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, a4a/]q4T
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, o 8fB
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? Cd Bsd
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? `vbd7i
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function z = zernfun(n,m,r,theta,nflag) YJ6y]r
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Z}X oWT2f
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N <[*%d~92z
% and angular frequency M, evaluated at positions (R,THETA) on the f&=WgITa
% unit circle. N is a vector of positive integers (including 0), and Kivr)cIG
% M is a vector with the same number of elements as N. Each element dWR-}>
% k of M must be a positive integer, with possible values M(k) = -N(k) `Zdeq.R]
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, adCTo
% and THETA is a vector of angles. R and THETA must have the same *8I+D>x
% length. The output Z is a matrix with one column for every (N,M) B|fh 4FNy
% pair, and one row for every (R,THETA) pair. $m
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% <R;t>~8x
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike M53{e;.kN
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), N~=,RPjq
% with delta(m,0) the Kronecker delta, is chosen so that the integral N<d0C
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, N}|<P[LW
% and theta=0 to theta=2*pi) is unity. For the non-normalized |qOoL*z
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. }q$6^y
% 7O.?I#
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% The Zernike functions are an orthogonal basis on the unit circle. bU3P;a(
% They are used in disciplines such as astronomy, optics, and L- '{
% optometry to describe functions on a circular domain. c6 f=r
% \Fh#CI
% The following table lists the first 15 Zernike functions. uGoySt&;(
% R>C^duos.
% n m Zernike function Normalization o[A y2"e?
% -------------------------------------------------- z~m{'O`
% 0 0 1 1 KfPYH\0
% 1 1 r * cos(theta) 2 $.5f-vQp
% 1 -1 r * sin(theta) 2 8*bEsc|
% 2 -2 r^2 * cos(2*theta) sqrt(6) c>$PLO^
% 2 0 (2*r^2 - 1) sqrt(3) mJ #|~I*Z-
% 2 2 r^2 * sin(2*theta) sqrt(6)
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% 3 -3 r^3 * cos(3*theta) sqrt(8) 1H-d<G0)
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) H^d2|E[D
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) hvFXYq_[O
% 3 3 r^3 * sin(3*theta) sqrt(8) @H83Ad
% 4 -4 r^4 * cos(4*theta) sqrt(10) 7Rq|N$y.3
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 39yp1
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Au\j6mB
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) IG(1h+5R(
% 4 4 r^4 * sin(4*theta) sqrt(10) }Sx+: N*
% -------------------------------------------------- /0_^Z2
% id?B<OM
% Example 1: G~+BO'U9'G
% v'e5j``=
% % Display the Zernike function Z(n=5,m=1) ob_*fP
% x = -1:0.01:1; /19ZyQw9
% [X,Y] = meshgrid(x,x); 2zPO3xL,
% [theta,r] = cart2pol(X,Y); [6u8EP0xM
% idx = r<=1; >^Z==1
% z = nan(size(X)); j3Yz=bsQ{c
% z(idx) = zernfun(5,1,r(idx),theta(idx)); w=Yc(Y:h
% figure uD0<|At/
% pcolor(x,x,z), shading interp dI%#cf1
% axis square, colorbar w9aLTLv-
% title('Zernike function Z_5^1(r,\theta)') |y%M";MI
% #,5v#|u|7
% Example 2: RG8Ek"D@
% FhFP M)[
% % Display the first 10 Zernike functions DGJt$o=&@
% x = -1:0.01:1; hMNC]
% [X,Y] = meshgrid(x,x); %+bw2;a6
% [theta,r] = cart2pol(X,Y); 6>d0i
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% idx = r<=1; 5*hA6Ex7
% z = nan(size(X)); =U`9_]~1c@
% n = [0 1 1 2 2 2 3 3 3 3]; &_o.:SL|
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ; !9-I%e
% Nplot = [4 10 12 16 18 20 22 24 26 28]; z#u<]] 5
% y = zernfun(n,m,r(idx),theta(idx)); 9`FPV`/
% figure('Units','normalized') j&|>Aa${
% for k = 1:10 xV\mS+#
% z(idx) = y(:,k); r^Mu`*x*
% subplot(4,7,Nplot(k)) ^ fqco9^;
% pcolor(x,x,z), shading interp 2'-!9!C
% set(gca,'XTick',[],'YTick',[]) 5<77o|
% axis square JBMJR
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) }{S pV
% end nsjrzO79L8
% Y7GHIzX
% See also ZERNPOL, ZERNFUN2. n1Fp$9%
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% Paul Fricker 11/13/2006 I=c}6
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% Check and prepare the inputs: * R_mvJlT
% ----------------------------- f}ES8Hh[
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) l|"SM6
error('zernfun:NMvectors','N and M must be vectors.') 48g`i
end 4iC=+YUn
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if length(n)~=length(m) *Hed^[sO
error('zernfun:NMlength','N and M must be the same length.') \Pt_5.bTs[
end VI(2/**
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n = n(:); }~+_|
m = m(:); `Qxdb1>mjY
if any(mod(n-m,2)) Nu4PY@m]C
error('zernfun:NMmultiplesof2', ... )9~-^V0A^>
'All N and M must differ by multiples of 2 (including 0).') t +h}hL
end T(q/$p&q
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if any(m>n) 3x3 =ke!
error('zernfun:MlessthanN', ... JL:\\JT.
'Each M must be less than or equal to its corresponding N.') Yue#
end
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if any( r>1 | r<0 ) CU_8
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error('zernfun:Rlessthan1','All R must be between 0 and 1.') 4)z*Vux
end /;V:<mekf
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ^#p+#_*V
error('zernfun:RTHvector','R and THETA must be vectors.') bc%N !d
end p)YI8nW
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r = r(:); mY0FewwTy
theta = theta(:); NKRI|'Y,
length_r = length(r); 9 6j*F,{
if length_r~=length(theta) yl UkVr
error('zernfun:RTHlength', ... &A)u!l Ue
'The number of R- and THETA-values must be equal.') bTJ l
end =b/:rSd$NA
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% Check normalization: QvvH/u
% -------------------- .e1Yd8
if nargin==5 && ischar(nflag) `HV~.C
isnorm = strcmpi(nflag,'norm'); 9Pjw<xt
if ~isnorm /4%ycr6
error('zernfun:normalization','Unrecognized normalization flag.') 6<
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end w+G+&ak<
else rlP?Uh
isnorm = false; Lf0Wc'9{
end m=Fk
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% U%H6jVE
% Compute the Zernike Polynomials &N|`Q(QXS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !r %u@[(
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% Determine the required powers of r: .xV^%e?H
% ----------------------------------- Jt|W%`X>D
m_abs = abs(m); NjP7?nXSx
rpowers = []; )L/o|%r!
for j = 1:length(n) ql2O%B.6?
rpowers = [rpowers m_abs(j):2:n(j)]; 3JXKpk?
end KreF\M%Ke
rpowers = unique(rpowers); ["ML&2|o
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% Pre-compute the values of r raised to the required powers, -EIfuh
% and compile them in a matrix: 8}>s{u;W
% ----------------------------- &)GlLpaT
if rpowers(1)==0 EB2 5N~7
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Fa-F`U@h(m
rpowern = cat(2,rpowern{:}); d[$YTw
rpowern = [ones(length_r,1) rpowern]; Xot2L{EIUE
else X\HP&;Wd
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); gSt'<v
rpowern = cat(2,rpowern{:}); z\r29IRh
end k.Q4oyei
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% Compute the values of the polynomials: xn,I<dL39
% -------------------------------------- xY$@^(Q\
y = zeros(length_r,length(n)); 3 Q~zli:
for j = 1:length(n) \Ws$@J-M
s = 0:(n(j)-m_abs(j))/2; :,1kSM%r
pows = n(j):-2:m_abs(j); _a-At
for k = length(s):-1:1 &7L g)PG
p = (1-2*mod(s(k),2))* ... 4)+L(KyB2
prod(2:(n(j)-s(k)))/ ... -cs$E2
-
prod(2:s(k))/ ... "HrZv+{
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... YW'l),Z
prod(2:((n(j)+m_abs(j))/2-s(k))); OoOr@5g
idx = (pows(k)==rpowers); Hwiftx
y(:,j) = y(:,j) + p*rpowern(:,idx); h7cE"m
end -cL wjI
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if isnorm 8Q\ T,C
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); vCsJnKqK
end }-2U,Xg[
end pu,|_N[xq8
% END: Compute the Zernike Polynomials +puF0]TR,i
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RE.t<VasP
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% Compute the Zernike functions: ;QqC c!b
% ------------------------------ p n(y4we
idx_pos = m>0; #bmbK{ [
idx_neg = m<0; #Z1
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z = y; $[{YE[a
if any(idx_pos) V6uh'2
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); @JU
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end ) $=!e%{
if any(idx_neg) E4qQ
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); N%fDgK
end Uo=_=.GQ
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% EOF zernfun CQ/ps,~M