{\VOU
error('zernfun:MlessthanN', ... #R"9(Q&
'Each M must be less than or equal to its corresponding N.') %CfJ.;BDNE
end C16MzrB}(N
Tc,Bv7:
if any( r>1 | r<0 ) cE/7B'cR
error('zernfun:Rlessthan1','All R must be between 0 and 1.') UAnq|NJO
end Zn1+} Z@I
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) <_uLf9ja
error('zernfun:RTHvector','R and THETA must be vectors.') ED"@!M`1
end mG7Wu{~=U
xYc)iH6&
r = r(:); w}G2m)(
theta = theta(:); Z{EHV7
length_r = length(r); -. L)-%wIV
if length_r~=length(theta) |]RV[S3v
error('zernfun:RTHlength', ... `i8osX[ &p
'The number of R- and THETA-values must be equal.') =2s5>Oz+
end 1B+MCt4
vpFN{UfD
% Check normalization: $/}*HWVZ
% -------------------- VE&
?Zd~
if nargin==5 && ischar(nflag) 'v*
=}k
isnorm = strcmpi(nflag,'norm'); 'BpK(PlUh
if ~isnorm k[6@\D-
error('zernfun:normalization','Unrecognized normalization flag.') AT<gV/1l
end A5!jrSyv
else oylY1~~}0K
isnorm = false; +&jWM-T"-
end _K)B
<P3r+ 1|R
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <t,uj.9_
% Compute the Zernike Polynomials k
sJz44
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% XrYMv
WT
Nb>|9nu
O
% Determine the required powers of r: R@5jEf
% ----------------------------------- L5(rP\B
m_abs = abs(m); 8Z4d<DIJ
rpowers = []; S5@/;T
for j = 1:length(n) HelC_%#^
rpowers = [rpowers m_abs(j):2:n(j)]; Mlb=,l
end F:%= u
=
rpowers = unique(rpowers); <GF)5QB
_b<Fz`V
% Pre-compute the values of r raised to the required powers, "FT5]h
% and compile them in a matrix: (sW:^0 p
% ----------------------------- 4;M
if rpowers(1)==0 mn{8"@Z
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); D%LqLLD
rpowern = cat(2,rpowern{:}); fGcAkEstT!
rpowern = [ones(length_r,1) rpowern]; (zWzF_v
else q]0a8[]3
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); x +!<_p
rpowern = cat(2,rpowern{:}); brb8C%j}9
end QUaz;kNC7
qBpv[m
% Compute the values of the polynomials: c,Zs.
kC
% -------------------------------------- vz)A~"E
y = zeros(length_r,length(n)); u'd+:uH
for j = 1:length(n) RsIEY5Q
s = 0:(n(j)-m_abs(j))/2; Tg&{P{$
pows = n(j):-2:m_abs(j); Y:^~KS=Uz
for k = length(s):-1:1 ;wbQTp2
p = (1-2*mod(s(k),2))* ... ~=Z&l
prod(2:(n(j)-s(k)))/ ... 0Tp?ED_
prod(2:s(k))/ ... O4@Ki4f3A%
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... G-G!c2o
prod(2:((n(j)+m_abs(j))/2-s(k))); gT<E4$I69
idx = (pows(k)==rpowers); zG[fPD
y(:,j) = y(:,j) + p*rpowern(:,idx); Y)N(uv6
end ;8JJ#ED
r[eZV"
if isnorm [@";\C_I
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); R I:x`do
end <.HHV91
end yH|ucN~k5S
% END: Compute the Zernike Polynomials WnLgpt2G
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =3{h9
z<+".sD'
% Compute the Zernike functions: K-)*S\<}
% ------------------------------ YUF!Y9!
idx_pos = m>0; 4adCMfP7.
idx_neg = m<0; m1gJ"k6
`j
; QR|v
z = y; -vGyEd7
if any(idx_pos) _R1UEE3M
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); N(dn"`8
end C N"Vw
if any(idx_neg) Fw+JhIVP
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); n #p6i
end [{Fr{La`D'
( iP,F]
% EOF zernfun