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， $*fMR,~t&
我输入10阶的n、m，r,theta为38025*1向量，最后得到的z是29525*10阶的矩阵， o&$A]ph8X
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？ r* Ca}Z
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ F7#J LE=
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function z = zernfun(n,m,r,theta,nflag) I
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 9 djk[ttA)
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N brUF6rQ
% and angular frequency M, evaluated at positions (R,THETA) on the 9x =Y^',5
% unit circle. N is a vector of positive integers (including 0), and TOQP' /
% M is a vector with the same number of elements as N. Each element {bY%# m
% k of M must be a positive integer, with possible values M(k) = -N(k) i=2N ;sAl
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, FU 4L6n
% and THETA is a vector of angles. R and THETA must have the same nAdf= D'P
% length. The output Z is a matrix with one column for every (N,M) b d!Y\OD
% pair, and one row for every (R,THETA) pair. 7- fb.V9
% 8KzkB;=n
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike * r7rZFS
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), L ~N460
% with delta(m,0) the Kronecker delta, is chosen so that the integral 1bwOm hkS
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, #o#H?Vo9b
% and theta=0 to theta=2*pi) is unity. For the non-normalized "3Y0`&:D
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 5` p.#
% GnJt0 {
% The Zernike functions are an orthogonal basis on the unit circle. 4BpZJ~(p
% They are used in disciplines such as astronomy, optics, and AFwdJte9e
% optometry to describe functions on a circular domain. `'7R,
% AH~ E )S
% The following table lists the first 15 Zernike functions. O?#7N[7
% ] Zh%DQ
% n m Zernike function Normalization SXP]%{@ R/
% -------------------------------------------------- :gFx{*xN/9
% 0 0 1 1 X 0+vXz{~g
% 1 1 r * cos(theta) 2 dG ?*y
% 1 -1 r * sin(theta) 2 \:LW(&[!
% 2 -2 r^2 * cos(2*theta) sqrt(6) KHvYUT Y
% 2 0 (2*r^2 - 1) sqrt(3) 8zW2zkv2|#
% 2 2 r^2 * sin(2*theta) sqrt(6) FGBbO\< /
% 3 -3 r^3 * cos(3*theta) sqrt(8) H3-hcx54T
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ~})e?q;b
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 5*u+q2\F
% 3 3 r^3 * sin(3*theta) sqrt(8) @-`*m+$U6
% 4 -4 r^4 * cos(4*theta) sqrt(10) 0?|<I{z2
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `C'H.g\>2Q
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) U- k`s[dv
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +X
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% 4 4 r^4 * sin(4*theta) sqrt(10) &s>Jb?_5Mx
% -------------------------------------------------- nKj7.,>;:<
% 2&J)dtqz
% Example 1: YKK*ER0
% -X6PRE5a2
% % Display the Zernike function Z(n=5,m=1) ]JQULE)
% x = -1:0.01:1; b4Ekqas
% [X,Y] = meshgrid(x,x); BDQsP$'6QT
% [theta,r] = cart2pol(X,Y); 4 s9LB
% idx = r<=1; nQ3A~ ()
% z = nan(size(X)); n|yO9:Uw<
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ] 7c=PC
% figure aw&,S"A@
% pcolor(x,x,z), shading interp $M:*T.3
% axis square, colorbar A?OQE9'
% title('Zernike function Z_5^1(r,\theta)') +R:(_:7
% {R{=+2K!|k
% Example 2: a(ZcmYzXU
% P@~yx#G
% % Display the first 10 Zernike functions 0jWVp- y
% x = -1:0.01:1; as = fCuJ
% [X,Y] = meshgrid(x,x); P16~Qj
% [theta,r] = cart2pol(X,Y); SSzIih@u
% idx = r<=1; b*lkBqs$
% z = nan(size(X)); yEy6 ]f+>+
% n = [0 1 1 2 2 2 3 3 3 3]; Q22 GIr
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; W[r>.7>?h
% Nplot = [4 10 12 16 18 20 22 24 26 28]; t"I77aZ$A
% y = zernfun(n,m,r(idx),theta(idx)); sV*H`N')S
% figure('Units','normalized') t sRdvFFq
% for k = 1:10 lH~[f
% z(idx) = y(:,k); G=bCNn<
% subplot(4,7,Nplot(k)) : .`2 ^
% pcolor(x,x,z), shading interp uCB=u[]y4
% set(gca,'XTick',[],'YTick',[]) 'dc#F3
% axis square j_j]"ew)
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) >y+B
% end X2"/%!65{
% + LJ73
!
% See also ZERNPOL, ZERNFUN2. ML p9y#
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% Paul Fricker 11/13/2006 :b!s2n!u
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% Check and prepare the inputs: b\,+f n
% ----------------------------- yaX
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) EKN~H$.
error('zernfun:NMvectors','N and M must be vectors.') |k9
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end r:
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if length(n)~=length(m) q1ma%eiN
error('zernfun:NMlength','N and M must be the same length.') #lO Mm9
end y|jq?M<A
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bW:!5"_{H
n = n(:); MpOc
m = m(:); ]I6 J7A[
if any(mod(n-m,2)) l Nv|M)I
error('zernfun:NMmultiplesof2', ... 3__-n V
'All N and M must differ by multiples of 2 (including 0).') 8 xMX
end dQG=G% W
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#p{ 4^
if any(m>n) HE\ K@3-
error('zernfun:MlessthanN', ... WfRXP^a
'Each M must be less than or equal to its corresponding N.') {\\T gs
end - !
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if any( r>1 | r<0 ) 23jwAsSo
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 7x8
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end o;R I*I
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) TJ* T:?>e
error('zernfun:RTHvector','R and THETA must be vectors.') q0 \6F^;M
end @KUWxFak
M'l ;:
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r = r(:); 8JUwf
theta = theta(:); (Awm9|. {+
length_r = length(r); wS3'?PRX
if length_r~=length(theta) D3 K8F@d
error('zernfun:RTHlength', ... 3= ;<$+I6
'The number of R- and THETA-values must be equal.') `wU!`\
end )WFr</z5bA
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% Check normalization: a-J.B.A$Z/
% -------------------- k== h|\|
if nargin==5 && ischar(nflag) ?Ss!e$jf
isnorm = strcmpi(nflag,'norm'); \lNN Msd&
if ~isnorm lk80#( :Z
error('zernfun:normalization','Unrecognized normalization flag.') Jfl!#UAD|n
end K"@M,8hb
else '}#9)}x!
isnorm = false; 3irl
( ;v
end 9(<@O%YU
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <QGXy=
% Compute the Zernike Polynomials 0Wp|1)ljA
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z<{Q aY$"
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% Determine the required powers of r: |
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% ----------------------------------- oe~b}:
m_abs = abs(m); #A8sLkY
rpowers = []; ( &x['IR
for j = 1:length(n) 6;5Ss?ep
rpowers = [rpowers m_abs(j):2:n(j)]; "5$B>S(Q
end Ny)X+2Ae
rpowers = unique(rpowers); ?!/kZM_ts
B4 }bVjs
IV) j1
% Pre-compute the values of r raised to the required powers, {H'Y `+
% and compile them in a matrix: lU8Hd|@-
% ----------------------------- 7 "D.L-H
if rpowers(1)==0 cj5+N M"
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ;i +#fQO7Q
rpowern = cat(2,rpowern{:}); x'R`.
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rpowern = [ones(length_r,1) rpowern]; 'H <\x
else 8, >P
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); u\nh[1)a)
rpowern = cat(2,rpowern{:}); "" ZQ/t\
end ,
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r" ,GC]
% Compute the values of the polynomials: S ByW[JE
% -------------------------------------- ` sU/& P
y = zeros(length_r,length(n)); S>1Iky|
for j = 1:length(n) K@hw.Xq"
s = 0:(n(j)-m_abs(j))/2; [j'X;tVX{
pows = n(j):-2:m_abs(j); K", N!koj
for k = length(s):-1:1
M \Kx'N
p = (1-2*mod(s(k),2))* ... G,w(d@
prod(2:(n(j)-s(k)))/ ... JqiP>4Uwm^
prod(2:s(k))/ ... v| 2T%y_
u
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... R{T$[ $6S
prod(2:((n(j)+m_abs(j))/2-s(k))); V3j= Kf
idx = (pows(k)==rpowers); bA->{OPkT
y(:,j) = y(:,j) + p*rpowern(:,idx); 5/Uy{Xt
end !%0 * z
, zY$8y]
if isnorm i
K? w6
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); kMd.h[X~
end H7:] ]j1
end N87B8rDl
% END: Compute the Zernike Polynomials B^9j@3Ux
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?6Y?a2 |
3m)y|$R
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% Compute the Zernike functions: |$b}L7_
% ------------------------------ s->^=dy
idx_pos = m>0; ^gnZ+`3
idx_neg = m<0; V~ 5jfcd
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H_Q+&9^/
z = y; wAW5
Z0D
if any(idx_pos) kZ3ThIk%
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 7Jho}5J
end D}X\Ca"h
if any(idx_neg) 3$9W%3
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); n6a`;0f[R
end W 6/yn
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% EOF zernfun LSr]S79N1
