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 t#-4edB,
function z = zernfun(n,m,r,theta,nflag) r&:yZN
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. l2wu>Ar7.
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N DiZv sc
% and angular frequency M, evaluated at positions (R,THETA) on the ="Azg8W
% unit circle. N is a vector of positive integers (including 0), and <$#^)]Ts
% M is a vector with the same number of elements as N. Each element :3J`+V}9;
% k of M must be a positive integer, with possible values M(k) = -N(k) ~(`MP<
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, E>2AG3)
% and THETA is a vector of angles. R and THETA must have the same 8|+@A1)&4
% length. The output Z is a matrix with one column for every (N,M) 1 .o0"
% pair, and one row for every (R,THETA) pair. {W%XSE
% ^?A>)?Sq
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ' 8Q}pp`
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), c`\qupnY
% with delta(m,0) the Kronecker delta, is chosen so that the integral mQ<Vwx0
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Z]5xy_La
% and theta=0 to theta=2*pi) is unity. For the non-normalized &0d5".|s
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. "{~^EQq,
% Y
7?q`
% The Zernike functions are an orthogonal basis on the unit circle. 8k.#4}fP
% They are used in disciplines such as astronomy, optics, and 4CS$%Cu\?w
% optometry to describe functions on a circular domain. w7\
\m9
% R[m+s=+
% The following table lists the first 15 Zernike functions. Kv#Q$$)r
% ,.;{J|4P
% n m Zernike function Normalization 9c5DEq
% -------------------------------------------------- Tq6\oIBkV
% 0 0 1 1 xsvJjs;=
% 1 1 r * cos(theta) 2 A-M6MW
% 1 -1 r * sin(theta) 2 @f,/ K1k
% 2 -2 r^2 * cos(2*theta) sqrt(6) ?]+!gz1
% 2 0 (2*r^2 - 1) sqrt(3) 5F]2.<i
% 2 2 r^2 * sin(2*theta) sqrt(6) ]9wTAb
% 3 -3 r^3 * cos(3*theta) sqrt(8) f>Tn#OW
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) uNqN &7g
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ,WAJ&
'^
% 3 3 r^3 * sin(3*theta) sqrt(8) 5UG"i_TC
% 4 -4 r^4 * cos(4*theta) sqrt(10) 5)->.* G*
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) tU>7jo[-p
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) $2Bll 5!]
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 'S9jMyZrZ
% 4 4 r^4 * sin(4*theta) sqrt(10) fEGnI\
% -------------------------------------------------- #;;A~d:V
% "wxyY^"
% Example 1: _! ?a9
% { /
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% % Display the Zernike function Z(n=5,m=1) V%`\x\Xat
% x = -1:0.01:1; 3XncEdy_
% [X,Y] = meshgrid(x,x); 2cZgG^
% [theta,r] = cart2pol(X,Y); i7&ay\+@
% idx = r<=1; [LV>z
% z = nan(size(X)); @jZ1WHS_a
% z(idx) = zernfun(5,1,r(idx),theta(idx)); A3J=,aRI_v
% figure UunZ/A$]m
% pcolor(x,x,z), shading interp .B!
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% axis square, colorbar -"x@ V7X
% title('Zernike function Z_5^1(r,\theta)') AyOy&]g
% 8}Q2!,9Q
% Example 2: meGLT/
% :8]y*j
% % Display the first 10 Zernike functions @z=L\e{
% x = -1:0.01:1; F^?DnZs
% [X,Y] = meshgrid(x,x); bu=RU
% [theta,r] = cart2pol(X,Y); B!4~A{
% idx = r<=1; g]d0B!Ar~
% z = nan(size(X)); ,y}~rYsP%
% n = [0 1 1 2 2 2 3 3 3 3]; R ^INl@(O
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 0aJcX)
% Nplot = [4 10 12 16 18 20 22 24 26 28]; O]oH}#5b
% y = zernfun(n,m,r(idx),theta(idx)); 4MCj*ok<
% figure('Units','normalized') iAt&927
% for k = 1:10 CbOCL~ "
% z(idx) = y(:,k); ~*e@^Nv)v
% subplot(4,7,Nplot(k)) _KZTY`/*
% pcolor(x,x,z), shading interp WM
]eb, 8q
% set(gca,'XTick',[],'YTick',[]) !1
Y[e^
% axis square YU\k D
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Sf2xI'
% end v]}\Ns/
% _s}`ohKvD
% See also ZERNPOL, ZERNFUN2. q RRvZhf
:*YnH&
% Paul Fricker 11/13/2006 1R7tnR@[u
>.uIp4@(
F'T.-lEO_d
% Check and prepare the inputs: WS%yV|e
% ----------------------------- g|tclBx
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) COHook(:
error('zernfun:NMvectors','N and M must be vectors.') /Zxq-9
end Q 87'zf
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if length(n)~=length(m) v[)8 1uY
error('zernfun:NMlength','N and M must be the same length.') beNy5~M$
end Tl1H2s=G-
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n = n(:); Ko|gH]B'
m = m(:); D2RvFlAXu
if any(mod(n-m,2)) `A-
error('zernfun:NMmultiplesof2', ... ]Qe"S>,?`
'All N and M must differ by multiples of 2 (including 0).') FuG;$';H75
end
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IsDwa qd|
if any(m>n) ZKM@U?PK
error('zernfun:MlessthanN', ... F3L+X5D.yu
'Each M must be less than or equal to its corresponding N.') t/l<X]o
end ,hm&]
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if any( r>1 | r<0 ) 4[)tO-v:Y
error('zernfun:Rlessthan1','All R must be between 0 and 1.')
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end ph~BxK )i6
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ` $q0fTz
error('zernfun:RTHvector','R and THETA must be vectors.') tq51;L
end I+31:#d
T`9nY!
r = r(:); 1-E utq
theta = theta(:); M`E}1WNQ?]
length_r = length(r); `Jh<8~1
if length_r~=length(theta) +k[w)7Q
error('zernfun:RTHlength', ... nj1PR`AE
'The number of R- and THETA-values must be equal.') <j3|Mh_(I
end >]uu?!PU
}daU/
% Check normalization: 9SJSUv:@
% -------------------- }_('3C,Ba
if nargin==5 && ischar(nflag) {qOqtkj
isnorm = strcmpi(nflag,'norm'); }(,{^".[}
if ~isnorm Z*-a=u%gl'
error('zernfun:normalization','Unrecognized normalization flag.') 9'@G7*Yn
end {WQ6=wGpS
else HJP~
lg
isnorm = false; i1'G_bo4F7
end oxdX2"WwU
Nr).*]g@~
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% KP7 {
% Compute the Zernike Polynomials UcH#J &r
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \
FJ ae
[B +:)i
% Determine the required powers of r: (/s~L*gF{
% ----------------------------------- z 7+>G/o
m_abs = abs(m); 6ud<U#\b&
rpowers = []; }D.\2x(J
for j = 1:length(n) 96P&+
rpowers = [rpowers m_abs(j):2:n(j)]; >s1?rC
end N;k )>
rpowers = unique(rpowers); $PAAmaigi
$?dQ^]<,
% Pre-compute the values of r raised to the required powers, /Gn0|]KI
% and compile them in a matrix: PB!XApTb
% ----------------------------- M|zTs\1I
if rpowers(1)==0 L&~' SC
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); D@:'*Z(
rpowern = cat(2,rpowern{:}); \9uK^oS
rpowern = [ones(length_r,1) rpowern]; B|,d
else 1
-C~C]&
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); FCWk8/
rpowern = cat(2,rpowern{:}); +S`cUn7
end 9!kp3x/`
<q>d@Foi
% Compute the values of the polynomials: j%Xa8$
% -------------------------------------- 6>
z{xYat
y = zeros(length_r,length(n)); yz5! >|EB
for j = 1:length(n) L#J2J$=
s = 0:(n(j)-m_abs(j))/2; vU]n0)<KB
pows = n(j):-2:m_abs(j); gS@<sO$d>
for k = length(s):-1:1 V!>j:"
p = (1-2*mod(s(k),2))* ... %QEyvl4
prod(2:(n(j)-s(k)))/ ... El: @l%
prod(2:s(k))/ ... 1iNMgA
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 9*huO#
prod(2:((n(j)+m_abs(j))/2-s(k))); y)a)VvU":
idx = (pows(k)==rpowers); @65xn)CD{
y(:,j) = y(:,j) + p*rpowern(:,idx); yn_.
end -ZyY95E<
m l@%H
if isnorm 8FZC0j.^DH
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); MLg{Y?@
end f-ceDn
end x<' $
% END: Compute the Zernike Polynomials cza_LO(
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 72.Msnn
{?2|rv)
% Compute the Zernike functions: !pkIaCxs
% ------------------------------ ';c 6
idx_pos = m>0;
3bR%#G%
idx_neg = m<0; R!lug;u#
ICr.Gwe3_
z = y; 0:<Y@#L
if any(idx_pos) EWgJ"WTF
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); wf &Jd:)4t
end 41s\^'^&
if any(idx_neg) 9 wbQ$>G9
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ZS;V?]\(
end C/#pK2xY
/fQ}Ls\
% EOF zernfun