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, m)tu~neM
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, MbRTOH
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? _Vr- bpAf
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? yJ $6vmQ
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function z = zernfun(n,m,r,theta,nflag) %10ONe}
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. x6UXd~
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% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N xuK"pS
% and angular frequency M, evaluated at positions (R,THETA) on the GTdoUSUq
% unit circle. N is a vector of positive integers (including 0), and HOP*QX8C%
% M is a vector with the same number of elements as N. Each element )^ah, ;(
% k of M must be a positive integer, with possible values M(k) = -N(k) B)JMughq_
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 5kiW@{m
% and THETA is a vector of angles. R and THETA must have the same $tmdE)"&
% length. The output Z is a matrix with one column for every (N,M) vE:*{G;Y
% pair, and one row for every (R,THETA) pair. uHg q"e
% 9J3fiA_
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike >yC=@Uq+
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), d_!Z /M,
% with delta(m,0) the Kronecker delta, is chosen so that the integral W+ S~__K
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, G4cgY|71
% and theta=0 to theta=2*pi) is unity. For the non-normalized i>Q!5
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ;'7(gAE
% `B)@
% The Zernike functions are an orthogonal basis on the unit circle. /$c87\
% They are used in disciplines such as astronomy, optics, and YYe G9yR
% optometry to describe functions on a circular domain. m/=nz.
% NrqJf-ldo
% The following table lists the first 15 Zernike functions. +{:uPY#1
% 53i]Q;k [
% n m Zernike function Normalization }DhqzKl
% -------------------------------------------------- Z 1HH0{q-A
% 0 0 1 1 QLd*f[n
% 1 1 r * cos(theta) 2 AF'<
% 1 -1 r * sin(theta) 2
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% 2 -2 r^2 * cos(2*theta) sqrt(6) Q~,Mzt"}W
% 2 0 (2*r^2 - 1) sqrt(3) up5f]:!
% 2 2 r^2 * sin(2*theta) sqrt(6) p!UR;xHI\
% 3 -3 r^3 * cos(3*theta) sqrt(8) (4YLUN&1O$
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) b$_81i
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) WNyW1?"
% 3 3 r^3 * sin(3*theta) sqrt(8) \,#$,dUXD
% 4 -4 r^4 * cos(4*theta) sqrt(10) c{M
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% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ( /]'e}
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) |}=eY?iXo
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) nR_Zrm
% 4 4 r^4 * sin(4*theta) sqrt(10) z<%P"
% -------------------------------------------------- Geq]wv8
% 9!( 8o
% Example 1: Aw#<: 6-
% 5u!\c(TJ+
% % Display the Zernike function Z(n=5,m=1) p@tg pFt
% x = -1:0.01:1; h( | T.
% [X,Y] = meshgrid(x,x); ?NMk|+
% [theta,r] = cart2pol(X,Y); T fLqxioqZ
% idx = r<=1; 4XpWDfa.}
% z = nan(size(X)); c1f"z1Z
% z(idx) = zernfun(5,1,r(idx),theta(idx)); a-NTA
% figure 2*Qv6
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% pcolor(x,x,z), shading interp zgb$@JC
% axis square, colorbar 94tfR$W;-
% title('Zernike function Z_5^1(r,\theta)') As,`($=
% Y1PR?c
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% Example 2: y'2|E+*V
% '`jGr+K,wU
% % Display the first 10 Zernike functions \g}]u(zg%
% x = -1:0.01:1; y7HFmGM
% [X,Y] = meshgrid(x,x); f?5>V
% [theta,r] = cart2pol(X,Y); (?4%Xtul1
% idx = r<=1; 9?l a5
% z = nan(size(X)); t`o"K
% n = [0 1 1 2 2 2 3 3 3 3]; n>'(d*[e&
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 7]VR)VA M
% Nplot = [4 10 12 16 18 20 22 24 26 28]; @A,8>0+
% y = zernfun(n,m,r(idx),theta(idx)); :kgh~mx5LF
% figure('Units','normalized') iH(7.?.r
% for k = 1:10
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% z(idx) = y(:,k); OUBGbld
% subplot(4,7,Nplot(k)) &=@{`2&
% pcolor(x,x,z), shading interp io#}z4"'qY
% set(gca,'XTick',[],'YTick',[]) Ln>!4i+-B)
% axis square D$ds[if$U,
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) C$w%!
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% end {nmG/dn{
% !ku}vTe
% See also ZERNPOL, ZERNFUN2. ('&lAn
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% Paul Fricker 11/13/2006 Nv5^2^Sc=
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% Check and prepare the inputs: w.m8SvS&b
% ----------------------------- 0z=KnQx"4
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) v-8>@s jy8
error('zernfun:NMvectors','N and M must be vectors.') Z
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end ASXGM0t
%2 r~
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if length(n)~=length(m) 3BMz{ny=
error('zernfun:NMlength','N and M must be the same length.') b**vUt\
end p(yv
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n = n(:); LW1 4 'A}
m = m(:); s#$t!F??9
if any(mod(n-m,2)) /H'- }C
error('zernfun:NMmultiplesof2', ... gPMR,TU
'All N and M must differ by multiples of 2 (including 0).') IyO0~Vx>
end vj?{={Y
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if any(m>n) }3z3GU8Q-
error('zernfun:MlessthanN', ... k0Vri$x
'Each M must be less than or equal to its corresponding N.') v`4w=!4
end fN2Sio:
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if any( r>1 | r<0 ) BK!Yl\I<
error('zernfun:Rlessthan1','All R must be between 0 and 1.') bm#5bhX\|
end
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Cmx<>7fN
error('zernfun:RTHvector','R and THETA must be vectors.') uEgR>X>
end $#=d@Nw_
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r = r(:); "~Fg-{jM%
theta = theta(:); \S h/<z
length_r = length(r); 19fa7E<
if length_r~=length(theta) >nkVZ;tL
error('zernfun:RTHlength', ... KS_+R@3Z
'The number of R- and THETA-values must be equal.') 8~U
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end $:s@nKgnD~
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%&r
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% Check normalization: +s#%\:Y M
% -------------------- ~W@dF~r
if nargin==5 && ischar(nflag) b`e_}^,c
isnorm = strcmpi(nflag,'norm'); J`g5Qn@S
if ~isnorm 21!X[)r
error('zernfun:normalization','Unrecognized normalization flag.') u(zgKoF9A
end :'DX
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else 5 3pW:`
isnorm = false; hk
!=ZE3
end APl]EV"l
T#*,ME7|m
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :^(y~q?
% Compute the Zernike Polynomials 1(;{w+nM
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8R)K$J$Hm
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% Determine the required powers of r: YI0l&'7
% ----------------------------------- %Za}q]?
m_abs = abs(m); :s_o'8z7L
rpowers = []; =Ji[ ;wy@
for j = 1:length(n) ztU"CRa8
rpowers = [rpowers m_abs(j):2:n(j)]; ltOS()[X
end 7"|Qmyb
rpowers = unique(rpowers); 6zM:p/
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j+3\I>
% Pre-compute the values of r raised to the required powers, F,vkk{Z>
% and compile them in a matrix: 7fqQ
% ----------------------------- [w}- )&c
if rpowers(1)==0 N:|``n>
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ^.J_ w
rpowern = cat(2,rpowern{:}); Kjbk
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rpowern = [ones(length_r,1) rpowern]; [xGwqa03
else 4lPO*:/
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); w*{{bISw|
rpowern = cat(2,rpowern{:}); _V3z!aI
end Fepsa;\sU
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% Compute the values of the polynomials: #j7&2L
% -------------------------------------- oY ~q^Y
y = zeros(length_r,length(n)); TQb/lY9*
for j = 1:length(n) ";dS~(~
s = 0:(n(j)-m_abs(j))/2; F7'MoH
pows = n(j):-2:m_abs(j); $mK;{9Z
for k = length(s):-1:1 6}Y==GPt
p = (1-2*mod(s(k),2))* ... *& w/*h$!
prod(2:(n(j)-s(k)))/ ... _'!qOt7D
prod(2:s(k))/ ... Lvt3S
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prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... .S:(O+#Gm
prod(2:((n(j)+m_abs(j))/2-s(k))); b
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idx = (pows(k)==rpowers); 0J?443AY
y(:,j) = y(:,j) + p*rpowern(:,idx); `[$>S
end <IIz-6*V
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if isnorm Lp%J:ogV`
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); p+Q 9?9
end F
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end B _ J2Bf
% END: Compute the Zernike Polynomials m>Z3p7!N}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8'E7Uj
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% Compute the Zernike functions: N5W!(h)
% ------------------------------ u~,hTY(%
idx_pos = m>0; !'!\>x$
idx_neg = m<0; "KF]s.
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3\]j4*i!
z = y; 5(2 C
if any(idx_pos) >'#vC]@
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); .|CoueH
end 'uzHI@i
if any(idx_neg) HjzAFXRG
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); (mbm',%- (
end =,6X_m
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% EOF zernfun z(,j)".