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, Mh/dpb\Z
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, DiwxXqY
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? >\=3:gb:
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ~8P!XAU56%
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function z = zernfun(n,m,r,theta,nflag) p#?7w
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. v}O30wE
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N v|%Z+w
% and angular frequency M, evaluated at positions (R,THETA) on the xLWwYK
% unit circle. N is a vector of positive integers (including 0), and _Wp{[TH
% M is a vector with the same number of elements as N. Each element dDGgvi|[Mz
% k of M must be a positive integer, with possible values M(k) = -N(k) vAh6+K.e
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, p&bROuw<T
% and THETA is a vector of angles. R and THETA must have the same -vR5BMy=
% length. The output Z is a matrix with one column for every (N,M) >
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% pair, and one row for every (R,THETA) pair. b8"?VS5-"
% TKY*`?ct
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike uU <=d
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Yu[ t\/
% with delta(m,0) the Kronecker delta, is chosen so that the integral w?wG(+X7
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, y7
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% and theta=0 to theta=2*pi) is unity. For the non-normalized ;q:zT\A
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. `V]5 sE]G
% -tHU6s,
% The Zernike functions are an orthogonal basis on the unit circle. MgOR2,cR
% They are used in disciplines such as astronomy, optics, and !^=*Jq>
% optometry to describe functions on a circular domain. 9N<<{rQ,F
% M/ni6%x
% The following table lists the first 15 Zernike functions. UAFwi%@!-q
% 5JCG2jqx0
% n m Zernike function Normalization CBOi`bEf
% -------------------------------------------------- + SFVv_n
% 0 0 1 1 WDc+6/<
% 1 1 r * cos(theta) 2 FsV'Cu@!U
% 1 -1 r * sin(theta) 2 ;WM"cJo9
% 2 -2 r^2 * cos(2*theta) sqrt(6) xtE_=5$~
% 2 0 (2*r^2 - 1) sqrt(3) Xg
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% 2 2 r^2 * sin(2*theta) sqrt(6) u7\J\r4,+
% 3 -3 r^3 * cos(3*theta) sqrt(8) +!z{5:
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) fA<[f
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) *4xat:@{{
% 3 3 r^3 * sin(3*theta) sqrt(8) TRQF^P3o
% 4 -4 r^4 * cos(4*theta) sqrt(10) ^G.Xc\^w:
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Q6AC(n@:FV
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) =aj/,Q]
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 8lb%eb]U
% 4 4 r^4 * sin(4*theta) sqrt(10) `m>*d!h=
% -------------------------------------------------- 7_Z#m (
% Q`D~5ci
% Example 1: %K`% *D
% LbG_ z =A
% % Display the Zernike function Z(n=5,m=1) Q/I!}C4
% x = -1:0.01:1; Jd(,/q
% [X,Y] = meshgrid(x,x); e~@[18
% [theta,r] = cart2pol(X,Y); fX.>9H[w@~
% idx = r<=1; sqJSSNt
% z = nan(size(X)); mc_ch$r!
% z(idx) = zernfun(5,1,r(idx),theta(idx)); lR[qqFR
% figure YZ7|K<
% pcolor(x,x,z), shading interp X4t s)>"d
% axis square, colorbar W#BM(I
% title('Zernike function Z_5^1(r,\theta)') @,u/w4
% /yF QeE
% Example 2: H nUYqhZS
% ?)[EO(D
% % Display the first 10 Zernike functions C;`XlQG `
% x = -1:0.01:1; w{uuSe
% [X,Y] = meshgrid(x,x); Jn3 An
% [theta,r] = cart2pol(X,Y); %Gj8F4{
% idx = r<=1; c`WHNky%j
% z = nan(size(X)); &S]@Ot<z
% n = [0 1 1 2 2 2 3 3 3 3]; no]z1D
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; _ozg_E
% Nplot = [4 10 12 16 18 20 22 24 26 28]; |-
rI@2`
% y = zernfun(n,m,r(idx),theta(idx)); D3^7y.u<)
% figure('Units','normalized') XC "'Q+
% for k = 1:10 i|}[A
% z(idx) = y(:,k); vR=6pl$|~~
% subplot(4,7,Nplot(k)) l)w Hl%p
% pcolor(x,x,z), shading interp 2aB^WY'tC
% set(gca,'XTick',[],'YTick',[]) d/|D<Sb[s
% axis square ;3@YZM'wt
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) OhmQ,
% end vRxM4O~"
% f<*Js)k
% See also ZERNPOL, ZERNFUN2. P=+nB*hG
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% Paul Fricker 11/13/2006 ]f: v,a
J}@z_^|"mJ
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% Check and prepare the inputs: M$d%p6Cv
% ----------------------------- P<2+L|X?}
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) <ggtjw S
error('zernfun:NMvectors','N and M must be vectors.') 6uKMCQ=h
end -0eq_+oQ
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if length(n)~=length(m) pge++Di
error('zernfun:NMlength','N and M must be the same length.') 7,MS '2nz
end lz0TK)kuC
A'K%WW*'U
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n = n(:); CWG6;NT6m
m = m(:); ,\d6VBP&
if any(mod(n-m,2)) >'5_Y]h4m|
error('zernfun:NMmultiplesof2', ... _#s=h_
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'All N and M must differ by multiples of 2 (including 0).') L0]_hxE?
end eo!zW
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if any(m>n) ror|R@;y
error('zernfun:MlessthanN', ... CGP3qHrXt
'Each M must be less than or equal to its corresponding N.') u!U"N*Y"
end 0T5=W U
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if any( r>1 | r<0 ) dg7=X{=9jv
error('zernfun:Rlessthan1','All R must be between 0 and 1.') $1zvgep
end <U9/InN0[
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) o}'bv
error('zernfun:RTHvector','R and THETA must be vectors.') omf Rs
end H{c?lT
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r = r(:); o{kbc5_
theta = theta(:); l\!-2 T6Y
length_r = length(r); M4LktR-[
if length_r~=length(theta) +P`(Rf"luu
error('zernfun:RTHlength', ... 0l#)fJo
'The number of R- and THETA-values must be equal.') =AEz9d ciS
end rf9_eP
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&&<9p;E
% Check normalization: QIn/,Yd
% -------------------- 5;TuVU.8Q
if nargin==5 && ischar(nflag) geefnb
isnorm = strcmpi(nflag,'norm'); p(m1O70C
if ~isnorm _0 snAt^iC
error('zernfun:normalization','Unrecognized normalization flag.') hc$@J}`
end >7U>Yh
else ]Lqt(c
isnorm = false; rn:!dV[
end jN+N(pIi.o
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% W;Jx<-#1
% Compute the Zernike Polynomials L1)@z8]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8Chu"PM%-J
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MYnH2w]
% Determine the required powers of r: 6vf\R*D|A
% ----------------------------------- g#K'6VK{
m_abs = abs(m); i(wgB\9i4
rpowers = []; 2#/p|$;Ec'
for j = 1:length(n) VzRx%j/i
rpowers = [rpowers m_abs(j):2:n(j)]; dj[apuiF
end VLg
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rpowers = unique(rpowers); L8vOB I7N
w3D]~&]
3rf#Q}"
% Pre-compute the values of r raised to the required powers, 9-bG<`v\E
% and compile them in a matrix: #G,XDW2"w
% ----------------------------- Hwe)Tsh e
if rpowers(1)==0 g>7Y~_}
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); re,.@${H
rpowern = cat(2,rpowern{:}); *R`MMm
rpowern = [ones(length_r,1) rpowern]; YirC*
else ;
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rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); X`\:_|
rpowern = cat(2,rpowern{:}); 4W\,y_Q o
end 8!h'j
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% Compute the values of the polynomials: dNVv4{S
% -------------------------------------- 0%)5.=6
y = zeros(length_r,length(n)); !Zw f
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for j = 1:length(n) 0v"&G<J
s = 0:(n(j)-m_abs(j))/2; D)&o8D`
pows = n(j):-2:m_abs(j); H^CilwD158
for k = length(s):-1:1 [7"}=9
p = (1-2*mod(s(k),2))* ... FyEDt@J
prod(2:(n(j)-s(k)))/ ... <qiICb)~
prod(2:s(k))/ ... ]u&dJL
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... `h;}3r#R{
prod(2:((n(j)+m_abs(j))/2-s(k))); g^o_\hp
idx = (pows(k)==rpowers); a|N0(C
y(:,j) = y(:,j) + p*rpowern(:,idx); A:Rw@B$
end qZG-Lh
2%]hYr;
if isnorm ixOw=!@
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); wt7.oKbW
end +X!+'>
end }g,X5v?W
% END: Compute the Zernike Polynomials &?$\Y,{
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~!
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@&/\r
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% Compute the Zernike functions: ,e ELRzjl
% ------------------------------ cy:;)E>/
idx_pos = m>0; owMuT^x?
idx_neg = m<0; Rx.
rj~
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z = y; ^~I @
spR4
if any(idx_pos) 1XnBK$`
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); M5+W$W
end $o+&Y5:
if any(idx_neg) G(i\'#5+
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); [u\CD sX
end RUrymkHFB
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Fi{mr*}
% EOF zernfun w:tGPort