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, F~l3?3ZV
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, ;_Rx|~!!
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? .PAkW2\#
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? =v;-{oN!
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function z = zernfun(n,m,r,theta,nflag) aF;TsB
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. {YGz=5 ^
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 'g=yJ
% and angular frequency M, evaluated at positions (R,THETA) on the xd .I5
% unit circle. N is a vector of positive integers (including 0), and +qz)KtJS
% M is a vector with the same number of elements as N. Each element M"9
zK[cz
% k of M must be a positive integer, with possible values M(k) = -N(k) UxS;m4
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, "BVz5?
% and THETA is a vector of angles. R and THETA must have the same yZ!Eu#81
% length. The output Z is a matrix with one column for every (N,M) +p cj8K%
% pair, and one row for every (R,THETA) pair. AV2q*
% W#lvH=y
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike y1
}d(%
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), x1}q!)e
% with delta(m,0) the Kronecker delta, is chosen so that the integral .e"jnP~
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Zgg 7pL)#c
% and theta=0 to theta=2*pi) is unity. For the non-normalized "pWdz}!
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. V.-?aXQ *
% no/]Me!j=
% The Zernike functions are an orthogonal basis on the unit circle. <#s-hQ
% They are used in disciplines such as astronomy, optics, and <HzAh<_@F
% optometry to describe functions on a circular domain. "FXS;Jf
% 0}^-, Q,
% The following table lists the first 15 Zernike functions. 9nG] .@H
% U1"t|KW8
% n m Zernike function Normalization ROjjN W`W
% -------------------------------------------------- &
9]KkY=
% 0 0 1 1 *g,?13Q_
% 1 1 r * cos(theta) 2 kK1qFe?]
% 1 -1 r * sin(theta) 2 LNN:GD)>
% 2 -2 r^2 * cos(2*theta) sqrt(6) cdfll+
% 2 0 (2*r^2 - 1) sqrt(3) LQh\j|e9
% 2 2 r^2 * sin(2*theta) sqrt(6) sTA/2d
% 3 -3 r^3 * cos(3*theta) sqrt(8) r2](~&i2
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) h#n8mtt&i
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) L$Leo6<3a
% 3 3 r^3 * sin(3*theta) sqrt(8) 6m.Ku13;
% 4 -4 r^4 * cos(4*theta) sqrt(10) j0%0yb{-^
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) RYV6hp)|
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) eFnsf}(Iy
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) L|2COX
% 4 4 r^4 * sin(4*theta) sqrt(10) $HXB !$d
% -------------------------------------------------- 2 Lamvf
% kR6 t
.
% Example 1: (wlsn6h
% XF7W'^
% % Display the Zernike function Z(n=5,m=1) !Q(xOc9>Ug
% x = -1:0.01:1; #pe{:f?
% [X,Y] = meshgrid(x,x); L~oFW'
% [theta,r] = cart2pol(X,Y); lQsQRp
% idx = r<=1; > 4ct[fW+
% z = nan(size(X)); avpw+M6+
% z(idx) = zernfun(5,1,r(idx),theta(idx)); !U#++Zig%
% figure \i,cL)HM
% pcolor(x,x,z), shading interp NI1HUUZz
% axis square, colorbar osd^SnL1/5
% title('Zernike function Z_5^1(r,\theta)') IP'igX
% +_gT|vlU
% Example 2: "pZ3
% h3kHI?jMWG
% % Display the first 10 Zernike functions ILi5WuOYX
% x = -1:0.01:1; NVjJ/
% [X,Y] = meshgrid(x,x); 2&Byq
% [theta,r] = cart2pol(X,Y); 0v@/I<
% idx = r<=1; N-rmk
% z = nan(size(X)); K7hf m%`N
% n = [0 1 1 2 2 2 3 3 3 3]; YF -w=Y6
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; j&/.[?K
% Nplot = [4 10 12 16 18 20 22 24 26 28]; aVP|:OAj
% y = zernfun(n,m,r(idx),theta(idx)); eCp| QSXE
% figure('Units','normalized') fl"y@;;#h
% for k = 1:10 >-w=7,?'?z
% z(idx) = y(:,k); UPKi/)C;
% subplot(4,7,Nplot(k)) lkfFAwnc
% pcolor(x,x,z), shading interp |nEVOy>'
% set(gca,'XTick',[],'YTick',[]) ^2r}_AX
% axis square s3-ktZ@
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) l;BX\S
% end ,8IAhQa
% 8sIrG
% See also ZERNPOL, ZERNFUN2. KupMndK
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% Paul Fricker 11/13/2006 16QbB;
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% Check and prepare the inputs: &ZE\@Vc
% ----------------------------- h_~|O[5|)
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) c,q"}nE8w
error('zernfun:NMvectors','N and M must be vectors.') %%~}Lw
end _?s %MNaX
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if length(n)~=length(m) |.C
error('zernfun:NMlength','N and M must be the same length.') )@qup _M@
end ^QAiySR`0
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n = n(:); C6d]tLE
m = m(:); ]&:b<]K3
if any(mod(n-m,2)) #jZ@l3
error('zernfun:NMmultiplesof2', ... mhk/>+hF
'All N and M must differ by multiples of 2 (including 0).') Q)S>VDLA
end C,r`I/;
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if any(m>n) Q0cY/'>4
error('zernfun:MlessthanN', ... xb>n&ym?
'Each M must be less than or equal to its corresponding N.') 23-t$y]
end C4{\@v}t
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if any( r>1 | r<0 ) zBt`L,^
error('zernfun:Rlessthan1','All R must be between 0 and 1.') poD\C;o"
end j`R<90~/
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Taf
n:Nw}
error('zernfun:RTHvector','R and THETA must be vectors.') U,<]J*b(@4
end 5r4gmy>
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r = r(:); 1n'$Ji7
theta = theta(:); 4TUtY:
length_r = length(r); A)hhnb0o
if length_r~=length(theta) s=N#CE
error('zernfun:RTHlength', ... uxOJ3
'The number of R- and THETA-values must be equal.') I1)-,/nEjg
end PW%1xHLfk
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% Check normalization: W|dpFh`
% -------------------- 94|yvh.B
if nargin==5 && ischar(nflag) ]U,CKJF%/
isnorm = strcmpi(nflag,'norm'); gg-};0P-
if ~isnorm 9?;@*x
error('zernfun:normalization','Unrecognized normalization flag.') B6bOEPQ
end r<*O
else s=d+GMa
isnorm = false; x(PKFn
end pe()f/Jx(
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $PSY:Zz
% Compute the Zernike Polynomials 4:vTxNs&S
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% u#`+[AC`
X JY5@I.
r6`\d k
% Determine the required powers of r: NZLXN
% ----------------------------------- 6b?`:$Cw3)
m_abs = abs(m); XOrcygb2
rpowers = []; XRa(sXA3
for j = 1:length(n) D_d|=i
rpowers = [rpowers m_abs(j):2:n(j)]; Ic'Q5kfM
end g nt45]@{
rpowers = unique(rpowers); } ^"0T-ua
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#?9Q{0e
% Pre-compute the values of r raised to the required powers, Kax#OYLpg
% and compile them in a matrix: &hayR_F9
% ----------------------------- 0G5'Y;8
if rpowers(1)==0 y%4 Gp
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); |olNA*4
rpowern = cat(2,rpowern{:}); '61i2\[lZQ
rpowern = [ones(length_r,1) rpowern]; S'o ]=&
else Xo Y7/&&
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); R<_?W#$j
rpowern = cat(2,rpowern{:}); ga-{!$b*
end :zlpfm2
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% Compute the values of the polynomials: $A`xhh[
% -------------------------------------- i\Yl
y = zeros(length_r,length(n)); ivfXat-
for j = 1:length(n) /xySwSmh3
s = 0:(n(j)-m_abs(j))/2; "u;YI=+
pows = n(j):-2:m_abs(j); iK!dr1:wSw
for k = length(s):-1:1 &]< 3~6n
p = (1-2*mod(s(k),2))* ... xP{-19s1]
prod(2:(n(j)-s(k)))/ ... xW>ySEf
prod(2:s(k))/ ... Z:@6Lv?CN
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... MiJ6 n[iv
prod(2:((n(j)+m_abs(j))/2-s(k))); WL l_'2h
idx = (pows(k)==rpowers); &~#iIk~%
y(:,j) = y(:,j) + p*rpowern(:,idx); G>%AZr{M
end t?{B_Bf
%c X"#+e
if isnorm d+6]u_J
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); mV?&%>*(f
end |SQ|qbe=
end jWvtv ng
% END: Compute the Zernike Polynomials o.Oq__ >$H
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0|XKd24BN
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% Compute the Zernike functions: 5 y
% ------------------------------ F;u_7OM
idx_pos = m>0; ;cKH1
idx_neg = m<0; cy|%sf`
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z = y; :.uk$jx
if any(idx_pos) yNa;\UF
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); `T"rG}c
end J}TfRrf
if any(idx_neg) YEv
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z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); S~)w\(r
end 5mgHlsDzu
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% EOF zernfun iW;i!,