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, ndr)3tuYu
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, gc##V]OD
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? ba8 6 N
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? md?b*
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function z = zernfun(n,m,r,theta,nflag) W@^J6sH
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. S`=n&'
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ^00{Hd6
% and angular frequency M, evaluated at positions (R,THETA) on the P'sfi>A
% unit circle. N is a vector of positive integers (including 0), and w#&z]O9r
% M is a vector with the same number of elements as N. Each element (_K_`5d;QI
% k of M must be a positive integer, with possible values M(k) = -N(k)
r@k"4ce-
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, gY!N3 *:
% and THETA is a vector of angles. R and THETA must have the same 5X0QxnnV
% length. The output Z is a matrix with one column for every (N,M) UgC)7
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% pair, and one row for every (R,THETA) pair. oE1M/*myS
% HMV)U{
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Rv<L#!;
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% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), m<{"}4'
% with delta(m,0) the Kronecker delta, is chosen so that the integral /YFa
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% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, _42Z={pZZq
% and theta=0 to theta=2*pi) is unity. For the non-normalized r!kLV )_
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. }~F~hf>s
% 9*\g`fWc}{
% The Zernike functions are an orthogonal basis on the unit circle. =2%VZE7Vm
% They are used in disciplines such as astronomy, optics, and L6+C]t}>6
% optometry to describe functions on a circular domain. lm$;:Roj*
% %G[/H.7s-
% The following table lists the first 15 Zernike functions. 0Gsu
% "]#'QuR
% n m Zernike function Normalization SNab
% -------------------------------------------------- (~&w-w3
% 0 0 1 1 26.)U r<F
% 1 1 r * cos(theta) 2 n(>C'<otj
% 1 -1 r * sin(theta) 2 zb :kanb-
% 2 -2 r^2 * cos(2*theta) sqrt(6) hm\\'_u
% 2 0 (2*r^2 - 1) sqrt(3) \0?$wIH?
% 2 2 r^2 * sin(2*theta) sqrt(6) 2JZdw
% 3 -3 r^3 * cos(3*theta) sqrt(8) qnJ50 VVW
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) {q,?<zBzu
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) tuLH}tkNY
% 3 3 r^3 * sin(3*theta) sqrt(8) ^I`a;
% 4 -4 r^4 * cos(4*theta) sqrt(10) T@P!L
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) J\=a gQ
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 3z3_7XI
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Y5Z!og
% 4 4 r^4 * sin(4*theta) sqrt(10) ;iU%Kt
% -------------------------------------------------- ]
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% }G'XkoI&
% Example 1: m5*[t7@%
% SkHYXe"]
% % Display the Zernike function Z(n=5,m=1) . I==-|
% x = -1:0.01:1; aGK@)&h$
% [X,Y] = meshgrid(x,x); -Sz_mr
% [theta,r] = cart2pol(X,Y); Wp[9beI*M
% idx = r<=1; o=_c2m
% z = nan(size(X)); ()\jCNLT
% z(idx) = zernfun(5,1,r(idx),theta(idx)); G\=_e8(
% figure H)>sTST(
% pcolor(x,x,z), shading interp OJ1tV% E
% axis square, colorbar W5SN I>|E
% title('Zernike function Z_5^1(r,\theta)') dv!r.
% BzN@gQo
% Example 2: >o/95xk2
% pRi<cO
% % Display the first 10 Zernike functions BBnq_w"a
% x = -1:0.01:1; ;:]\KJm}?
% [X,Y] = meshgrid(x,x); Y#HI;Y^RP
% [theta,r] = cart2pol(X,Y); HBiBv-=,
% idx = r<=1; mgQIhXH5L
% z = nan(size(X)); Ef@,hX
% n = [0 1 1 2 2 2 3 3 3 3]; 5 1dSFr<#
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ,_ .v_
% Nplot = [4 10 12 16 18 20 22 24 26 28]; vt1lR5
% y = zernfun(n,m,r(idx),theta(idx)); O{]9hm(tN
% figure('Units','normalized') x({C(Q'O
% for k = 1:10 *Y6xvib9*
% z(idx) = y(:,k); L/Vx~r`P
% subplot(4,7,Nplot(k)) 2@khSWV
% pcolor(x,x,z), shading interp ke%pZ7{u
% set(gca,'XTick',[],'YTick',[]) F)Oe9x\/
% axis square 2k5/SV
X
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) vmX"+sHz$]
% end Y)|N"f;
% 27A!\pn
% See also ZERNPOL, ZERNFUN2. %d;ezY '2
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% Paul Fricker 11/13/2006 T
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% Check and prepare the inputs: V diJ>d[
% ----------------------------- GTl
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) dl~|Izm
error('zernfun:NMvectors','N and M must be vectors.') -e]7n*}H$
end e0HfP v_
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if length(n)~=length(m) 7cTk@Gq
error('zernfun:NMlength','N and M must be the same length.') H/fUM
end *rh,"Zo
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n = n(:); 5:.{oSy7n
m = m(:); >I"V],d!6
if any(mod(n-m,2)) u bW]-U=T
error('zernfun:NMmultiplesof2', ... p&b5% 4P
'All N and M must differ by multiples of 2 (including 0).') 9KuD(EJS
end tJ0NPI56yP
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if any(m>n) r'7LR
error('zernfun:MlessthanN', ... &[[K"aM1
'Each M must be less than or equal to its corresponding N.') SPkn3D6
end z@ 35NZn
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if any( r>1 | r<0 ) #q\x$
error('zernfun:Rlessthan1','All R must be between 0 and 1.') %;xOB^H^
end 5Wx~ZQZ
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) |a^U]
error('zernfun:RTHvector','R and THETA must be vectors.') w n|]{Ww35
end @OpNHQat9
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r = r(:); :mcYZPX#
theta = theta(:); Xd
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length_r = length(r); C#0Qd%
if length_r~=length(theta) s#9Ui#[=h
error('zernfun:RTHlength', ... #'baPqdO
'The number of R- and THETA-values must be equal.') 5s{j=.O
end (qMj-l
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% Check normalization: b\7iY&.C|
% -------------------- pKG<Nvgz&
if nargin==5 && ischar(nflag) @C_KV0i
isnorm = strcmpi(nflag,'norm'); ,5
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if ~isnorm B=f,QU
error('zernfun:normalization','Unrecognized normalization flag.') -e GL) M
end q'[}9e`Q
else O*6n$dUj3
isnorm = false; K$ }a8rH
end "_UdBG
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +0wT!DZW\=
% Compute the Zernike Polynomials &
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% am1[9g8L
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% Determine the required powers of r: n{0Ld -zH
% ----------------------------------- ZFm`UXS
m_abs = abs(m); +avMX&%
rpowers = []; ?4H#G)F
for j = 1:length(n) f_ ^1J
rpowers = [rpowers m_abs(j):2:n(j)]; `>(W"^
end eDI=nSo
rpowers = unique(rpowers); e> rRTN
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% Pre-compute the values of r raised to the required powers, Rgy-OA
% and compile them in a matrix: BAj-akc f
% ----------------------------- T VmH
if rpowers(1)==0 2zSG&",2D
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); M,5j5<7
rpowern = cat(2,rpowern{:}); S{]7C?4`
rpowern = [ones(length_r,1) rpowern]; +yob)%
else :# E*Y8-
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); <:>SGSE9
rpowern = cat(2,rpowern{:}); wFh8?Z3u_
end n%^ LPD
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% Compute the values of the polynomials: !rDdd%Z
% -------------------------------------- rPNb\Ri
y = zeros(length_r,length(n)); f*{
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for j = 1:length(n) vr^~yEr
s = 0:(n(j)-m_abs(j))/2; d,vNem-Z*L
pows = n(j):-2:m_abs(j); /^{BUo
for k = length(s):-1:1 D-Vai#Cd
p = (1-2*mod(s(k),2))* ... ]r!>{
prod(2:(n(j)-s(k)))/ ... #o/H~Iv
prod(2:s(k))/ ... Snly UP~P
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 6Tw#^;q-
prod(2:((n(j)+m_abs(j))/2-s(k))); 'TC/vnM
idx = (pows(k)==rpowers); GDhE[of
y(:,j) = y(:,j) + p*rpowern(:,idx); `(+o=HsD
end 1axQ)},o@p
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if isnorm 7'-Lp@an
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); r)9Dy,
end B_U{ s\VY
end /){KOCBl;
% END: Compute the Zernike Polynomials UtB6V)YI
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% OdWou|Gz
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% Compute the Zernike functions: Z@8vL
% ------------------------------ R3)57OyV
idx_pos = m>0; e~ aqaY~}
idx_neg = m<0; XoLJ L]+?
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zl-2$}<a
z = y; a07@C
if any(idx_pos) )VCzn~uf
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); kg][qn|>J]
end N"/-0(9[
if any(idx_neg) G2LK]
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); I1X/Lj=
end ^JZ^>E~
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% EOF zernfun nVu&/