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 7QFEQ}
function z = zernfun(n,m,r,theta,nflag) je%12DM
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 1nmWL0
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ,"ZlY}!Gn
% and angular frequency M, evaluated at positions (R,THETA) on the (k45k/PAP
% unit circle. N is a vector of positive integers (including 0), and 6*Qpq7Ml
% M is a vector with the same number of elements as N. Each element i i
Y[
% k of M must be a positive integer, with possible values M(k) = -N(k) =0Sa
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, @]4 s&;
% and THETA is a vector of angles. R and THETA must have the same 'M/&bu r
% length. The output Z is a matrix with one column for every (N,M) s:H1v&t,<
% pair, and one row for every (R,THETA) pair. + k:?;ZG
% WKML#U]5T
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ocUu
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), SO"P3X
% with delta(m,0) the Kronecker delta, is chosen so that the integral @I:&ozy }=
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, (1vS)v
$L
% and theta=0 to theta=2*pi) is unity. For the non-normalized "(GeW286k
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. =G6@:h=
% nX'.'3
% The Zernike functions are an orthogonal basis on the unit circle. !y.7"G*
% They are used in disciplines such as astronomy, optics, and r>o6}Mx$
% optometry to describe functions on a circular domain. :f:C*mYvu
% Z0KA4O$eL
% The following table lists the first 15 Zernike functions. [j39A`t7
o
% Hy'&x?F6
% n m Zernike function Normalization "?-s
Qn
% -------------------------------------------------- Tr)[q>
% 0 0 1 1 ~~mQ
% 1 1 r * cos(theta) 2 l:HuG!
% 1 -1 r * sin(theta) 2 )-gyDA
% 2 -2 r^2 * cos(2*theta) sqrt(6) M:E#}(
% 2 0 (2*r^2 - 1) sqrt(3) <D}k@M
Z
% 2 2 r^2 * sin(2*theta) sqrt(6) j/&7L@Y
% 3 -3 r^3 * cos(3*theta) sqrt(8) XlPy(>
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) T8LwDqio
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ,H8Pmn?
% 3 3 r^3 * sin(3*theta) sqrt(8) Dlp::U*N'
% 4 -4 r^4 * cos(4*theta) sqrt(10) pP&~S<[
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Xob##{P3
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) bql6Z1l
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) SrIynO
% 4 4 r^4 * sin(4*theta) sqrt(10) m{|n.b
% -------------------------------------------------- Zlhr0itf
% '1<QK
% Example 1: ; V8 =B8w
% X@rAe37h+
% % Display the Zernike function Z(n=5,m=1) lKcnM3n
% x = -1:0.01:1; XT)@)c7j
% [X,Y] = meshgrid(x,x); %o>1$f]
% [theta,r] = cart2pol(X,Y); e!#:h4I
% idx = r<=1; wB@A?&UY
% z = nan(size(X)); u}$3.]-.?T
% z(idx) = zernfun(5,1,r(idx),theta(idx)); $1YnQgpT
% figure S3w? X
% pcolor(x,x,z), shading interp +}]xuYzo
% axis square, colorbar qW*)]s)z
% title('Zernike function Z_5^1(r,\theta)') [/FIY!nC?
% PYGHN
T
% Example 2: oVdmgmT.Y
% zK v}J
% % Display the first 10 Zernike functions wbTw\b=
% x = -1:0.01:1; V.qB3V$
% [X,Y] = meshgrid(x,x); $|KbjpQ
% [theta,r] = cart2pol(X,Y); Jc*A\-qC.
% idx = r<=1; 8I%1
`V
% z = nan(size(X)); 4?`7XJ0a
% n = [0 1 1 2 2 2 3 3 3 3]; q-'zZ#
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; tP3Upw"U
% Nplot = [4 10 12 16 18 20 22 24 26 28]; raCxHY
% y = zernfun(n,m,r(idx),theta(idx)); {8eNQ-4I
% figure('Units','normalized') %VgR *
% for k = 1:10 74_ji!
% z(idx) = y(:,k); B4%W,F:@
% subplot(4,7,Nplot(k)) ~_Aclm?
% pcolor(x,x,z), shading interp 0[^f9NZ>-
% set(gca,'XTick',[],'YTick',[]) :0/I2:
% axis square !U@[lBW
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) sNWj+T
% end 0=NB[eG
% YIfbcR5
% See also ZERNPOL, ZERNFUN2. B--`=@IRf"
\7RP6o
% Paul Fricker 11/13/2006 wNn6".S
Xh5
z8
}0:=)e
% Check and prepare the inputs: j:g/[_0s
% ----------------------------- u?!p[y6
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Gmc0yRN
error('zernfun:NMvectors','N and M must be vectors.') z'
@F@k6
end =73wngw
7C=t19&R'
if length(n)~=length(m) HghNI
error('zernfun:NMlength','N and M must be the same length.') Hc71 .rqS
end JHcC}+H[
%%*t{0!H+
n = n(:); w1[F]|
m = m(:); rQU;?[y
if any(mod(n-m,2)) ^j@,N&W:lG
error('zernfun:NMmultiplesof2', ... >#SQDVFf
'All N and M must differ by multiples of 2 (including 0).') HA| YLj?|g
end vNP,c]:%
EI'(
if any(m>n) LbnR=B!
error('zernfun:MlessthanN', ... IL\#!|>
'Each M must be less than or equal to its corresponding N.') p tMysYT'
end .- {B
o@}Jd0D4
if any( r>1 | r<0 ) P'[w9'B
error('zernfun:Rlessthan1','All R must be between 0 and 1.') A >Js`s
end jlItPdCv
0EOpK%{
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ZhW>H
error('zernfun:RTHvector','R and THETA must be vectors.') _&P![o)x
end 3eD#[jkAI;
%c):^;6p
r = r(:); |dK_^~;o
theta = theta(:); '6WaG
hvO
length_r = length(r); n>{>3?
if length_r~=length(theta) SBs_rhe
error('zernfun:RTHlength', ... '~2;WF0h
'The number of R- and THETA-values must be equal.') Y6f0 ?lB
end z>~Hc8*]3
:`25@<*u
% Check normalization: \)pk/
% -------------------- 52=?!
JM
if nargin==5 && ischar(nflag) ^8-CUH\
isnorm = strcmpi(nflag,'norm'); qlO(z5Ak
if ~isnorm Z3)1!|#Q
error('zernfun:normalization','Unrecognized normalization flag.') iXeywO2nP
end 4 QD.'+L
else YvR MUT
isnorm = false; 1t6VS 3
end wpO-cJ!,
D3N\$ D
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% gq!|0
% Compute the Zernike Polynomials /aP4'U8ov
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% crG+BFi
Nw*
>$v
% Determine the required powers of r: B[}#m'Lv
% ----------------------------------- C[z5&
x2
m_abs = abs(m); ]25 x X
rpowers = []; U:"E:Bxz;m
for j = 1:length(n) NLf6}
rpowers = [rpowers m_abs(j):2:n(j)]; >d%;+2
end r$<[`L+6
rpowers = unique(rpowers); hKj"Lb9]
&N.D!7X
% Pre-compute the values of r raised to the required powers, w-LMV>+6|
% and compile them in a matrix: |5^tp
% ----------------------------- 9q(*'rAm
if rpowers(1)==0 -AWL :<
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); LR|L P)I
rpowern = cat(2,rpowern{:}); :A9G>qg
rpowern = [ones(length_r,1) rpowern]; hi^@969
else d ]R&mp|'
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 'tm%3`
F
rpowern = cat(2,rpowern{:}); ~ (I'm[
end &;I=*B~kE$
;Sl]8IZ
% Compute the values of the polynomials: Ev+m+
% -------------------------------------- ~`~mnlN
y = zeros(length_r,length(n)); FwKT_XkY
for j = 1:length(n) '7Q5"M'
s = 0:(n(j)-m_abs(j))/2; R-5EztmLae
pows = n(j):-2:m_abs(j); ] ;"blB
for k = length(s):-1:1 /Sy:/BQ
p = (1-2*mod(s(k),2))* ... J0K25w
prod(2:(n(j)-s(k)))/ ... ;w--fqxVl
prod(2:s(k))/ ... ancs
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... *c9/ I
prod(2:((n(j)+m_abs(j))/2-s(k))); Kw_> X&GcJ
idx = (pows(k)==rpowers);
_8]hn[
y(:,j) = y(:,j) + p*rpowern(:,idx); <_(UAv
end {kVhht]X
9=D09@A%e
if isnorm W(.q.Sx>
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); a$-:F$z
end KVQ|l,E,
/
end AM?62
% END: Compute the Zernike Polynomials <Wqk5mR
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RHe'L36W
(nL''#Ka
% Compute the Zernike functions: fg}&=r
% ------------------------------ ` 9iB`<
idx_pos = m>0; ] /w:5o#
idx_neg = m<0; b8o}bm{s
C5k\RS9
z = y; 33/aYy
if any(idx_pos) SY &)?~C
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ,j^z];
end $w%n\t>B
if any(idx_neg) uv>T8(w
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); fZ8at
end ^6c=[N$aW
U5_1-wV
% EOF zernfun