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， mSLA4[4{
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， 8C]K36q
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？ X-LCIT|1
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ n2} (Pt.
K 8#MQR2@
@~ k 4,dJ
EyR ~VKbJ'
scX'>\w&c
function z = zernfun(n,m,r,theta,nflag) \4Uhc3
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ! yqe z
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 94^)Ar~O
% and angular frequency M, evaluated at positions (R,THETA) on the F
t/yPv
% unit circle. N is a vector of positive integers (including 0), and 6_|iXs(&
% M is a vector with the same number of elements as N. Each element [wU e"{
% k of M must be a positive integer, with possible values M(k) = -N(k) z]_2lx2e
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, j9gn7LS
% and THETA is a vector of angles. R and THETA must have the same /]j^a:#"6t
% length. The output Z is a matrix with one column for every (N,M) ;)~}/nR<a
% pair, and one row for every (R,THETA) pair. JfD-CoQS'
% e}dGK=`
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike (
jA C Lo
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 4>^LEp
% with delta(m,0) the Kronecker delta, is chosen so that the integral !/nXEjW?
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, "<Ozoo1&w
% and theta=0 to theta=2*pi) is unity. For the non-normalized &~mJ
).*
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 8\WV.+
% W( pq _H'
% The Zernike functions are an orthogonal basis on the unit circle. [|".j#ZlK
% They are used in disciplines such as astronomy, optics, and Fn>KdoByN
% optometry to describe functions on a circular domain. /H4Z.|@
% n%|og^\ 0
% The following table lists the first 15 Zernike functions. 'tTUro1~
% \SSHj ONX
% n m Zernike function Normalization 9 |D*}OY>
% -------------------------------------------------- vB7 4r]'F
% 0 0 1 1 ai% *s&0/Y
% 1 1 r * cos(theta) 2 O2 2Q
g
% 1 -1 r * sin(theta) 2 BG |m 5f
% 2 -2 r^2 * cos(2*theta) sqrt(6) 5P Zzaz<
% 2 0 (2*r^2 - 1) sqrt(3) JBhM*-t(M1
% 2 2 r^2 * sin(2*theta) sqrt(6) vA3wn><
% 3 -3 r^3 * cos(3*theta) sqrt(8) = N&5]Z
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) M*!WXQlud
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) `j1oxJm
% 3 3 r^3 * sin(3*theta) sqrt(8) [Dhqyjq
% 4 -4 r^4 * cos(4*theta) sqrt(10) u6nO\.TTtY
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) H )51J:4
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) H*j!_>W
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) HQvJ*U4++
% 4 4 r^4 * sin(4*theta) sqrt(10) GO?hB4 9T
% -------------------------------------------------- 7 }: +Yx
% 3CzF@t;5
% Example 1: li hIPMU
% NnH]c+
% % Display the Zernike function Z(n=5,m=1) w73?E#8
% x = -1:0.01:1; _tUh*"e&
% [X,Y] = meshgrid(x,x); [#=IKsO'R6
% [theta,r] = cart2pol(X,Y); ]A ;.}1'
% idx = r<=1; O8 OAXRt/Y
% z = nan(size(X)); 7nm'v'\u+V
% z(idx) = zernfun(5,1,r(idx),theta(idx)); &szYa-K*
% figure :8K}e]!c1
% pcolor(x,x,z), shading interp b)@D@K"5
% axis square, colorbar @9eN\b%I^H
% title('Zernike function Z_5^1(r,\theta)') 2 x>7>;>
% U9ZuD40\
% Example 2: fy]c=:EmD
% 2X<%BFsE
% % Display the first 10 Zernike functions |kH.o=
% x = -1:0.01:1; SJ91(K
% [X,Y] = meshgrid(x,x); 'hE'h?-7
% [theta,r] = cart2pol(X,Y); a:8 MoH 4
% idx = r<=1; cZJ5L>ox
% z = nan(size(X)); =A! r ZG
% n = [0 1 1 2 2 2 3 3 3 3]; 8>@J W]
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ) ,CKuq>
% Nplot = [4 10 12 16 18 20 22 24 26 28]; nK32or3
% y = zernfun(n,m,r(idx),theta(idx)); 55 '
% figure('Units','normalized') U
shIQh
% for k = 1:10 DK
eB%k
% z(idx) = y(:,k); NR ny]!
% subplot(4,7,Nplot(k)) CuD}Uo+u
% pcolor(x,x,z), shading interp Nc &J%a
% set(gca,'XTick',[],'YTick',[]) r;xy/*%Mtj
% axis square 9dw*
++
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) c<, LE@ V
% end d<+hQ\B F,
% ]J [d8S5
% See also ZERNPOL, ZERNFUN2. .uVd '
}zks@7 kf
=[8K#PZ$w
% Paul Fricker 11/13/2006 U(0FL6sPC
$|xSM2
7Q]c=i cg
5D.Sg;\
JO`r)_
% Check and prepare the inputs: gROK4'j6y
% ----------------------------- m ";..V
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) PWZd<
error('zernfun:NMvectors','N and M must be vectors.') 'da
'WZG
end ^GAJ9AF@(
DI8<0.L
q8&l%-d`
if length(n)~=length(m) H<[~V0=
error('zernfun:NMlength','N and M must be the same length.') `vMhrn
end 5VP 0Xa ~
3 rB0 H
yq49fEgc@U
n = n(:); F@k}p-e~
m = m(:); y] c1x=x
if any(mod(n-m,2)) Yb-{+H8{ J
error('zernfun:NMmultiplesof2', ... oz>2P.7
'All N and M must differ by multiples of 2 (including 0).') }^iqhUvT F
end |4g0@}nr+W
(5 @H
EwX&Cj".
if any(m>n) w8> h6x "
error('zernfun:MlessthanN', ... 5e$1KN`
'Each M must be less than or equal to its corresponding N.') );':aX j
end tH)j EY9
h Fik>B#!
GkX Se)#p
if any( r>1 | r<0 ) ;RJ
8h
x
error('zernfun:Rlessthan1','All R must be between 0 and 1.') h# "$W;(
end >>
"gb/x,
V0v,s^\H
Kc?4q= 7q
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 7M~sol[ *
error('zernfun:RTHvector','R and THETA must be vectors.') w^ut,`yW R
end jtlRom}
cu($mjC@T
} 1e4u{
r = r(:); 7n6g;8xE
theta = theta(:); Hl0"
zS[
length_r = length(r); jZz Tnmm&?
if length_r~=length(theta) _yv# v_Z
error('zernfun:RTHlength', ... 5K ;E*s,
'The number of R- and THETA-values must be equal.') 2^=.j2
end 4_k N';a4Q
#M16qOEw
/W$i8g
% Check normalization: *$ g!/,
% -------------------- 8Rwk
o6x
if nargin==5 && ischar(nflag) d]] z )
isnorm = strcmpi(nflag,'norm'); *u1q7JFQk
if ~isnorm X n$ZA -
error('zernfun:normalization','Unrecognized normalization flag.') 7bzm5w@v
end NC%hsg^0/
else nf/?7~3?[
isnorm = false; SOhM6/ID2/
end +Cw_qS"=
= 'NV3by
L "|Bm{Run
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% n(J>'Z
% Compute the Zernike Polynomials b& +zAt.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Dz: +.
@k
^obuMQ;
t\K
(zE
% Determine the required powers of r: p0bWzIH
% ----------------------------------- Bzrn mz5S
m_abs = abs(m); z}ar$}T
rpowers = []; ]8 \I{LR
for j = 1:length(n) R J{$`d
rpowers = [rpowers m_abs(j):2:n(j)]; +gX,r$bX
end Nnl 3r@
rpowers = unique(rpowers); yov~'S9
=EP`,zqn$9
< /{Zb.
% Pre-compute the values of r raised to the required powers, qh ~bX
i!
% and compile them in a matrix: Cy]=Y
% ----------------------------- Cf=H~&`Z
if rpowers(1)==0 jW\:+Taq
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ~\_VWXXvIW
rpowern = cat(2,rpowern{:}); ;o!p9MEpz;
rpowern = [ones(length_r,1) rpowern]; 1.cP3k l
else 'RMUjJ-!
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); B`B =bn+4
rpowern = cat(2,rpowern{:}); %+(AKZu:
end }`B
.(3n
(' 5?-
OOqT 0w N
% Compute the values of the polynomials: <
'5~p$
% -------------------------------------- KdR4<qVV}
y = zeros(length_r,length(n)); N `|A
for j = 1:length(n) &:&89<C'
s = 0:(n(j)-m_abs(j))/2;
e^Zm09J
pows = n(j):-2:m_abs(j); W`;;f Je
for k = length(s):-1:1 ^3$l!>me
p = (1-2*mod(s(k),2))* ... /|
v.A\ :
prod(2:(n(j)-s(k)))/ ... Jj- \Eb?
prod(2:s(k))/ ... OyZR&,q
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... uQ5h5Cfz
prod(2:((n(j)+m_abs(j))/2-s(k))); DXLXGvcM
idx = (pows(k)==rpowers); N)X Tmh2v|
y(:,j) = y(:,j) + p*rpowern(:,idx); IL].!9
end !DZ=`a?y
z'q~%1t
if isnorm OIrm9D #
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); p R'J4~
end ,n/]ALz>~
end f ^$,;
% END: Compute the Zernike Polynomials ^PZ[;F40
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1B~O!']N<
=?RI`}vw_H
x0
)V
o]r
% Compute the Zernike functions: .k,j64
r
% ------------------------------ p;+O/'/j
idx_pos = m>0; =} `d
idx_neg = m<0; v~nKO?{
ku]5sd >b
y(COB6r
z = y; ${ {4L ?7
if any(idx_pos) ,{_i{WV
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); C*Vm} |)
end ;k gP:n
if any(idx_neg) * dB eb
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 9-42A7g^C
end 'c35%? ]
T2e-RR
(T%F^s5D
% EOF zernfun # A/OGi
