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 6}c!>n['
function z = zernfun(n,m,r,theta,nflag) rOEBL|P0
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. fQ!W)>mi
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N u R5h0Fi
% and angular frequency M, evaluated at positions (R,THETA) on the ,f,+) C$
% unit circle. N is a vector of positive integers (including 0), and bVN?7D(
% M is a vector with the same number of elements as N. Each element w;AbJCv2
% k of M must be a positive integer, with possible values M(k) = -N(k) f]?&R c2C
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, D5bi)@G7z
% and THETA is a vector of angles. R and THETA must have the same 55<!H-zt
% length. The output Z is a matrix with one column for every (N,M) z%+rI
% pair, and one row for every (R,THETA) pair. 4%_c9nat
% $kmY[FWu?
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike `uusUw-Gf
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 5-({z%:P
% with delta(m,0) the Kronecker delta, is chosen so that the integral hDUU_.q)D
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, eA?|X|
% and theta=0 to theta=2*pi) is unity. For the non-normalized p}gA8o
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. y<R5}F
% Gkfzb>_V]
% The Zernike functions are an orthogonal basis on the unit circle. L5KcI
% They are used in disciplines such as astronomy, optics, and 0Db=/sJ>
% optometry to describe functions on a circular domain. x00'wY|
% if\`M'3Xx
% The following table lists the first 15 Zernike functions. Em{;l:;(W
% x.|sCqx
% n m Zernike function Normalization Rx&.,gzj[
% -------------------------------------------------- N;!!*3a9=
% 0 0 1 1 j8^#698X
% 1 1 r * cos(theta) 2 u:W/6QS
% 1 -1 r * sin(theta) 2 "66#F
% 2 -2 r^2 * cos(2*theta) sqrt(6) a7u*d`3X=
% 2 0 (2*r^2 - 1) sqrt(3) ;tA$
x!5]
% 2 2 r^2 * sin(2*theta) sqrt(6) +N2ILE8[<
% 3 -3 r^3 * cos(3*theta) sqrt(8) {dE(.Z?]!#
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) DOkuT/+
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) wzoT!-_X
% 3 3 r^3 * sin(3*theta) sqrt(8) :h3U^
% 4 -4 r^4 * cos(4*theta) sqrt(10) !>Q\Y`a,*
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^4\0,>
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) oGg<s3;UND
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) MMD=4;X
% 4 4 r^4 * sin(4*theta) sqrt(10) [Ran/D\.
% -------------------------------------------------- Tl]yl$
% P;'ZdZ(SLu
% Example 1: RwrRN+&s\
% uocFOlU0n
% % Display the Zernike function Z(n=5,m=1) KV6D0~
% x = -1:0.01:1; #(+V&<K
% [X,Y] = meshgrid(x,x); V;J3lV<
% [theta,r] = cart2pol(X,Y); W.D>$R2
% idx = r<=1; WI&}94w
% z = nan(size(X)); OmfHrlA
% z(idx) = zernfun(5,1,r(idx),theta(idx)); m,]9\0GUd
% figure zq ?xY`E
% pcolor(x,x,z), shading interp a gL@A
% axis square, colorbar mC(YO y
% title('Zernike function Z_5^1(r,\theta)') EaL>~:j
% {/aHZ<I&^h
% Example 2: Y!Io @{f
% "}-S%v`)z
% % Display the first 10 Zernike functions QJjk#*?,|
% x = -1:0.01:1; ,\RR@~u'
% [X,Y] = meshgrid(x,x); 4HGS
% [theta,r] = cart2pol(X,Y); QX=x^(M$m
% idx = r<=1; -m
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% z = nan(size(X)); # M!1W5#
% n = [0 1 1 2 2 2 3 3 3 3]; ,]n~j-X
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; pNmWBp|ER
% Nplot = [4 10 12 16 18 20 22 24 26 28]; M&Ln'BC
% y = zernfun(n,m,r(idx),theta(idx)); >XM]UdP
% figure('Units','normalized') *_}0vd
% for k = 1:10 #<u;.'R
% z(idx) = y(:,k); C'Y2kb
% subplot(4,7,Nplot(k)) !<~cjgdx
% pcolor(x,x,z), shading interp /J&DYxl":
% set(gca,'XTick',[],'YTick',[]) b8vZ^8tBV
% axis square i*!2n1c[
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) |pq9i)e&
% end WA:r4V
% n:k4t
% See also ZERNPOL, ZERNFUN2. SQx&4R.
n;>=QG
-v
% Paul Fricker 11/13/2006 .`v%9-5v
=]"I0G-s!
m_`%#$s}
% Check and prepare the inputs: b&LAk-}[
% ----------------------------- ?0+g.,9
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) E7aG&K
error('zernfun:NMvectors','N and M must be vectors.') =1,1}OucP
end Sw5-^2x0'
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if length(n)~=length(m) (vchZn#
error('zernfun:NMlength','N and M must be the same length.') hv\Dz*XTs0
end x.] tGS
*-Vr=e<8
n = n(:); GCfVH?Vx
m = m(:); /m 7~-~$V
if any(mod(n-m,2)) be5N{lPT@;
error('zernfun:NMmultiplesof2', ... .sFN[>)
'All N and M must differ by multiples of 2 (including 0).') &Vgpv#&Cfx
end WBm)Q#1:
*vvm8ik
if any(m>n) }@tgc?CD
error('zernfun:MlessthanN', ... 1)zXv
'Each M must be less than or equal to its corresponding N.') ~{vB2
end J1~E*t^
.V3e>8gw3
if any( r>1 | r<0 ) wEJzLFCn
error('zernfun:Rlessthan1','All R must be between 0 and 1.') BNI)y@E^X
end jiLJiYMg
Zzz94`
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Z,Us<du
error('zernfun:RTHvector','R and THETA must be vectors.') (+/d*4
end ecQ,DOX|b
xk7Dx}
r = r(:); _:p-\Oo.
theta = theta(:); i*@PywT"i3
length_r = length(r); L/]
(pXEp
if length_r~=length(theta) 9|v%bO
error('zernfun:RTHlength', ... uN>5Eh&=Pf
'The number of R- and THETA-values must be equal.') aW{5m@p{"
end ACZK]~Y'N*
>!a- "
% Check normalization: a'dlAda
% -------------------- C"_ Roir?
if nargin==5 && ischar(nflag) ;B[(~LCyT
isnorm = strcmpi(nflag,'norm'); .Y^cs+-o
if ~isnorm Z*UVbyC
error('zernfun:normalization','Unrecognized normalization flag.') <'SS IMr
end *h3iAcM8
else 7C,giCYU
isnorm = false; }*xjO/Ey
end $u yx
hwJ>IQ1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Gsb^gd
% Compute the Zernike Polynomials 9:-7.^`P
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @|Yn~PwKs
vlE]RB
% Determine the required powers of r: 2{vAs
% ----------------------------------- [wnp]'+!
m_abs = abs(m); >$E;."a
rpowers = []; [w|Klq5
for j = 1:length(n) _ezRE"F5
rpowers = [rpowers m_abs(j):2:n(j)]; $/;K<*O$
end '@Rk#=85Z
rpowers = unique(rpowers); BI %XF
9{
vB{iw}Hi!
% Pre-compute the values of r raised to the required powers, ~?HK,`0h>
% and compile them in a matrix: {B4qeG5
% ----------------------------- "`4ky]
if rpowers(1)==0 (tg9"C
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Ddpcov
rpowern = cat(2,rpowern{:}); 2b^Fz0
w4
rpowern = [ones(length_r,1) rpowern]; \U>&W
else 2Ki_d
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); S)j(%g
rpowern = cat(2,rpowern{:}); 09jE7g @X}
end Y<irNp9
~~-VScG&
% Compute the values of the polynomials: #fns3=/H
% -------------------------------------- [X!w@d= i
y = zeros(length_r,length(n)); gK({InOP
for j = 1:length(n) w]{c*4o
s = 0:(n(j)-m_abs(j))/2; PgT8
1u
pows = n(j):-2:m_abs(j); 111A e*U
for k = length(s):-1:1 H)7v$A,5%
p = (1-2*mod(s(k),2))* ... /]!2k9u\
prod(2:(n(j)-s(k)))/ ... igk<]AwxS
prod(2:s(k))/ ... P@vUQ
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... BOG.[?yx
prod(2:((n(j)+m_abs(j))/2-s(k))); $Vq5U9-
idx = (pows(k)==rpowers); WK(X/!1/k
y(:,j) = y(:,j) + p*rpowern(:,idx); 8{2
end &s vg<UZ
DR}I+<*%aD
if isnorm b&&l
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); B#jnM~fJz
end uMZ~[Sz
end n>j2$m1[
% END: Compute the Zernike Polynomials DlE, aYB
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I@/
G#3Zr
pQ:^ ziwa3
% Compute the Zernike functions: .G!xcQ`?
% ------------------------------ S,AxrQc
idx_pos = m>0; "}*D,[C5e
idx_neg = m<0; b2UDP W
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z = y;
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if any(idx_pos) NOXP}M
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); DMG~56cTO,
end '!7>*<
if any(idx_neg) Nyy&'\`!
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); _Ik?WA_;
end W?.469yy
&3Zb?
% EOF zernfun