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， ?#: ']q
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， s-Yu(X2
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？ 5~DKx7P!Z
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ !w%c= V]tV
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function z = zernfun(n,m,r,theta,nflag) fDB. r$|d
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~< UYJc
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N lrKT ?siB
% and angular frequency M, evaluated at positions (R,THETA) on the ,~Xe #e M
% unit circle. N is a vector of positive integers (including 0), and NR_3nt^h
% M is a vector with the same number of elements as N. Each element x+6z9{O
% k of M must be a positive integer, with possible values M(k) = -N(k) ]] 0 M
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, + ,)k@OI
% and THETA is a vector of angles. R and THETA must have the same yQj J-g(.
% length. The output Z is a matrix with one column for every (N,M) F O_}9 <s
% pair, and one row for every (R,THETA) pair. y4p"LD5%^
% c)^A|{,G
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike sB *dv06b0
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), oi7k#^
% with delta(m,0) the Kronecker delta, is chosen so that the integral Zh;}Q( w
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, &0Y
|pY
% and theta=0 to theta=2*pi) is unity. For the non-normalized (9aOET>GG
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. i{$P.i/&
% JC~sz^>p\
% The Zernike functions are an orthogonal basis on the unit circle. LA\3 ,Uv
% They are used in disciplines such as astronomy, optics, and CUpRtE8@[_
% optometry to describe functions on a circular domain. z-We>KX
% i H-,l
% The following table lists the first 15 Zernike functions. U
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% C T|0KB&
% n m Zernike function Normalization 5TuwXz 1v
% -------------------------------------------------- 9x4z m
% 0 0 1 1 y,&[OrCm^\
% 1 1 r * cos(theta) 2 lc5(^ ~
% 1 -1 r * sin(theta) 2 w ]T_%mdk
% 2 -2 r^2 * cos(2*theta) sqrt(6) jA:'P~`Hj
% 2 0 (2*r^2 - 1) sqrt(3) jsgDJ}
% 2 2 r^2 * sin(2*theta) sqrt(6) _7:Bxx4B
% 3 -3 r^3 * cos(3*theta) sqrt(8) %4x0^<k~
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ~x'8T!M{
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ~YW;'
% 3 3 r^3 * sin(3*theta) sqrt(8) 8 NNh8k#6
% 4 -4 r^4 * cos(4*theta) sqrt(10) cOpe6H6,bz
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,|\\C 6s
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) wo (O +L/w
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |-/@3gPO
% 4 4 r^4 * sin(4*theta) sqrt(10) jMI30
% -------------------------------------------------- Q
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% UJL'4 t/
% Example 1: AG vhSd7
% |;Jt *
_
% % Display the Zernike function Z(n=5,m=1) Q/Z>w+zh#
% x = -1:0.01:1; W!X Buk-
% [X,Y] = meshgrid(x,x); _IU5HT}2
% [theta,r] = cart2pol(X,Y); TeZu*c
% idx = r<=1; ^hZ0"c
% z = nan(size(X)); .c<U5/
% z(idx) = zernfun(5,1,r(idx),theta(idx)); _/P ;`@
% figure rBL2A
% pcolor(x,x,z), shading interp => 'j_|
% axis square, colorbar Iy*Q{H3[
% title('Zernike function Z_5^1(r,\theta)') 16I[z+RG
% IpzU=+h
% Example 2: xgNV0;g,
% 45%D^~2~F
% % Display the first 10 Zernike functions INk|NEX
% x = -1:0.01:1; fYZd:3VdC
% [X,Y] = meshgrid(x,x); DU*q hW`X
% [theta,r] = cart2pol(X,Y); yNwSiZE X
% idx = r<=1; @0H0!9'
% z = nan(size(X)); $1s>efP -
% n = [0 1 1 2 2 2 3 3 3 3]; GYJ j$'
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; YT[=o}jS
% Nplot = [4 10 12 16 18 20 22 24 26 28];
M 54czo=l
% y = zernfun(n,m,r(idx),theta(idx)); [\Aws^fD_
% figure('Units','normalized') vYLsp Z;S
% for k = 1:10 +B +cN[d
% z(idx) = y(:,k); jc>B^mq x
% subplot(4,7,Nplot(k)) rB( Q)N
% pcolor(x,x,z), shading interp 8>vNa
% set(gca,'XTick',[],'YTick',[]) :D 2GLq *\
% axis square J z&dC
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) FoYs <aER
% end $'!n4}$}
% ~tW<]l7
% See also ZERNPOL, ZERNFUN2. ^l
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1v3
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% Paul Fricker 11/13/2006 _E-{*,7bZS
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n@C#,v#^0
f D_3lbiL(
% Check and prepare the inputs: BjjuZN&
% ----------------------------- /
)[\+Nc
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) f::^zAV
error('zernfun:NMvectors','N and M must be vectors.') yVPFH~1@\
end |D'!.$7%
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e cvZwL
if length(n)~=length(m) =''*'a- P
error('zernfun:NMlength','N and M must be the same length.') =:xJZy$
end m^/>C -&C
b-c6.aKf|
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n = n(:); #:MoZw`rlw
m = m(:); [ >j.x 2=
if any(mod(n-m,2)) <x$f D37
error('zernfun:NMmultiplesof2', ... xw1,Wbu]
'All N and M must differ by multiples of 2 (including 0).') %$_?%X0=t
end 9Y2(.~w6X
.Z 17X_
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if any(m>n) ;gRPTk$X3
error('zernfun:MlessthanN', ... q} g0-Da
'Each M must be less than or equal to its corresponding N.') vL "n oLs
end =7 -k D3
%uP/v\l
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if any( r>1 | r<0 ) {+9t!'
error('zernfun:Rlessthan1','All R must be between 0 and 1.') N =8CVI
end IeIv k 55
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!) d
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 7:.!R^5H
error('zernfun:RTHvector','R and THETA must be vectors.') Z3Xgi~c
end G6"4JTW O
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r = r(:); itO1ROmu
theta = theta(:); VOmS>'$
length_r = length(r); KZ [:o,jp>
if length_r~=length(theta) yN}< l%
error('zernfun:RTHlength', ... =G rg
'The number of R- and THETA-values must be equal.') xt XK3[s
end z7*mT}Q
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6S Srkj }U
% Check normalization: B8V85R
% -------------------- 1a},(ZcdX
if nargin==5 && ischar(nflag) vRO`hG H
isnorm = strcmpi(nflag,'norm'); hN1{?PQ
if ~isnorm xSjs+Y;Mu
error('zernfun:normalization','Unrecognized normalization flag.') !z?:Y#P3
end ;cWFh4_
else < ZeZq
isnorm = false; &R}2/Mt
end f Aeq(tI=
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %|?PG i@5
% Compute the Zernike Polynomials %X Wb|-=
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MqswYK-s
mINir-
lem\P_V)
% Determine the required powers of r: =)XC"kU p
% ----------------------------------- #c2JWDH1F
m_abs = abs(m); EXbhyg
rpowers = []; & Pzr)W(
for j = 1:length(n) y+a]?`2
rpowers = [rpowers m_abs(j):2:n(j)]; v!? >90a
end ,7|2K &C5
rpowers = unique(rpowers); c5tCw3$ t
nrI-F,1
1x4{~g\
% Pre-compute the values of r raised to the required powers, C+c;UzbD
% and compile them in a matrix: ]1n
=O"vE
% ----------------------------- 2UjQ!g`
if rpowers(1)==0 Y[ t oN9,
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); /*{s1Zcb
rpowern = cat(2,rpowern{:}); x AR9* <-
rpowern = [ones(length_r,1) rpowern]; .
[\S=K|/
else H!dg(d^
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); skn];%[v\
rpowern = cat(2,rpowern{:}); dJd(m&.|N
end =P<7tsSuoK
cYp]zn+6
`+Ojh>"*z*
% Compute the values of the polynomials: `G^MTDp?L+
% -------------------------------------- (Puag*
y = zeros(length_r,length(n)); %,G0)t
for j = 1:length(n) k9?f E
s = 0:(n(j)-m_abs(j))/2; F2RU7o'f.
pows = n(j):-2:m_abs(j); 3 ]}wZY0
for k = length(s):-1:1 G4eY}3F7,4
p = (1-2*mod(s(k),2))* ... hbv>Jjd
prod(2:(n(j)-s(k)))/ ... E"$AOM?(*i
prod(2:s(k))/ ... z>X<Di&x)
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... -5t . 1/
prod(2:((n(j)+m_abs(j))/2-s(k))); |/-H:\5
idx = (pows(k)==rpowers); PF?tEw_WB
y(:,j) = y(:,j) + p*rpowern(:,idx); d@-bt s&3
end # Y;_W;#
8n^v,s >
if isnorm fB3W} dr
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); qkN{l88
end oO8V0VE\
end L<**J\=7M
% END: Compute the Zernike Polynomials z tLP {q#
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [vs5e3B)
Wp<4F 6C$@
\-B8 `ah
% Compute the Zernike functions: w(KB=lA2
% ------------------------------ t)Mi,ljY[
idx_pos = m>0; h{&}p-X&[
idx_neg = m<0; 9,`eYAu
-_RMiGM?T
P~y %
z = y; m:,S1V_jl
if any(idx_pos) 5g3D}F>OJ
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); !!4` #Z0+#
end & A%*sD6
if any(idx_neg) >H q)1o
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); HTz&h#)JQ
end ~;A36M-[.
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l |c#
% EOF zernfun P<@V
