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 ]?c9;U
function z = zernfun(n,m,r,theta,nflag) E/OfkL*\
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. W<Ri(g-
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 7fE U5@
% and angular frequency M, evaluated at positions (R,THETA) on the .:r
l<.
% unit circle. N is a vector of positive integers (including 0), and zPm|$d
% M is a vector with the same number of elements as N. Each element vLI'Z)\
% k of M must be a positive integer, with possible values M(k) = -N(k) Xnc?oT+
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, f0M5^
% and THETA is a vector of angles. R and THETA must have the same BMi5F?Q'G
% length. The output Z is a matrix with one column for every (N,M) !KC4[;Y
% pair, and one row for every (R,THETA) pair. Y+)qb);
% *jC Hv
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike (! a;}V<7
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), $&Lw 2 c0
% with delta(m,0) the Kronecker delta, is chosen so that the integral JIatRc?g
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, me@k~!e"z
% and theta=0 to theta=2*pi) is unity. For the non-normalized 1 EL#T&
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?uh%WN6nU]
% ,,8'29yEq
% The Zernike functions are an orthogonal basis on the unit circle. U5:5$T,C
% They are used in disciplines such as astronomy, optics, and {&TP&_|H
% optometry to describe functions on a circular domain. YgV" *~
% hm,H3pN
% The following table lists the first 15 Zernike functions. __%){j6
% XcFu:B
% n m Zernike function Normalization z"\<GmvB
% -------------------------------------------------- 'r'+$D7
% 0 0 1 1 Sc14F
Fs
% 1 1 r * cos(theta) 2 q"0_Px9P
% 1 -1 r * sin(theta) 2 6DVHJ+WTV
% 2 -2 r^2 * cos(2*theta) sqrt(6) ,8[R0wsBaz
% 2 0 (2*r^2 - 1) sqrt(3) +OaBA>Jh9
% 2 2 r^2 * sin(2*theta) sqrt(6) c8h71Cr
% 3 -3 r^3 * cos(3*theta) sqrt(8) lk4U/:
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 7hlzuZob+y
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) E>c*A40=.n
% 3 3 r^3 * sin(3*theta) sqrt(8) D4jZh+_|S
% 4 -4 r^4 * cos(4*theta) sqrt(10) Esdv+f}4;
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) wd*V,ZN7
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) nTv^][
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |33_="
% 4 4 r^4 * sin(4*theta) sqrt(10) o*5b]XWw
% -------------------------------------------------- `3^%ft~l
% Z{^Pnit
% Example 1: o0kKf+[
% Bo4iX,zu
% % Display the Zernike function Z(n=5,m=1) Ow0( q^H<
% x = -1:0.01:1; <YAs0
% [X,Y] = meshgrid(x,x); th|'t}bWV
% [theta,r] = cart2pol(X,Y); =zW`+++3
% idx = r<=1; yRWZ/,9x
% z = nan(size(X)); jwp?eL!7
% z(idx) = zernfun(5,1,r(idx),theta(idx)); x-T7
tr&(
% figure 5Z>+NKQ
% pcolor(x,x,z), shading interp _iH:>2p 5R
% axis square, colorbar :gM_v?sy
% title('Zernike function Z_5^1(r,\theta)') Ask~
% T5eJIc3a"
% Example 2: .2
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% g$U7bCHG
% % Display the first 10 Zernike functions v*&WqVg
% x = -1:0.01:1; _N"c,P0
% [X,Y] = meshgrid(x,x); &; [0.:;
% [theta,r] = cart2pol(X,Y); Tffdm
% idx = r<=1; Of;$
VK'
% z = nan(size(X)); [Qn=y/._r
% n = [0 1 1 2 2 2 3 3 3 3]; ;F:Qz^=.a
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; :+<GJj_d+
% Nplot = [4 10 12 16 18 20 22 24 26 28]; i9^m;Y)^I
% y = zernfun(n,m,r(idx),theta(idx)); }g"K\x:Z
% figure('Units','normalized') oz'^.+uvE
% for k = 1:10 m^;A]0h+
% z(idx) = y(:,k); |?LUt@r;
% subplot(4,7,Nplot(k)) ]GiDfYs7%
% pcolor(x,x,z), shading interp K 5AArI
% set(gca,'XTick',[],'YTick',[]) uDMyO<\
% axis square Bg}(Sy
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) `aM8L
% end \GCT3$
% G3D!ifho.#
% See also ZERNPOL, ZERNFUN2. *40Z}1ng
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% Paul Fricker 11/13/2006 pW5PF)([
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84c[ Z
% Check and prepare the inputs: }/VSIS@Z
% ----------------------------- -O6\!Wo=-
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) *oru;=D@8
error('zernfun:NMvectors','N and M must be vectors.') tVHQ$jJY%
end @l?2",
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if length(n)~=length(m) +\66; 7]s
error('zernfun:NMlength','N and M must be the same length.') oI9-jW
end &\Yd)#B/
x=3+@'
n = n(:); ^ =RSoR
m = m(:); D,SL_*r{
if any(mod(n-m,2)) 'p4b8:X
error('zernfun:NMmultiplesof2', ... UpqDGd7M
'All N and M must differ by multiples of 2 (including 0).') y0
qq7Dmu
end lPn&,\9@~
(=w ff5U
if any(m>n) M5l*D'GE]
error('zernfun:MlessthanN', ... *Bx'g|
u
'Each M must be less than or equal to its corresponding N.') &:Sb$+z
end jIL$hqo
;aUI3n%
if any( r>1 | r<0 ) UdX aC= Q
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ;/ao3Q
end Xj;5i
Vq
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 0= -D
error('zernfun:RTHvector','R and THETA must be vectors.')
q
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end -O>*`
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f n'N^
r = r(:); 2s8(r8 AI
theta = theta(:); Y\ G^W8
length_r = length(r); -cnlj
if length_r~=length(theta) gb@ |\n
error('zernfun:RTHlength', ... 8Zvh"Z?
'The number of R- and THETA-values must be equal.') `-)Fx<e
end o!M*cyq
1@A*Jj[R%
% Check normalization: parC~)b_
% -------------------- w\3'wD!
if nargin==5 && ischar(nflag) {>=#7e-]
isnorm = strcmpi(nflag,'norm'); YK|Y^TU^
if ~isnorm !YEU<9
error('zernfun:normalization','Unrecognized normalization flag.') &_y+hV{
end 7<c&)No;
else 1">]w2je:
isnorm = false; /WIHG0D
end G q
r(.
blA]z!FU
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7&9'=G
% Compute the Zernike Polynomials UT7".1H
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @X6|[r&Z
!T3Esv
% Determine the required powers of r: 6'N!)b^-
% ----------------------------------- ZW|VAn'>
m_abs = abs(m); |d1%N'Ll
rpowers = []; dc0Ro,
for j = 1:length(n) 84*Fal~Som
rpowers = [rpowers m_abs(j):2:n(j)]; Epm=&6zf
end v`$9;9
rpowers = unique(rpowers); ^y"$k
nNff~u)I
% Pre-compute the values of r raised to the required powers, W[3)B(Vq<E
% and compile them in a matrix: __V6TDehJ$
% ----------------------------- x 1"ikp}
if rpowers(1)==0 GX
lFS#`
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); S
Yvifgp
rpowern = cat(2,rpowern{:}); l@om2|B
rpowern = [ones(length_r,1) rpowern]; :1wMGk
else B1A5b=6G<
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); -zVa[&
rpowern = cat(2,rpowern{:}); 2;`"B|-T
end U<
p kg
0R2 AhA#
% Compute the values of the polynomials: 3rZ" T
% -------------------------------------- 1XO*yZF
y = zeros(length_r,length(n)); ?%h JZm;
for j = 1:length(n) 8D:{05
s = 0:(n(j)-m_abs(j))/2; -$4%@Z
pows = n(j):-2:m_abs(j); f. =4p^
for k = length(s):-1:1 c])b?dJ*
p = (1-2*mod(s(k),2))* ... G?]E6R
prod(2:(n(j)-s(k)))/ ... $0Y&r]'
prod(2:s(k))/ ... "/?*F\5
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ${ ~UA6
prod(2:((n(j)+m_abs(j))/2-s(k))); [#td
idx = (pows(k)==rpowers); >1tGQ
cg
y(:,j) = y(:,j) + p*rpowern(:,idx); J7.bFW'
end zY|]bP[NEH
K`FgU7g{
if isnorm Sh]x`3 ).
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); kI3-G~2
end .so{ RI
end zHB{I(q
% END: Compute the Zernike Polynomials Y(SgfWeK@1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% y+?tUSPP
@X/S
h:
% Compute the Zernike functions: Rhx7eU#&
% ------------------------------ !o4xI?
idx_pos = m>0; xM;gF2
idx_neg = m<0; h{sW$WA
%~ecrQ;
z = y; q'2PG@
if any(idx_pos) tTyu,%/m
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); $_Qo
end 1qUdj[Bj
if any(idx_neg) 2>O2#53ls0
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); =,[46 ;q
end GKY:"q&h
Whd4-pR8
% EOF zernfun