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, /@@?0xjX
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, CmdPa!4)
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? %S<))G
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? =H?^G[ y
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function z = zernfun(n,m,r,theta,nflag) 1HUe8m[#3
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. W/u_<\
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ;TboS-Y
% and angular frequency M, evaluated at positions (R,THETA) on the 6<No_x |_
% unit circle. N is a vector of positive integers (including 0), and Za7!n{?0
% M is a vector with the same number of elements as N. Each element
!qTP
% k of M must be a positive integer, with possible values M(k) = -N(k) fDwqu.K
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, RM#.-gW
% and THETA is a vector of angles. R and THETA must have the same '3TfW61]
% length. The output Z is a matrix with one column for every (N,M) +HoCG;C{
% pair, and one row for every (R,THETA) pair. GP_%.fO\M
% @[~j|YH}
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike >z k6{kC
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), % E8s>D
% with delta(m,0) the Kronecker delta, is chosen so that the integral eNr2-R
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 0">9n9
% and theta=0 to theta=2*pi) is unity. For the non-normalized 3#Xv))w1
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. _cd=PZhI
% h&x;#.SYK
% The Zernike functions are an orthogonal basis on the unit circle. jk1mP6'P|
% They are used in disciplines such as astronomy, optics, and "`
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% optometry to describe functions on a circular domain. )V9wU1.
% (*Q8!"D^6
% The following table lists the first 15 Zernike functions. [y(<1]i-a
% F\-oZ#g
% n m Zernike function Normalization r(I&`kF<
% -------------------------------------------------- AhQsv.t
% 0 0 1 1 dI~{0)s
% 1 1 r * cos(theta) 2 T5>'q;jM
% 1 -1 r * sin(theta) 2 =Iy khrS
% 2 -2 r^2 * cos(2*theta) sqrt(6) ^-%O
% 2 0 (2*r^2 - 1) sqrt(3) ij02J`w:Ra
% 2 2 r^2 * sin(2*theta) sqrt(6) !~te&ccPE
% 3 -3 r^3 * cos(3*theta) sqrt(8) {r_x\VC=p
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ||'A9
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) j~#v*qmDU
% 3 3 r^3 * sin(3*theta) sqrt(8) Wn5xX5H C
% 4 -4 r^4 * cos(4*theta) sqrt(10) 6gB;m$:fV
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #=czqZw
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) sH :_sOV*
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )uy2,`z
% 4 4 r^4 * sin(4*theta) sqrt(10) 0t -=*7w%
% -------------------------------------------------- R'h.lX
% BZk0B?
% Example 1: &cT@MV5
% :F
pt>g
% % Display the Zernike function Z(n=5,m=1) j:[#eC
% x = -1:0.01:1; Jf@~/!m}'
% [X,Y] = meshgrid(x,x); i=\`f& B
% [theta,r] = cart2pol(X,Y); k<k@Tlo
% idx = r<=1; Bu7aeBP
% z = nan(size(X)); 5wa!pR\c
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Kk 6i
% figure }!jn%@_y@
% pcolor(x,x,z), shading interp jGtu>|Gj
% axis square, colorbar pZ&?uo67_
% title('Zernike function Z_5^1(r,\theta)') Us4#O&
% @@#(<[S\B
% Example 2: z;PF%F
% dd!Q[]$ }
% % Display the first 10 Zernike functions LmjGU[L,@
% x = -1:0.01:1; f|&,SI ?
% [X,Y] = meshgrid(x,x); ZW`wA2R0
% [theta,r] = cart2pol(X,Y); Z6_fI
% idx = r<=1; M+Eg{^ q`
% z = nan(size(X)); H*h4D+Kxv
% n = [0 1 1 2 2 2 3 3 3 3]; mZ#h p}\.
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; O.$OLK;v
% Nplot = [4 10 12 16 18 20 22 24 26 28]; I0} G,
q
% y = zernfun(n,m,r(idx),theta(idx)); f<*-;
% figure('Units','normalized') kB]*2o9-3
% for k = 1:10 !]=S A &
% z(idx) = y(:,k); g!!:o(k
% subplot(4,7,Nplot(k)) epxbTJfc
% pcolor(x,x,z), shading interp YI+o:fGC5
% set(gca,'XTick',[],'YTick',[]) %)P)Xb
% axis square 1NQU96
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ;
nYR~~
% end [k
% (?#"S67
% See also ZERNPOL, ZERNFUN2. x1`zD*{
`_ )5K u}
tJ Mm
% Paul Fricker 11/13/2006 dS;Ui]/J
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% Check and prepare the inputs: TJz}
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% ----------------------------- _!^2A3c<
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) `2@f=$B
error('zernfun:NMvectors','N and M must be vectors.') aHBM9 %gV
end 5 IFc"
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if length(n)~=length(m) X oh@ (%
error('zernfun:NMlength','N and M must be the same length.') \Vl)q>K_h
end ![/ QW
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n = n(:); >o!5)\F
m = m(:); u~\ NL{
if any(mod(n-m,2)) =[IKwmCX
error('zernfun:NMmultiplesof2', ... `{'h+v`
'All N and M must differ by multiples of 2 (including 0).') |#x]/AXa0/
end 9[Xe|5?c
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if any(m>n) gutf[Ksu
error('zernfun:MlessthanN', ... Ct386j><
'Each M must be less than or equal to its corresponding N.') R6qC0@*
end 9DaoMOPEI
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if any( r>1 | r<0 ) okLheF
error('zernfun:Rlessthan1','All R must be between 0 and 1.') uAv'%/
end !sav~dB)
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) a'B 5m]%
error('zernfun:RTHvector','R and THETA must be vectors.') \zV'YeG
end );L +)UV
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r = r(:); jQY^[A
theta = theta(:); A:,R.P>`C
length_r = length(r); |5me }!C
if length_r~=length(theta) W Z^u%Z
error('zernfun:RTHlength', ... KhPDkD-
'The number of R- and THETA-values must be equal.') k~pbXA*u
end 4Q^i"jT
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% Check normalization: JbW!V Y
% -------------------- psB9~EU&Q
if nargin==5 && ischar(nflag) f<P>IE
isnorm = strcmpi(nflag,'norm'); Tg/rV5@ka
if ~isnorm W0KSLxM
error('zernfun:normalization','Unrecognized normalization flag.') y<n<uZ;
end uqK[p^{
else DK }1T
isnorm = false; 21.N+H'
end 7G9o%!D5
%!p/r`
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% E/uKzzD9
% Compute the Zernike Polynomials 8u bb~ B;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }ygxmb^@Z
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?\Lf=[
% Determine the required powers of r: 'EsdYx5C
% ----------------------------------- iM{UB=C
m_abs = abs(m); K 6HH_T
rpowers = []; (vr
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for j = 1:length(n) ,P$Crs[
rpowers = [rpowers m_abs(j):2:n(j)]; $_b^p=
end ~Is-^k)y
rpowers = unique(rpowers); ulxy 4] h
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% Pre-compute the values of r raised to the required powers, e?eX9yA7F
% and compile them in a matrix: .GNl31f0
% ----------------------------- Gt5'-Hyo
if rpowers(1)==0 ICXz(?a
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); :gacP?
rpowern = cat(2,rpowern{:}); 7P7d[KP<
rpowern = [ones(length_r,1) rpowern]; g'{hp:
else D}7G|gX1
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Hp?uYih0
rpowern = cat(2,rpowern{:}); L'$;;eM4
end fDIKR[B
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% Compute the values of the polynomials: ?&GV~DYxA
% -------------------------------------- +q@g
y = zeros(length_r,length(n)); LFV',1+
for j = 1:length(n) ?^W`7H F%0
s = 0:(n(j)-m_abs(j))/2; fN{JLp
pows = n(j):-2:m_abs(j); !ie'}|c
for k = length(s):-1:1 jbK<"T5
p = (1-2*mod(s(k),2))* ... g7nqe~`{
prod(2:(n(j)-s(k)))/ ... Zi~-m]9U
prod(2:s(k))/ ... @8s:,Y_
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... k&8&