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， au0)yg*V1
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， F9-xp7 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？ LT # *n r
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ <:>a51HBX
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function z = zernfun(n,m,r,theta,nflag) vY }/CBmg
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~ hYG%
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N /R 2:Js
% and angular frequency M, evaluated at positions (R,THETA) on the VT;$:>! +
% unit circle. N is a vector of positive integers (including 0), and om;jXf}A
% M is a vector with the same number of elements as N. Each element hPD2/M
% k of M must be a positive integer, with possible values M(k) = -N(k) RzFv ``g
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, c o@Q
% and THETA is a vector of angles. R and THETA must have the same
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% length. The output Z is a matrix with one column for every (N,M) I:bi8D6
% pair, and one row for every (R,THETA) pair. ~Ci|G3BW
% iHWl%]7sN
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike D{ @x
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), k +&LOb 7
% with delta(m,0) the Kronecker delta, is chosen so that the integral tE=P9 \4
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ZIkXy*<(
% and theta=0 to theta=2*pi) is unity. For the non-normalized | u7vY/
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. h6 8sQd
% /&cb`^"U^
% The Zernike functions are an orthogonal basis on the unit circle. b":cj:mxL
% They are used in disciplines such as astronomy, optics, and LIirOf~e;!
% optometry to describe functions on a circular domain. 5Y _)%u
% :hCp@{
% The following table lists the first 15 Zernike functions. cZ%weQa#N)
% ()=
% n m Zernike function Normalization UR:cBr
% -------------------------------------------------- I[@}+p0
% 0 0 1 1 !1w= _
% 1 1 r * cos(theta) 2 |SQ5 Sb
% 1 -1 r * sin(theta) 2 .E"hsGH9h
% 2 -2 r^2 * cos(2*theta) sqrt(6) Ql3hq.E
% 2 0 (2*r^2 - 1) sqrt(3) Y!Wz7
C
% 2 2 r^2 * sin(2*theta) sqrt(6) j <Lj1 P3
% 3 -3 r^3 * cos(3*theta) sqrt(8) 9Ze TS~i
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 7M=`Z{=9
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ]'EtLFv)
% 3 3 r^3 * sin(3*theta) sqrt(8) W;eHDQ|
% 4 -4 r^4 * cos(4*theta) sqrt(10) Jf YO|,
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) T&fqn!i
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) X GbtmmQG
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Fp'k{
% 4 4 r^4 * sin(4*theta) sqrt(10) ?8) _,
% -------------------------------------------------- xQ-]Iw5
% NYm2fFPc
% Example 1: E,>/6AU
% @K3<K (
% % Display the Zernike function Z(n=5,m=1) (kK6=Mrf
% x = -1:0.01:1; (6L[eWuTn
% [X,Y] = meshgrid(x,x); 9~ SfZ,(
% [theta,r] = cart2pol(X,Y); GxuFO5wz
% idx = r<=1; wtu WzHrF
% z = nan(size(X)); cX 9
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% z(idx) = zernfun(5,1,r(idx),theta(idx)); 4S`2")V
% figure 7D@O:yO
% pcolor(x,x,z), shading interp V<ziJ7H/
% axis square, colorbar ^%VMp>s
% title('Zernike function Z_5^1(r,\theta)') `p|{(g'
% 2bPrND\P=
% Example 2: :-fCyF)EI
% W`*S?QGzl@
% % Display the first 10 Zernike functions Q"h/o"-h
% x = -1:0.01:1; 3<88j&9
% [X,Y] = meshgrid(x,x);
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% [theta,r] = cart2pol(X,Y); Jlj=FA`
% idx = r<=1; MN}@EQvW==
% z = nan(size(X)); C4 H M
% n = [0 1 1 2 2 2 3 3 3 3]; EC<g7 _0F
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; sk5h_[tK
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 7q&Ru|T33
% y = zernfun(n,m,r(idx),theta(idx)); jeFX?]Q
% figure('Units','normalized') rwWs\ ~.H
% for k = 1:10 U3}r.9/
% z(idx) = y(:,k); Y6~/H
% subplot(4,7,Nplot(k)) w+)MrB-}
% pcolor(x,x,z), shading interp Rq-BsMX!A
% set(gca,'XTick',[],'YTick',[]) j7IX"O%f\
% axis square z@R: ~
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) %5?qS`/c(
% end
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% /o Q ^j'v
% See also ZERNPOL, ZERNFUN2. 8=Xy19<;t
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% Paul Fricker 11/13/2006 A>8"8=C
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% Check and prepare the inputs: }pxMO? h$
% ----------------------------- xdGmiHN
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) FR"yGx#$
error('zernfun:NMvectors','N and M must be vectors.') ];P$w.0
end N j4=
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if length(n)~=length(m) z UN&L7D
error('zernfun:NMlength','N and M must be the same length.') P(D 0ru
end CT(VV6I\
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n = n(:); |)`<D
m = m(:); E_ #MQ;n
if any(mod(n-m,2)) }i0(^"SoXZ
error('zernfun:NMmultiplesof2', ... lM oi5q
'All N and M must differ by multiples of 2 (including 0).') lJ1_Zs `
end |+K3\b
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if any(m>n) ?"-% >y@w
error('zernfun:MlessthanN', ... ,kS3Ioj
'Each M must be less than or equal to its corresponding N.') U\dq
Mp#Wy
end YL*yiZ9
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if any( r>1 | r<0 ) r$7fw}'I
error('zernfun:Rlessthan1','All R must be between 0 and 1.') /<ODP6Yy;
end G>"=Af(t?Y
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) |e49F
error('zernfun:RTHvector','R and THETA must be vectors.') qbcaiU`-^"
end vU= +
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;>%@
r = r(:); 36MqEUjyB
theta = theta(:); 3Ov? kWFO
length_r = length(r); u~ [=5r
if length_r~=length(theta) Lso4Z Z;
error('zernfun:RTHlength', ... qI (<5Wxl
'The number of R- and THETA-values must be equal.') W\f u0^
end ,n )f=q*%
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% Check normalization: M18<d1*
% --------------------
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if nargin==5 && ischar(nflag) (|Zah1k&]
isnorm = strcmpi(nflag,'norm'); o!bIaeEaU
if ~isnorm i|M^QKvF
error('zernfun:normalization','Unrecognized normalization flag.') vq(ElX TO
end r5#8 V zr
else vSyR%
j
isnorm = false; /O@TqH
end h zv4+1Wd[
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Lmp_8q-Ej
% Compute the Zernike Polynomials
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Q:rQ;/b0/
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% Determine the required powers of r: LHJ}I5zv
% ----------------------------------- '#Yqs/V
m_abs = abs(m); 8Qm%T7]UFb
rpowers = [];
AW[_k%
for j = 1:length(n) :U>[*zE4&
rpowers = [rpowers m_abs(j):2:n(j)]; I;u1mywd
end "CH3\O\
rpowers = unique(rpowers); Ng =_#<
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% Pre-compute the values of r raised to the required powers, j*XhBWE?
% and compile them in a matrix: VgBZ@*z(x
% ----------------------------- ?^f= 7e8]
if rpowers(1)==0 0-VC$)S
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ]; CTr 0
rpowern = cat(2,rpowern{:}); n\/ JNzd3
rpowern = [ones(length_r,1) rpowern]; B:?MMXB
else v%|S)^c?:
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); F0i`HO{
rpowern = cat(2,rpowern{:}); } ={T Vs^
end #.KVT#% ~{
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V DN@=/
% Compute the values of the polynomials: k/lU]~PE
% -------------------------------------- 8?
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y = zeros(length_r,length(n)); j
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for j = 1:length(n) vGIe"$hNh
s = 0:(n(j)-m_abs(j))/2; s+omCr|H;A
pows = n(j):-2:m_abs(j); ].(l^W
for k = length(s):-1:1 gL/D | =
p = (1-2*mod(s(k),2))* ... W08rGY
prod(2:(n(j)-s(k)))/ ... eI #b%h
prod(2:s(k))/ ... "kdmqvTHK0
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... #uc9eh}CWO
prod(2:((n(j)+m_abs(j))/2-s(k))); 8c%Sd'+Pt
idx = (pows(k)==rpowers); O3*}L2 j@
y(:,j) = y(:,j) + p*rpowern(:,idx); 9P 7^*f:E
end l(~i>iQ
4
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if isnorm 1+Z@4 ;fk
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 9-`P\/
end (p?7-~6|:
end 8hZY Z /T
% END: Compute the Zernike Polynomials exP:lO_0n
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% gXb
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% Compute the Zernike functions: Tu[I84
% ------------------------------ P/ XO5`
idx_pos = m>0; ?cvV~&$gc
idx_neg = m<0; {^ jRV@
l'Kx#y$
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z = y; H '_ v
if any(idx_pos) s9\N{ar#
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); />0
Bm`A
end ;i>(r ;ZM
if any(idx_neg) q L-Ni
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); }fqy vI
end 04E
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\; FE@
% EOF zernfun ny 'wS
