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， 3Iua*#<m,
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， 5*E]ETo@R
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？ M V<^ !W
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ 5G::wuxk
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function z = zernfun(n,m,r,theta,nflag) 'r4/e-`pK
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 3) 8Q S
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N vBRQp&YwX
% and angular frequency M, evaluated at positions (R,THETA) on the R,gR;Aarw
% unit circle. N is a vector of positive integers (including 0), and K:!"+q
% M is a vector with the same number of elements as N. Each element }
uO); k5H
% k of M must be a positive integer, with possible values M(k) = -N(k) 4S5,w(6N
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, * $yR*}A
% and THETA is a vector of angles. R and THETA must have the same 1s%#$ 7
% length. The output Z is a matrix with one column for every (N,M) V}bjK8$$
% pair, and one row for every (R,THETA) pair. w0 ht
% wr5AG<%(
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ?a8^1:
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), @AG n{q
% with delta(m,0) the Kronecker delta, is chosen so that the integral r) HHwh{9
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, i8`Vv7LF
% and theta=0 to theta=2*pi) is unity. For the non-normalized lU @]@_<
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. q o p^;~
% e]`[yf
% The Zernike functions are an orthogonal basis on the unit circle. d_CKP"TA
% They are used in disciplines such as astronomy, optics, and ?h.wK
% optometry to describe functions on a circular domain. Y -pzy ']4
% >XK
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% The following table lists the first 15 Zernike functions. ;hY S6
% Rd2qe /
% n m Zernike function Normalization `Zf^E
>)
% -------------------------------------------------- |y&*MTfV4L
% 0 0 1 1 6""G,"B
% 1 1 r * cos(theta) 2 aIJt0;
% 1 -1 r * sin(theta) 2 hHN'w73z
% 2 -2 r^2 * cos(2*theta) sqrt(6) i'4B3
% 2 0 (2*r^2 - 1) sqrt(3) (}a8"]Z
% 2 2 r^2 * sin(2*theta) sqrt(6) {wO 3< 9
% 3 -3 r^3 * cos(3*theta) sqrt(8) g,}_G3[j0m
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) HIQ _%L4]
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Qc gRAo+u
% 3 3 r^3 * sin(3*theta) sqrt(8) F5?m6`g?
% 4 -4 r^4 * cos(4*theta) sqrt(10) }'""(,2
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) b6}H$Sx~
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) FB
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% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ' qWALu
% 4 4 r^4 * sin(4*theta) sqrt(10) uZc`jNc\
% -------------------------------------------------- .P;*D ws
% v 0
}@
% Example 1: H18Tn !RDS
% } E0,z
% % Display the Zernike function Z(n=5,m=1) 74H )|Dkx
% x = -1:0.01:1; "{tg8-a4)
% [X,Y] = meshgrid(x,x); 0*rQ3 Z
% [theta,r] = cart2pol(X,Y); [<-
% idx = r<=1; L$9 . 8W
% z = nan(size(X)); qDv93
% z(idx) = zernfun(5,1,r(idx),theta(idx)); AB F"~=aL
% figure :E _g"_
% pcolor(x,x,z), shading interp s>0't
% axis square, colorbar l ;JA8o\ x
% title('Zernike function Z_5^1(r,\theta)') x$IX5:E#e
% d{XO /YQw
% Example 2: _9JhL:cY
% &{>cZh}\
% % Display the first 10 Zernike functions /e7O$L)
% x = -1:0.01:1; lp<g \
% [X,Y] = meshgrid(x,x); +s,Qmmb7)
% [theta,r] = cart2pol(X,Y); nN*w~f"
% idx = r<=1; ;u;# g
% z = nan(size(X));
f#?fxUH~
% n = [0 1 1 2 2 2 3 3 3 3]; e?opkq\f
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 'XZ) !1N
% Nplot = [4 10 12 16 18 20 22 24 26 28]; MOsl_^c
% y = zernfun(n,m,r(idx),theta(idx)); BnCbon )
% figure('Units','normalized') ])L
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% for k = 1:10 9A} # 6
% z(idx) = y(:,k); F">Qpgt
% subplot(4,7,Nplot(k)) "ul {d(K3
% pcolor(x,x,z), shading interp 2ggdWg7z
% set(gca,'XTick',[],'YTick',[]) IqC]! H0
% axis square 29!q!g |
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) K@#(*."
% end odPL {XFj
% Fb^:V4<T
% See also ZERNPOL, ZERNFUN2. 6xWe=QGE
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c%&: 6QniZ
% Paul Fricker 11/13/2006 LM}Ib.
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c}y [[EX
% Check and prepare the inputs: I3 ,= 0z
% ----------------------------- c:-!'l$ !
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) |_`E1Y}}
error('zernfun:NMvectors','N and M must be vectors.') i|T)p_y(!a
end ] T;EdK-
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DTa N"{
if length(n)~=length(m) LXEfPLS
error('zernfun:NMlength','N and M must be the same length.') 3
|hHR
end /[Z,MG
3=Cc.a/3
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n = n(:); fRtUvC-#H
m = m(:); O9EKRt
if any(mod(n-m,2)) JcbwD lUb
error('zernfun:NMmultiplesof2', ... >S'17D
'All N and M must differ by multiples of 2 (including 0).') 5] HS^II"
end blTo5N LX
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if any(m>n) GlbySD@
error('zernfun:MlessthanN', ... Q\ cjPc0y
'Each M must be less than or equal to its corresponding N.') \|E^v6E%0
end 4$*%gL;f^
$% 1vW=d
))&;}2{
if any( r>1 | r<0 ) Hm$=h>rY9[
error('zernfun:Rlessthan1','All R must be between 0 and 1.') A j2OkD
end Xlb0/T<g!
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& _"]5/"(
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) WHjUR0NZ
error('zernfun:RTHvector','R and THETA must be vectors.') M@?xa/E6 4
end \/1<E?Q
f
}; f#^gz'
X LL/4 )
r = r(:); b@S Cn9
theta = theta(:); 3'^k$;^
length_r = length(r); \ gLHi~
if length_r~=length(theta) -5T=:2M
error('zernfun:RTHlength', ... 2Z3('?\z~
'The number of R- and THETA-values must be equal.') tI7: 5 Cm
end emdoA:w+
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% Check normalization: c o9 .wB@
% -------------------- Kt"BE j
if nargin==5 && ischar(nflag) GKo K7qH\J
isnorm = strcmpi(nflag,'norm'); E;1Jh(58)b
if ~isnorm /)dF K~
error('zernfun:normalization','Unrecognized normalization flag.') f-5:wM&
end mZx&Xez_G
else
u$-U*r
isnorm = false; s 1e:v+B]
end %-<'QYYP
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ds&e|VSH;
% Compute the Zernike Polynomials '3<fsK=
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Pv<24:a o
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cCa|YW^j
% Determine the required powers of r: 7 ^Ns&Q
% ----------------------------------- ( ZY@$''
m_abs = abs(m); ^D< r
rpowers = []; 4E+hRKuo,
for j = 1:length(n) }J7zTj~{
rpowers = [rpowers m_abs(j):2:n(j)]; m+#iR}*1L
end z k O<-w
rpowers = unique(rpowers); xCYE
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i: Zm*+Gi
F35#dIs`&
% Pre-compute the values of r raised to the required powers, (sQr X{~
% and compile them in a matrix: 'sxNDnGg
% ----------------------------- .VkbYK
if rpowers(1)==0 >P /][MT
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); jaa "~5TO8
rpowern = cat(2,rpowern{:}); Bf$_XG3
rpowern = [ones(length_r,1) rpowern]; v*9<c{ a
else K X}Rr 7a
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); P9S2?Q
rpowern = cat(2,rpowern{:}); :<Fe
end 4WE6fJ2X
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% Compute the values of the polynomials: &`fh E N
% -------------------------------------- i,|0@Vy
y = zeros(length_r,length(n)); ~j-cS
J3
for j = 1:length(n) cetvQAGXY
s = 0:(n(j)-m_abs(j))/2; on8WQf' A#
pows = n(j):-2:m_abs(j); h(F<h_
for k = length(s):-1:1 8@PX7!9
p = (1-2*mod(s(k),2))* ... gd0Vp Xf'
prod(2:(n(j)-s(k)))/ ... Q7g>4GZC
prod(2:s(k))/ ... .S%0
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ; /0 Q1-
prod(2:((n(j)+m_abs(j))/2-s(k))); )
/v 6l
idx = (pows(k)==rpowers); } =v )Js
y(:,j) = y(:,j) + p*rpowern(:,idx); [2dn\z28
end e {KByFl
4&HXkRs:
if isnorm W,K%c=
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ?]rPRV
end l9Q(xuhv
end E7Ulnvd
% END: Compute the Zernike Polynomials !=y] Sv~h
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *A~
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0x9 x@gF
4pln5v=
% Compute the Zernike functions: i@][rdhT
% ------------------------------ k2xHH$+{#=
idx_pos = m>0; jM &r{^(
idx_neg = m<0; 2>\v*adG
dV Q- k
gZEi]/8_
z = y; 4Td{;Y="yF
if any(idx_pos) ^0#; YOk
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); $46{<4.
end ]7Fs$y.
if any(idx_neg) !J =sk4T
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); oxj3[</'k
end zvvP81$W
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% EOF zernfun r|*&GHo L
