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 u7^(?"x
function z = zernfun(n,m,r,theta,nflag) P3`$4p?
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 7UY4* j|[C
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ~;?<OOt|wG
% and angular frequency M, evaluated at positions (R,THETA) on the xL1Li]fM!'
% unit circle. N is a vector of positive integers (including 0), and }NoP(&ebz*
% M is a vector with the same number of elements as N. Each element VP>*J`'H
% k of M must be a positive integer, with possible values M(k) = -N(k) ,cL;,YN
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, )l$}plT4
% and THETA is a vector of angles. R and THETA must have the same y+T[="W
% length. The output Z is a matrix with one column for every (N,M) ;}iB9 Tl
% pair, and one row for every (R,THETA) pair. "!D y[J
% 'AX5V-t
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike m3BL
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), O>pv/Ns
% with delta(m,0) the Kronecker delta, is chosen so that the integral Yb-{+H8{J
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, oz>2P.7
% and theta=0 to theta=2*pi) is unity. For the non-normalized X3a 9-
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. .=9WY_@SZ
% ;:j1FOj
% The Zernike functions are an orthogonal basis on the unit circle. zxx\jpBBk
% They are used in disciplines such as astronomy, optics, and |dqHpogh
% optometry to describe functions on a circular domain. OtoM
% vjS=ZinN"
% The following table lists the first 15 Zernike functions. ;<N:! $p
% uf90
% n m Zernike function Normalization 'Gqv`rq&
% -------------------------------------------------- %2T
i
Rb
% 0 0 1 1 |bz%SB
% 1 1 r * cos(theta) 2 3PGAUQR#"q
% 1 -1 r * sin(theta) 2 ^l|b>z"0ao
% 2 -2 r^2 * cos(2*theta) sqrt(6)
>iae2W`
% 2 0 (2*r^2 - 1) sqrt(3) chKK9SC+|
% 2 2 r^2 * sin(2*theta) sqrt(6) y7M{L8{0
% 3 -3 r^3 * cos(3*theta) sqrt(8) +x=)Kp>
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) cd1G.10
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) HB8s[]A:D
% 3 3 r^3 * sin(3*theta) sqrt(8) vo`&
% 4 -4 r^4 * cos(4*theta) sqrt(10) }VZExqm)
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) HK8sn1j
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 6ki2/ Q
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) s91[@rh/
% 4 4 r^4 * sin(4*theta) sqrt(10) @2/|rq
% -------------------------------------------------- 1*9.K'
% ?}Z t&(#
% Example 1: \O;2^
% (_zlCHB
% % Display the Zernike function Z(n=5,m=1) vUJ;D
% x = -1:0.01:1; -p&u=
% [X,Y] = meshgrid(x,x); 8^>c_%e}
% [theta,r] = cart2pol(X,Y); ]~I+d/k
d
% idx = r<=1; ve
ysW(z
% z = nan(size(X)); bu|.Jw"
% z(idx) = zernfun(5,1,r(idx),theta(idx)); +ODua@ULFB
% figure nf/?7~3?[
% pcolor(x,x,z), shading interp SOhM6/ID2/
% axis square, colorbar "0PrdZMx
% title('Zernike function Z_5^1(r,\theta)') \]V:>=ry>
% IibrZ/n6
% Example 2: Q2VF+g,
% 1j$\ 48Z
% % Display the first 10 Zernike functions \~l_w
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% x = -1:0.01:1; &)mZ~cPU3
% [X,Y] = meshgrid(x,x); 9p qsr~
% [theta,r] = cart2pol(X,Y); ZpVkgX4
% idx = r<=1; ZOqS"3j! j
% z = nan(size(X)); :J`@@H
% n = [0 1 1 2 2 2 3 3 3 3]; -!Myw&*\V
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; %hsCB
.r>|
% Nplot = [4 10 12 16 18 20 22 24 26 28]; e4tIO
% y = zernfun(n,m,r(idx),theta(idx)); ;Zd_2CZ
% figure('Units','normalized') b$,Hlh,^
% for k = 1:10 z6iKIw
$
% z(idx) = y(:,k); 2+gbMd4n
% subplot(4,7,Nplot(k)) HE,L8S
% pcolor(x,x,z), shading interp qh~bX
i!
% set(gca,'XTick',[],'YTick',[]) T+v*@#iJ_
% axis square iPTQqx-m$7
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ;>v.(0FE6
% end {R!yw`#^B
% |6*Bu1
% See also ZERNPOL, ZERNFUN2. CJ\a7=*i
)x|;%.8FX7
% Paul Fricker 11/13/2006 NS[eQ_rT
zl@^[km{
%+(AKZu:
% Check and prepare the inputs: /l*v *tl
% ----------------------------- eWcqf/4?"
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ep"[;$Eb
error('zernfun:NMvectors','N and M must be vectors.') _J
l(:r\%
end 0SIC=p=J
a{]=BY oL
if length(n)~=length(m) \)6glAtN
error('zernfun:NMlength','N and M must be the same length.') ?bB>}:~j)
end VI2lwE3
Tn}`VW~
n = n(:); )hZ7`"f,ZN
m = m(:); fwFJe(.
if any(mod(n-m,2)) D~6[C:m
error('zernfun:NMmultiplesof2', ... uQ5h5Cfz
'All N and M must differ by multiples of 2 (including 0).') DXLXGvcM
end N)X Tmh2v|
r<UVO$N
if any(m>n) k&dXK
error('zernfun:MlessthanN', ... ,MCTb '=G
'Each M must be less than or equal to its corresponding N.') z'q~%1t
end f$o^Xu
IOl_J>D]F
if any( r>1 | r<0 ) fu"cX;
error('zernfun:Rlessthan1','All R must be between 0 and 1.') TEC^|U`G
end U**8^:*y#:
F^yW3|Sb
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Y !<m8\
error('zernfun:RTHvector','R and THETA must be vectors.') ^[?y 2A:
end h6h6B.\Ld
(;l@d|g
r = r(:); kTb$lLG\xk
theta = theta(:); D\Ak-$kJ^
length_r = length(r); GcVQz[E
if length_r~=length(theta) 68tyWd}
error('zernfun:RTHlength', ... d51lTGH7Z
'The number of R- and THETA-values must be equal.') iq; |
i!
end |
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0'Pjnk-i
% Check normalization: 0JlNUO5Nt
% -------------------- VgHO&vU
if nargin==5 && ischar(nflag) s6 yvq#:
isnorm = strcmpi(nflag,'norm'); ],]Rv#`
if ~isnorm %B%_[<B
error('zernfun:normalization','Unrecognized normalization flag.') cJo%j -AM
end /Y0~BQC7!
else 0?7yM:!l
isnorm = false; -n _Y.~
end H/D=$)3op
P<]U
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% J>Ar(p
% Compute the Zernike Polynomials AFAg3/
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $J7V]c*-b
,!:c6F+
% Determine the required powers of r: FOiwA.:0
% ----------------------------------- ?H=YJK$k
m_abs = abs(m); k~tEUsv
rpowers = []; Qte5E}V`
for j = 1:length(n) .(@=L1C<}J
rpowers = [rpowers m_abs(j):2:n(j)]; :aNjh
end {T4_Xn -I
rpowers = unique(rpowers); z$1RD)TQB
0+>g/>
% Pre-compute the values of r raised to the required powers, A0{xt*g
% and compile them in a matrix: zj`c%9N+
% -----------------------------
'LYDJ~
if rpowers(1)==0 #/G!nN #
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); iXWHI3
rpowern = cat(2,rpowern{:}); g257jarkMF
rpowern = [ones(length_r,1) rpowern]; Ik:G5m<ta
else j\uZo.Ot+
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); F-;J N
rpowern = cat(2,rpowern{:}); ?@"@9na
end 6N@=*0kh-
q%3VcR$J
% Compute the values of the polynomials: /~"-q
% -------------------------------------- n_QuuUB
y = zeros(length_r,length(n)); g0,~|.
for j = 1:length(n) xg p)G!
s = 0:(n(j)-m_abs(j))/2; ~^F]t$rz
pows = n(j):-2:m_abs(j); FWW4n_74
for k = length(s):-1:1 Q7+WV`&
p = (1-2*mod(s(k),2))* ... 3!P^?[p3
prod(2:(n(j)-s(k)))/ ... ktU:Uq
prod(2:s(k))/ ... | R,dsBd
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 8{4'G$6
prod(2:((n(j)+m_abs(j))/2-s(k))); RRO@r}A!y
idx = (pows(k)==rpowers); >{^_]phlb
y(:,j) = y(:,j) + p*rpowern(:,idx); cj>@Jx}]M
end Sm/8VSY
`gl?y;xC
if isnorm HYl+xH'.j
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); "@x(2(Y&
end :V9Q<B^
end ]@U?hD
% END: Compute the Zernike Polynomials S]H[&o1o
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xM_#FxJb
2^XmtT
% Compute the Zernike functions: L4iWR/&
% ------------------------------ ckX8eg!f
idx_pos = m>0; }wf8y
idx_neg = m<0; #c|l|Xvq2
}cz58%
z = y; L`t786
(M
if any(idx_pos) SrA6}kS
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); IsE&k2 SD
end ~cr iZI/
if any(idx_neg) |1wZ`wGZ:L
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); UB@(r86d
end {JWixbA
i?_Q@uA~<:
% EOF zernfun