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, aT4I sPA?_
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, YsO3( HS
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? sU(<L0
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? hbdB67,
r2%Qk
F MX^k
uc{s\_
E-jJ!>&K
function z = zernfun(n,m,r,theta,nflag) WA6reZ
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. `h%K8];<6f
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N |.z4 VJi4
% and angular frequency M, evaluated at positions (R,THETA) on the W7W(jMH
% unit circle. N is a vector of positive integers (including 0), and IG.!M@_
% M is a vector with the same number of elements as N. Each element hG~HV{6
% k of M must be a positive integer, with possible values M(k) = -N(k) _z=ytt9D
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ::p%R@?
% and THETA is a vector of angles. R and THETA must have the same ?o1QjDG
% length. The output Z is a matrix with one column for every (N,M) Avww@$
% pair, and one row for every (R,THETA) pair. $D='NzE/
% p;qFMzyS9
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike qeDXG
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), @;G%7&ps
% with delta(m,0) the Kronecker delta, is chosen so that the integral XXw>h4hl
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, j.!5&^;u4
% and theta=0 to theta=2*pi) is unity. For the non-normalized \kZ@2.pN
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 99\lZ{f(
% z }Lf]w?
% The Zernike functions are an orthogonal basis on the unit circle. nx(jYXVT
% They are used in disciplines such as astronomy, optics, and \sAkKPI
% optometry to describe functions on a circular domain. ]eUD3WUe>q
% ]z!Df\I
% The following table lists the first 15 Zernike functions. Mp QsM-iW
% EQe$~}[
% n m Zernike function Normalization q[Tl#*P?y
% -------------------------------------------------- )<%CI#s#
% 0 0 1 1 [!C!R$AMa
% 1 1 r * cos(theta) 2 rB-R(2
CCN
% 1 -1 r * sin(theta) 2 AC\y|X8-
% 2 -2 r^2 * cos(2*theta) sqrt(6) Y <`X$
% 2 0 (2*r^2 - 1) sqrt(3) :%gM
Xsb
% 2 2 r^2 * sin(2*theta) sqrt(6) #eF,* d
% 3 -3 r^3 * cos(3*theta) sqrt(8) ^M1jv(
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) n%;4Fm?
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) LSR0yCU
% 3 3 r^3 * sin(3*theta) sqrt(8) /2''EF';
% 4 -4 r^4 * cos(4*theta) sqrt(10) Es- =0gpK
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ]XcWGQv~
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) GTi=VSGqF
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) f9OY>|a9
% 4 4 r^4 * sin(4*theta) sqrt(10) xU2i&il^!
% -------------------------------------------------- Z`f?7/"B
% p' 6h9/
% Example 1: yf[1?{iVo
% 7|"l/s9,
% % Display the Zernike function Z(n=5,m=1) gL~3z'$
% x = -1:0.01:1; \x<,Ma=D
% [X,Y] = meshgrid(x,x); ^I9U<iNIL
% [theta,r] = cart2pol(X,Y); &1Y7Ne
% idx = r<=1; H?eG5
% z = nan(size(X)); @HTs.4
% z(idx) = zernfun(5,1,r(idx),theta(idx));
nI4oQE
% figure :L6%57
% pcolor(x,x,z), shading interp qfdL *D
% axis square, colorbar GPizR|}h
% title('Zernike function Z_5^1(r,\theta)') L8f_^
*,
% }
@
[!%hE
% Example 2: NiEz3ODSi
% y<*\D_J
% % Display the first 10 Zernike functions n^rbc;}
% x = -1:0.01:1; ~c55LlO>
% [X,Y] = meshgrid(x,x); #S]O|$&*
% [theta,r] = cart2pol(X,Y); nVr V6w
% idx = r<=1; 0$NzRPbH
% z = nan(size(X)); Y
O|hwhe_
% n = [0 1 1 2 2 2 3 3 3 3]; ,gW$m~\
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; me F.
% Nplot = [4 10 12 16 18 20 22 24 26 28]; -tx%#(?wH
% y = zernfun(n,m,r(idx),theta(idx));
'SXLnoeTa
% figure('Units','normalized') ^$mCF%e8H
% for k = 1:10 q,_EHPc
% z(idx) = y(:,k); tKeozV[V
% subplot(4,7,Nplot(k)) lfG',hlI;
% pcolor(x,x,z), shading interp z8r?C
% set(gca,'XTick',[],'YTick',[]) xXnSo0`LF
% axis square {MN6JGb|'
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) V)4?y9xZv
% end Bio QV47B
% ~}/_QlX` K
% See also ZERNPOL, ZERNFUN2. Hq~SRc~
J7`;l6+Gb
NGRXNh+
% Paul Fricker 11/13/2006 8Ht=B,7T
1FG"Ak}D
APJFy@l}
z
=\ENG|x#
tR4+]K
% Check and prepare the inputs: xIV#}z0
% ----------------------------- |MN2v[y
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) [S-#}C?~
error('zernfun:NMvectors','N and M must be vectors.') + rM]RFi
end 3g56[;Up?
WRRR "Q$
RQu[FZT,
if length(n)~=length(m) N/b$S@
error('zernfun:NMlength','N and M must be the same length.') 6-\'
*5r
end hD7vjg&Z
&h.?~Ri
/!.]Y8yEH
n = n(:);
;5
m = m(:); 1bDAi2 H
if any(mod(n-m,2)) EMxMJ=
error('zernfun:NMmultiplesof2', ... I.>8p]X
'All N and M must differ by multiples of 2 (including 0).') 3[?;s}61
end YG5mzP<T
ooCfr?E
~Y;Z5e=
if any(m>n) fN21[Jv3
error('zernfun:MlessthanN', ... Y4lN xvY
'Each M must be less than or equal to its corresponding N.') eht>4)
end 90-s@a3B-j
]3Ibl^J
jK%Lewq
if any( r>1 | r<0 ) meXwmO
error('zernfun:Rlessthan1','All R must be between 0 and 1.') sPl3JP&s
end >5TXLOYZ
YN7OQqa
" YOl6n
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) U7e2NES
error('zernfun:RTHvector','R and THETA must be vectors.') 3qDbfO[
end )c 79&S
m( %PZ*s
V&/Cb&~Uw
r = r(:); b.8T<@a
theta = theta(:); (^_INy*
length_r = length(r); |Ho}
D~
if length_r~=length(theta) (yeWArQ
error('zernfun:RTHlength', ... L)S
V?FBx
'The number of R- and THETA-values must be equal.') aWP9i&
end 7{k?"NF
OB^j
b8
gNkBHwv
% Check normalization: K$s{e0
79
% -------------------- >svx
8CT
if nargin==5 && ischar(nflag) Z^%HDB9^
isnorm = strcmpi(nflag,'norm'); TN08,:k
if ~isnorm "5Z5x%3I
error('zernfun:normalization','Unrecognized normalization flag.') 4af^SZ)l
end v`Ja Bn
else _Kh8
<$h
isnorm = false; v-"nyy-&Z
end /YvwQ
-zzT:C
9%0^fhrJ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V^><
=DNE
% Compute the Zernike Polynomials Q)8I(*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G
c,
u?>8`]r
?+%bEZ`
% Determine the required powers of r: 5Q8s{WQ
% ----------------------------------- n;:C{5
m_abs = abs(m); =+[`9
rpowers = []; ~at:\h4:
for j = 1:length(n) 0bSnD|#I
rpowers = [rpowers m_abs(j):2:n(j)]; v_pFI8Cz)
end I=
cayR
rpowers = unique(rpowers); t8.3
8.7lc2aX
r029E-
% Pre-compute the values of r raised to the required powers, ZqjLZ9?q
% and compile them in a matrix: &]A0=h2{P*
% ----------------------------- 'TA
!JB+
if rpowers(1)==0 <7gv<N6BQf
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); b?,=|H
rpowern = cat(2,rpowern{:}); R+=wSG ]
rpowern = [ones(length_r,1) rpowern]; 9ESV[
else 5v=e(Ph+
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); `joyHKZI.
rpowern = cat(2,rpowern{:}); kP^=
end g'2;///
N&|,!Cu
{M\n
% Compute the values of the polynomials: 9oG)\M.6w
% -------------------------------------- VtGZB3
y = zeros(length_r,length(n)); IABF_GwF
for j = 1:length(n) R D?52\
s = 0:(n(j)-m_abs(j))/2; O]j<