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 "dFuQB
function z = zernfun(n,m,r,theta,nflag) a{!
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. O;SD90
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N PJ'.s
% and angular frequency M, evaluated at positions (R,THETA) on the UO8./%'
% unit circle. N is a vector of positive integers (including 0), and $#HUxwx4
% M is a vector with the same number of elements as N. Each element rhO8 v
% k of M must be a positive integer, with possible values M(k) = -N(k) rRd8W}B
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 0|OmQ\SQ
% and THETA is a vector of angles. R and THETA must have the same '/GZ,~q
% length. The output Z is a matrix with one column for every (N,M) ~/1eF7
% pair, and one row for every (R,THETA) pair. BV512+M
% 5 $:
q
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike z]0UW\S/
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), A"no!AN
% with delta(m,0) the Kronecker delta, is chosen so that the integral [LrA_N
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, &46Ro|XE`
% and theta=0 to theta=2*pi) is unity. For the non-normalized 3`>nQ4zC
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. WG(%Pkowv
% @Yzc?+x
% The Zernike functions are an orthogonal basis on the unit circle. "&N1$$
% They are used in disciplines such as astronomy, optics, and QP1bm]QYA
% optometry to describe functions on a circular domain. V8IEfU
% NiO|Aki{
% The following table lists the first 15 Zernike functions. Bw8&Amxx:
% Ilv
_.
% n m Zernike function Normalization G8Sx;Xi
% -------------------------------------------------- D$/*Z5Z)]
% 0 0 1 1 e=w.7DSE
% 1 1 r * cos(theta) 2 Yn1CU
% 1 -1 r * sin(theta) 2 K!onV3mR
% 2 -2 r^2 * cos(2*theta) sqrt(6) r-IG.ym3
% 2 0 (2*r^2 - 1) sqrt(3) 4&/m>%r
% 2 2 r^2 * sin(2*theta) sqrt(6) &s<'fSI
% 3 -3 r^3 * cos(3*theta) sqrt(8) HT6+OK(~dJ
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) fk
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) m9m]q&hx
% 3 3 r^3 * sin(3*theta) sqrt(8) ^.;
x
% 4 -4 r^4 * cos(4*theta) sqrt(10) Q2HULz{
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) r4(Cb_
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Sn~|<Vf
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #+6t|
% 4 4 r^4 * sin(4*theta) sqrt(10) 4KCJ(<p|
% -------------------------------------------------- a~"<lzu|$
% (v$$`zh
% Example 1: M}*#{UV2
% Ri&?uCCM
% % Display the Zernike function Z(n=5,m=1) /ng+IC3
% x = -1:0.01:1; -|&5aH]
% [X,Y] = meshgrid(x,x); %9P)Okq
% [theta,r] = cart2pol(X,Y);
cnwpd%]o
% idx = r<=1; ~Y /55uC
% z = nan(size(X)); E#A}J:
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ^lCQHz
% figure %?~`'vYoi
% pcolor(x,x,z), shading interp r%^J3
% axis square, colorbar 6m!%X GZT
% title('Zernike function Z_5^1(r,\theta)') (XJ0?;js=
% *cnxp-)ub
% Example 2: <4QOjW
% .P>-Fh,_p
% % Display the first 10 Zernike functions f0,,<ib.w
% x = -1:0.01:1; >7^i>si
% [X,Y] = meshgrid(x,x); q*B(ZG
% [theta,r] = cart2pol(X,Y); |.c|\e z/
% idx = r<=1; Lavm
% z = nan(size(X)); Z_Z; g]|!
% n = [0 1 1 2 2 2 3 3 3 3]; M4m90C;dq
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; }9,^=g-
% Nplot = [4 10 12 16 18 20 22 24 26 28]; MEZc/Ru-[
% y = zernfun(n,m,r(idx),theta(idx)); &@anv.D
% figure('Units','normalized') c _faW
% for k = 1:10 g<"k\qs7
% z(idx) = y(:,k); Jf|6 FQo&
% subplot(4,7,Nplot(k)) E8QY6 gKF
% pcolor(x,x,z), shading interp :4,
OA
% set(gca,'XTick',[],'YTick',[]) /"*eMe!=
% axis square [J71aH
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @p}"B9h*^
% end bPHqZ*f
% wqyrs|P
% See also ZERNPOL, ZERNFUN2. uh_2yw_
2UGnRZ8:1Y
% Paul Fricker 11/13/2006 lImg+r T{
16N+
zjVQ \L
% Check and prepare the inputs: <h7FS90S
% ----------------------------- !^EdB}@yS
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) %Z#s9QC
error('zernfun:NMvectors','N and M must be vectors.') Im* ~6[
end "o
^cv
+~~&FO2
if length(n)~=length(m) hn@T ]k
error('zernfun:NMlength','N and M must be the same length.') 6YCFSvA#/
end &bO5+[
~&?{hd.
n = n(:); Xob,jo}a
m = m(:); Z1t?+v+Ro*
if any(mod(n-m,2)) e<;^P(g`E
error('zernfun:NMmultiplesof2', ... SpB\kC"K
'All N and M must differ by multiples of 2 (including 0).') KS6H`Mm}/
end fEB>3hI
">pt,QV
if any(m>n) _ Db05:r@
error('zernfun:MlessthanN', ... =oPc\VYW
'Each M must be less than or equal to its corresponding N.') aN/0'V|&ym
end ^*fZ
1 ynjDin<
if any( r>1 | r<0 ) zAxscDf'
error('zernfun:Rlessthan1','All R must be between 0 and 1.') GvCB3z
end ]U8VU
M#PutrH
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) -!kfwJg8N(
error('zernfun:RTHvector','R and THETA must be vectors.') .<Lbv5m
end v,] &[`
.%'$3=/oe
r = r(:); B?G!~lQ)o
theta = theta(:); !7KSNwGu
length_r = length(r); m<DiYxK
if length_r~=length(theta) L`M.Htm8
error('zernfun:RTHlength', ... *yx&4)Or
'The number of R- and THETA-values must be equal.') PU6Sa-fQ2,
end L:(>ON
7 q%|-`#
% Check normalization: *61+Fzr
% -------------------- d\R]>
if nargin==5 && ischar(nflag) r[TTG0|
isnorm = strcmpi(nflag,'norm'); \VTNXEw*G
if ~isnorm G q" [5r"
error('zernfun:normalization','Unrecognized normalization flag.') .=nx5yz
end SREe,
e\
else &s|a\!>l
isnorm = false; k[6xuyY]
end dHtbl\6
.)zX<~,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c~1X/,biA
% Compute the Zernike Polynomials @"\j]ZEnY
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7HpfHqJ7
Y~</vz+H
% Determine the required powers of r: kbxy^4"X
% ----------------------------------- A@W/
m_abs = abs(m); *7ggw[~
rpowers = []; Gg\805L@
for j = 1:length(n) .!kO2/:6
rpowers = [rpowers m_abs(j):2:n(j)]; Jf/X3\0N7
end ~is$Onf99#
rpowers = unique(rpowers); h|MTE~
%4%$NdU"
% Pre-compute the values of r raised to the required powers, }[[
% and compile them in a matrix: eu]t.Co[X
% ----------------------------- ^+ hJ& 9W
if rpowers(1)==0 Ls<.&3X2
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ZwV`} 2{
rpowern = cat(2,rpowern{:}); T\G2B*fGd
rpowern = [ones(length_r,1) rpowern]; d"1DE
else Nvlfi8.
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); wz,T7L
rpowern = cat(2,rpowern{:}); GAZw4dz
end Q}a,+*N.
<*g!R!
% Compute the values of the polynomials: ^k'?e"[gTs
% -------------------------------------- _::q
S!
y = zeros(length_r,length(n)); Y=%SK8]Q;
for j = 1:length(n) D*>EWlZ
s = 0:(n(j)-m_abs(j))/2; aXoD{zA
pows = n(j):-2:m_abs(j); uG|d7LS,%
for k = length(s):-1:1 \WDL?(G<
p = (1-2*mod(s(k),2))* ... y U-^w^4
prod(2:(n(j)-s(k)))/ ... )wmG&"qsP
prod(2:s(k))/ ... [|=#~(yYQ
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Qg7rkRia
prod(2:((n(j)+m_abs(j))/2-s(k))); /D]V3|@E
idx = (pows(k)==rpowers); ,~R`@5+
y(:,j) = y(:,j) + p*rpowern(:,idx); P <$)v5f
end eb])=
SNV[KdvP*
if isnorm ,Zpc vK/S
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 4k
HFfc
end 8sDbvVh1F
end cB;:}Q08#
% END: Compute the Zernike Polynomials <K~> :4c
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +0w~Skd,
!besMZ
% Compute the Zernike functions: I^M%+\
% ------------------------------ U{IY
F{;@
idx_pos = m>0; fZ9EE3
idx_neg = m<0; p19[qy~.
d},IQ,Az:Z
z = y; Vvth,
if any(idx_pos) h\oAW?^
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 0{ZYYB&"~J
end A9*( O)
if any(idx_neg) FS3MR9
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); A`=;yD
end 7,i}M
o -< 5<
% EOF zernfun