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， >Tl W]st
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， p29yaM
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？ V & mH#k
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ Mf ;|z0 UX
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function z = zernfun(n,m,r,theta,nflag) H 4 }%;m%
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. eC +" mhB
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N E I=N aq
% and angular frequency M, evaluated at positions (R,THETA) on the tL={ y*
% unit circle. N is a vector of positive integers (including 0), and 't0+:o">:
% M is a vector with the same number of elements as N. Each element s.R-<Y 3
% k of M must be a positive integer, with possible values M(k) = -N(k) d%#!nq{vd
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, qLQ <1>u
% and THETA is a vector of angles. R and THETA must have the same o [bE
% length. The output Z is a matrix with one column for every (N,M) t
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% pair, and one row for every (R,THETA) pair. n*vTVt)dJ
% ~|y^\U@
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Ge^zX$.'
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), )h>\05|T
% with delta(m,0) the Kronecker delta, is chosen so that the integral 7K>D@O
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, QQg8+{>
% and theta=0 to theta=2*pi) is unity. For the non-normalized E ;BP N
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. %Y cx C0S[
% vU_d=T%$
% The Zernike functions are an orthogonal basis on the unit circle. }J ei$0x
% They are used in disciplines such as astronomy, optics, and .>m H]/]m
% optometry to describe functions on a circular domain. X(Y#9 N"
% e2] 4a3
% The following table lists the first 15 Zernike functions. e/"yGQu
% oUJj 5iu}
% n m Zernike function Normalization ADv^e JJ|
% -------------------------------------------------- u* t,i`
% 0 0 1 1 S=0"f}Jo.
% 1 1 r * cos(theta) 2 mR{CVU
% 1 -1 r * sin(theta) 2 n S_Ta
% 2 -2 r^2 * cos(2*theta) sqrt(6) _BZ1Vnv
% 2 0 (2*r^2 - 1) sqrt(3) &8[ZN$Xe"
% 2 2 r^2 * sin(2*theta) sqrt(6) G(U 9rJ9
% 3 -3 r^3 * cos(3*theta) sqrt(8) O1GDugZ
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ?QCmSK=L
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) nNt*} k
% 3 3 r^3 * sin(3*theta) sqrt(8) )E'Fke
% 4 -4 r^4 * cos(4*theta) sqrt(10) /y}"M
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -PNi^
K_
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Q~Ay8L+
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,:D=gQ@`
% 4 4 r^4 * sin(4*theta) sqrt(10) V ]79vC
% -------------------------------------------------- 9T(L"9r-e
% 96(R'^kNX
% Example 1: x(L(l=^"
% r55qmPhg
% % Display the Zernike function Z(n=5,m=1) ]dvPx^`d{
% x = -1:0.01:1; OF c\fW#
% [X,Y] = meshgrid(x,x); `NBbTQtgO
% [theta,r] = cart2pol(X,Y); K&=D-50%
% idx = r<=1; >\V6+ $cNp
% z = nan(size(X)); WfTD7?\dw
% z(idx) = zernfun(5,1,r(idx),theta(idx)); b2N6L2~V
% figure \+-zRR0
% pcolor(x,x,z), shading interp rwi w
Rh
% axis square, colorbar RFw(]o,9cR
% title('Zernike function Z_5^1(r,\theta)') ~12_D'8D[
% S_J,[#&
% Example 2: t/ }L36@+
% \tY"BC4.
% % Display the first 10 Zernike functions >lr hHU
% x = -1:0.01:1; {m[s<A(
% [X,Y] = meshgrid(x,x); 4SgF,ac3r
% [theta,r] = cart2pol(X,Y); B$rTwR"( -
% idx = r<=1; }91*4@B7
% z = nan(size(X)); O|QUNr9
% n = [0 1 1 2 2 2 3 3 3 3]; |6aJwe+*
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ; e@gO
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Fh K&@@_
% y = zernfun(n,m,r(idx),theta(idx)); ~g6"'Cya?k
% figure('Units','normalized') e@E17l-
% for k = 1:10 +b^ ]Pz5
% z(idx) = y(:,k); @Mm/C?#*O
% subplot(4,7,Nplot(k)) i}v9ut]B
% pcolor(x,x,z), shading interp t8QRi! \=
% set(gca,'XTick',[],'YTick',[]) c!It ^ *
% axis square CE uWw:)
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) T-'~? [v
% end &)n_]R#)
% 7h %4]
% See also ZERNPOL, ZERNFUN2. K UKACUL
OO n X`
#uSK#>H_!
% Paul Fricker 11/13/2006 8-m
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ks*Y9D*=
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>{C\H.N
% Check and prepare the inputs: ?7
\\e ;j}
% ----------------------------- Tzzq#z&F
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) WK0C
error('zernfun:NMvectors','N and M must be vectors.') p~&BChBl!=
end =7% o E[
WF2NG;f=
} g|nz8
if length(n)~=length(m) E V2 )
error('zernfun:NMlength','N and M must be the same length.') 2?W7I/F
end |Y},V_@d
%y&]' A
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n = n(:); ib$nc2BP b
m = m(:); {hQ6K)s
if any(mod(n-m,2)) <xo-Fv
error('zernfun:NMmultiplesof2', ... z
x@$RS+]
'All N and M must differ by multiples of 2 (including 0).') ; Y"N6%
end MV0Lq:# N
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if any(m>n) 9M a0^_
error('zernfun:MlessthanN', ... O /Rhf[7v*
'Each M must be less than or equal to its corresponding N.') uj r(K=E
end tnz+bX 26
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if any( r>1 | r<0 ) c>HK9z{
error('zernfun:Rlessthan1','All R must be between 0 and 1.') fY,|o3#
end x[( ?#
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ;apzAF
error('zernfun:RTHvector','R and THETA must be vectors.') 8z2Rry
w
end ?+0GfIV
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n6MM5h/#r
r = r(:); C [uOReo
theta = theta(:); WC,+Cn e
length_r = length(r); _:g&,2bc
if length_r~=length(theta) k |YWOy@D~
error('zernfun:RTHlength', ... &QNY,Pj
'The number of R- and THETA-values must be equal.') zR;X*q"T$4
end
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*Dmx&F=3,5
% Check normalization: "*z_O
% -------------------- K_/zuTy
if nargin==5 && ischar(nflag) _ oFs #kW
isnorm = strcmpi(nflag,'norm');
\
%=9
if ~isnorm Q=uR Kh
error('zernfun:normalization','Unrecognized normalization flag.') /)
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end |X19fgk
else *sw7niw
isnorm = false; S4^N^lQ ]
end 23!;}zHp
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6 9K{+|
% Compute the Zernike Polynomials n 5-)/R[z
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +rXF{@
l
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p d[ncL
% Determine the required powers of r: V'Kgdj
% ----------------------------------- )D&M2CUw"f
m_abs = abs(m); V/d/L3p
rpowers = []; )E#2J$TD
for j = 1:length(n) fBn"kr;
rpowers = [rpowers m_abs(j):2:n(j)]; l_tw<`Ep
end I<td1Y1q
rpowers = unique(rpowers); %+>s#Q2d
$~0Q@):
4+$b~ u
% Pre-compute the values of r raised to the required powers, U9y|>P\)T
% and compile them in a matrix: /cr}N%HZB
% ----------------------------- D]a:@x`+Bz
if rpowers(1)==0 N,dT3we
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); WEg6Kz
rpowern = cat(2,rpowern{:}); YTQt3=1ii
rpowern = [ones(length_r,1) rpowern]; }9HmTr|
else kum#^^4G|
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 'ly? P8h
rpowern = cat(2,rpowern{:}); %@ a8P
end L4u;|-znw
" xmP6=1
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% Compute the values of the polynomials: SzwQOs*
% -------------------------------------- 2~[@ _
y = zeros(length_r,length(n)); v~3B:k:?l
for j = 1:length(n) *L6P Le
s = 0:(n(j)-m_abs(j))/2; tM-^<V&
pows = n(j):-2:m_abs(j); >d"3<S ; b
for k = length(s):-1:1 w=]Ks'C]
p = (1-2*mod(s(k),2))* ... D4eTTfQ
prod(2:(n(j)-s(k)))/ ... wbDM5%
prod(2:s(k))/ ... '{ I_\~*
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... a,F&`Wg
prod(2:((n(j)+m_abs(j))/2-s(k))); ;*ix~taL%
idx = (pows(k)==rpowers); |7,L`utp
y(:,j) = y(:,j) + p*rpowern(:,idx); e^4 p%
end L Mi:%i%\
M.-"U+#aD
if isnorm }+o:j'jB
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 2?m.45`
end @`tXKP$so
end |@ ,|F:h<M
% END: Compute the Zernike Polynomials j'[m:/
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% w -Nhs6
t }IkK=f
I;5R2" 3
% Compute the Zernike functions: ? D,=37
% ------------------------------ O# wpbrJ
idx_pos = m>0; vZ/6\ Cz
idx_neg = m<0; x!\ONF5$
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WN( ymcdYB
z = y; 08X_}97#WF
if any(idx_pos) AL$&|=C-$
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Vpy 2\wZWb
end '$4O!YI9@
if any(idx_neg) G}5 #l
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); t8^m `W
end ~~/xR s
eh1Q7 ~
m}>F<;hQ
% EOF zernfun go+Q~NV
