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， M('cG
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， %.R_[.W
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？ -ijQT B
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ ]-aeoa#
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function z = zernfun(n,m,r,theta,nflag) g ~@0p7]Y
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 5yQv(<~*G
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N SS;QPWRZ
% and angular frequency M, evaluated at positions (R,THETA) on the ?s5zTT0U>$
% unit circle. N is a vector of positive integers (including 0), and FZ/l
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% M is a vector with the same number of elements as N. Each element <nj [=C4v
% k of M must be a positive integer, with possible values M(k) = -N(k) Sn/~R|3XA7
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, $E4W{ad2jW
% and THETA is a vector of angles. R and THETA must have the same QW f)5S
% length. The output Z is a matrix with one column for every (N,M) 0\jOg
% pair, and one row for every (R,THETA) pair. Tf"DpA!_
% L&'0d$Tg8
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 0 n,5"B
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ys;e2xekg
% with delta(m,0) the Kronecker delta, is chosen so that the integral K0\a+ 6kh
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, %1]2+_6
% and theta=0 to theta=2*pi) is unity. For the non-normalized O` dob&C
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Co19^g*
% N) h>Ie
% The Zernike functions are an orthogonal basis on the unit circle. =3oz74O[
% They are used in disciplines such as astronomy, optics, and C<fNIc~.
% optometry to describe functions on a circular domain. K. R2)o`
% +c\s%Gzrh
% The following table lists the first 15 Zernike functions. @U:T}5)wc
% lrh6lt)
% n m Zernike function Normalization 3_T'T zQ u
% -------------------------------------------------- 4ij`
% 0 0 1 1 #Y}Hh7.<
% 1 1 r * cos(theta) 2 [NvEX Td
% 1 -1 r * sin(theta) 2 =O)JPo&iwY
% 2 -2 r^2 * cos(2*theta) sqrt(6) {zUc*9
% 2 0 (2*r^2 - 1) sqrt(3) i&>,aiH @
% 2 2 r^2 * sin(2*theta) sqrt(6) #fGb M!3p
% 3 -3 r^3 * cos(3*theta) sqrt(8) ^l^_ K)tw*
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) #G.3a]p}"
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) oJ8_hk<Va8
% 3 3 r^3 * sin(3*theta) sqrt(8) D-3/?"n
% 4 -4 r^4 * cos(4*theta) sqrt(10) !Y]}& pUP
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) fsjA7)/
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) y=vH8D]%X
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) YC=BP5^
% 4 4 r^4 * sin(4*theta) sqrt(10) ;*W]]4fy
% -------------------------------------------------- qW7"qw=
% 4&ea*w
% Example 1: aG}9 Z8D
% pN0c'COy^
% % Display the Zernike function Z(n=5,m=1) &" tce6&
% x = -1:0.01:1; a
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% [X,Y] = meshgrid(x,x); mZ;yk(
% [theta,r] = cart2pol(X,Y); I9F[b#'Pn
% idx = r<=1; G< jpJ
% z = nan(size(X)); ,uKvE`H
% z(idx) = zernfun(5,1,r(idx),theta(idx)); N0v d>b
% figure E;4Ns
% pcolor(x,x,z), shading interp b7AuKY{L
% axis square, colorbar U*&ZQw
% title('Zernike function Z_5^1(r,\theta)') 50DPzn
% 4(aesZ8h
% Example 2: K%=n \ Y
% l IFt/
% % Display the first 10 Zernike functions Ab2g),;c
% x = -1:0.01:1; uAvs
% [X,Y] = meshgrid(x,x); =|U2 }U;
% [theta,r] = cart2pol(X,Y); ZHC sv]l
% idx = r<=1; k@8#By l|
% z = nan(size(X)); 3yKI2en "
% n = [0 1 1 2 2 2 3 3 3 3]; 9uS 7G *
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; oo Z-T>$
% Nplot = [4 10 12 16 18 20 22 24 26 28]; owMH
% y = zernfun(n,m,r(idx),theta(idx)); <,E*,&0W
% figure('Units','normalized') ,#wVqBEk
% for k = 1:10 YQ]H3GA
% z(idx) = y(:,k); s3+O=5
% subplot(4,7,Nplot(k)) {-Y_8@&
% pcolor(x,x,z), shading interp <;6 ])
% set(gca,'XTick',[],'YTick',[]) L\Jl'r|
% axis square r0 X2cc
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) QhGg^h%6
% end HQ
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% *(vq-IE\$
% See also ZERNPOL, ZERNFUN2. `>sqP aD
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% Paul Fricker 11/13/2006 hL` zV
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% Check and prepare the inputs: aydal 9M
% ----------------------------- NdNfai
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) s{k\1 P(G}
error('zernfun:NMvectors','N and M must be vectors.') I)Lb"
end wi9|
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if length(n)~=length(m) [*vN`AfE
error('zernfun:NMlength','N and M must be the same length.') Pu^~]^W)
end *(`. h\+
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n = n(:); f
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m = m(:); ^97ZH)Ww
if any(mod(n-m,2)) jkP70Is
error('zernfun:NMmultiplesof2', ... 3E Zw F
'All N and M must differ by multiples of 2 (including 0).') _B1uE2j9
end +wUhB\F
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if any(m>n) Q#Y3%WF
error('zernfun:MlessthanN', ... ->hxHr`!%a
'Each M must be less than or equal to its corresponding N.') .cF$f4>2
end cx,A.Lc
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if any( r>1 | r<0 ) :'dc=C
error('zernfun:Rlessthan1','All R must be between 0 and 1.') M([H\^\:
end I.u,f:Fl'
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) oPy zk7{
error('zernfun:RTHvector','R and THETA must be vectors.') 8@aS9 th$
end 4) 3pa *
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r = r(:); 7h]R{ _
theta = theta(:); ' r n;|K
length_r = length(r); \r%Vgne-g
if length_r~=length(theta) <PN;D#2bh
error('zernfun:RTHlength', ... Ql@yN@V
'The number of R- and THETA-values must be equal.') LQ @|M.$ A
end aTh%oBrtP
_ <a)\UR
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% Check normalization: N`Xnoehu
% -------------------- Kg`x9._2
if nargin==5 && ischar(nflag)
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isnorm = strcmpi(nflag,'norm'); W79A4l<
if ~isnorm /mwr1GU
error('zernfun:normalization','Unrecognized normalization flag.') {}o>ne nx\
end 1ysLZ;K
else \ui^
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isnorm = false; /eRtj:9 M
end |~uzQU7
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% z83:a) U
% Compute the Zernike Polynomials M y"!j,Up
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% z#J/*712
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% Determine the required powers of r: DVoV:pk
% ----------------------------------- `/JR}g{O
m_abs = abs(m); ; 9 &1JX
rpowers = []; 06@ 0r
for j = 1:length(n) T :S{ 3
rpowers = [rpowers m_abs(j):2:n(j)]; sR>;h /
end . 02(O
rpowers = unique(rpowers); g}
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p@ 8^gc
% Pre-compute the values of r raised to the required powers, mYjiiql~
% and compile them in a matrix: WUW b 5xA
% ----------------------------- q\b
?o!# _
if rpowers(1)==0 d)GkXl l1D
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); e=]oh$]
rpowern = cat(2,rpowern{:}); jbhJ;c :
rpowern = [ones(length_r,1) rpowern]; Go+xL/f
else 3Ra\2(bR
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); G2+)R^FSC
rpowern = cat(2,rpowern{:}); 8P<UO
end YV3TxvX MR
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% Compute the values of the polynomials: .FbZVY c]
% -------------------------------------- [OoH5dD
y = zeros(length_r,length(n)); G E~(N N
for j = 1:length(n) @2~O^5[>
s = 0:(n(j)-m_abs(j))/2; aC8,Y$>?E`
pows = n(j):-2:m_abs(j); }M9DqZ;I
for k = length(s):-1:1 &^ 3~=$
p = (1-2*mod(s(k),2))* ... .f !]@"\
prod(2:(n(j)-s(k)))/ ... @Wx`l) b
prod(2:s(k))/ ... /,!7jF:
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 3n84YX{
prod(2:((n(j)+m_abs(j))/2-s(k))); L >Ez-
idx = (pows(k)==rpowers); rGn5Q V
y(:,j) = y(:,j) + p*rpowern(:,idx); ngkeJ)M0$
end vBnK u
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if isnorm a._^E/EV
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); W]!@Zlal
end zA'gb'MmW
end D#/%*|
% END: Compute the Zernike Polynomials f.$a FOn
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c6Yf"~TD0
= 8$0 $d
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% Compute the Zernike functions: RlslF9f
% ------------------------------ {Ukc D+.Y
idx_pos = m>0; K?FX<PT
idx_neg = m<0; _8x'GK
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z = y; {+/
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if any(idx_pos) PV]k3&y
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ||'i\X|[
end l C /Hib
if any(idx_neg) BS-:dyBw
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); u>t|X}JH
end %<=w [*i
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% EOF zernfun 9+'@
