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， `<<9A\Y-f
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， AOcUr)
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？ z +wegF
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ a+k3w zJ
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function z = zernfun(n,m,r,theta,nflag) \k=%G_W
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 0
.T5%
_ /
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N LqJV
% and angular frequency M, evaluated at positions (R,THETA) on the 0Db=/sJ>
% unit circle. N is a vector of positive integers (including 0), and wEI?
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% M is a vector with the same number of elements as N. Each element FdEUZ[IT`{
% k of M must be a positive integer, with possible values M(k) = -N(k) XA. 1Y)
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 3bo
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% and THETA is a vector of angles. R and THETA must have the same a wQGu,<N
% length. The output Z is a matrix with one column for every (N,M) HP<a'| r
% pair, and one row for every (R,THETA) pair. |{ZdA r.;
% FBouXu #
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike lm&^`Bn)
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), r #w 7qEtD
% with delta(m,0) the Kronecker delta, is chosen so that the integral 0xCe6{86
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, x=x%F;
% and theta=0 to theta=2*pi) is unity. For the non-normalized +t g${3ti_
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. mO]dP;,
% K~3Y8ca
% The Zernike functions are an orthogonal basis on the unit circle. > MRu oJ
% They are used in disciplines such as astronomy, optics, and ? }`mQ <~
% optometry to describe functions on a circular domain. r6aIW8
% L
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% The following table lists the first 15 Zernike functions. 6jm/y@|F!
% Z }>;@c
% n m Zernike function Normalization *a{WJbau]
% -------------------------------------------------- @PQd6%@
% 0 0 1 1 7,alZ"%W
% 1 1 r * cos(theta) 2 :1g pbfW
% 1 -1 r * sin(theta) 2 #(+V&< K
% 2 -2 r^2 * cos(2*theta) sqrt(6) b^}U^2S%
% 2 0 (2*r^2 - 1) sqrt(3) ; }ThBb3
% 2 2 r^2 * sin(2*theta) sqrt(6) -3b_}by
% 3 -3 r^3 * cos(3*theta) sqrt(8) o^owv(
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) wHx_lsY;
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) dShGIH?
% 3 3 r^3 * sin(3*theta) sqrt(8) ^4<&"ao o
% 4 -4 r^4 * cos(4*theta) sqrt(10) Q~' \oWz
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) A>FWvlLw'm
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) oY; C[X
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `P: [.hRu
% 4 4 r^4 * sin(4*theta) sqrt(10) %CgV:.,K
% -------------------------------------------------- 3%Q9521
% fuF{8-ua
% Example 1: ]TcQGW@'
% U. $Th_
% % Display the Zernike function Z(n=5,m=1) ^/x\HGrw
% x = -1:0.01:1; x1E;dbOZ
% [X,Y] = meshgrid(x,x); m] -cRf)9
% [theta,r] = cart2pol(X,Y); Vu E$-)&)
% idx = r<=1; uAoZ&8D6
% z = nan(size(X)); !3DY#
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 2vsV :LS.
% figure ;, \!&o6
% pcolor(x,x,z), shading interp AA=eWg
% axis square, colorbar $ye>;Ek
% title('Zernike function Z_5^1(r,\theta)') 88?O4)c
% zE/\2F$
% Example 2: =2} kiLKO
% 3Z#WAhfS:
% % Display the first 10 Zernike functions t&EY$'c
% x = -1:0.01:1; wg\ p&avvb
% [X,Y] = meshgrid(x,x); fd>&RbUp
% [theta,r] = cart2pol(X,Y); +#< Z/
% idx = r<=1; Ve)BF1YG
% z = nan(size(X)); [/n@BK
% n = [0 1 1 2 2 2 3 3 3 3]; ja&m-CFK
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; )1 HWD]>4
% Nplot = [4 10 12 16 18 20 22 24 26 28]; L*vKIP<EMM
% y = zernfun(n,m,r(idx),theta(idx)); _ F|}=^Z`
% figure('Units','normalized') T"gk^.
% for k = 1:10 r=54@`O!
% z(idx) = y(:,k); U)aftH
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% subplot(4,7,Nplot(k)) B_b5&M@
% pcolor(x,x,z), shading interp &CN(PZv
% set(gca,'XTick',[],'YTick',[]) s2 :Vm\
% axis square K1]3zLnS
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) B##X94aTT
% end #V#!@@c;?
% 've[Mx
% See also ZERNPOL, ZERNFUN2. #reW)P>
?N!kYTR%}
b`=g# B|
% Paul Fricker 11/13/2006 ~(GN Y5
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% Check and prepare the inputs: N>] J$[j
% ----------------------------- lmL$0{Yr
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~<s =yjTu+
error('zernfun:NMvectors','N and M must be vectors.') O7uCTB+
end ,w BfGpVb
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if length(n)~=length(m) <yBa5m@/
error('zernfun:NMlength','N and M must be the same length.') u |.7w 2
end D>HbJCG4^
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n = n(:); J.M&Vj:
m = m(:); woBx609Aak
if any(mod(n-m,2)) X ,^([$
error('zernfun:NMmultiplesof2', ... 1<_/Qu>V
'All N and M must differ by multiples of 2 (including 0).') +{I" e,Nk
end [ ;sTl~gC
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if any(m>n) p3{Ff5FZ
error('zernfun:MlessthanN', ... 8"ZS|^#
'Each M must be less than or equal to its corresponding N.') \hBzP^*"n
end ; D/6e6
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if any( r>1 | r<0 ) %9Z0\
a)[
error('zernfun:Rlessthan1','All R must be between 0 and 1.') K5 BL4N
end Q9xb7)G
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'=#fELMW
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) =y )K er
error('zernfun:RTHvector','R and THETA must be vectors.') N)R5#JX
end }f? [m&<
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r = r(:); BK6
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theta = theta(:); q^Oj/w s
length_r = length(r); OHsA]7S
if length_r~=length(theta) pq&[cA_w
error('zernfun:RTHlength', ... c"Vp 5lo0
'The number of R- and THETA-values must be equal.')
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end t H.L_< N
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% Check normalization: PM#3N2?|E
% -------------------- kROIVO1|`
if nargin==5 && ischar(nflag) Z${eDl6i
isnorm = strcmpi(nflag,'norm'); uW=G1 *n-
if ~isnorm ]77f`<q<}!
error('zernfun:normalization','Unrecognized normalization flag.') \U>&W
end 2Ki _d
else S)j( %g
isnorm = false; L/C~l3
end Mb 4"bDBsl
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P~ =yT W
% Compute the Zernike Polynomials aK@
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z_ m<x!
x;z=[eE
'o#oRK{#
% Determine the required powers of r: p'2IlQ\
% ----------------------------------- F=1 #qo<?
m_abs = abs(m); 'g ,h
rpowers = []; ;<m`mb4x[
for j = 1:length(n) d!0rq4v7
rpowers = [rpowers m_abs(j):2:n(j)]; %
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end pr z CO w
rpowers = unique(rpowers); -8Mb~Hfl0
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n{sk
% Pre-compute the values of r raised to the required powers, 4Zwbu
% and compile them in a matrix: e7xBi!I)~
% ----------------------------- |`#fX (=
if rpowers(1)==0 $KGMAg/H
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); j _N<aX
rpowern = cat(2,rpowern{:}); &TQ~!ZMOR"
rpowern = [ones(length_r,1) rpowern]; 0 h*Le
else Jl`^`Yv
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); R2sG'<0B0
rpowern = cat(2,rpowern{:}); "}*D,[C5e
end b2UDP W
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% Compute the values of the polynomials: NOXP}M
% -------------------------------------- PD0&ep1h7G
y = zeros(length_r,length(n)); CMW4Zqau*
for j = 1:length(n) _Ik?WA_;
s = 0:(n(j)-m_abs(j))/2; tSJ#
pows = n(j):-2:m_abs(j); uo]xC+^
for k = length(s):-1:1 %(/E
`
p = (1-2*mod(s(k),2))* ... ^ WO3,
prod(2:(n(j)-s(k)))/ ... e>Z& 0lV:
prod(2:s(k))/ ... T3{~f
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... $5 JeN{B
prod(2:((n(j)+m_abs(j))/2-s(k))); i3N{D t
idx = (pows(k)==rpowers); y&,|+h
y(:,j) = y(:,j) + p*rpowern(:,idx); Gd%i?(U,R
end m.m6 .
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if isnorm '@.6Rd 8
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); #:gl+
end & mO n]
end ,X^3.ILz
% END: Compute the Zernike Polynomials 1 #,4P 1"
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s;OGb{H7
rC^ 5Z
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% Compute the Zernike functions: qS/}aDk&
% ------------------------------ ))|d~m
idx_pos = m>0; SZ9Oz -?
idx_neg = m<0; .h=n [`RB
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z = y; m1pA]}Y/5o
if any(idx_pos) A[+ )PkR
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); mufGv%U2
end qhx M O[f
if any(idx_neg) w{*kbGB8s7
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); FE!jN-#
end 8j#S+=l>
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% EOF zernfun Ow/,pC >V
