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， ?=@Q12R)X
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， ofrlTw&o
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？ szsZFyW )+
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ fdH'z:Xao
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function z = zernfun(n,m,r,theta,nflag) ]#7Y @Yo
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. : c/=fWM%
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N my\oC^/9
% and angular frequency M, evaluated at positions (R,THETA) on the [@FeRIu8
% unit circle. N is a vector of positive integers (including 0), and WO*WAP)n
% M is a vector with the same number of elements as N. Each element nTtt$I@hW
% k of M must be a positive integer, with possible values M(k) = -N(k) fN%5D z-e
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, \ g[f4xAV
% and THETA is a vector of angles. R and THETA must have the same {j=hQL3
% length. The output Z is a matrix with one column for every (N,M) KZ
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% pair, and one row for every (R,THETA) pair. jeuNTDjeL
% i$ZpoM
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike H><mcah
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ZxeE6&#M^w
% with delta(m,0) the Kronecker delta, is chosen so that the integral yURh4@
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, i. dAL)V
% and theta=0 to theta=2*pi) is unity. For the non-normalized +n~rM'^4/
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ps;o[gB@5
% A kQFb2|ir
% The Zernike functions are an orthogonal basis on the unit circle. -Aym+ N9
% They are used in disciplines such as astronomy, optics, and J1ro\"
% optometry to describe functions on a circular domain. V^5k> `A
% <.B > LU
% The following table lists the first 15 Zernike functions. M,U=zNPnk
% cZ2,
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% n m Zernike function Normalization "=TTsxyM6P
% -------------------------------------------------- #w?%&,Kp
% 0 0 1 1 A(sx5Ynp
% 1 1 r * cos(theta) 2 jJQfCOD$
% 1 -1 r * sin(theta) 2 { rJF)\2
% 2 -2 r^2 * cos(2*theta) sqrt(6) &$Ip$"H
% 2 0 (2*r^2 - 1) sqrt(3) nPX'E`ut-V
% 2 2 r^2 * sin(2*theta) sqrt(6) *8eh%3_$h
% 3 -3 r^3 * cos(3*theta) sqrt(8) _q4d gi z
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) b 020U>)v
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) (S 3kP5:F
% 3 3 r^3 * sin(3*theta) sqrt(8) ' g! _Flk
% 4 -4 r^4 * cos(4*theta) sqrt(10) Jj!tRZT
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) < 1%XN
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) _Ws k3AP
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) X_S]8Aa
% 4 4 r^4 * sin(4*theta) sqrt(10) t"Rf67
% -------------------------------------------------- |N.q[>^R
% -@?>nLQb
% Example 1: a9JJuSRC
% DQX x }%Px
% % Display the Zernike function Z(n=5,m=1) U1tPw`0h
% x = -1:0.01:1; t7%Bv+Uo
% [X,Y] = meshgrid(x,x); j|8{Vy qd
% [theta,r] = cart2pol(X,Y); X "59`Yh
% idx = r<=1; @!HMd{ r
% z = nan(size(X)); ptL}F~
% z(idx) = zernfun(5,1,r(idx),theta(idx)); BnY|t2r
% figure znpZ0O\!
% pcolor(x,x,z), shading interp FOyfk$
% axis square, colorbar yAkN2
% title('Zernike function Z_5^1(r,\theta)') %Ne>'252y
% 2*E<G|-F
% Example 2: K4L#%KUPW
% R.$Y1= U6
% % Display the first 10 Zernike functions e%7P$.
% x = -1:0.01:1; UsKn4Kh
% [X,Y] = meshgrid(x,x); 5 : >
% [theta,r] = cart2pol(X,Y); *3oQS"8
% idx = r<=1; wpMQ 7:j
% z = nan(size(X)); 8j +;Xlh
% n = [0 1 1 2 2 2 3 3 3 3]; +/8?+1E ^
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; I4ct``Di
% Nplot = [4 10 12 16 18 20 22 24 26 28]; !t{!.
% y = zernfun(n,m,r(idx),theta(idx)); !.N=Y;@lY
% figure('Units','normalized') oK[,xqyA
% for k = 1:10 ^?`,f>`M
% z(idx) = y(:,k); LM`#S /h
% subplot(4,7,Nplot(k)) $
$+z^%'_
% pcolor(x,x,z), shading interp 2Rt ZTn
% set(gca,'XTick',[],'YTick',[]) o?8 j *]
% axis square g
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% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) CBHWMetJ*
% end ~<R~ Q:T
% 5<
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% See also ZERNPOL, ZERNFUN2. SX8%F:<.
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% Paul Fricker 11/13/2006 Pk2 "\y@q/
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% Check and prepare the inputs: pp _ddk
% ----------------------------- %%u4( ' =
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) >?x Vr
error('zernfun:NMvectors','N and M must be vectors.') pYQs|5d
end _"TG:RP
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if length(n)~=length(m) \hX^Cn=6
error('zernfun:NMlength','N and M must be the same length.') fTcRqov
end ]t<%>Z$
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n = n(:); Ha\ hQ'99
m = m(:); nV1,
):kh
if any(mod(n-m,2)) BJTljg( {o
error('zernfun:NMmultiplesof2', ... I}{eYXh
'All N and M must differ by multiples of 2 (including 0).') -z94>}Z=
end z""(M4
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if any(m>n) MDt4KD+bZ
error('zernfun:MlessthanN', ... Po[zzj>m
'Each M must be less than or equal to its corresponding N.') =n&83MYX
end 1owoh,V6
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if any( r>1 | r<0 ) FUqiP(A
error('zernfun:Rlessthan1','All R must be between 0 and 1.') vF 1$$7 k
end uNDkK o<M
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) |,crQ'N'
error('zernfun:RTHvector','R and THETA must be vectors.') vJ s /ett
end G})mw
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r = r(:); OduTg^R
theta = theta(:); WJWrLu92\U
length_r = length(r); }I0^nv1
if length_r~=length(theta) LGkKR{ep(
error('zernfun:RTHlength', ... }#1{Ghs S
'The number of R- and THETA-values must be equal.') >Ww F0W9?
end qYs6P LC
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% Check normalization: {z7{ta
% -------------------- 8,Z0J
if nargin==5 && ischar(nflag) m[XN,IE#u
isnorm = strcmpi(nflag,'norm'); 0ni5 :tYy
if ~isnorm g o@}r< B$
error('zernfun:normalization','Unrecognized normalization flag.') {_JLmyaerZ
end &DV'%h>i=
else 4 KKNw 9L)
isnorm = false; c W2:D$Pe
end ),_bDI L+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &nss[w$%C
% Compute the Zernike Polynomials A@4Cfb@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "#() 4.9
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% Determine the required powers of r: ,Dj ZDw
% ----------------------------------- 0WFZx
Ad "
m_abs = abs(m); n.)- aRu[
rpowers = []; E_z@\z MB
for j = 1:length(n) A, os rv
rpowers = [rpowers m_abs(j):2:n(j)]; N=kACEo
end t%%I .zIV7
rpowers = unique(rpowers); Y+N87C<
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% Pre-compute the values of r raised to the required powers, ` mi!"pm w
% and compile them in a matrix: la-+ `
% ----------------------------- x8H)m+AW
if rpowers(1)==0 >&TktQO_T
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); }5gQZ'ys'
rpowern = cat(2,rpowern{:}); -%A6eRShk
rpowern = [ones(length_r,1) rpowern]; ,/KHKLY7
else z<ek?0?yS
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 9: Y\D.M
rpowern = cat(2,rpowern{:}); REJ}T:
end 3+Q6<MS
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% Compute the values of the polynomials: 5H79) n>
% -------------------------------------- Zqao4
y = zeros(length_r,length(n)); E,;nx^`!l
for j = 1:length(n) *6 h.#$\
s = 0:(n(j)-m_abs(j))/2; mb#)w`<
pows = n(j):-2:m_abs(j); D -jew &B
for k = length(s):-1:1 ]Kf HuYjM
p = (1-2*mod(s(k),2))* ... ?;$g, 2n
prod(2:(n(j)-s(k)))/ ... b `2~
prod(2:s(k))/ ... ( GeJBw,Q
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... &^}w| J?
prod(2:((n(j)+m_abs(j))/2-s(k))); eRf 8'-"#-
idx = (pows(k)==rpowers); j>6{PDaT
y(:,j) = y(:,j) + p*rpowern(:,idx); U;^{uQJ+,
end TiOvrp 7B
zIL.R#|D=
if isnorm l6O2B/2j
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); :{sX8U%
end WN0^h Dc-
end ZK; HW
% END: Compute the Zernike Polynomials k~?@~xm,R
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >Nov9<p
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xGA%/dy,;
% Compute the Zernike functions: 2 @ad! h
% ------------------------------ i^n&K:6
idx_pos = m>0; ]t,ppFC#
idx_neg = m<0; | o?@Eh
;%U`P8b!
$ }&6p6|
z = y; _ K9jj
if any(idx_pos) /g_}5s-Z
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); n>@( gDq
end ThHK1{87X}
if any(idx_neg) uv@4/M`
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ]-O:| q>]
end 7==Uoy*O
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% EOF zernfun rUj]6j=e
