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 <Dx]b*H
function z = zernfun(n,m,r,theta,nflag) 0Io'bF
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 6"c1;P!4
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N t+,4Ya|Xj
% and angular frequency M, evaluated at positions (R,THETA) on the 5TBp'7 /s~
% unit circle. N is a vector of positive integers (including 0), and "MIq.@8ra
% M is a vector with the same number of elements as N. Each element AamVms
% k of M must be a positive integer, with possible values M(k) = -N(k) l5+gsEux]
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, bs9aE<j
% and THETA is a vector of angles. R and THETA must have the same e*+FpW@
% length. The output Z is a matrix with one column for every (N,M) ,!V]jP)
% pair, and one row for every (R,THETA) pair. p8s:g~ W
% ]"c+sMW
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike tO_H!kP
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Y(\T-
bI
% with delta(m,0) the Kronecker delta, is chosen so that the integral !*2%"H*
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, #W.vX?-'0
% and theta=0 to theta=2*pi) is unity. For the non-normalized Qb8KPpd
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 2_Wg!bq
% 6#j$GH *
% The Zernike functions are an orthogonal basis on the unit circle. 0&ByEN99
% They are used in disciplines such as astronomy, optics, and SI:U0gUc
% optometry to describe functions on a circular domain. 7iJ&6=/
% JQ:Ri
% The following table lists the first 15 Zernike functions. AmwWH7,g
% X(jVRr_m9
% n m Zernike function Normalization Hi_G
% -------------------------------------------------- 'qdPw%d
% 0 0 1 1 K[chjp!$l
% 1 1 r * cos(theta) 2 ogFKUD*h&>
% 1 -1 r * sin(theta) 2 uxg9yp@|
% 2 -2 r^2 * cos(2*theta) sqrt(6) M MzGd:0b
% 2 0 (2*r^2 - 1) sqrt(3) 8q`$y$06Dk
% 2 2 r^2 * sin(2*theta) sqrt(6) Mg#j3W}]
% 3 -3 r^3 * cos(3*theta) sqrt(8) yqSs,vz
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) (M
=Y&M'f
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Y!6/[<r$~k
% 3 3 r^3 * sin(3*theta) sqrt(8) u *
% 4 -4 r^4 * cos(4*theta) sqrt(10) 1 nvTce
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) vzF5xp.
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) s:00yQ
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) smG>sEp2
% 4 4 r^4 * sin(4*theta) sqrt(10) %+ZJhHT
% -------------------------------------------------- +i\&6HGK;-
% iJnU%
% Example 1: iTW? W\d
% yT{8d.Rh
% % Display the Zernike function Z(n=5,m=1) (;VVCAoy
% x = -1:0.01:1; ,]}?.g
% [X,Y] = meshgrid(x,x); E,n}HiAz7V
% [theta,r] = cart2pol(X,Y); K/ &?VIi`z
% idx = r<=1; H A}f,),G
% z = nan(size(X)); `si#aU
% z(idx) = zernfun(5,1,r(idx),theta(idx)); *&AfR8x_z
% figure ylKmj]A
% pcolor(x,x,z), shading interp /v095H@
% axis square, colorbar c:83LZ
% title('Zernike function Z_5^1(r,\theta)') -/]W+[
% nN$Y(2ZN
% Example 2: XWJwJ
% ( 6(x'ByT
% % Display the first 10 Zernike functions @DW[Z`X
% x = -1:0.01:1; ?=GXqbS"
% [X,Y] = meshgrid(x,x); 5 ,0d
% [theta,r] = cart2pol(X,Y); +.RKi!
% idx = r<=1; crO@?m1
% z = nan(size(X)); |}){}or
% n = [0 1 1 2 2 2 3 3 3 3]; JO14KY*%
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; m~Ld~I"
% Nplot = [4 10 12 16 18 20 22 24 26 28]; QrApxiw
% y = zernfun(n,m,r(idx),theta(idx)); p2PY@d}}.
% figure('Units','normalized') k7tYa;C
% for k = 1:10 w@2Vts
% z(idx) = y(:,k); Cw5%\K$=
% subplot(4,7,Nplot(k)) ,mPnQ?
% pcolor(x,x,z), shading interp BF{w)=@/'
% set(gca,'XTick',[],'YTick',[]) = sAn,ri
% axis square zU6a'tP
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) UEak^Mm;=2
% end 3Q/#T1@
% JcJmds
% See also ZERNPOL, ZERNFUN2. _wb0'xoK"
Ba\6?K
% Paul Fricker 11/13/2006 2A4FaBq"
~.PP30'
R E1/"[t
% Check and prepare the inputs: Li 2Zndp
% ----------------------------- M(|
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) w",?
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error('zernfun:NMvectors','N and M must be vectors.') TG
n-7 88
end '2hbJk
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if length(n)~=length(m) ,n2i@?NHZ
error('zernfun:NMlength','N and M must be the same length.') 0;,IKXK6X
end dQy>Nmfy
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n = n(:); s!/TU{8J
m = m(:); 7iuQ9q^&
if any(mod(n-m,2)) T~sTBGcv
error('zernfun:NMmultiplesof2', ... P`U<7xF~
'All N and M must differ by multiples of 2 (including 0).') ryO$6L
end C@o%J.9"#
4VN aq<8
if any(m>n) 3`9{T>
error('zernfun:MlessthanN', ... /E wGW
'Each M must be less than or equal to its corresponding N.') \^*<
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end *X%m@KLIKv
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if any( r>1 | r<0 ) b(GFMk
error('zernfun:Rlessthan1','All R must be between 0 and 1.') jc_\'Gr+[
end b7C
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fbZibcQ%k
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) QzGV.Mt2
error('zernfun:RTHvector','R and THETA must be vectors.') tXF]t
end E_$z`or
4{9d#[KW
r = r(:); l#3($QV,
theta = theta(:); G,&%VQ3P>
length_r = length(r); =fc:6JR
if length_r~=length(theta) \d.F82
error('zernfun:RTHlength', ... yI:#
|w|
'The number of R- and THETA-values must be equal.') ? y},,
end V6iL5&
>L((2wfiN
% Check normalization: @-.? B
% -------------------- mkvvNm3
if nargin==5 && ischar(nflag)
Ex@`O+
isnorm = strcmpi(nflag,'norm'); y_F}s9wj
if ~isnorm @^nu#R
error('zernfun:normalization','Unrecognized normalization flag.') @%tXFizh
end M%Ku5X6:/
else LR)&
[{Kk
isnorm = false; >AD=31lq
end }|8*sk#[
g+q@i{Yn
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .I?@o8'x
% Compute the Zernike Polynomials A,i()R'I
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% lXrD!1F
lpQP"%q
% Determine the required powers of r: P1 +"v*
% ----------------------------------- fhp)S",
m_abs = abs(m); 74vmt<Q
rpowers = []; wN]J8Ir
for j = 1:length(n) -@%%*YI>
rpowers = [rpowers m_abs(j):2:n(j)]; y<r}"TAf-
end W|Ldu;#
rpowers = unique(rpowers); f~& a-
O?K./So&
% Pre-compute the values of r raised to the required powers, eVy2|n9rH
% and compile them in a matrix: |:iEfi]j
% ----------------------------- ryD%i"g<
if rpowers(1)==0 m$UvFP1>u1
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); /MMtTB
H
rpowern = cat(2,rpowern{:}); OS7RQw1
rpowern = [ones(length_r,1) rpowern]; vx 0UoKX
else ?_4^le[;
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); a#(U2OP
rpowern = cat(2,rpowern{:}); 7s>a2
end \d68-JS@~
#;j9}N
% Compute the values of the polynomials: Z}Cqd?_')
% -------------------------------------- 3l:XhLOj
y = zeros(length_r,length(n)); w-FnE}"l
for j = 1:length(n) v+q<BYq
s = 0:(n(j)-m_abs(j))/2; Y5TS>iEE]
pows = n(j):-2:m_abs(j); L4974E?S
for k = length(s):-1:1 l)}t,!M6
p = (1-2*mod(s(k),2))* ... eqzTQen8q
prod(2:(n(j)-s(k)))/ ... X\2_;zwf
prod(2:s(k))/ ... ,7/
_T\d<
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... k&Jo"[i&WO
prod(2:((n(j)+m_abs(j))/2-s(k))); tP'GNsq+m
idx = (pows(k)==rpowers); >[K?fJ$+
y(:,j) = y(:,j) + p*rpowern(:,idx); 2;(W-]V?
end ]6~k4
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if isnorm w!,QxrOV~
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); v&0d$@6/U
end B3b,F #
end #tz8{o?ebN
% END: Compute the Zernike Polynomials qzdaN5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fGO*%)
E`E'<"{Yd
% Compute the Zernike functions: _Xh=&(/8@
% ------------------------------ kyAs'R@z
idx_pos = m>0; !LSs9_w
idx_neg = m<0; ,VG9)K1K
2ij/N%l
z = y; BR3mAF
if any(idx_pos) 0VG=?dq
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 75 Fp[Q-
end YRa4W.&Yn
if any(idx_neg) Sr7@ buF
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); nZW4} ~0j
end &q>h*w4O
&wGg6$
% EOF zernfun