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, RIXUzKLO
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, N :E7rtT,M
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? ]|cL+|':y
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? _h#SP+>
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function z = zernfun(n,m,r,theta,nflag) IaDc hI
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. rYI9?q
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N '2+Rb7V
% and angular frequency M, evaluated at positions (R,THETA) on the i*`; /x'+
% unit circle. N is a vector of positive integers (including 0), and # [c`]v
% M is a vector with the same number of elements as N. Each element
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% k of M must be a positive integer, with possible values M(k) = -N(k) @}&o(q1M0
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, y:Ycn+X.
% and THETA is a vector of angles. R and THETA must have the same HhfuHZ<
% length. The output Z is a matrix with one column for every (N,M) Yc+0OBH[
% pair, and one row for every (R,THETA) pair. #8.%YG
% 0(fN
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike _aOisN{
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ]kC/b^~+m
% with delta(m,0) the Kronecker delta, is chosen so that the integral 9N^&~O|1
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, r0=Aru5n
% and theta=0 to theta=2*pi) is unity. For the non-normalized ;5 W|#{I
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. so h3d
% 3| 5Af
% The Zernike functions are an orthogonal basis on the unit circle. g0w<vD`<g
% They are used in disciplines such as astronomy, optics, and D.G+*h@ g
% optometry to describe functions on a circular domain. MrIo.
% e6{}hiM
% The following table lists the first 15 Zernike functions. F5Tah{
% z@hlN3dg
% n m Zernike function Normalization "i$Avm
% -------------------------------------------------- U[9`:aV;
% 0 0 1 1 Hf
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% 1 1 r * cos(theta) 2 Qn>0s
% 1 -1 r * sin(theta) 2 pNFL;k+p}
% 2 -2 r^2 * cos(2*theta) sqrt(6) @A(*&PU>j
% 2 0 (2*r^2 - 1) sqrt(3) :4|W;Lkd!
% 2 2 r^2 * sin(2*theta) sqrt(6) eaQ)r?M
% 3 -3 r^3 * cos(3*theta) sqrt(8) @$ E&H`da
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) e=KA|"vxh
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ajF-T=5
% 3 3 r^3 * sin(3*theta) sqrt(8) 3QSP](W-(
% 4 -4 r^4 * cos(4*theta) sqrt(10) |}paa
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 9W$FX
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 9j458Yd4*
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) l v]TE"
% 4 4 r^4 * sin(4*theta) sqrt(10) ]Bw2> 6W
% -------------------------------------------------- FJl#NOp&
% R.Xh&@f`
% Example 1: N(0G!sTI
% fw@n[u{~
% % Display the Zernike function Z(n=5,m=1) Q:$<`K4)
% x = -1:0.01:1; wowv>!N!X-
% [X,Y] = meshgrid(x,x); G"&9u2 k
% [theta,r] = cart2pol(X,Y); YUdCrb9F
% idx = r<=1; d0YN:lJc
% z = nan(size(X)); f]H[uzsV
% z(idx) = zernfun(5,1,r(idx),theta(idx)); *"#62U6
% figure E/@w6uIK[
% pcolor(x,x,z), shading interp LU5e!bP
% axis square, colorbar 9u";%5 4
% title('Zernike function Z_5^1(r,\theta)') >h>X/a(=~
% zSMNk AM
% Example 2: !P7&{I,e
% f Co- ony
% % Display the first 10 Zernike functions zJNiAc
% x = -1:0.01:1; D4%5T>^LW[
% [X,Y] = meshgrid(x,x); wS"[m>.{v
% [theta,r] = cart2pol(X,Y); 5tI4m#y2
% idx = r<=1; qQC<oR
% z = nan(size(X)); p$dVGvM(
% n = [0 1 1 2 2 2 3 3 3 3]; 9dl\`zlA*
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Vrl)[st!;I
% Nplot = [4 10 12 16 18 20 22 24 26 28]; i8A{DMc,U
% y = zernfun(n,m,r(idx),theta(idx)); Gv(bD6Rz
% figure('Units','normalized') t_1a.Jv
% for k = 1:10 +grIw#j
% z(idx) = y(:,k); ^Nl)ocHv!
% subplot(4,7,Nplot(k)) BG!;9Z{u
% pcolor(x,x,z), shading interp a=bP
% set(gca,'XTick',[],'YTick',[]) ;=piJ%k
% axis square ]O2ku^yM
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) .8[B
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% end qUX
% y&T(^EA;
% See also ZERNPOL, ZERNFUN2. 'j>+eA>
z,/0e@B >
e R"XXF0u
% Paul Fricker 11/13/2006 5`CPaJT$
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% Check and prepare the inputs: [V'3/#Z
% ----------------------------- ??tyz4$;
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) "4N%I
error('zernfun:NMvectors','N and M must be vectors.') FtbqZN[
end 6||zwwk'.
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if length(n)~=length(m) xo#&&/6
error('zernfun:NMlength','N and M must be the same length.') _%#Q
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end 1.WdxMpW9
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n = n(:); Oq~>P!=
m = m(:); 0&$+ CWSM
if any(mod(n-m,2)) ]Cd1&
error('zernfun:NMmultiplesof2', ... f&=y\uP]
'All N and M must differ by multiples of 2 (including 0).') (XYYbP
end }}Ah-QU
!%b.k6%>w
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if any(m>n) OoOKr
error('zernfun:MlessthanN', ... ~J1;Z0}#
'Each M must be less than or equal to its corresponding N.') gNr/rp9A$m
end Sqj'2<~W
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if any( r>1 | r<0 ) %m[
:},
error('zernfun:Rlessthan1','All R must be between 0 and 1.') (pXZ$R:
end cF{5[?wS
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) bv$g$
error('zernfun:RTHvector','R and THETA must be vectors.') Hb5^+.xur
end cQEK>aAd
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r = r(:); xXQ#?::m
theta = theta(:); 'T@K$xL8
length_r = length(r); t{?U NW
if length_r~=length(theta)
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error('zernfun:RTHlength', ... 8XtZF,Du
'The number of R- and THETA-values must be equal.') %Y8#I3jVJ
end ~5$V8yfx h
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% Check normalization: *=i|E7Irg
% -------------------- +jD?h-]
if nargin==5 && ischar(nflag) _U)BOE0o
isnorm = strcmpi(nflag,'norm'); m}w~ d /
if ~isnorm J^[>F{8!n
error('zernfun:normalization','Unrecognized normalization flag.') C!xq p
end hEAt4z0P
else _ +Ww1f
isnorm = false; g[fCvWm#d
end [f["9(:
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% O--
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% Compute the Zernike Polynomials |T7 < !
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% n[4F\I>
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@dE|UZ=(
% Determine the required powers of r: %RA8M-
d
% ----------------------------------- MB|+F
m_abs = abs(m); j|3p.Cy
rpowers = []; fis**f0
for j = 1:length(n) xZAc~~9tD
rpowers = [rpowers m_abs(j):2:n(j)]; K(RG:e~R0i
end n%PHHu
rpowers = unique(rpowers); /CX_@%m}e=
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% Pre-compute the values of r raised to the required powers, y7aBF13Kl
% and compile them in a matrix: Sz4YPl
% ----------------------------- _?Zg$7VJ
if rpowers(1)==0 Cv{>|g#
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Ut4cli&cC
rpowern = cat(2,rpowern{:}); :lz@G4=C
rpowern = [ones(length_r,1) rpowern]; '5zolp%St
else PR?Ls{}p\
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); e m`z=JGG
rpowern = cat(2,rpowern{:}); xaQ]Vjw
end b%<-(o/
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% Compute the values of the polynomials: In)8AK(Hw
% -------------------------------------- /Zw^EM6c
y = zeros(length_r,length(n)); ;w";s$
for j = 1:length(n) [#$: X+lw
s = 0:(n(j)-m_abs(j))/2; F9(*MP|
pows = n(j):-2:m_abs(j); t_1(Ex
for k = length(s):-1:1 ?EF[OyE
p = (1-2*mod(s(k),2))* ... U{(B)dFTH
prod(2:(n(j)-s(k)))/ ... MKIX(r(|
prod(2:s(k))/ ... @]ydWd
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... SQ7Ws u>T@
prod(2:((n(j)+m_abs(j))/2-s(k))); -[A4B)
idx = (pows(k)==rpowers); qP? V{N
y(:,j) = y(:,j) + p*rpowern(:,idx); q_PxmPE@3v
end \fG?j@Qx
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if isnorm !}7FC>Cx
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); @-y.Y}k#$~
end KSsv~!3Yf
end QiBo]`)%
% END: Compute the Zernike Polynomials ^PDz"L<*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }gw
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% Compute the Zernike functions: T:$ a
x
% ------------------------------ l1*qDzb
idx_pos = m>0; ]6)^+(zU
idx_neg = m<0; 4^h_n1A
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z = y; "%$jl0i_c
if any(idx_pos) HD^ Ou5YB
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 1#LXy%^tO
end 5~GHAi
if any(idx_neg) Q/'jwyj_
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ia#Z$I6
end .}'49=c
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% EOF zernfun z|,YO6(L