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 r6Vw!^]8u8
function z = zernfun(n,m,r,theta,nflag) \VIY[6sn\M
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 5QXU"kWH
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N QaEiP n~
% and angular frequency M, evaluated at positions (R,THETA) on the jCtk3No
% unit circle. N is a vector of positive integers (including 0), and Bx}"X?%S
% M is a vector with the same number of elements as N. Each element +?3RC$jyw
% k of M must be a positive integer, with possible values M(k) = -N(k) UJp'v_hN
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, #
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% and THETA is a vector of angles. R and THETA must have the same Rl0"9D87z
% length. The output Z is a matrix with one column for every (N,M) .j,xh )v"
% pair, and one row for every (R,THETA) pair. y_W?7S
% B [YyA
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Cb<7?),vK
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), !.V_?aYi8
% with delta(m,0) the Kronecker delta, is chosen so that the integral cy
mC?8<
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ,3}+t6O"
% and theta=0 to theta=2*pi) is unity. For the non-normalized &Q"vXs6Gt
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 3I}AA.h'00
% !~F oy F
% The Zernike functions are an orthogonal basis on the unit circle. "#0P*3-c
% They are used in disciplines such as astronomy, optics, and {df;R|8l
% optometry to describe functions on a circular domain.
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% 3HP
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a
% The following table lists the first 15 Zernike functions. af6<w.i
% 6 mLC{X[
% n m Zernike function Normalization mP15PZ
% -------------------------------------------------- # Dgkl
% 0 0 1 1 B[8RBTsA
% 1 1 r * cos(theta) 2 G='`*_$
% 1 -1 r * sin(theta) 2 1z2v[S&pk
% 2 -2 r^2 * cos(2*theta) sqrt(6) V#b*:E.cA
% 2 0 (2*r^2 - 1) sqrt(3) >#mKM%T2MJ
% 2 2 r^2 * sin(2*theta) sqrt(6) T$r/XAs
% 3 -3 r^3 * cos(3*theta) sqrt(8) xZ2 1iQeN
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) N@k'
s
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) yCkWuU9
% 3 3 r^3 * sin(3*theta) sqrt(8) \J?&XaO=
% 4 -4 r^4 * cos(4*theta) sqrt(10) q\!"FDOl4
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Dqwd=$2%
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ]!P6Z?
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 5M)B
% 4 4 r^4 * sin(4*theta) sqrt(10) ^_G#JJ\@$
% -------------------------------------------------- ~v/`
`s
% qx >Z@o
% Example 1: CP"5E?dcK
% Z9% u,Cb
% % Display the Zernike function Z(n=5,m=1) l1 08.ao
% x = -1:0.01:1; $`0^E#Nl
% [X,Y] = meshgrid(x,x); ~/SLGyu
% [theta,r] = cart2pol(X,Y); ^HP$r*
% idx = r<=1; T=V{3v@zs
% z = nan(size(X)); g_tEUaiK
% z(idx) = zernfun(5,1,r(idx),theta(idx)); g0/R\
% figure 3~WI3ZIR
% pcolor(x,x,z), shading interp \KpJIHkBRy
% axis square, colorbar 4TU\SP8sM
% title('Zernike function Z_5^1(r,\theta)') !m_y@~pV#u
% MB>4Y]rtU
% Example 2: yl' IL#n]r
% d@Bd*iI<
% % Display the first 10 Zernike functions J$jLGy& '
% x = -1:0.01:1; sKiy1Ww
% [X,Y] = meshgrid(x,x); g;o5m}
% [theta,r] = cart2pol(X,Y); n~w[ajC/
% idx = r<=1; bccf4EyQ
Y
% z = nan(size(X)); c(3idO*R)
% n = [0 1 1 2 2 2 3 3 3 3]; <Z~Nz>'r
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; yQu/({D
% Nplot = [4 10 12 16 18 20 22 24 26 28]; <7ag=IgDy
% y = zernfun(n,m,r(idx),theta(idx)); Gh{9nM_\"
% figure('Units','normalized') K;\fJ2ag
% for k = 1:10 Pa|*Jcr
% z(idx) = y(:,k); ZL!5dT&@W
% subplot(4,7,Nplot(k)) T0@<u
% pcolor(x,x,z), shading interp a{ByU%
% set(gca,'XTick',[],'YTick',[]) ]wbV1Y"
% axis square cUi6 On1C
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) VeFfkg4
% end 6(A"5B=\
% =7~;*Ts
% See also ZERNPOL, ZERNFUN2. OCqknA
h:z$uG
% Paul Fricker 11/13/2006 G&6`?1k
fE>JoQs38
?6MUyH]a
% Check and prepare the inputs: 7Z}T!HFMr
% ----------------------------- 8k Sb92
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +rrA>~
error('zernfun:NMvectors','N and M must be vectors.') O6q5qA
end _t X1z^
mI^S% HT
if length(n)~=length(m) { ux'9SA
error('zernfun:NMlength','N and M must be the same length.') vhU
$GG8
end -7I%^u
%wJ>V-\e
n = n(:); 1yc$b+TH
m = m(:); j3
@Q
if any(mod(n-m,2)) `Z2-<:]6&a
error('zernfun:NMmultiplesof2', ... e&<=+\ul
'All N and M must differ by multiples of 2 (including 0).') 2rf#Bq?7
end U'} [:h~)
~>%% kQt
if any(m>n) xCu\ jc)2
error('zernfun:MlessthanN', ... B|AIl+y
'Each M must be less than or equal to its corresponding N.') /5f=a
end @[ '?AsO
CT=5V@_u\
if any( r>1 | r<0 ) f_. 0 uM
error('zernfun:Rlessthan1','All R must be between 0 and 1.') !,DA`Yt
end BL\H@D
1HRcEzA
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Gx%f&H~Z^
error('zernfun:RTHvector','R and THETA must be vectors.') Oj7).U0;#
end ]#FQde4]5
;l@Ge`&u
r = r(:); t0ZaI E
theta = theta(:); !3*%-8bp
length_r = length(r); Oh7wyQiV
if length_r~=length(theta) J>0RN/38o
error('zernfun:RTHlength', ... T'14OU2N{Y
'The number of R- and THETA-values must be equal.') 6s:
end '"V]>)
7C@m(oK
% Check normalization: xI5zP?
_v
% -------------------- ^%33&<mB}
if nargin==5 && ischar(nflag) ,Mn?h\
isnorm = strcmpi(nflag,'norm'); R+=Xr<`%U|
if ~isnorm l]5!$N*
error('zernfun:normalization','Unrecognized normalization flag.') S^SF!k=
end Ec!R3+
else _&$nJu
isnorm = false; Ke\FzZ]
end 69``j{Z+
;E\ e.R
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tj" EUqKQ
% Compute the Zernike Polynomials ) !l1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \.`{nq
<IQ}j^u-F
% Determine the required powers of r: J~5+=V7OV
% ----------------------------------- l`E KL2n
m_abs = abs(m); kNUNh[
rpowers = []; -lI6!a^
for j = 1:length(n) =K6{AmG$
rpowers = [rpowers m_abs(j):2:n(j)]; ']>/$[!
end 1lHBg
rpowers = unique(rpowers); $"{I|UFC
v,)vW5jGI
% Pre-compute the values of r raised to the required powers, e>_Il']Mb
% and compile them in a matrix: Z}r9jM
% ----------------------------- I oC}0C7
if rpowers(1)==0 XCE<].w
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 2PVQSwW:
rpowern = cat(2,rpowern{:}); R-BN}ZS
rpowern = [ones(length_r,1) rpowern]; $7&t`E)qY
else NYF
7Ep; _
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 20BU;D3
rpowern = cat(2,rpowern{:}); M}!E :bv'
end >L88`
`g,i`<
% Compute the values of the polynomials: e\H1IR3
% -------------------------------------- '<hgc
y = zeros(length_r,length(n)); Vg1MA
for j = 1:length(n) Jnq}SUev
s = 0:(n(j)-m_abs(j))/2; 1(m[L=H5>
pows = n(j):-2:m_abs(j); 2[Bw+<YA`
for k = length(s):-1:1 bBXUD;$
p = (1-2*mod(s(k),2))* ... sj% \lq
prod(2:(n(j)-s(k)))/ ... w?A6S-z
prod(2:s(k))/ ... ,gn**E
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... uBxs`'C
prod(2:((n(j)+m_abs(j))/2-s(k))); <FU1|
idx = (pows(k)==rpowers); 'FmnlC1
y(:,j) = y(:,j) + p*rpowern(:,idx); v\Xyz
)
end #TG.weTC
fTV}IP
if isnorm :pg]0X;
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); -jL10~/
end 8H2A<&3i
end `:;fc
% END: Compute the Zernike Polynomials U
jB5Xks
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% HT=-mwa_]
2vX!j!_
% Compute the Zernike functions: iig@$
i#
% ------------------------------ fk?(mxx"
idx_pos = m>0; Wx F0LhM
idx_neg = m<0; hGlRf_{
>R2o7~
z = y; _J33u3v
if any(idx_pos) `ouCQ]tKz
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); }#QYZ nR
end 3`DwKv`+
if any(idx_neg) Jnf@u
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); aj@<4A=;
end E0<$zP}V}F
SW*Yu{
% EOF zernfun