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， )D82N`c2\i
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， aCLq k'
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？ &q|K!5[k
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ qXjxNrK
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3 gj+%%!G\
function z = zernfun(n,m,r,theta,nflag) g^ i&gNDx
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. p`#R<K
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N h.s+)fl\
% and angular frequency M, evaluated at positions (R,THETA) on the t\j *}# S
% unit circle. N is a vector of positive integers (including 0), and VD]zz
^
% M is a vector with the same number of elements as N. Each element 1s@+;QUib
% k of M must be a positive integer, with possible values M(k) = -N(k) Z @@K[$
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Eue~Y+K*b
% and THETA is a vector of angles. R and THETA must have the same wtV#l4
% length. The output Z is a matrix with one column for every (N,M) 9
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% pair, and one row for every (R,THETA) pair. x`IEU*z#
% 4^OY
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% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike bl(RyA gA
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), U\<?z Dw
% with delta(m,0) the Kronecker delta, is chosen so that the integral &7wd?)s
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 6 ez<g
Uf
% and theta=0 to theta=2*pi) is unity. For the non-normalized <)-Sj,
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 5vZ^0yFQ
% :s6o"VkW
% The Zernike functions are an orthogonal basis on the unit circle. U, - 39mr
% They are used in disciplines such as astronomy, optics, and Wo RZW%
% optometry to describe functions on a circular domain. z4]api(xZ
% Gvqxi|
% The following table lists the first 15 Zernike functions. `& sH-d4v
% V0XvJ
% n m Zernike function Normalization )fSOi| |C
% -------------------------------------------------- Nf "r4%M<6
% 0 0 1 1 {"QNJq#:
% 1 1 r * cos(theta) 2 8j % Tf;
% 1 -1 r * sin(theta) 2 ^ t g<K
% 2 -2 r^2 * cos(2*theta) sqrt(6) '>s sqBnI
% 2 0 (2*r^2 - 1) sqrt(3) p\ZNy\N^
% 2 2 r^2 * sin(2*theta) sqrt(6) sAD}#Zw$
% 3 -3 r^3 * cos(3*theta) sqrt(8) 28J^D MOW
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Y@ksQ_u
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 6U, O*WJ%e
% 3 3 r^3 * sin(3*theta) sqrt(8) NI
[
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% 4 -4 r^4 * cos(4*theta) sqrt(10) bNNr]h8y-
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4'A!; ]:
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 5 VAK:eB
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) '>0fWBs
% 4 4 r^4 * sin(4*theta) sqrt(10) ],a 5)kV
% -------------------------------------------------- 1@1U/ss1
% MgrLSKLT
% Example 1: d]6#m'U
% aV|hCN~
% % Display the Zernike function Z(n=5,m=1) g Psi
% x = -1:0.01:1; &wCg\j_c
% [X,Y] = meshgrid(x,x); ?fjuh}Q5h
% [theta,r] = cart2pol(X,Y); q $tUH)0
% idx = r<=1; '*w00
% z = nan(size(X)); 7 Vo$(kj
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ?D*/*Gk{
% figure ~%=MpQ3
% pcolor(x,x,z), shading interp v`zJb00 DT
% axis square, colorbar o`P %&
% title('Zernike function Z_5^1(r,\theta)') K& 70{r
% ^ ALly2
% Example 2: \BZhf?9U
% Y>G@0r BG
% % Display the first 10 Zernike functions \$e)*9)
% x = -1:0.01:1; ,>-< (Qi
% [X,Y] = meshgrid(x,x); _FVcx7l!u
% [theta,r] = cart2pol(X,Y); ~r`9+b[9{
% idx = r<=1; TQ*1L:X7M&
% z = nan(size(X)); uPG4V 2
% n = [0 1 1 2 2 2 3 3 3 3]; DSk/q-'u
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; M
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% Nplot = [4 10 12 16 18 20 22 24 26 28]; 5$&%r e!{Z
% y = zernfun(n,m,r(idx),theta(idx)); s1NKLt
% figure('Units','normalized') 2h1C9n%j9
% for k = 1:10 K]0:?h;%Ld
% z(idx) = y(:,k); *HO}~A%Lx
% subplot(4,7,Nplot(k)) ruzspS
% pcolor(x,x,z), shading interp `t9?=h!
% set(gca,'XTick',[],'YTick',[]) O_ DtvjI'
% axis square TDNQu_E
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) pd7NF-KD
% end J/GSceHF
% WP+oFkw>
% See also ZERNPOL, ZERNFUN2. 5Z\#0":e
%i-c0|,T4
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% Paul Fricker 11/13/2006 ~ga`\% J
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% Check and prepare the inputs: U($dx.`v#
% ----------------------------- X+} 1
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Q[I=T&
error('zernfun:NMvectors','N and M must be vectors.') W"'iIh)z
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end I'iGt~4$
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if length(n)~=length(m) (+ 3Wgl+]/
error('zernfun:NMlength','N and M must be the same length.') A"D ,Kg
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end .!,z:l$Kh
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n = n(:);
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m = m(:); *"n vX2i z
if any(mod(n-m,2)) "7V2lu
error('zernfun:NMmultiplesof2', ... BT"42#7_
'All N and M must differ by multiples of 2 (including 0).') [YT>*BH ?
end 9Z'8!$LYg
` `e$AS
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if any(m>n) qt
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error('zernfun:MlessthanN', ... [7S } g
'Each M must be less than or equal to its corresponding N.') 4NG?_D5&
end Ii_ojQP-z
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if any( r>1 | r<0 ) GF%314Xu
error('zernfun:Rlessthan1','All R must be between 0 and 1.') UHxE)]J
end e0@Y#7N62
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) )Ocl=H|=
error('zernfun:RTHvector','R and THETA must be vectors.') /BV03B
end mAW, ?h
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r = r(:); Y + \%
theta = theta(:); '@CR\5 @
length_r = length(r); Gkv{~ ?95
if length_r~=length(theta) )V:]g\t
error('zernfun:RTHlength', ... 5-0{+R5v
'The number of R- and THETA-values must be equal.') [[ Y0
end - !L"')
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% Check normalization: njnDW~Snb
% -------------------- 1=a>f "cyf
if nargin==5 && ischar(nflag) 0`A~HH }
isnorm = strcmpi(nflag,'norm'); ZwerDkd
if ~isnorm
pzgSg[ |
error('zernfun:normalization','Unrecognized normalization flag.') $aPfGZ<i
end ]
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else "0k8IVwp
isnorm = false; {$^DM ANDx
end Mz;[ +p
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,SiY;(b=\
% Compute the Zernike Polynomials _fP&&}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]a3iEA2 (
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% Determine the required powers of r: ~/ ilx# d
% ----------------------------------- f5= =";eP
m_abs = abs(m); H'UR8 %
rpowers = []; 'EfR|7m
for j = 1:length(n) t"YNgC ^
rpowers = [rpowers m_abs(j):2:n(j)]; d/e|'MPX
end b (I2m
rpowers = unique(rpowers); ?
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% Pre-compute the values of r raised to the required powers, "AhTH.ZP
% and compile them in a matrix: !GQ\"Ufs>
% ----------------------------- l?)ZJ3]a
if rpowers(1)==0 n%\
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rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); BiZ=${y
rpowern = cat(2,rpowern{:}); ^p/Ob'!
rpowern = [ones(length_r,1) rpowern]; ^@_m "^C
else /Y2/!mU</
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 3o|I[!2.
rpowern = cat(2,rpowern{:}); 'iY*6<xS<
end c$QX )V
!;!~n`
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% Compute the values of the polynomials: ?`Y\)'}
% -------------------------------------- }/,CbKi,+
y = zeros(length_r,length(n)); 02k4 N%
for j = 1:length(n) gxGrspqg
s = 0:(n(j)-m_abs(j))/2; Q!FLR>8
pows = n(j):-2:m_abs(j); UP{j5gR:_
for k = length(s):-1:1 M8b4NF_&
p = (1-2*mod(s(k),2))* ... WmQ 01v
prod(2:(n(j)-s(k)))/ ... nD2, !71
prod(2:s(k))/ ... Px>va01n
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... pBC<u
prod(2:((n(j)+m_abs(j))/2-s(k))); h`} 3h<
8
idx = (pows(k)==rpowers); LN_OD5gZ
y(:,j) = y(:,j) + p*rpowern(:,idx); iY bX
end @ E53JKYhY
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if isnorm ='E$-_
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Sm2>'C
end Fequm +
end do
^RF<G
% END: Compute the Zernike Polynomials $,hwU3RVxc
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ? QDWuPhN
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% Compute the Zernike functions: ^SfS~G Q
% ------------------------------ BD#.-xWV
idx_pos = m>0; te4= S
idx_neg = m<0; '~wpP=<yyF
v}v 5
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z = y; GW2 ')}g
if any(idx_pos) U~2 `P
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); k,Zm GllQ]
end yO>V/5`
if any(idx_neg) dy>|c j
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); C+MSVc
end )DUL)S
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[[)_BmS5r
% EOF zernfun 6b Z[Kt
