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, C4|H5H
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, ZJOO*S
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? O6b.oS'-
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 4v#A#5+O E
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function z = zernfun(n,m,r,theta,nflag) "rcV?5?v~
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. zC WN,K`
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 0GcOI}
% and angular frequency M, evaluated at positions (R,THETA) on the )
B[S4K2
% unit circle. N is a vector of positive integers (including 0), and MNH-SQB |
% M is a vector with the same number of elements as N. Each element _{mG\*q
% k of M must be a positive integer, with possible values M(k) = -N(k) $sb `BS
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, @WuG8G
% and THETA is a vector of angles. R and THETA must have the same 4=ZN4=(_[
% length. The output Z is a matrix with one column for every (N,M) ,Ad{k
% pair, and one row for every (R,THETA) pair. %!V =noo
% F>"B7:P1:Q
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike wQrD(Dv(yA
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), f=Kt[|%'e
% with delta(m,0) the Kronecker delta, is chosen so that the integral 43/!pW
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, DX<xkS[P
% and theta=0 to theta=2*pi) is unity. For the non-normalized S!R:a>\
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Rqun}v}
% ke5_lr(
% The Zernike functions are an orthogonal basis on the unit circle. l/6(V:
% They are used in disciplines such as astronomy, optics, and {AO`[
% optometry to describe functions on a circular domain. 2-DJ3OL]k
% Vv.q{fRvYB
% The following table lists the first 15 Zernike functions. j)lgF:
% Kc{~Q
% n m Zernike function Normalization 3.?B')
% -------------------------------------------------- 1fcyGZq
% 0 0 1 1 |&\cr\T\r
% 1 1 r * cos(theta) 2 xi! R[xr1
% 1 -1 r * sin(theta) 2 Wfj*)j
Q
% 2 -2 r^2 * cos(2*theta) sqrt(6) ~.TKzh'eB
% 2 0 (2*r^2 - 1) sqrt(3) 5dEek7wnf
% 2 2 r^2 * sin(2*theta) sqrt(6) TuMD+^x
% 3 -3 r^3 * cos(3*theta) sqrt(8) j(`V&S
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) I.'sK9\Zp
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) V1\x.0Fs
% 3 3 r^3 * sin(3*theta) sqrt(8) 1"
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% 4 -4 r^4 * cos(4*theta) sqrt(10) gpe-)hD@R
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) }OLBEhGs
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Jk=d5B
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Fhbp,CX4p
% 4 4 r^4 * sin(4*theta) sqrt(10) ?KXgG'!!
% -------------------------------------------------- 4e9'yi
% =y1/V'2E
% Example 1: M{M?#Q
% tCbnB
% % Display the Zernike function Z(n=5,m=1) bcE%EQ
% x = -1:0.01:1; z9P;HGuZ
% [X,Y] = meshgrid(x,x); DX4"}w
% [theta,r] = cart2pol(X,Y); XjV,wsZ=
% idx = r<=1; w@ \quy:
% z = nan(size(X)); JnBg;D|)@
% z(idx) = zernfun(5,1,r(idx),theta(idx));
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% figure * 57y.](w
% pcolor(x,x,z), shading interp cT,5xp"a
% axis square, colorbar pk2}]jx"
% title('Zernike function Z_5^1(r,\theta)') +}@6V4BRn
% ,L,?xvWG
% Example 2: @Z%I g
% h]#bPb
% % Display the first 10 Zernike functions "\u_gk{g
% x = -1:0.01:1; 8A3!XA
% [X,Y] = meshgrid(x,x); nLv"ON~
% [theta,r] = cart2pol(X,Y); WMXk-?v4
% idx = r<=1; VS_xC$X!S
% z = nan(size(X)); tx01*2]pX
% n = [0 1 1 2 2 2 3 3 3 3]; L?p,Sy<RI
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; a]u1_ $)
% Nplot = [4 10 12 16 18 20 22 24 26 28]; F3V_rE<
% y = zernfun(n,m,r(idx),theta(idx)); =#ls<Zo:
% figure('Units','normalized') 4'ymPPY
% for k = 1:10 iPoDesp
% z(idx) = y(:,k); jM DG
% subplot(4,7,Nplot(k)) ;\N${YIn
% pcolor(x,x,z), shading interp 8I*WVa$l
% set(gca,'XTick',[],'YTick',[]) ??.9`3CYo
% axis square Ib665H7w
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) `VxfAV?}
% end yvz2eAXa
% ,
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% See also ZERNPOL, ZERNFUN2. _LUTIqlvi
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% Paul Fricker 11/13/2006 h%T$m_
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#`9D,+2iB%
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% Check and prepare the inputs: ntT~_Ba8;u
% ----------------------------- ]C
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) h"~GaI
error('zernfun:NMvectors','N and M must be vectors.') E5}wR(i,4
end ;\5^yDv[e
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if length(n)~=length(m) 8+7=yN(
error('zernfun:NMlength','N and M must be the same length.') &J~%Nt
end :jp4 !0w
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n = n(:); D9|?1+Kc
m = m(:); + ^9;<>P
if any(mod(n-m,2)) =_/,C
error('zernfun:NMmultiplesof2', ... 4&c7^ 4w~
'All N and M must differ by multiples of 2 (including 0).') FOU^Wcop%
end =5-|H;da
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if any(m>n) jGtoc,\X
error('zernfun:MlessthanN', ... Q*+_%n1
/
'Each M must be less than or equal to its corresponding N.') )@]Y1r4U
end p|D-ez8
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@fH?y Z=>
if any( r>1 | r<0 ) ){.J`X5r
error('zernfun:Rlessthan1','All R must be between 0 and 1.') YC
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end {|jG_
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 9x~qcH%
error('zernfun:RTHvector','R and THETA must be vectors.') W~1MeAI
end
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r = r(:); btC.EmX
theta = theta(:); 2_ x~y|<9
length_r = length(r); hk O)q|1
if length_r~=length(theta) U-$ B"w &
error('zernfun:RTHlength', ... %DQ.f*%
'The number of R- and THETA-values must be equal.') GMZj@q
end h%Nbx:vKk
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% Check normalization: GA{Q6]B
% -------------------- A|BvRZd
if nargin==5 && ischar(nflag) J!QzF)$4J
isnorm = strcmpi(nflag,'norm'); $h Isab_
if ~isnorm }@pe`AF^
error('zernfun:normalization','Unrecognized normalization flag.') GB+U>nf
end XB &-k<C
else RoXU>a:nS
isnorm = false; xi6Fs, 2S
end Xf.w(-
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% iaR^] |7_
% Compute the Zernike Polynomials _ "ysJ&
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LDL#*g
Zcg=a_
%$
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% Determine the required powers of r: lA39$oJ
% ----------------------------------- 8KpG0DC
m_abs = abs(m); |5}{4k~9J
rpowers = []; 2#nn}HEOC
for j = 1:length(n) /Xi:k
rpowers = [rpowers m_abs(j):2:n(j)]; jZ <*XX
end ^P-!pK*
rpowers = unique(rpowers); =>6Z"LD(
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% Pre-compute the values of r raised to the required powers, K \}xb2s
% and compile them in a matrix: Rww"Z=F
% ----------------------------- 5:f}bW*
if rpowers(1)==0 l\5}\9yS
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); d]h[]Su/?
rpowern = cat(2,rpowern{:}); -t
%.I=|
rpowern = [ones(length_r,1) rpowern]; W K#lE&V3
else H7)(<6b,z
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); `3r *Ae
rpowern = cat(2,rpowern{:}); io:?JnQSA
end ?x]T&S{
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;W#G<M&n'
% Compute the values of the polynomials: +bd/*^
% -------------------------------------- J6Mm=bO5
y = zeros(length_r,length(n)); SZc6=^$
for j = 1:length(n) ltHC+8aZ
s = 0:(n(j)-m_abs(j))/2; a2iaP
pows = n(j):-2:m_abs(j); -4b9(
for k = length(s):-1:1 W.o
W=<
p = (1-2*mod(s(k),2))* ... NS=puo
prod(2:(n(j)-s(k)))/ ... =#1iio&
prod(2:s(k))/ ... ms3Ec`i9
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... LL-MZ~ZB
prod(2:((n(j)+m_abs(j))/2-s(k))); 8Md*9E#J("
idx = (pows(k)==rpowers); hdN3r{
y(:,j) = y(:,j) + p*rpowern(:,idx); \C*?a0!:Z}
end e&F,z=XJ}
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if isnorm pz?.(AmU\
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); QsI>_<r
end +S|y)W8
end 2NsI3M4$8
% END: Compute the Zernike Polynomials b#k$/A@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% n?aogdK$V
LtH;#Q
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% Compute the Zernike functions: ImklM7A
% ------------------------------ Lc*i[J<s
idx_pos = m>0; 4jis\W}%L3
idx_neg = m<0; y"!+Fus9
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tTF/$`Q#*
z = y; tb&{[|O^
if any(idx_pos) kYxn5+~
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); >F,~ QHcz
end .knRH^
if any(idx_neg) ]Rnr>_>x;
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 5H==m~
end Tp[ub(/;7
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% EOF zernfun !K(0)~u