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 V^/h;/!^
function z = zernfun(n,m,r,theta,nflag) \rw'QAi8r
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. &;uGIk>s
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N xc3Ov9`8%
% and angular frequency M, evaluated at positions (R,THETA) on the !VJT"Ds_
% unit circle. N is a vector of positive integers (including 0), and }RC.Q`b
% M is a vector with the same number of elements as N. Each element VC_3 ll]vr
% k of M must be a positive integer, with possible values M(k) = -N(k) (_s!,QUe
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, jS5t?0
% and THETA is a vector of angles. R and THETA must have the same AOvH&9**
% length. The output Z is a matrix with one column for every (N,M) +E""8kW- Z
% pair, and one row for every (R,THETA) pair. DbPBgD>Q
% ul5::
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 7+A-7ci
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), *`\4j*$^
% with delta(m,0) the Kronecker delta, is chosen so that the integral 8&`T<ECq>
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, cSH tl<UY
% and theta=0 to theta=2*pi) is unity. For the non-normalized $AL|d[[T[
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. %o SfL;W7
% Q
xj|lr
% The Zernike functions are an orthogonal basis on the unit circle. 6w .iEb
% They are used in disciplines such as astronomy, optics, and <7RkM
% optometry to describe functions on a circular domain. EP%
M8
% [\w>{
% The following table lists the first 15 Zernike functions. "~i#9L/H
% ni 02N3R
% n m Zernike function Normalization plz=G}Y
% -------------------------------------------------- * Kp ^al
% 0 0 1 1 9Rt(G_'
% 1 1 r * cos(theta) 2 y+~Aw"J}
% 1 -1 r * sin(theta) 2 % 'L=
% 2 -2 r^2 * cos(2*theta) sqrt(6) JqH.QnKcv
% 2 0 (2*r^2 - 1) sqrt(3) ]>]H:NEq
% 2 2 r^2 * sin(2*theta) sqrt(6) U%SNROj
% 3 -3 r^3 * cos(3*theta) sqrt(8) ~jrU#<'G9
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Vv*5{_
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) a}+_Yo(Q
% 3 3 r^3 * sin(3*theta) sqrt(8) 9BgQoK@
% 4 -4 r^4 * cos(4*theta) sqrt(10) Xb07 l3UG
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)
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% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) >=RHE@
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^,\se9=(
% 4 4 r^4 * sin(4*theta) sqrt(10) g]hn@{[
% -------------------------------------------------- W1M/Z[h6)5
% BRQ5
% Example 1: jl?y}
% 70 DQ/b
% % Display the Zernike function Z(n=5,m=1) A5J#x6@
% x = -1:0.01:1; $F==n4)
% [X,Y] = meshgrid(x,x); N'1 [t
% [theta,r] = cart2pol(X,Y); v(WL 3[y;
% idx = r<=1;
61 HqBa
% z = nan(size(X)); kv`3Y0R-"
% z(idx) = zernfun(5,1,r(idx),theta(idx)); %>QSeX
% figure \?Oa}&k$F8
% pcolor(x,x,z), shading interp ZpP6Q
% axis square, colorbar m$e@<~To
% title('Zernike function Z_5^1(r,\theta)') TTjjyZ@
% N6 Cc%,
% Example 2: -ZMl[;OM
% uc
`rt"
% % Display the first 10 Zernike functions cVt$#A)
% x = -1:0.01:1; 9H Bx[2&
% [X,Y] = meshgrid(x,x); U*#E aL
% [theta,r] = cart2pol(X,Y); sRI=TE]s
% idx = r<=1; 'J<zVD}0
% z = nan(size(X)); ~s^6Q#Z9|
% n = [0 1 1 2 2 2 3 3 3 3]; i2 Iu2
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Mdq'> <ajL
% Nplot = [4 10 12 16 18 20 22 24 26 28]; /:];2P6#X
% y = zernfun(n,m,r(idx),theta(idx)); MZ#2WP)F
% figure('Units','normalized') 1F%*k &R
% for k = 1:10 _O'rZ5}&
% z(idx) = y(:,k); nHL>}Yg
% subplot(4,7,Nplot(k)) G;.u>92r|
% pcolor(x,x,z), shading interp XcW3IO
% set(gca,'XTick',[],'YTick',[]) O#ajoE
% axis square xo@/k
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}'])
7qdl,z
% end 5EZr"[8M
% n@8{FoF
% See also ZERNPOL, ZERNFUN2. >5Rw~
A-NC,3
% Paul Fricker 11/13/2006 j-\^
}K.&
xn#I7]]G
t7&
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% Check and prepare the inputs: 5|H(N}S_
% ----------------------------- Ib<+m%Ac
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +]2~@=<@
error('zernfun:NMvectors','N and M must be vectors.') 5^R#e(mr
end Kwmo)|7uPU
H< 3b+Sg
if length(n)~=length(m) [}Yci:P_ +
error('zernfun:NMlength','N and M must be the same length.') eT
\Q
end i}12mjF
5 s2}nIe
n = n(:); Y
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m = m(:); BwA~*5TFu
if any(mod(n-m,2)) n!,TBCNX
error('zernfun:NMmultiplesof2', ... {ca^yHgGy
'All N and M must differ by multiples of 2 (including 0).') ~.=HN}E
end IOsDVIXL\
Nd!=3W5?
if any(m>n) :BiR6>1:
error('zernfun:MlessthanN', ... ))-M+CA
'Each M must be less than or equal to its corresponding N.') (B4A$t
end Hm[!R:HW,S
bao5^t}
if any( r>1 | r<0 ) X "r$,~
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ?v*7!2;
end v[=E f
rm;"98~zJ?
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Tm,L?Jh
error('zernfun:RTHvector','R and THETA must be vectors.') 833t0Ml1A/
end -s84/E4Y*
+m},c-,=$w
r = r(:); E^ti!4{<
theta = theta(:); !!pi\J?sk
length_r = length(r); uw&,pq
if length_r~=length(theta) d|HM
error('zernfun:RTHlength', ... s: .XF|e{
'The number of R- and THETA-values must be equal.') q.Mck9R7
end +VFwYdW,
qf {B
% Check normalization: +F6_P
% -------------------- c.> (/
if nargin==5 && ischar(nflag) lt"*y.%@b
isnorm = strcmpi(nflag,'norm'); Q";eyYdOL
if ~isnorm `cRB!w=KHV
error('zernfun:normalization','Unrecognized normalization flag.') s$G8`$+i1
end NGzqiu"J
else YA8~O5
isnorm = false; F'Vl\qPt
end x/^zNO\1
*a.*Ha
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (pg9cM]NA
% Compute the Zernike Polynomials @=1``z#
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,_-*/- 7;8
1W7BN~p14
% Determine the required powers of r: I(S6DkU
% ----------------------------------- md
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m_abs = abs(m); .`RC,R`C
rpowers = []; m^+~pC5
for j = 1:length(n) AXI:h"so
rpowers = [rpowers m_abs(j):2:n(j)]; w\4m-Z{
end MPc=cLv
rpowers = unique(rpowers); tYa*%|!v
T`;M!-)2
% Pre-compute the values of r raised to the required powers, y?hW#l~#X
% and compile them in a matrix: }A^,y
% ----------------------------- GjG3aqP&!
if rpowers(1)==0 8B9zo&
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); rpWy 6oD
rpowern = cat(2,rpowern{:}); _
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rpowern = [ones(length_r,1) rpowern]; r/ f;\w7
else >$F]Ss)$
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); _XPc0r:?>
rpowern = cat(2,rpowern{:}); tsD^8~
t|h
end 6Fb~`J~s
!}7m^
% Compute the values of the polynomials: s9>!^MzBK
% -------------------------------------- kRPg^Fw"Vw
y = zeros(length_r,length(n)); \:7EKzQ
for j = 1:length(n) +3CMfYsr8
s = 0:(n(j)-m_abs(j))/2; A@r,A?(
pows = n(j):-2:m_abs(j); N R{:4zJT
for k = length(s):-1:1 T(DE^E@a
p = (1-2*mod(s(k),2))* ... 4N&}hOM'S
prod(2:(n(j)-s(k)))/ ... E
.5xzY
prod(2:s(k))/ ... e+TNG &_
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... (0S7
prod(2:((n(j)+m_abs(j))/2-s(k))); "N_?yA#(j
idx = (pows(k)==rpowers); ^p/mJ1/s7
y(:,j) = y(:,j) + p*rpowern(:,idx); 70eN]OY
end F^-4Pyq@
1\uS~RR
if isnorm 5JXLfYTUI
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 7j8_O@_
end =UY@,*q:c
end ,d#4Ib
% END: Compute the Zernike Polynomials .MlE1n'
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S y~ 1U
'>BHwc
% Compute the Zernike functions: {'%=tJ[YX
% ------------------------------ ";]m]PRAam
idx_pos = m>0; jC%I]#!n
idx_neg = m<0; h>?OWI
, fn=%tiUk
z = y; }{J8U2])k
if any(idx_pos) oLoa71Q}
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); FBsw\P5w
end sTDBK!9I
if any(idx_neg) m`~ Qr~
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); vNIQc "\-
end MZ'HMYed
2X`M&)"X
% EOF zernfun