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, D:1@1Jr
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, <q'l7S
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? s<s}6|Z
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? DiFYVR<@
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function z = zernfun(n,m,r,theta,nflag) (~GQncqa
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. uuC ["Z
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N .^Sglo
% and angular frequency M, evaluated at positions (R,THETA) on the ubcB<=xb
% unit circle. N is a vector of positive integers (including 0), and -&1(~7
% M is a vector with the same number of elements as N. Each element D'g,<-ahl
% k of M must be a positive integer, with possible values M(k) = -N(k) v675C# l(
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, .XJ'2yKof
% and THETA is a vector of angles. R and THETA must have the same H7zN|NdNw
% length. The output Z is a matrix with one column for every (N,M) {&=+lr_h?
% pair, and one row for every (R,THETA) pair. V`Cyx^P
% Q^(CqQo!<
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 8xPt1Sotq[
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 4q}+8F`0F
% with delta(m,0) the Kronecker delta, is chosen so that the integral =;rLv7(a
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, F]aoTy
% and theta=0 to theta=2*pi) is unity. For the non-normalized V}jGxt0
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Mog>W&U
% Q|'f3\
% The Zernike functions are an orthogonal basis on the unit circle. 2q~.,vpP
% They are used in disciplines such as astronomy, optics, and u<-)C)z
% optometry to describe functions on a circular domain. uvId],dQ5
% e\%,\uV}
% The following table lists the first 15 Zernike functions. K:,V>DL
% (` *BZ_
% n m Zernike function Normalization \|HEe{nA
% -------------------------------------------------- #Rw!a#CX.
% 0 0 1 1 jIol`WX
% 1 1 r * cos(theta) 2 R#T-o,m
% 1 -1 r * sin(theta) 2 ;b<w'A_1
% 2 -2 r^2 * cos(2*theta) sqrt(6) TSB2]uH
% 2 0 (2*r^2 - 1) sqrt(3) &jE\D^>ko
% 2 2 r^2 * sin(2*theta) sqrt(6) F.[%0b E
% 3 -3 r^3 * cos(3*theta) sqrt(8) Tagf7tw4
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) _@DOH2lXJ
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) TnF~'RZYb
% 3 3 r^3 * sin(3*theta) sqrt(8) >8f~2dH2%
% 4 -4 r^4 * cos(4*theta) sqrt(10) y )QLR<wf
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) nu0pzq\6
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) [:8\F#KW
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) z`{sD]
% 4 4 r^4 * sin(4*theta) sqrt(10) FZ"n6hWA
% -------------------------------------------------- eZ'8JU]
% ,lZ19B?WP
% Example 1: Z-iU7 O
% `Fd
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% % Display the Zernike function Z(n=5,m=1) 8 v/H;65
% x = -1:0.01:1; B)0/kY7c
% [X,Y] = meshgrid(x,x); 'S`l[L:.8
% [theta,r] = cart2pol(X,Y); ;uBGB
h<
% idx = r<=1; 6S`_L
% z = nan(size(X)); tOIqX0dWd
% z(idx) = zernfun(5,1,r(idx),theta(idx)); x[0T$
% figure htBA.eQ
% pcolor(x,x,z), shading interp 7^gO>2~
% axis square, colorbar JipNI8\r
% title('Zernike function Z_5^1(r,\theta)') Z/Rp?Jz\j/
% IiPX`V>RC
% Example 2: y``\^F
% UqK.b}s
% % Display the first 10 Zernike functions `<7\Zl
% x = -1:0.01:1; S\GWMB!oF
% [X,Y] = meshgrid(x,x); M':-f3aT%
% [theta,r] = cart2pol(X,Y); E7X6RB b
% idx = r<=1; cYSn
% z = nan(size(X)); F2N"aQ&
% n = [0 1 1 2 2 2 3 3 3 3]; 'O<b'}-A
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; MBWoPK
% Nplot = [4 10 12 16 18 20 22 24 26 28]; )D8op;Fn
% y = zernfun(n,m,r(idx),theta(idx)); f_c\uN@f
% figure('Units','normalized') h FU8iB`Q
% for k = 1:10 l.}PxZ
% z(idx) = y(:,k); +7.|1x;C
% subplot(4,7,Nplot(k)) @Jd&[T27Lr
% pcolor(x,x,z), shading interp &[G)YD
% set(gca,'XTick',[],'YTick',[]) ,rB(WKU
% axis square iw )gNQ%z4
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 2S8;=x}/
% end }B0[S_mw
% +XWTu!
% See also ZERNPOL, ZERNFUN2. }&0LoW/
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% Paul Fricker 11/13/2006 8m+~HSIR
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% Check and prepare the inputs: qiz(k:\o
% ----------------------------- 8m0*89HEu
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Snkb^Kt
error('zernfun:NMvectors','N and M must be vectors.') Uu7]`U l
end Xt$qjtVM
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if length(n)~=length(m) ilL%
error('zernfun:NMlength','N and M must be the same length.') h0F=5| B
end gSFZ>v*6
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]!ai?z%cK#
n = n(:); 4Sh8w%s
m = m(:); 4)iP%%JH
if any(mod(n-m,2)) aen%
error('zernfun:NMmultiplesof2', ... H9WYt#
'All N and M must differ by multiples of 2 (including 0).') -mO#HZ Iq
end <zXG}JuL@T
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if any(m>n) &4OOW;,?<
error('zernfun:MlessthanN', ... R+!U.:-yz
'Each M must be less than or equal to its corresponding N.') P5my]4|x
end tav@a)
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%U{6 `m
if any( r>1 | r<0 ) / =9Y(v
error('zernfun:Rlessthan1','All R must be between 0 and 1.') #?)6^uTW
end ;bwBd:Y
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) &0kr[Ik.
error('zernfun:RTHvector','R and THETA must be vectors.') k
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end faOiNR7;h
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r = r(:); ;MeY@*"{
theta = theta(:); @PM<pEve
length_r = length(r); =cRmaD
if length_r~=length(theta) cn}15JHdR
error('zernfun:RTHlength', ... A\?t^T
'The number of R- and THETA-values must be equal.') ?Tc|3U
end 4-
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% Check normalization: OcUj_Zd
% -------------------- E^J &?-
if nargin==5 && ischar(nflag) -aBhN~
isnorm = strcmpi(nflag,'norm'); z#G\D5yX[*
if ~isnorm xD*Zcw(vj~
error('zernfun:normalization','Unrecognized normalization flag.') qGq]E`O
end }Rz,}^B
else n
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isnorm = false; MR|A_e^x
end i'<hT
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% gZ&4b'XS,
% Compute the Zernike Polynomials e!0xh
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% oaha5aWH
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% Determine the required powers of r: F4~OsgZ'N
% ----------------------------------- Pz*BuL<
m_abs = abs(m); `'|6b5`2j
rpowers = []; 41/civX>V
for j = 1:length(n) sT =|"H?
rpowers = [rpowers m_abs(j):2:n(j)]; L[PqEN\i
end vE`;1UA}
rpowers = unique(rpowers); tX%
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,X|FyO(p
% Pre-compute the values of r raised to the required powers, 8p829
% and compile them in a matrix: *CGHp8
% ----------------------------- #IGcQY
if rpowers(1)==0 o_\vudXK
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); R6X2d\l#
rpowern = cat(2,rpowern{:}); oeKl\cgFx
rpowern = [ones(length_r,1) rpowern]; IZdWEbN1
else D(Z#um8n
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); DNj<:Pdd)
rpowern = cat(2,rpowern{:}); CD`6R.
end g_ep
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% Compute the values of the polynomials:
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% -------------------------------------- Q(x=;wf5r
y = zeros(length_r,length(n)); qPi $kecx
for j = 1:length(n) f-^*p
s = 0:(n(j)-m_abs(j))/2; >9XG+f66E
pows = n(j):-2:m_abs(j); m.6uLaD"!}
for k = length(s):-1:1 $Vp&7OC]
p = (1-2*mod(s(k),2))* ... .z$UNB(!M
prod(2:(n(j)-s(k)))/ ... i:N-Q)<Q*)
prod(2:s(k))/ ... ,h%n5R$:
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... !1S!)#
prod(2:((n(j)+m_abs(j))/2-s(k))); %iPIgma
idx = (pows(k)==rpowers); ~eTp( XG
y(:,j) = y(:,j) + p*rpowern(:,idx); aiX4;'$x!
end {|%^'lS
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if isnorm 4]zn,g?&
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); B4*,]lS?
end 41B.ZE+*qd
end W|;`R{<I%
% END: Compute the Zernike Polynomials e7iQG@i7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;E{@)X..|
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% Compute the Zernike functions: MC-Z6l2
% ------------------------------ ,:
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idx_pos = m>0; J#w=Z>oz <
idx_neg = m<0; j^Qk\(^#IV
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z = y; R))4J
if any(idx_pos) cWQ &zc
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); (.z0.0W
end a{;+_J3S
if any(idx_neg) jA@
uV,w
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); _MQh<,Z8
end .GYdC'
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% EOF zernfun $+{o*