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， />;1 }
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， Y 3r m')c
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？ Eq ^k @
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ 2 <w uzP|
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function z = zernfun(n,m,r,theta,nflag) WS8m^~S@\
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. VO3& ! uOd
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N }\}pSq W
% and angular frequency M, evaluated at positions (R,THETA) on the wXp
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% unit circle. N is a vector of positive integers (including 0), and <qN0Q7
% M is a vector with the same number of elements as N. Each element Xn-GSW3{
% k of M must be a positive integer, with possible values M(k) = -N(k) <y=VDb/
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 9K~2!<
% and THETA is a vector of angles. R and THETA must have the same HXhz |s0
% length. The output Z is a matrix with one column for every (N,M)
02:]
% pair, and one row for every (R,THETA) pair. :S}!i?n
% *"` dO9Yf_
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike $,q~ q^0
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ! TY9\8JzV
% with delta(m,0) the Kronecker delta, is chosen so that the integral G\G TS}u[
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, i`/_^Fndyu
% and theta=0 to theta=2*pi) is unity. For the non-normalized /
pzdX%7
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 5=tvB,Ux4
% `rsPIOu
% The Zernike functions are an orthogonal basis on the unit circle. x@I *(I
% They are used in disciplines such as astronomy, optics, and w~a^r]lPW
% optometry to describe functions on a circular domain. tGnBx)J|
% a AZS^S4v
% The following table lists the first 15 Zernike functions. BDSZ '
% CI"7* z_
% n m Zernike function Normalization \O5`R-
% -------------------------------------------------- XL@i/5C[
% 0 0 1 1 Vy0s%k
% 1 1 r * cos(theta) 2 #j
-bT4!
% 1 -1 r * sin(theta) 2 $X8(OS5d'
% 2 -2 r^2 * cos(2*theta) sqrt(6) p3ox%4
% 2 0 (2*r^2 - 1) sqrt(3) r (xh5{^x
% 2 2 r^2 * sin(2*theta) sqrt(6) ZC 7R f
% 3 -3 r^3 * cos(3*theta) sqrt(8) 1oD,E!+^d
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) MTo<COp($
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) s. I%[kada
% 3 3 r^3 * sin(3*theta) sqrt(8) ntbl0Sk
% 4 -4 r^4 * cos(4*theta) sqrt(10) x F:
O6KL
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) "* W:
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) |D+"+w/
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) I|69| ^
% 4 4 r^4 * sin(4*theta) sqrt(10) u~n*P``{
% -------------------------------------------------- W1'F)5(?7
% a5=8zO#%g
% Example 1: <WFA3
% ,Oa-AF/p
% % Display the Zernike function Z(n=5,m=1) )[RLCZ
% x = -1:0.01:1; n% zW6}
% [X,Y] = meshgrid(x,x); +\ g/KbV7
% [theta,r] = cart2pol(X,Y); 0Jz H dz
% idx = r<=1; %@
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% z = nan(size(X)); Q ^X
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ap=m5h27
% figure `i5U&K. 7
% pcolor(x,x,z), shading interp W Ll_;BgN
% axis square, colorbar FsQeyh>
% title('Zernike function Z_5^1(r,\theta)') .j?`U[V%a
% 873$EiyXR
% Example 2: O
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% p=%Vo@*]
% % Display the first 10 Zernike functions S :)Aj6>6
% x = -1:0.01:1; Py* ( %
% [X,Y] = meshgrid(x,x); HT&CbEa4'
% [theta,r] = cart2pol(X,Y); ~hK7(K
% idx = r<=1; m(CW3:|
% z = nan(size(X)); .nN=M># /
% n = [0 1 1 2 2 2 3 3 3 3]; p F kA,
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; HV O
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% Nplot = [4 10 12 16 18 20 22 24 26 28]; },58B
% y = zernfun(n,m,r(idx),theta(idx)); sd4eJ
% figure('Units','normalized') I\ e?v`e
% for k = 1:10 {!!df.h
% z(idx) = y(:,k); D4,kGU@
% subplot(4,7,Nplot(k)) |vW(;j6
% pcolor(x,x,z), shading interp gc(Gc vdB\
% set(gca,'XTick',[],'YTick',[]) LXYpP- E
% axis square c%1k'Q
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) mGx! {v~i&
% end HYVSi 3[
% j"(o>b v7
% See also ZERNPOL, ZERNFUN2. +&hhj~ I.
$VEG1]/svp
^(z7?T
% Paul Fricker 11/13/2006 1Q_ C
1ocd$)B|}
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$,KUH
% Check and prepare the inputs: TljN!nv]
% ----------------------------- t^ _0w[
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) S1jI8 #z}_
error('zernfun:NMvectors','N and M must be vectors.') cr GFU?8
end )Ve-) rZ
|-Rg].
3V=(P.A Tm
if length(n)~=length(m) OAigq6[,
error('zernfun:NMlength','N and M must be the same length.') 6Gt~tlt:L
end $ti*I;)h4
N:R6
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=}
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n = n(:); 5cc;8i
m = m(:); Pjz_KO/
if any(mod(n-m,2)) D5]AL5=Xt2
error('zernfun:NMmultiplesof2', ... qHwHP 1
'All N and M must differ by multiples of 2 (including 0).') GMk\
l
end w+A:]SU
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if any(m>n) 2MT_5j5[N
error('zernfun:MlessthanN', ... FHz tF$Z
'Each M must be less than or equal to its corresponding N.') `
=ocr8c
end sk ' < K5~
#h,7dz.d
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if any( r>1 | r<0 ) #Z;6f{yWf
error('zernfun:Rlessthan1','All R must be between 0 and 1.') W& M=%
end XKp$v']u
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) oZM6%-@qi
error('zernfun:RTHvector','R and THETA must be vectors.') $qz(9M(m#
end yH`4 sd
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{;:QY 1Q T
r = r(:); \R"} =7
theta = theta(:); X?6E0/r&9
length_r = length(r); XOOWrK7O
if length_r~=length(theta) mT] +wi&
error('zernfun:RTHlength', ... j[E8C$lW
'The number of R- and THETA-values must be equal.') c L+-- $L
end v %?y5w
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tKr.{#)
% Check normalization: A%Ov.~&\G
% -------------------- }Iyr u3M][
if nargin==5 && ischar(nflag) t1LIZ5JY
isnorm = strcmpi(nflag,'norm'); 3o).8b_3g
if ~isnorm i oIOyj
error('zernfun:normalization','Unrecognized normalization flag.') M<Gr~RKmAn
end b*;zdGX.A9
else Fe: M'.
isnorm = false; _'eG
end {J aulg
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ark+Df/
% Compute the Zernike Polynomials KOQiX?'
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jCJbmEfo9@
%_kXC~hH_
^A&i$RRO
% Determine the required powers of r: g&79?h4UXQ
% ----------------------------------- trl:\m
m_abs = abs(m); s=[Tm}[
rpowers = []; fPW|)e"
for j = 1:length(n) Y 6NoNc]h
rpowers = [rpowers m_abs(j):2:n(j)]; Nu/D$m 'PY
end fG *1A\t]
rpowers = unique(rpowers); tEU}?k+:j)
\hlQu{q.
Gk y
e
% Pre-compute the values of r raised to the required powers, 3CKd[=-Z
% and compile them in a matrix: Ffv v8x
% ----------------------------- ?M W *`U
if rpowers(1)==0 "7]YvZYu0
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);
<>|&%gmz
rpowern = cat(2,rpowern{:}); {2A| F{7>
rpowern = [ones(length_r,1) rpowern]; S1Z~-i*w
else gY],U4_:p
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ] "ZL<?3g
rpowern = cat(2,rpowern{:}); |JUb 1|gi
end uTWij4)a
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;
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% Compute the values of the polynomials: iRbTH}4i
% -------------------------------------- U<pG P
y = zeros(length_r,length(n)); [lU0TDq
for j = 1:length(n) |UudP?E
s = 0:(n(j)-m_abs(j))/2; U-U^N 7
pows = n(j):-2:m_abs(j); T[ ~8u9/
for k = length(s):-1:1 gI~ 4A,
p = (1-2*mod(s(k),2))* ... @Cnn8Y&'
prod(2:(n(j)-s(k)))/ ... 8 !R +wy
prod(2:s(k))/ ... { r.KY
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... nV[0O8p2Md
prod(2:((n(j)+m_abs(j))/2-s(k))); 34D7qR
idx = (pows(k)==rpowers); #5Q?Q~E@
y(:,j) = y(:,j) + p*rpowern(:,idx); >5O#_ ?
end TO,XN\{y
bOB<m4
if isnorm "k ;j@
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); IIZu&iZo\
end *mv Dh9v
end ~o <+tL
% END: Compute the Zernike Polynomials / LH#
3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mY)Y47iL
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% Compute the Zernike functions:
E43Gk!/|(
% ------------------------------ 5qe6/E@
idx_pos = m>0; A"Prgf
eT
idx_neg = m<0; u|.c?fW'3
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z = y; <FkoWN
if any(idx_pos) 2\b 2W _
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); u|G&CV#r
end nfld j33*
if any(idx_neg) >~ %EB?8
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); rfz\DvV d
end wU"0@^k]<
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% EOF zernfun brdfj E 8
