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, (wIzat
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, 9EDfd NN
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? 00v&lQBW
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? X[
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function z = zernfun(n,m,r,theta,nflag) 'o|30LzYgQ
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. L^2FQti>
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N r.3/F[.
% and angular frequency M, evaluated at positions (R,THETA) on the S5~VD?O,
% unit circle. N is a vector of positive integers (including 0), and f` =CpO*
% M is a vector with the same number of elements as N. Each element Gj"7s8(/K|
% k of M must be a positive integer, with possible values M(k) = -N(k) (?_S6HE
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, VE+IKj!VG0
% and THETA is a vector of angles. R and THETA must have the same mxb(<9O
% length. The output Z is a matrix with one column for every (N,M) H
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% pair, and one row for every (R,THETA) pair. R\o<7g-|
% ee%fqVQ8P
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 0/S_e)U
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), R|O8RlH
% with delta(m,0) the Kronecker delta, is chosen so that the integral C<KrMRWh^
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, (WJ${OW
% and theta=0 to theta=2*pi) is unity. For the non-normalized JkW9D)6
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Ee`1F#c
% WVVJ
% The Zernike functions are an orthogonal basis on the unit circle. t]7&\ihZi~
% They are used in disciplines such as astronomy, optics, and X[f=h=|
% optometry to describe functions on a circular domain. !r#?C9Sq
% LPMU8Er
% The following table lists the first 15 Zernike functions. \
[a%('}
% oc8:r
% n m Zernike function Normalization N<QXmgqx
% -------------------------------------------------- O_Oj|'bBC
% 0 0 1 1 [9Ss#~
% 1 1 r * cos(theta) 2 &u#&@J
% 1 -1 r * sin(theta) 2 LpR3BP@At
% 2 -2 r^2 * cos(2*theta) sqrt(6) PO6&bIr
% 2 0 (2*r^2 - 1) sqrt(3) xg)v0y~
% 2 2 r^2 * sin(2*theta) sqrt(6) E b=}FuV
% 3 -3 r^3 * cos(3*theta) sqrt(8) LX^u_Iu
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ]`Oo%$Ue
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 2WU@*%sk"
% 3 3 r^3 * sin(3*theta) sqrt(8) 5 ~TdD6}
% 4 -4 r^4 * cos(4*theta) sqrt(10) jBegh9KHq
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) R
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% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ],P;WPU
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,3@#F/c3i~
% 4 4 r^4 * sin(4*theta) sqrt(10) e^.Fa59
% -------------------------------------------------- =i~}84>
% Ei2'[PK
% Example 1: K)J(./
% =$]uoA
% % Display the Zernike function Z(n=5,m=1) E9;|'Vy<E
% x = -1:0.01:1; \Gc+WpS(
% [X,Y] = meshgrid(x,x); !Q#{o^{Y~
% [theta,r] = cart2pol(X,Y); 9<KAXr#
% idx = r<=1; rF]h$Z8o
% z = nan(size(X)); -wjN"g<
% z(idx) = zernfun(5,1,r(idx),theta(idx)); *4V=z#
% figure F7Zwh5W
% pcolor(x,x,z), shading interp -|E!e.^7:
% axis square, colorbar aG^4BpIP
% title('Zernike function Z_5^1(r,\theta)') ;<leKcvhQ&
% St e=&^
% Example 2: 9/nn)soC3
% \EVBwE,
% % Display the first 10 Zernike functions =Q.^c.sw
% x = -1:0.01:1; V,$0p1?J
% [X,Y] = meshgrid(x,x); je!-J8{
% [theta,r] = cart2pol(X,Y); v8y1b%
% idx = r<=1; ]C) 4
% z = nan(size(X)); {7)st
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% n = [0 1 1 2 2 2 3 3 3 3]; M@5?ZZ4L
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; p\bDY
% Nplot = [4 10 12 16 18 20 22 24 26 28]; |`cKD >
% y = zernfun(n,m,r(idx),theta(idx)); ^k9kJ+x^S2
% figure('Units','normalized') }K&7%N4LZ
% for k = 1:10 3g >B"t
% z(idx) = y(:,k); &}A[x1x06)
% subplot(4,7,Nplot(k)) [D!jv"
% pcolor(x,x,z), shading interp Rj4|Q:XG
% set(gca,'XTick',[],'YTick',[]) 1;{Rhu7*
k
% axis square hRCed4qA
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) zzyHoZJP
% end RO.k]x6
% ll C#1
% See also ZERNPOL, ZERNFUN2. >"C,@cN}B
Ry'= ke
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% Paul Fricker 11/13/2006 m&?#;J|B$
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% Check and prepare the inputs: tQjLOv+?=
% ----------------------------- k3uit+ge}
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) `|/<\
error('zernfun:NMvectors','N and M must be vectors.') 'nwx9]q
end O)jWZOVp >
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if length(n)~=length(m) /bi}'H+#
error('zernfun:NMlength','N and M must be the same length.') }yz (xH
end `I3r3WyA
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n = n(:); fP.
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m = m(:); (Kv#m
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if any(mod(n-m,2)) k<"oiCE
error('zernfun:NMmultiplesof2', ... [Lzw#XE
'All N and M must differ by multiples of 2 (including 0).') *#C+iAF|)'
end ~FN9 [aJF+
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if any(m>n) wf:OK[r9
error('zernfun:MlessthanN', ... dzDqZQY$
'Each M must be less than or equal to its corresponding N.') 1
=M ?GDc
end nuw70*ell
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if any( r>1 | r<0 ) f>N!wgo[
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 3 yB!M
end `nZ )>
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) zc;|fHW~O
error('zernfun:RTHvector','R and THETA must be vectors.') )s%[T-uKi
end TL}++e
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r = r(:); c1E'$-
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theta = theta(:); PEc=\?
length_r = length(r); j'HZ\_
if length_r~=length(theta) -}KC=,]vh
error('zernfun:RTHlength', ... FW21 U<
'The number of R- and THETA-values must be equal.') [rSR:V?"a
end
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% Check normalization: pP*a
% -------------------- ;,?KI$K
if nargin==5 && ischar(nflag) ;{U@qQD7
isnorm = strcmpi(nflag,'norm'); :gep:4&u
if ~isnorm 2(#7[mgPI
error('zernfun:normalization','Unrecognized normalization flag.') %3ICI
end f PM8f
else *q-['"f
isnorm = false; TztAZ2C
end @n{JM7ctJ
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% AMyIAZnYq)
% Compute the Zernike Polynomials w%JTTru
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (A~/ '0/
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% Determine the required powers of r: F*3j.lI
% ----------------------------------- K>DRJz
m_abs = abs(m); !BOY@$Y
rpowers = []; c+hQSm|bf)
for j = 1:length(n) O8j_0
rpowers = [rpowers m_abs(j):2:n(j)]; qa0 yg8,<
end 8[E!E)4M
rpowers = unique(rpowers); &C"L
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% Pre-compute the values of r raised to the required powers, 6{^E{go
% and compile them in a matrix: *fn*h[pV&
% ----------------------------- 9*{[buZX
if rpowers(1)==0 9mmCp&~Z
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); X ><?F|#7T
rpowern = cat(2,rpowern{:}); rjp-Fw~1w
rpowern = [ones(length_r,1) rpowern]; d;>#Sxf
else `CgaS#
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); rU2%dkTa
rpowern = cat(2,rpowern{:}); :[hgxJu+
end {6_|/KE9_
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% Compute the values of the polynomials: oaoU _V
% -------------------------------------- gT#&"aP5S
y = zeros(length_r,length(n)); w[IE
for j = 1:length(n) S&b*rA02zp
s = 0:(n(j)-m_abs(j))/2; #nK>Z[
pows = n(j):-2:m_abs(j); %\H|B0
for k = length(s):-1:1 ](wvu(y\E
p = (1-2*mod(s(k),2))* ...
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prod(2:(n(j)-s(k)))/ ... ^7 &5
z&o
prod(2:s(k))/ ... t ]_VG
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 32/MkuY^u
prod(2:((n(j)+m_abs(j))/2-s(k))); 2E)wpgUc?e
idx = (pows(k)==rpowers); JAQb{KefdO
y(:,j) = y(:,j) + p*rpowern(:,idx); S/ODqL|
end %Ntcvp)
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if isnorm O]1y0BOQ
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); C0}IE,]
end v,4pp@8rv
end f-E("o
% END: Compute the Zernike Polynomials &'}RrW-s
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s1h/}
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% Compute the Zernike functions: NT+?#0I
% ------------------------------ @]-jl}:]
idx_pos = m>0; 8$;=Uf,x
idx_neg = m<0; \0vr>C
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z = y; cxr=k%~}J
if any(idx_pos) +E.GLn2/
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); qpE&go=k'
end V&\[)D'c
if any(idx_neg) ;bLEL"x%
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); !`M|C?b
end
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% EOF zernfun =yyp?WmC8