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， v5\5:b {/
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， '3 b'moy
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？ 61w
({F
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ N 8iLI`
` {qt4zd0
jU- a a+
6>]w1
H
B!
P/?
function z = zernfun(n,m,r,theta,nflag) Ci4; e
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. .8->n aj|
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N g4u 6#.m(
% and angular frequency M, evaluated at positions (R,THETA) on the y 2)W"PuG
% unit circle. N is a vector of positive integers (including 0), and Z9.0#Jnu
% M is a vector with the same number of elements as N. Each element /xSFW7d1
% k of M must be a positive integer, with possible values M(k) = -N(k) &1Cs'
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, gyb99c,)
% and THETA is a vector of angles. R and THETA must have the same {
V) `6
% length. The output Z is a matrix with one column for every (N,M) U\u07^h[
% pair, and one row for every (R,THETA) pair. \Si p
% zW\ s{
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike !6l* Jc3
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), `^] D;RfE
% with delta(m,0) the Kronecker delta, is chosen so that the integral S@'%dN6e
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, /Kh,
% and theta=0 to theta=2*pi) is unity. For the non-normalized ]}kw'&
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. =Oq *9=v|
% 16>D?;2o(
% The Zernike functions are an orthogonal basis on the unit circle. d@p#{ -
% They are used in disciplines such as astronomy, optics, and vz~Oi
% optometry to describe functions on a circular domain. y Vp,)T9
% rvlvk"
% The following table lists the first 15 Zernike functions. 1Au+X3
% R+U$;r8l
% n m Zernike function Normalization +TyN;e
% -------------------------------------------------- x5 CM P%}d
% 0 0 1 1 u>]3?ty`
% 1 1 r * cos(theta) 2 tS >^x
% 1 -1 r * sin(theta) 2 M\/hK2J# #
% 2 -2 r^2 * cos(2*theta) sqrt(6) ="5 D}%
% 2 0 (2*r^2 - 1) sqrt(3) < :Mz2 Rg
% 2 2 r^2 * sin(2*theta) sqrt(6) y%X! l(gQ
% 3 -3 r^3 * cos(3*theta) sqrt(8) O5aXa_A_u
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ]
j8bv3
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) yx|{:Li!
% 3 3 r^3 * sin(3*theta) sqrt(8) j !w{
% 4 -4 r^4 * cos(4*theta) sqrt(10) ha Y]gmC
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /y$ Fw9R;
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ``P9fd
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 33EF/k3vW
% 4 4 r^4 * sin(4*theta) sqrt(10) x(cv }#}S8
% -------------------------------------------------- !: m.-TE
% K"x_=^,Yu*
% Example 1: NhCucSU<K
% I/XSW #
% % Display the Zernike function Z(n=5,m=1) 9=~ZA{0J
% x = -1:0.01:1; 1f<R,>
% [X,Y] = meshgrid(x,x); n| {#5#
% [theta,r] = cart2pol(X,Y); @,n)1*{P
% idx = r<=1;
oX8EY l
% z = nan(size(X)); TIxOMY y
% z(idx) = zernfun(5,1,r(idx),theta(idx)); \yu7,v
% figure D*Zj oU
% pcolor(x,x,z), shading interp l'/`2Y1
% axis square, colorbar vUVFW'-
% title('Zernike function Z_5^1(r,\theta)') F Gx)?
% } L)[>
% Example 2: u\Ylo.)b
% L7 Hv )
% % Display the first 10 Zernike functions ",.f
% x = -1:0.01:1; kqm (D#
% [X,Y] = meshgrid(x,x); DH
yv^
% [theta,r] = cart2pol(X,Y); Q(]m 1\a
% idx = r<=1; k9f|R*LM
% z = nan(size(X)); h @Ea5x
% n = [0 1 1 2 2 2 3 3 3 3]; CYLab5 A
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; [9${4=Kq
% Nplot = [4 10 12 16 18 20 22 24 26 28]; b9RH sr]V
% y = zernfun(n,m,r(idx),theta(idx)); v I I{i
% figure('Units','normalized') XQ]vJQYIR
% for k = 1:10 +7} ^Y} (
% z(idx) = y(:,k); $j .;$~F
% subplot(4,7,Nplot(k)) hNM 8H
% pcolor(x,x,z), shading interp n82tZpn
% set(gca,'XTick',[],'YTick',[]) [M[ <'+^*
% axis square ()IZ7#kL?
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) -0d9 ,,c
% end v?OVhV
% pE&G]ZC
% See also ZERNPOL, ZERNFUN2. \Q|-Npw
3e&+[j
`P; r[j"
% Paul Fricker 11/13/2006 RJ'[m~yl5X
"-$}GUK?Z
@ W>@6E
c$ !?4z_.
q4[}b-fF
% Check and prepare the inputs: P|<V0
Vs.
% ----------------------------- 1`K-f
m)
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) `!vqT 3p,
error('zernfun:NMvectors','N and M must be vectors.') qU !dg
end &T,|?0>~=J
Pp-N2t86#2
Xe %J{
if length(n)~=length(m) bg i_QB#k\
error('zernfun:NMlength','N and M must be the same length.') ?Fl}@EA#M
end &))d],tJX
N.JR($N$
{Nl?
n = n(:); ]Z civnN#
m = m(:); 'z.
GAR
if any(mod(n-m,2)) gB'ajX=OA/
error('zernfun:NMmultiplesof2', ... -`PziG l@<
'All N and M must differ by multiples of 2 (including 0).') ] zol?
end ed,A'S= d
B?z2@,
e"t0 rScA
if any(m>n) EfEgY|V0
error('zernfun:MlessthanN', ... Z< i }XCE
'Each M must be less than or equal to its corresponding N.') .7.lr[$g
end YGo?%.X
qS vV |G
|#2WN-
if any( r>1 | r<0 ) IE`3I#v
error('zernfun:Rlessthan1','All R must be between 0 and 1.') =y][j+WH
end (SyD)G\rj
h ik.qK
+-j-)WU?,
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Fa%1] R
error('zernfun:RTHvector','R and THETA must be vectors.') -Q n-w3~&
end sG`x |%t
=x8[%+
]bY|> q
r = r(:); #hxYB
theta = theta(:); g{Hgs
length_r = length(r); iwK.* 07+
if length_r~=length(theta) dEJqgp}\p
error('zernfun:RTHlength', ... <N vw*yA
'The number of R- and THETA-values must be equal.') E {orezP
end M@cFcyk K
.^wpf S
|{ N{VK
% Check normalization: e34>q:#5l
% -------------------- qq5X3K2&
if nargin==5 && ischar(nflag) Pf[E..HF*d
isnorm = strcmpi(nflag,'norm'); XDY]LAV
if ~isnorm g$K\rA
error('zernfun:normalization','Unrecognized normalization flag.') KoERg&fY
end {XOl &
else v $]B;;[A
isnorm = false; O-(V`BZe
end =LaE EL
pa!BJ]~
Gm|-[iUTG]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B->AY.&j
% Compute the Zernike Polynomials _9h$8(wjn
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8FuxN2
wo@ T @Ve~
Pu3oQD ldV
% Determine the required powers of r: %hVR|K|J
% ----------------------------------- 8qyEHUN2q
m_abs = abs(m); sM-,95H
rpowers = []; Wlc&QOfF
for j = 1:length(n) |oPRP1F-;e
rpowers = [rpowers m_abs(j):2:n(j)]; E#J})cPzw
end fu5L)P^T
rpowers = unique(rpowers); a:cci?cb
bT ,_=7F
p [Po*c.b
% Pre-compute the values of r raised to the required powers, @su<h\)
% and compile them in a matrix: iXMJ1\!q\|
% ----------------------------- i\sBey ND"
if rpowers(1)==0 8c9HJ9 vk
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 8<L{\$3HP|
rpowern = cat(2,rpowern{:}); jo e)b
rpowern = [ones(length_r,1) rpowern]; b > D
else fmW{c mr|
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Jy(G
A
rpowern = cat(2,rpowern{:}); yx]9rD1cz
end YlrN ^rO
ZwUBeyxS=c
;5tOQ&p%v
% Compute the values of the polynomials: `\UY5 n72
% -------------------------------------- Bv<g Vt
y = zeros(length_r,length(n)); L8`v
for j = 1:length(n) 0ID9=:J
s = 0:(n(j)-m_abs(j))/2; =~;~hZj
pows = n(j):-2:m_abs(j); 0/GBs~P
for k = length(s):-1:1 ng%[yY
p = (1-2*mod(s(k),2))* ... ^%7(
prod(2:(n(j)-s(k)))/ ... e0`z~z]6&
prod(2:s(k))/ ... 'n'>+W:
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... aKj|gwo!
prod(2:((n(j)+m_abs(j))/2-s(k))); mh3S?Uc
idx = (pows(k)==rpowers); /yI4; :/
y(:,j) = y(:,j) + p*rpowern(:,idx); O*~,L6# }
end Pxr/*X
CTNL->
if isnorm &s".hP 6
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); NH/A`Wm
end nm5DNpHk
end 9S%5 Z>
% END: Compute the Zernike Polynomials ve
d]X!
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <st <oR'
z8X7Y
>+SA
KL_ /f
% Compute the Zernike functions: ^C'S-2nGH
% ------------------------------ v5M4Rs&t
idx_pos = m>0; E; a,].
idx_neg = m<0; =o 9s?vOJ
I-R 7+o
!8G)` '
z = y; uyYV_Q0~;
if any(idx_pos) H7+"BWc
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Q 5AS N"_
end L3%frIUd
if any(idx_neg) ogFo/TKM
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 4t[7 lL`Z
end `]5qIKopL
*p(_="J,
:H&Q!\a
% EOF zernfun vFk@
