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 {)K](S
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function z = zernfun(n,m,r,theta,nflag) {8NwFN.
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. d$;/T('
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N s'_,:R\VM>
% and angular frequency M, evaluated at positions (R,THETA) on the PCfo
% unit circle. N is a vector of positive integers (including 0), and Ttv9"z
% M is a vector with the same number of elements as N. Each element 4Nmea-!*
% k of M must be a positive integer, with possible values M(k) = -N(k) \3PE+$
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ]HvZ$
% and THETA is a vector of angles. R and THETA must have the same AZZRa69=
% length. The output Z is a matrix with one column for every (N,M) 0\a8}b||
% pair, and one row for every (R,THETA) pair. G?V"SU.
% %%g-GyP
1
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike h[=nx^
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), YL5>V$i
% with delta(m,0) the Kronecker delta, is chosen so that the integral .RRlUWu
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, p#H]\P'
% and theta=0 to theta=2*pi) is unity. For the non-normalized vD=%`G[m
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. j Selop>N
% *_)E6Y?9
% The Zernike functions are an orthogonal basis on the unit circle. MEU[%hty_
% They are used in disciplines such as astronomy, optics, and |f NMs
% optometry to describe functions on a circular domain. {j6g@Vd6lx
% vg^Myn
% The following table lists the first 15 Zernike functions. #@_1fE
% |< N frz
% n m Zernike function Normalization v*P[W_.
% -------------------------------------------------- x N`T
% 0 0 1 1 .C5@QKU
% 1 1 r * cos(theta) 2 |NEd@
% 1 -1 r * sin(theta) 2 .[f;(WR
% 2 -2 r^2 * cos(2*theta) sqrt(6) 4r*Pa(;y
% 2 0 (2*r^2 - 1) sqrt(3) f9']
jJ+
% 2 2 r^2 * sin(2*theta) sqrt(6) .xpmp6-
% 3 -3 r^3 * cos(3*theta) sqrt(8) k|#Zy,
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ?~)Ak`=
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ~n]NyVFP
% 3 3 r^3 * sin(3*theta) sqrt(8) R{<Y4C2~
% 4 -4 r^4 * cos(4*theta) sqrt(10) BW71 s
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) t:9
ZCu ay
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) FaWl,} ]
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) v>at/ef
% 4 4 r^4 * sin(4*theta) sqrt(10) 3'@&c?Fye
% -------------------------------------------------- $-w5o`e
% [.U^Wrd
% Example 1: t F/nah
% (9z|a,
% % Display the Zernike function Z(n=5,m=1) GYqJ!,
% x = -1:0.01:1; BkT-m'I?
% [X,Y] = meshgrid(x,x); 9cOx@c+/
% [theta,r] = cart2pol(X,Y); 5bBCpNa
% idx = r<=1; %O /d4
% z = nan(size(X)); I Tn;m
% z(idx) = zernfun(5,1,r(idx),theta(idx)); _m7co :
% figure 6UI>GQ
% pcolor(x,x,z), shading interp LR\zy8y]
% axis square, colorbar ZeTL$E[E}
% title('Zernike function Z_5^1(r,\theta)') N
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% xA9V$# d|
% Example 2: ._ih$=
% 5Jw"{V?Ak
% % Display the first 10 Zernike functions h60\ Y 8
% x = -1:0.01:1; >p |yf.G
% [X,Y] = meshgrid(x,x); j ]HE>
% [theta,r] = cart2pol(X,Y); Zsk?QS FE
% idx = r<=1; CK Mv7
% z = nan(size(X)); pVz pN8!
% n = [0 1 1 2 2 2 3 3 3 3]; (uT^Nn9L=
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; CKN8z
% Nplot = [4 10 12 16 18 20 22 24 26 28]; q]+)c2M
% y = zernfun(n,m,r(idx),theta(idx)); zP|*(*
% figure('Units','normalized') :f]!O@.~
% for k = 1:10 um}N%5GAa
% z(idx) = y(:,k); qSR?,G
% subplot(4,7,Nplot(k)) X}?ESjZJ
% pcolor(x,x,z), shading interp @>CG3`?}
% set(gca,'XTick',[],'YTick',[]) )BB%4=u@~.
% axis square xBt<Yt"
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) +/}_%Cf8
% end Fu
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% iBS0rT_
% See also ZERNPOL, ZERNFUN2. L77EbP`P
}JH`'&3
% Paul Fricker 11/13/2006 @[0jFjK
VlV)$z_
WRY~fM
% Check and prepare the inputs: gTuX *7w
% ----------------------------- 6yp+h
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) oX#9RW/ >I
error('zernfun:NMvectors','N and M must be vectors.') 9yDFHz w
end ,NDxFy;d
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if length(n)~=length(m) @n>{&^-c
error('zernfun:NMlength','N and M must be the same length.') BQuRHi IV
end wYa0hNd
?U$}Rsk{#
n = n(:); 0|GpZuGO9
m = m(:); oq243\?Y
if any(mod(n-m,2)) U* 4{"
error('zernfun:NMmultiplesof2', ... q?1yE@th
'All N and M must differ by multiples of 2 (including 0).') o\:$V
end 9ec0^T
GPMrs)J*!
if any(m>n) wd"TM
error('zernfun:MlessthanN', ... Mo~ki"9.
'Each M must be less than or equal to its corresponding N.') BZ2nDW*%
end /5jKX 5r
jjYM3LQcdP
if any( r>1 | r<0 ) G^ K*+
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 8>2&h
end xp~YIeSg
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Th_Q
owk
error('zernfun:RTHvector','R and THETA must be vectors.') #`r(zI[
end OA!R5sOz"
tx gvVQ
r = r(:); F
ZM2
theta = theta(:); ]v<d0"2
length_r = length(r); ^zKt{a
if length_r~=length(theta) `D4oAx d9
error('zernfun:RTHlength', ... iJEB?y
'The number of R- and THETA-values must be equal.') _w\Y{(k
end c{^i$
G OH
% Check normalization: 56"#Syj
% -------------------- ,I/2.Q})[
if nargin==5 && ischar(nflag) VjC*(6<Gj
isnorm = strcmpi(nflag,'norm'); ?rky6
if ~isnorm Nvi Fq
error('zernfun:normalization','Unrecognized normalization flag.') 0`V3s]%iu
end @< wYT$
else xq#U4E
isnorm = false;
{VS''Lv
end B:B8"ODV
w 9/nVu
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^).WW
% Compute the Zernike Polynomials H&~5sEGa
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dK[*
N[#iT&@T}/
% Determine the required powers of r: "xL;(Fqu
% ----------------------------------- =X)Q7u".7
m_abs = abs(m); X\o/i\ C}
rpowers = []; ~8XX3+]z:X
for j = 1:length(n) pp*bqY
rpowers = [rpowers m_abs(j):2:n(j)]; ;Fx')
end R.91v4J
rpowers = unique(rpowers); JZWgr&O<
MFf05\aDu
% Pre-compute the values of r raised to the required powers, 'bZMh9|
% and compile them in a matrix: V"w`!
% ----------------------------- $&ex\_W
if rpowers(1)==0
#;5[('&[
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); IXbdS9,>F
rpowern = cat(2,rpowern{:}); nYI/&B{p
rpowern = [ones(length_r,1) rpowern]; 4 *Bp
else (45NZBs
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 3'?h;`v\Lo
rpowern = cat(2,rpowern{:}); C/{nr-V3u
end @SKO~?7T
sN6 0o 7.
% Compute the values of the polynomials: IyrZez
% -------------------------------------- w{_e"N
y = zeros(length_r,length(n)); 2$o2.$i81
for j = 1:length(n) d9`3EP)n
s = 0:(n(j)-m_abs(j))/2; 3~cS}N T
pows = n(j):-2:m_abs(j); :5TXA
for k = length(s):-1:1 z*Myokhf
p = (1-2*mod(s(k),2))* ... H arFo
prod(2:(n(j)-s(k)))/ ... ?l)}E
prod(2:s(k))/ ... C1ZFA![
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... X{0ax.
prod(2:((n(j)+m_abs(j))/2-s(k))); hEyX~f
idx = (pows(k)==rpowers); Y{%4F%Oy
y(:,j) = y(:,j) + p*rpowern(:,idx); UgF) J
end m1^dT_7Z
WHlD%u
if isnorm K[iY{
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); e8~62O^
end <7vI h0
end D)H?=G
% END: Compute the Zernike Polynomials j\XX:uU_
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b5iIV1g
4@/ q_*3o
% Compute the Zernike functions: [(D}%+2
% ------------------------------ *Gk<"pEeS
idx_pos = m>0; 9s;!iDFn
idx_neg = m<0; H]%mP|
<Z\MZ&{k{*
z = y; bqZ?uvc3
if any(idx_pos) "@c';".|
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ef;&Y>/
end r6O7&Me<
if any(idx_neg) syWv'Y[k?
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); SX_kr^#
end %4|n-`:
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% EOF zernfun