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, kV&9`c+
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, mdbp8,O
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? p_2pU)%
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? Bv9kSu9'~
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function z = zernfun(n,m,r,theta,nflag) 'Ot,H_pE
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. }#`:Qb \U
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N }< 5F
% and angular frequency M, evaluated at positions (R,THETA) on the K#mOSY;}
% unit circle. N is a vector of positive integers (including 0), and 8g~EL{'
% M is a vector with the same number of elements as N. Each element E JK0
% k of M must be a positive integer, with possible values M(k) = -N(k) Pbu{'y3J
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, V416g |lBO
% and THETA is a vector of angles. R and THETA must have the same FjFMR
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% length. The output Z is a matrix with one column for every (N,M) ) R2XU
% pair, and one row for every (R,THETA) pair. 3Q By\1h.
% ;_?MX/w|&
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike #{J,kcxS
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), qu|i;WZE
% with delta(m,0) the Kronecker delta, is chosen so that the integral DcD{*t?x
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 1zxq^BI
% and theta=0 to theta=2*pi) is unity. For the non-normalized oG oK,
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. GqKsK
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% ExBUpDQc
% The Zernike functions are an orthogonal basis on the unit circle.
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% They are used in disciplines such as astronomy, optics, and ]wVk+%e
% optometry to describe functions on a circular domain. ZWUP^V
% 9N8I
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% The following table lists the first 15 Zernike functions. #*%q'gyHT
% 4Xj4|Rw%
% n m Zernike function Normalization 0(TTw(;
% --------------------------------------------------
nY%5cJ`"
% 0 0 1 1 UUe#{6Jx_
% 1 1 r * cos(theta) 2 XGrue6ya
% 1 -1 r * sin(theta) 2 YDJ4c;37
% 2 -2 r^2 * cos(2*theta) sqrt(6) &a0r%L()X
% 2 0 (2*r^2 - 1) sqrt(3) 'tgKe!-@
% 2 2 r^2 * sin(2*theta) sqrt(6) 6IcNZ!j98
% 3 -3 r^3 * cos(3*theta) sqrt(8) O[^%{'
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) A^ \.Z4=d"
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) :mppv8bh
% 3 3 r^3 * sin(3*theta) sqrt(8) Jju#iwb
% 4 -4 r^4 * cos(4*theta) sqrt(10) (N-RIk73/O
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) pKUP2m`MW
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) n/d`qS
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) g=L]S-e
% 4 4 r^4 * sin(4*theta) sqrt(10) SLL3v,P(7
% --------------------------------------------------
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% ||7x;2e
% Example 1: ;bzX%f?|G
% @$^bMIj@W
% % Display the Zernike function Z(n=5,m=1) y&~w2{a
% x = -1:0.01:1; \>. LW9
% [X,Y] = meshgrid(x,x); 6fo3:P*O
% [theta,r] = cart2pol(X,Y); `4?~nbz
% idx = r<=1; =ac_,]z
% z = nan(size(X)); 2&mGT&HAVA
% z(idx) = zernfun(5,1,r(idx),theta(idx)); /1=4"|q>h'
% figure Q#I"_G&{
% pcolor(x,x,z), shading interp IY'=DePd
% axis square, colorbar 3rW|kkn
% title('Zernike function Z_5^1(r,\theta)') \W5O&G-C
% 8`>h}Q$
% Example 2: +d}E&=p_
% 96cJ8I8
% % Display the first 10 Zernike functions PX:'/{V
% x = -1:0.01:1; \uqjs+
% [X,Y] = meshgrid(x,x); S_MyoXV
% [theta,r] = cart2pol(X,Y); g,tjm(
% idx = r<=1; w27KI]%(
% z = nan(size(X)); 6k{2 +P
% n = [0 1 1 2 2 2 3 3 3 3]; mYN7kYR}<`
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; r`y ezbG
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 1d"Z>k:mn
% y = zernfun(n,m,r(idx),theta(idx)); Ei}/iBG@
% figure('Units','normalized') J?@DGp+t
% for k = 1:10 ,j;m!V
% z(idx) = y(:,k); c .3ZXqpI;
% subplot(4,7,Nplot(k)) ZX!r1*c
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% pcolor(x,x,z), shading interp kE>0M9EdH
% set(gca,'XTick',[],'YTick',[]) fqX"Lus `=
% axis square 3`d}~v{
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 'FlJpA}
% end s4Sd>D7
% [Aj Q#;#Q
% See also ZERNPOL, ZERNFUN2. WG*t::NN
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% Paul Fricker 11/13/2006 [${
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% Check and prepare the inputs: 'E/*d2CDM(
% ----------------------------- 6:GTD$Uz.
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) UDhG :
error('zernfun:NMvectors','N and M must be vectors.') B]m@:|Q
end :q8b;*:
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if length(n)~=length(m) ~<<nz9}o_
error('zernfun:NMlength','N and M must be the same length.') q)H1pwxD
end =U8a ?0
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n = n(:); 3iw{SEY
m = m(:); }kw/W#)J
if any(mod(n-m,2)) Um1[sMc{au
error('zernfun:NMmultiplesof2', ... IG(?xf\C
'All N and M must differ by multiples of 2 (including 0).') mj|)nOd
end A7(hw~+@
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if any(m>n) m(], r})
error('zernfun:MlessthanN', ... `_b`kzJ
'Each M must be less than or equal to its corresponding N.') uwRr LF
end <0yE
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if any( r>1 | r<0 ) xs\!$*R
error('zernfun:Rlessthan1','All R must be between 0 and 1.') OB[o2G <0
end USFDy
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 2X:4CC%5
error('zernfun:RTHvector','R and THETA must be vectors.') R!l:O=[<
end *Z m^
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r = r(:); d
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theta = theta(:); hp{OL< 2M
length_r = length(r); gM [w1^lj
if length_r~=length(theta) F4<O2!V
error('zernfun:RTHlength', ... A
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'The number of R- and THETA-values must be equal.') Ed9Z9
end h3T9"w[
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% Check normalization: 0afei4i~N
% -------------------- ]xguBh ]
if nargin==5 && ischar(nflag) rP!#RzL
isnorm = strcmpi(nflag,'norm'); s7oT G!
if ~isnorm bT
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error('zernfun:normalization','Unrecognized normalization flag.') upeU52@\
end 6U^\{<h_c
else zG e'*Qei
isnorm = false; >vuY+o;B
end ljKrj
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% e^lWR] v
% Compute the Zernike Polynomials ~+Z{Q25R
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'ejvH;V3i
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% Determine the required powers of r: RT>{*E<I
% ----------------------------------- V138d?Mm
m_abs = abs(m); Mwgu93?
rpowers = []; G;f/Tch
for j = 1:length(n) rp5(pV7*
rpowers = [rpowers m_abs(j):2:n(j)]; F @Te@n
end "*,XL
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rpowers = unique(rpowers); %F kMv
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% Pre-compute the values of r raised to the required powers, M>M`baM1
% and compile them in a matrix: zD3mX<sw
% -----------------------------
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if rpowers(1)==0 <(vCiH9~P
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); w,R[C\#J
rpowern = cat(2,rpowern{:}); \;rYo.+
rpowern = [ones(length_r,1) rpowern]; !~Q2|r
else H5D*|42
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ^<X@s1^#
rpowern = cat(2,rpowern{:}); .rPn5D Y
end nI0[;'Hn,
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pHoEa7:
% Compute the values of the polynomials: w,Ee>cV]a
% -------------------------------------- XT;u<aJs
y = zeros(length_r,length(n)); r[?1
for j = 1:length(n) b=3H
s = 0:(n(j)-m_abs(j))/2; C{2xHd/*
pows = n(j):-2:m_abs(j); M4xi1M#%
for k = length(s):-1:1 =!m}xdTP
p = (1-2*mod(s(k),2))* ... )Fb>8<%
prod(2:(n(j)-s(k)))/ ... s|y:UgD
prod(2:s(k))/ ... 0zY(:;X
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... xnE|Umz
prod(2:((n(j)+m_abs(j))/2-s(k))); TNJG#8 n%Y
idx = (pows(k)==rpowers); g R
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y(:,j) = y(:,j) + p*rpowern(:,idx); C;(t/zh
end @(C1_
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if isnorm YIR
R=qpn
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); J~(Wf%jM~
end L],f3<
end Q]o C47(
% END: Compute the Zernike Polynomials XR!us/U`a
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZIdA\_c
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% Compute the Zernike functions: ;O<9|?
% ------------------------------ qF iLh9=D
idx_pos = m>0; xooY'El*#
idx_neg = m<0; OxGE%R,
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z = y; q64k7<C,
if any(idx_pos) ?uMQP NYs
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); E\; ikX&1
end moVbw`T
if any(idx_neg) w{k)XY40sW
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); &F}"Z(B<wK
end .vG,fuf8
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];%0qb
% EOF zernfun q$G,KRy/