下面这个函数大家都不会陌生，计算zernike函数值的，并根据此可以还原出图像来， f=k_U[b4>
我输入10阶的n、m，r,theta为38025*1向量，最后得到的z是29525*10阶的矩阵， z a%gD
这个，跟我们用zygo干涉仪直接拟合出的36项zernike系数，有何关系呢？ u6nO\.TTtY
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ rJZR8bo
H*j!_>W
cY5w,.Q/ !
]p8 zT|bv
7s0\`eX o/
function z = zernfun(n,m,r,theta,nflag) 3v@h&7<E
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 0iYo&q'n
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N lZAXDxhnT
% and angular frequency M, evaluated at positions (R,THETA) on the Rh} }8 sv
% unit circle. N is a vector of positive integers (including 0), and 5?MaKNm }
% M is a vector with the same number of elements as N. Each element ]_BH"ng}
% k of M must be a positive integer, with possible values M(k) = -N(k) ZDG~tCh=@
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, yk y% +@2q
% and THETA is a vector of angles. R and THETA must have the same e2e! "kEF
% length. The output Z is a matrix with one column for every (N,M) G9^xv
% pair, and one row for every (R,THETA) pair. IRGcE&m
% :8K}e]!c1
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike q<j9l'dHG
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), \TZSn1isZX
% with delta(m,0) the Kronecker delta, is chosen so that the integral @9eN\b%I^H
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 2 x>7>;>
% and theta=0 to theta=2*pi) is unity. For the non-normalized dz?On\66
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. lE gjv,
% T|8:_4/l
% The Zernike functions are an orthogonal basis on the unit circle. 0 N"N$f
% They are used in disciplines such as astronomy, optics, and lb$_$+@Vr
% optometry to describe functions on a circular domain. [YP{%1*RM
% 55 '
% The following table lists the first 15 Zernike functions. U
shIQh
% DK
eB%k
% n m Zernike function Normalization NR ny]!
% -------------------------------------------------- O wuc9
% 0 0 1 1 #}Yrxf
% 1 1 r * cos(theta) 2 &<x .D]FA]
% 1 -1 r * sin(theta) 2 e!PB3I
% 2 -2 r^2 * cos(2*theta) sqrt(6) %&_^I*
% 2 0 (2*r^2 - 1) sqrt(3) w
>2sr^!y
% 2 2 r^2 * sin(2*theta) sqrt(6) S)g:+P
% 3 -3 r^3 * cos(3*theta) sqrt(8) 6I: 6+n
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Unv'm5/L
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) _P=+\ [|y
% 3 3 r^3 * sin(3*theta) sqrt(8) d#TA20`
% 4 -4 r^4 * cos(4*theta) sqrt(10) n\)1Bz
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `LNh amp
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) j g//I<D
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) u7^(?"x
% 4 4 r^4 * sin(4*theta) sqrt(10) ~|9VVeE
% -------------------------------------------------- 9Vqy<7i1
% V y$*v
% Example 1: O!%T<2i3
% 76 "4Q!
% % Display the Zernike function Z(n=5,m=1) 4d%0a%Z
% x = -1:0.01:1; ,cL; ,YN
% [X,Y] = meshgrid(x,x); 2,dWD<h
% [theta,r] = cart2pol(X,Y); (:qc[,m
% idx = r<=1; =w}JAEE|(i
% z = nan(size(X)); ,,BP}f+l$
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 6F!B*lr
% figure 9Q^cE\j
% pcolor(x,x,z), shading interp l_/(J)|a
% axis square, colorbar FLs$
% title('Zernike function Z_5^1(r,\theta)') @ J&korU
% C+uW]]~I)
% Example 2: t))MZw&@
% m0 As t<u
% % Display the first 10 Zernike functions EwX&Cj".
% x = -1:0.01:1; w8> h6x "
% [X,Y] = meshgrid(x,x); 5e$1KN`
% [theta,r] = cart2pol(X,Y); );':aX j
% idx = r<=1; tH)j EY9
% z = nan(size(X)); h Fik>B#!
% n = [0 1 1 2 2 2 3 3 3 3]; GkX Se)#p
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; C&>*~
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Bp_R"DS7A
% y = zernfun(n,m,r(idx),theta(idx)); BaW4 s4u
% figure('Units','normalized') _<LL@IX
% for k = 1:10 B Z|A&;
% z(idx) = y(:,k); g&c ~grD
% subplot(4,7,Nplot(k)) / n_s"[I4
% pcolor(x,x,z), shading interp z,4mg6gt
% set(gca,'XTick',[],'YTick',[]) gT_KOO0n
% axis square dgF%&*Il]O
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) $GFR7YC 7
% end ;5b d<N
% i-Rn,}v
% See also ZERNPOL, ZERNFUN2. ey=KA t
H:]cBk^[,
P2a5<#_ |
% Paul Fricker 11/13/2006 [K.1 X=O}
>4jE[$p]"
Bj1 % }B
LXK!4(xa W
/j$=?Rp
% Check and prepare the inputs: GeTk/tU
% ----------------------------- a&x:_vv
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) OQ&N] P2p
error('zernfun:NMvectors','N and M must be vectors.') VFL^-tXnA^
end 9Q%lS
>Ua'*
^~;i a7V&2
if length(n)~=length(m) N! 7}B
error('zernfun:NMlength','N and M must be the same length.') WHY /x /$
end R~4X?@ZB
80*hi)ux[
cx$IWQf2
n = n(:); 3$nK
m = m(:); Sp80xV_B
if any(mod(n-m,2)) Y/kq!)u;%L
error('zernfun:NMmultiplesof2', ... x/umwT,o v
'All N and M must differ by multiples of 2 (including 0).') D#b*M)X"
end \;)g<TwL
E 7fQ9]
a)JXxst
if any(m>n) =Z
error('zernfun:MlessthanN', ... fz=?QEG
'Each M must be less than or equal to its corresponding N.') #m.e9MU
end }_]AQN$'G
TC?B_;a
C7F Qc {
if any( r>1 | r<0 ) sQa;l]O:NC
error('zernfun:Rlessthan1','All R must be between 0 and 1.') iPTQqx-m$7
end ,Y/B49
V.P<>~W
D cvul4Q
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) TVD~Ix
error('zernfun:RTHvector','R and THETA must be vectors.') E$)| Kv^
end b&U1^{(
}tW-l*\U
L/" }; VI
r = r(:); D*%am|QL
theta = theta(:); G%erh}0~
length_r = length(r); H2s:M
if length_r~=length(theta) X_TjJmc
error('zernfun:RTHlength', ... 35& ^spb
'The number of R- and THETA-values must be equal.') &u.{]Yjx
end KS$t
e^Zm09J
:5NMgR.d
% Check normalization: ^3$l!>me
% -------------------- /|
v.A\ :
if nargin==5 && ischar(nflag) c* {6T}VZr
isnorm = strcmpi(nflag,'norm'); _RbfyyaN
if ~isnorm * ): |WDR
error('zernfun:normalization','Unrecognized normalization flag.') 9 (N
end 1Z# $X`
else v A/SrX.
isnorm = false; o&?c,FwN
end :\OSHs<M
,Xn2xOP
&kg^g%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5*YoK)2J
% Compute the Zernike Polynomials
,&h v x
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Hf`i~6
0\dmp'j]
PM\Ju]
% Determine the required powers of r: }>xwiSF?
% ----------------------------------- KZppQ0
m_abs = abs(m); DK IH{:L7
rpowers = []; u\*9\ G
for j = 1:length(n) RQ,# TbAe
rpowers = [rpowers m_abs(j):2:n(j)]; $Ll9ak}
end [3m\~JtS
rpowers = unique(rpowers); *
65/gG8>
z#tIa
o<Zlm)"%1
% Pre-compute the values of r raised to the required powers, ]01`r/ ->\
% and compile them in a matrix: { *yvvb
% ----------------------------- 3( BL
if rpowers(1)==0 *()['c#CC
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ] ,]Rv# `
rpowern = cat(2,rpowern{:}); %B%_[<B
rpowern = [ones(length_r,1) rpowern]; T~[:oil
else OI blBQ!
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); +4?Lwp 'q
rpowern = cat(2,rpowern{:}); 6 4_}"f U
end UQl?_ [G
.vu7$~7
t+?Bb7p,H
% Compute the values of the polynomials: N<)CG,/w[M
% -------------------------------------- M) bQvjj
y = zeros(length_r,length(n)); FuHBzBoM=
for j = 1:length(n) ';I} 6N
s = 0:(n(j)-m_abs(j))/2; X7*F~LFr j
pows = n(j):-2:m_abs(j); ;+ hh|NiQ
for k = length(s):-1:1 ~apt, hl
p = (1-2*mod(s(k),2))* ... [3Q0KCZ0(
prod(2:(n(j)-s(k)))/ ... ,->ihxf
prod(2:s(k))/ ... c^r8<KlI9
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 7[m+r:y
prod(2:((n(j)+m_abs(j))/2-s(k))); xs"i_se
idx = (pows(k)==rpowers); ]es|%j 2
y(:,j) = y(:,j) + p*rpowern(:,idx); <XeDJ8
'
end 2/?Zp=|j\
~fXNj-'RW
if isnorm uKJ:)oyaCP
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); iuV4xyp
end `c G ks
end jX7K- L
% END: Compute the Zernike Polynomials O/~T+T%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TNu % _
34
[c?0Q3F
l# 0zHBc
% Compute the Zernike functions: eb_.@.a
% ------------------------------ ('z=/"(l
idx_pos = m>0; Z518J46o
idx_neg = m<0; QV[&2&&^<<
FWW4n_74
6qZQ 20h
z = y; ?_]Y8f
if any(idx_pos) s\ *p |vc
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); e9p/y8gC
end [MeivrJ+
if any(idx_neg) Il =6t
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)');
eXl?f_9
end c^|8qvS $
}u^bTR?3
2zj`
H9
% EOF zernfun Ijh RSrCv
