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 P4Wd=Xoz6
function z = zernfun(n,m,r,theta,nflag) v;jrAND
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. xLq+njH E
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N dax|4R
% and angular frequency M, evaluated at positions (R,THETA) on the ~d){7OG
% unit circle. N is a vector of positive integers (including 0), and irgjq/&d
% M is a vector with the same number of elements as N. Each element [uZU p*.V
% k of M must be a positive integer, with possible values M(k) = -N(k) q>!T*BQ
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 9]7+fu
% and THETA is a vector of angles. R and THETA must have the same DlfXzKn;
% length. The output Z is a matrix with one column for every (N,M) &> }MoB
% pair, and one row for every (R,THETA) pair. A7~)h}~
% _ 4Hf?m7z
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ?W%3>A
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), B~yD4^
% with delta(m,0) the Kronecker delta, is chosen so that the integral Y13IrCA2
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, efZdtrKgy
% and theta=0 to theta=2*pi) is unity. For the non-normalized SS(jjpe&,
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. YWd:Ok0
% B=|yjA'Fg
% The Zernike functions are an orthogonal basis on the unit circle. u\smQhQGE
% They are used in disciplines such as astronomy, optics, and q2&&n6PYW
% optometry to describe functions on a circular domain. z8vFQO\I"
% \`|,wLgH
% The following table lists the first 15 Zernike functions. 7o0ej#
% *l_1T4]S
% n m Zernike function Normalization F2>o"j2
% -------------------------------------------------- e[>(L% QV+
% 0 0 1 1 )u3<lpoTy
% 1 1 r * cos(theta) 2 ;2#H M^Mu
% 1 -1 r * sin(theta) 2 aH,0+ |
% 2 -2 r^2 * cos(2*theta) sqrt(6) @fbvu_-].
% 2 0 (2*r^2 - 1) sqrt(3) nb(#;3DQ
% 2 2 r^2 * sin(2*theta) sqrt(6) x\I9J4Q
% 3 -3 r^3 * cos(3*theta) sqrt(8) q\d'}:kfu
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) oV,>u5:B
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) pd>EUdbrp&
% 3 3 r^3 * sin(3*theta) sqrt(8) h#;fBQ]
% 4 -4 r^4 * cos(4*theta) sqrt(10) [<8<+lH=P
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )k0bP1oGS
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Vu;tU.
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (O/hu3
% 4 4 r^4 * sin(4*theta) sqrt(10) |Z#)1K
% -------------------------------------------------- *kZJ
% [4PG_k[uTJ
% Example 1: k<8:
% #H M0s~^w&
% % Display the Zernike function Z(n=5,m=1) 9~Q.[ A
% x = -1:0.01:1; }SUe 4r&4}
% [X,Y] = meshgrid(x,x); sL+/Eeb` c
% [theta,r] = cart2pol(X,Y); U%w?muJW
% idx = r<=1; l$)pCo
% z = nan(size(X)); "4n_MV>p
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Tet,mzVuu
% figure j~Rh_\>Q
% pcolor(x,x,z), shading interp V"T;3@N/4
% axis square, colorbar V..m2nQj
% title('Zernike function Z_5^1(r,\theta)') Kax85)9u
% {jggiMwo.v
% Example 2: d=H C;T)
% :+!hR4Z~\;
% % Display the first 10 Zernike functions F-UY~i8
% x = -1:0.01:1; zx0{cNPK5
% [X,Y] = meshgrid(x,x); w9i1ag
% [theta,r] = cart2pol(X,Y); ]>*Z 1g;
% idx = r<=1; :mY(d6#A>
% z = nan(size(X)); \u",bMQF
% n = [0 1 1 2 2 2 3 3 3 3]; +4B>gS[ F
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; !mq+Oz~
% Nplot = [4 10 12 16 18 20 22 24 26 28]; w9c
% y = zernfun(n,m,r(idx),theta(idx)); DFqXZfjm
% figure('Units','normalized') L!-T`R8'c
% for k = 1:10 "m/0>UU0
% z(idx) = y(:,k); xjv?Z"X
% subplot(4,7,Nplot(k)) j
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% pcolor(x,x,z), shading interp bWjW_$8
% set(gca,'XTick',[],'YTick',[]) -zG/@.
% axis square won%(n,HT
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) !63x^# kg
% end >(~;V;
% y*|"!FK
% See also ZERNPOL, ZERNFUN2. GZ*cV3Y`&
MP0gLi
% Paul Fricker 11/13/2006 S : 9zz
f>l}y->-Ug
NqlG= pu
% Check and prepare the inputs: /,GDG=ra
% ----------------------------- [ V/*{Z
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) b~%(5r.
error('zernfun:NMvectors','N and M must be vectors.') ~hYG%
end F7cv`i?2."
g2w0#-
if length(n)~=length(m) AdR}{:ia
error('zernfun:NMlength','N and M must be the same length.') lN{-}f;TN
end |;Jcf3e(
V\X.AGc
n = n(:); Fag%#jxI
m = m(:); o;_v'
if any(mod(n-m,2)) $WrDZU 2z
error('zernfun:NMmultiplesof2', ... Z5_U D
'All N and M must differ by multiples of 2 (including 0).') b!ot%uZZ
end ([tbFI}A
m7g; psg
if any(m>n) WPCaxA+l
error('zernfun:MlessthanN', ... hSo\
'Each M must be less than or equal to its corresponding N.') G W|~sE +
end <gQw4
X0Xs"--}
if any( r>1 | r<0 ) "*XR'9~7
error('zernfun:Rlessthan1','All R must be between 0 and 1.') e ST8>r
end zF3fpEKe
/wH]OD{
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ]rXRon='
error('zernfun:RTHvector','R and THETA must be vectors.') QJ-6aB
end Jc(tV(z
yA
\C3r'
r = r(:); YPFjAQ
theta = theta(:); @/E5$mX`
length_r = length(r); \ C~Y
if length_r~=length(theta) NuLQkf)
error('zernfun:RTHlength', ... \h,S1KmIBD
'The number of R- and THETA-values must be equal.') E@Q+[~H }
end [#M^:Q
rpQB#
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% Check normalization: ^e8~eL+
% -------------------- 4}gqtw:
if nargin==5 && ischar(nflag) .@gv}`>
isnorm = strcmpi(nflag,'norm'); w=e~
M
if ~isnorm Qpe&_.&RE
error('zernfun:normalization','Unrecognized normalization flag.') Ca0~K42~
end K
p~x
else ~OAS T
isnorm = false; 1|q$Wn:*
end oV&AJ=|\
7=aF-;X3jj
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @K3<K(
% Compute the Zernike Polynomials (kK6=Mrf
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (6L[eWuTn
~?H _?}e
% Determine the required powers of r: $*\[I{Zau}
% ----------------------------------- )lTkqz8v
m_abs = abs(m); c7[|x%~
rpowers = []; `h+ sSIko
for j = 1:length(n) Fi14_{
rpowers = [rpowers m_abs(j):2:n(j)]; >Ke4lO"
end am]$`7R5d
rpowers = unique(rpowers); *[) b}?
5<0&y3
% Pre-compute the values of r raised to the required powers, Ugp[Ugr
% and compile them in a matrix: "\Zsr6y
% ----------------------------- hl(M0cxEWP
if rpowers(1)==0 cz~Fz;)2{N
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); _{_ybXG|
rpowern = cat(2,rpowern{:}); uosFpa
rpowern = [ones(length_r,1) rpowern]; `b=?z%LuT
else se:]F/
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 4onRO!G,
rpowern = cat(2,rpowern{:}); vUk <z*
end Gc^w,n[E
PO%Z.ol9
% Compute the values of the polynomials: }te\)
Yk.N
% -------------------------------------- a^hDxeG
y = zeros(length_r,length(n)); S zR7:U
for j = 1:length(n) MDZ,a0?4t
s = 0:(n(j)-m_abs(j))/2; kAsYh4[
pows = n(j):-2:m_abs(j); <5%x3e"7u
for k = length(s):-1:1 wR@&C\}9
p = (1-2*mod(s(k),2))* ... %5?qS`/c(
prod(2:(n(j)-s(k)))/ ... 56Z 1jN^U
prod(2:s(k))/ ... b) "bX}
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ^oDC F
prod(2:((n(j)+m_abs(j))/2-s(k))); a/A$
MXZ_
idx = (pows(k)==rpowers); ?W:YS82
y(:,j) = y(:,j) + p*rpowern(:,idx); _WO*N9Iz
end %JF.m$-
3J%(2}{y
if isnorm :s`~m;Y9?
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); !ba /]A/
end ~xZFm
end `CP#S7W^
% END: Compute the Zernike Polynomials d:cs8f4>
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "#anL8
q,w8ca4~y
% Compute the Zernike functions: owM3Gz%?UA
% ------------------------------ ,Dd
)=
idx_pos = m>0; 'id]<<F
idx_neg = m<0; 4iMo&E<
"Qj;pqR
z = y; K: hZ
if any(idx_pos) |3j'HN5S
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); lf3QMr+
end )!M %clm.
if any(idx_neg) pB*8D
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ^x8*]Sz#x
end ' P5ttI#|
xXkP(^ Y
% EOF zernfun