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， 2aw&YZ&X o
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， u%/fx~t$
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？ g]c[O*NTL
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ le +R16Z
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function z = zernfun(n,m,r,theta,nflag) YJg,B\z}
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. GZS1zTwBL
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N h&.wo !
% and angular frequency M, evaluated at positions (R,THETA) on the &AVpLf:?
% unit circle. N is a vector of positive integers (including 0), and T9)n Q[
% M is a vector with the same number of elements as N. Each element 'i;|c
% k of M must be a positive integer, with possible values M(k) = -N(k) )#|<w9uec
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, <!-sZ_qq
% and THETA is a vector of angles. R and THETA must have the same KrVcwAcq|1
% length. The output Z is a matrix with one column for every (N,M) ih,%i4<}6m
% pair, and one row for every (R,THETA) pair. ~R$~&x (b
% SG}V[Glk
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike G2 2NQ~w8
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), iovfo2!hD
% with delta(m,0) the Kronecker delta, is chosen so that the integral 0]QRsVz+
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, %75xr9yOP
% and theta=0 to theta=2*pi) is unity. For the non-normalized b2 _Yu^
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. alh >"9~!
% aQ^umrj@?9
% The Zernike functions are an orthogonal basis on the unit circle. CQel3Jtt.
% They are used in disciplines such as astronomy, optics, and Fh v/[j^X
% optometry to describe functions on a circular domain. Mb3}7 @/[
% /@AEJ][$
% The following table lists the first 15 Zernike functions. xtPLR/Z
% oH0X<'
% n m Zernike function Normalization M/x >51<
% -------------------------------------------------- J N^ &S
% 0 0 1 1 j!7`]
% 1 1 r * cos(theta) 2 Hf'G8vW
% 1 -1 r * sin(theta) 2 ,+`61J3W
% 2 -2 r^2 * cos(2*theta) sqrt(6) .@
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% 2 0 (2*r^2 - 1) sqrt(3) G?f\>QSZ
% 2 2 r^2 * sin(2*theta) sqrt(6) Kv sh
% 3 -3 r^3 * cos(3*theta) sqrt(8) s9dO,FMs0t
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) J=.`wZQkS
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) dz~co Z9
% 3 3 r^3 * sin(3*theta) sqrt(8) b jAnaya
% 4 -4 r^4 * cos(4*theta) sqrt(10) GgaTn!mJt
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,-x!$VqS
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) tm7u^9]
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1&fc1uYB4
% 4 4 r^4 * sin(4*theta) sqrt(10) y_xnai
% -------------------------------------------------- u%'\UmE w
% SIBoCs5
% Example 1: JS}{ %(B
% ~|wbP6</:-
% % Display the Zernike function Z(n=5,m=1) ?"?6,;F(4
% x = -1:0.01:1; s@M Yc@k
% [X,Y] = meshgrid(x,x); zP6.xp3
% [theta,r] = cart2pol(X,Y); V\(: @0"
% idx = r<=1; @EE."T9
% z = nan(size(X)); eIl]oC 7*
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 0%!rx{f#\
% figure -v6M<
% pcolor(x,x,z), shading interp P0 `Mdk371
% axis square, colorbar ;3_l@dP"
% title('Zernike function Z_5^1(r,\theta)') (98Nzgxgx}
% eY{+~|KZ
% Example 2: >
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% P{Q=mEQ
% % Display the first 10 Zernike functions 9&RFO$WH
% x = -1:0.01:1; F I"`DMb}
% [X,Y] = meshgrid(x,x); ~ %B<
% [theta,r] = cart2pol(X,Y); r\ nx=
% idx = r<=1; m S k5u 7
% z = nan(size(X)); 5k|9gICyd*
% n = [0 1 1 2 2 2 3 3 3 3]; #+$Q+Z|6k
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; y4+ ;z2' >
% Nplot = [4 10 12 16 18 20 22 24 26 28]; k+1 |I)z
% y = zernfun(n,m,r(idx),theta(idx)); e 8'wG{3A
% figure('Units','normalized') KR7@[
% for k = 1:10 A. UUW
% z(idx) = y(:,k); ;-Um Y}MU
% subplot(4,7,Nplot(k)) \QU^>2 3
% pcolor(x,x,z), shading interp ko5V9Drc
% set(gca,'XTick',[],'YTick',[]) 2w)-\/j }
% axis square };'\~g,1
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) vM_:&j_?``
% end lsN~*q?~]
% u.rY#cS,-R
% See also ZERNPOL, ZERNFUN2. <3,<\ub
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% Paul Fricker 11/13/2006 2P c%fuC
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% Check and prepare the inputs: ^k<o T'89
% ----------------------------- %,ngRYxT#
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) -GLMmZJt
error('zernfun:NMvectors','N and M must be vectors.') Ali 9pvE
end 7t.!lh5G%
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if length(n)~=length(m) ytNO*XoR
error('zernfun:NMlength','N and M must be the same length.') =_0UD{"_0
end ]r_;dY a
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n = n(:); U4G`ZK v(!
m = m(:); .KdyJ6o
if any(mod(n-m,2)) % \i9p]=
error('zernfun:NMmultiplesof2', ... 10H)^p%3+
'All N and M must differ by multiples of 2 (including 0).') H:"ma S\I
end "O(9 m.CZ
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if any(m>n) 'fPdpnJ<
error('zernfun:MlessthanN', ... qoAj]
")
'Each M must be less than or equal to its corresponding N.') '}R i`
end w|Nz_3tI
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r;m)nRu
if any( r>1 | r<0 ) n9s iX
error('zernfun:Rlessthan1','All R must be between 0 and 1.') VsA'de!V4[
end V %Sy"IG
^%` wJ.c
hdVdcn M
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) -1J[n0O.
error('zernfun:RTHvector','R and THETA must be vectors.') fNrgdfo
end H8 "@ iE,
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r = r(:); GaSPJt
theta = theta(:); ~,*b }O
length_r = length(r); <mAhr
if length_r~=length(theta) L QjsOo
error('zernfun:RTHlength', ... MR}Agu#LG
'The number of R- and THETA-values must be equal.') FHV-BuH5
end P
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% Check normalization: DY^ ;EZ!hb
% -------------------- xNbPsoK
if nargin==5 && ischar(nflag) A,4fEmWM
isnorm = strcmpi(nflag,'norm'); ~s5SZK*
if ~isnorm 2p " WTd
error('zernfun:normalization','Unrecognized normalization flag.') :>=\. \
end *BR ^U$,e
else [Jv@J\
isnorm = false; ,N0#!<}4
end &?(?vDFfZ
q`r**N+zn
/E\%>wv
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Jke k-m
% Compute the Zernike Polynomials p a# IJ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Hhh0 T>gi
z[;z>8|c
f`Fi# EKT
% Determine the required powers of r: w`5xrqt@
% ----------------------------------- (P$H <FtH
m_abs = abs(m); $ ,
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rpowers = []; ,S!az N=
for j = 1:length(n) eow'K
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rpowers = [rpowers m_abs(j):2:n(j)]; y`=]T>X&x
end \P6$mh\T
rpowers = unique(rpowers); L+q/){Dd(
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% Pre-compute the values of r raised to the required powers, sQ\8>[]
% and compile them in a matrix: is-7
j7;
% ----------------------------- =y!$/(H
if rpowers(1)==0 }1upi=+ aE
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); {yExQbN
rpowern = cat(2,rpowern{:}); hzvd t
rpowern = [ones(length_r,1) rpowern]; <Sr
else f=9|b
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); SBS3?hw
rpowern = cat(2,rpowern{:}); \7'+h5a
end aYSCw 3C<
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% Compute the values of the polynomials: tw9f%p
% -------------------------------------- sjV!5Z
y = zeros(length_r,length(n)); HK Un`ng
for j = 1:length(n) sdo [D
s = 0:(n(j)-m_abs(j))/2; ;N?]eM}yf
pows = n(j):-2:m_abs(j); $F5 b
for k = length(s):-1:1 #%h-[/
p = (1-2*mod(s(k),2))* ... K>@+m
prod(2:(n(j)-s(k)))/ ... Bn &Ws
prod(2:s(k))/ ... &e X!#nQ_.
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... s|y "WDyx5
prod(2:((n(j)+m_abs(j))/2-s(k))); |0f>aZ
idx = (pows(k)==rpowers); V 6,H}k
y(:,j) = y(:,j) + p*rpowern(:,idx); Ev}C<z k*
end "L&#lfOKG
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if isnorm ^0 \
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); j=r P:#
end /x
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end XLrwxj0
% END: Compute the Zernike Polynomials /$p6'1P8
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [UWd W
%#xaA'?
[
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% Compute the Zernike functions: #ZrHsf P
% ------------------------------ lUMS;H(
idx_pos = m>0; 7\ s"o&G
idx_neg = m<0; KJaXg;,H
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=E#%'/ A;c
z = y; J`].:IOh
if any(idx_pos) 8&qZ0GLaT
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ;"~
fZ2$U
end eEv@}1~
if any(idx_neg) HOJs[mqB%
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); /n{omx
end EWkLXU6t
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% EOF zernfun '9
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