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, +"VXw2R_e
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,
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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\@U3|Q5
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 4BJ w+EV8
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function z = zernfun(n,m,r,theta,nflag) 1HeE$
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. YF)c.Q0
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N \*30E<;C_
% and angular frequency M, evaluated at positions (R,THETA) on the 0He^r
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% unit circle. N is a vector of positive integers (including 0), and &[[Hfs2:-]
% M is a vector with the same number of elements as N. Each element PC& (1kJ
% k of M must be a positive integer, with possible values M(k) = -N(k) (_Rl
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% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, S|_"~Nd=
% and THETA is a vector of angles. R and THETA must have the same KtaoU2s
% length. The output Z is a matrix with one column for every (N,M) b2hXFwPe
% pair, and one row for every (R,THETA) pair. S\6.vw!'
% S8;5|ya
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike |p*s:*TJp
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), AN+S6t
% with delta(m,0) the Kronecker delta, is chosen so that the integral vgKdhN2kI
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, bqQR";
% and theta=0 to theta=2*pi) is unity. For the non-normalized v(Q-RR
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 3Sn#
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% Ym9~/'%]
% The Zernike functions are an orthogonal basis on the unit circle. f<Yg_ TG
% They are used in disciplines such as astronomy, optics, and E7@m& R
% optometry to describe functions on a circular domain. }IV=qW,
% ^x}k1F3
% The following table lists the first 15 Zernike functions. 4R9y~~+
% 77%I%<#
% n m Zernike function Normalization OJ<V<=MYZ
% -------------------------------------------------- {br6*
% 0 0 1 1 ?rQIUP{D7
% 1 1 r * cos(theta) 2 P:m6:F@hO
% 1 -1 r * sin(theta) 2 +\25ynM
% 2 -2 r^2 * cos(2*theta) sqrt(6) Ji0FHa_
% 2 0 (2*r^2 - 1) sqrt(3) nZ#0L`@"Y
% 2 2 r^2 * sin(2*theta) sqrt(6) *NoixV1>
% 3 -3 r^3 * cos(3*theta) sqrt(8) h:<?)g~U
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) eJ60@N\A
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) jJe?pT]o
% 3 3 r^3 * sin(3*theta) sqrt(8) J|DY
/v
% 4 -4 r^4 * cos(4*theta) sqrt(10) R-1C#R[
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) n?8xRaEf
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Z<[:v2
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) X 3(*bj>P
% 4 4 r^4 * sin(4*theta) sqrt(10) {w<"jw&2
% -------------------------------------------------- /(DnMHn\
% ]Tn""3#1g
% Example 1: Ev0=m;@_
% !5>PZ{J
% % Display the Zernike function Z(n=5,m=1) u Qz!of%x
% x = -1:0.01:1; 4.q^r]m*
% [X,Y] = meshgrid(x,x); *Jg&:(#}<J
% [theta,r] = cart2pol(X,Y); $Sd pF-'
% idx = r<=1; >ui;B$=
% z = nan(size(X)); 0uJ??4N9
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Z^#u n
% figure (E7C9U*
% pcolor(x,x,z), shading interp +*x9$LSD
% axis square, colorbar B$_-1^L
e
% title('Zernike function Z_5^1(r,\theta)') jXYjs8Iy
% gh.+}8="
% Example 2: y`J8hawp
% mIv}%hD
% % Display the first 10 Zernike functions |eP5iy wg
% x = -1:0.01:1;
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% [X,Y] = meshgrid(x,x); O+ xzM[[
% [theta,r] = cart2pol(X,Y); ]+T$D
% idx = r<=1; )Qh*@=$-
% z = nan(size(X)); mQ^SpK #
% n = [0 1 1 2 2 2 3 3 3 3]; q;QE(}.g
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; z(1`Iy
M
% Nplot = [4 10 12 16 18 20 22 24 26 28]; {ukQBu#}<
% y = zernfun(n,m,r(idx),theta(idx)); #S"s8wdD
% figure('Units','normalized') -b=Aj8h
% for k = 1:10 t/h,-x
% z(idx) = y(:,k); Sn[/'V^$a
% subplot(4,7,Nplot(k)) @oQ"FLF.
% pcolor(x,x,z), shading interp =!IoL7x
% set(gca,'XTick',[],'YTick',[]) (9v%66y
% axis square deCi\n
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) `pfRY!
% end ^n*:zmD
% Dfy=$:Q
% See also ZERNPOL, ZERNFUN2. W;|%)D)y
UD ;UdehC
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% Paul Fricker 11/13/2006 {pC$jd>T
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% Check and prepare the inputs: 0e(4+:0
% ----------------------------- 3(_:"?x A
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) z[0tM&pv
error('zernfun:NMvectors','N and M must be vectors.') $0Un'"`S
end kzC4V
#?'@?0<6
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if length(n)~=length(m) \Yh*ywwP#
error('zernfun:NMlength','N and M must be the same length.') s\0,@A
end 2Mj_wc
t\f[->f
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n = n(:); wxy@XN"/i+
m = m(:); EF'8-*
if any(mod(n-m,2)) vK$wc~
error('zernfun:NMmultiplesof2', ... 2Q;rSe._`
'All N and M must differ by multiples of 2 (including 0).') A+(+PfU
end \s7/`
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if any(m>n) l YpoS
error('zernfun:MlessthanN', ... A[m<xtm5K
'Each M must be less than or equal to its corresponding N.') %JI*)K1WI
end <7`U1DR=
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if any( r>1 | r<0 ) AOef1^S=
error('zernfun:Rlessthan1','All R must be between 0 and 1.') :KS"&h{ SY
end .9vt<<Kwh
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) K9+\Z
error('zernfun:RTHvector','R and THETA must be vectors.') hx ^ l
end _}
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r = r(:); FcyFE~>2
theta = theta(:); . Ctd$
length_r = length(r);
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if length_r~=length(theta) t :~,7
error('zernfun:RTHlength', ... {u4AOM=)
'The number of R- and THETA-values must be equal.') @U9`V&])F[
end =,8nfJ+x
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tl`x/
% Check normalization: q>.C5t'Qx
% -------------------- -Ua&/Yd/}
if nargin==5 && ischar(nflag) )&l5I4CIf
isnorm = strcmpi(nflag,'norm'); aLlHR_
if ~isnorm z<gII~%
error('zernfun:normalization','Unrecognized normalization flag.') ]GD&EQ
end $LiBJ~vV<
else Wl}J=
isnorm = false; IkO[R1K
end J0B*V0'zR
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k{U[ U1j
% Compute the Zernike Polynomials E&f/*V^
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r_kaS
als
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% Determine the required powers of r: A(H2Gt
D
% ----------------------------------- `G%h=rr^c
m_abs = abs(m); 2sp4Mm
rpowers = []; 8U}+9
for j = 1:length(n) AQ,"):ofvT
rpowers = [rpowers m_abs(j):2:n(j)]; C_yNSD
end 8dCRSU
rpowers = unique(rpowers); Wr-I~>D%_
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% Pre-compute the values of r raised to the required powers, }$g"|;<ha
% and compile them in a matrix: \:+ NVIN
% ----------------------------- fIJX5)D
if rpowers(1)==0 M^Tm{`O!
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); db&!t!#,
rpowern = cat(2,rpowern{:}); WD! " $
rpowern = [ones(length_r,1) rpowern]; /U-+ClZi@
else |<O^M q
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); <{@ D^L6h
rpowern = cat(2,rpowern{:}); ^Cvt^cI
end vP=H 2P
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% Compute the values of the polynomials: U@D\+T0
% -------------------------------------- 57O|e/2
y = zeros(length_r,length(n)); $4qM\3x0,
for j = 1:length(n) B I=57
s = 0:(n(j)-m_abs(j))/2; fRq+pUxU
pows = n(j):-2:m_abs(j); s_^N=3Si
for k = length(s):-1:1 o{QV'dgu
p = (1-2*mod(s(k),2))* ... sB$" mJ
prod(2:(n(j)-s(k)))/ ... Q)lD2
prod(2:s(k))/ ... Z
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... %UhLCyC/
prod(2:((n(j)+m_abs(j))/2-s(k))); e/#6qCE
idx = (pows(k)==rpowers); J^S!GG'gb
y(:,j) = y(:,j) + p*rpowern(:,idx); kD7'BP/#
end TjI&8#AWBA
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if isnorm ?%#no{9
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); K\zb+
end k8@bQ"#b
end AEDBr <
% END: Compute the Zernike Polynomials Zg0nsNA
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `^
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% Compute the Zernike functions: Q7{{r&|t&
% ------------------------------ C' {B
idx_pos = m>0; wXZ9@(^
idx_neg = m<0; gm=C0Sp?
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z = y; f`_{SU"3
if any(idx_pos) "]Uj _d
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); />pAZa
end :>Qu;Z1P
if any(idx_neg) IXlk1tHN4I
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ~4O3~Y_+GN
end 5rc3jIXc{|
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MPn/"Fij$
% EOF zernfun -B!
a
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