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, C6'*/wq
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, <`xRqe:&9
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? +%#MrNM'
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? nn)`eR&
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function z = zernfun(n,m,r,theta,nflag) d0A\#H_&
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. `,-hG
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N sMfFm@\ N
% and angular frequency M, evaluated at positions (R,THETA) on the L.0} UXd
% unit circle. N is a vector of positive integers (including 0), and *%N7QyO`I
% M is a vector with the same number of elements as N. Each element OHP3T(Q5
% k of M must be a positive integer, with possible values M(k) = -N(k) KBr5bcm4u
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, cUZ!;*
% and THETA is a vector of angles. R and THETA must have the same 7c29Ua~[
% length. The output Z is a matrix with one column for every (N,M) f1Yv hvWL
% pair, and one row for every (R,THETA) pair. XW~ BEa
% g2aT`=&Z
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike gl9pgY1ni
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 8XYD
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% with delta(m,0) the Kronecker delta, is chosen so that the integral XY!{ g(
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, *H%0Gsk
% and theta=0 to theta=2*pi) is unity. For the non-normalized $
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% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. =MJ-s;raq
% v#:+n+y\z
% The Zernike functions are an orthogonal basis on the unit circle. Krp
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% They are used in disciplines such as astronomy, optics, and vNC$f(cQ
% optometry to describe functions on a circular domain. yp[<9%Fi
% ez^*M:K
% The following table lists the first 15 Zernike functions. BP[CR1Gs
% @Z9>3'2]A
% n m Zernike function Normalization &?/N}g@K
% -------------------------------------------------- ;z.6'EYMG
% 0 0 1 1 3@PUg(M
% 1 1 r * cos(theta) 2 A*\o
c
% 1 -1 r * sin(theta) 2 `W"a!,s2
% 2 -2 r^2 * cos(2*theta) sqrt(6) Gg=aK~q6
% 2 0 (2*r^2 - 1) sqrt(3) R<n8M"B
% 2 2 r^2 * sin(2*theta) sqrt(6) xta}4:d-Y
% 3 -3 r^3 * cos(3*theta) sqrt(8) ;g:
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% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ,zAK3d&hj
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) .)iO Du
% 3 3 r^3 * sin(3*theta) sqrt(8) RNv{n
mf
% 4 -4 r^4 * cos(4*theta) sqrt(10) U6c)"^\
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) &Pu+(~'Q
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 5nM kd/
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1.Haf
% 4 4 r^4 * sin(4*theta) sqrt(10) 4*0:bhhhf_
% -------------------------------------------------- x r-;,W
% z$9@j2
% Example 1: `Mg8]H~
% e1^fUOS
% % Display the Zernike function Z(n=5,m=1) Z{Vxr*9oO
% x = -1:0.01:1; Dh
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% [X,Y] = meshgrid(x,x); :d;[DYFLxb
% [theta,r] = cart2pol(X,Y); I*^5'N'
% idx = r<=1; ;M}itM
% z = nan(size(X)); n11LxGwk
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Qo'yS"g<9)
% figure ;$vLq&(}
% pcolor(x,x,z), shading interp rlR
!&
% axis square, colorbar mbF(tSy
% title('Zernike function Z_5^1(r,\theta)') By2s ']bw
% D,c!#(v cK
% Example 2: 3Bejp+xX
% cb+l"FI7
% % Display the first 10 Zernike functions *3;UAfHv
% x = -1:0.01:1; 24//21m
% [X,Y] = meshgrid(x,x); y|^EGnaE
% [theta,r] = cart2pol(X,Y); @zo7.'7P
% idx = r<=1; Ffnk1/Zy
% z = nan(size(X)); E_~x==cb
% n = [0 1 1 2 2 2 3 3 3 3]; tE[H8
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; a>U6Ag<
% Nplot = [4 10 12 16 18 20 22 24 26 28]; fb23J|"
% y = zernfun(n,m,r(idx),theta(idx)); GMz8B-vk
% figure('Units','normalized') 'qjX$]H
% for k = 1:10 [q1Unm
% z(idx) = y(:,k); {-HDkG' 8
% subplot(4,7,Nplot(k)) flP>@i:e6
% pcolor(x,x,z), shading interp \]I
% set(gca,'XTick',[],'YTick',[]) 6a*83G,k
% axis square GzdRG^vN
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) C,"=}z1P
% end HMV)U{
% )|pU.K9qZ
% See also ZERNPOL, ZERNFUN2. *hF^fxLbl
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% Paul Fricker 11/13/2006 ]boE{R!I
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% Check and prepare the inputs: &tj0M.-
% ----------------------------- j0x5@1`6G
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 9Kbw
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error('zernfun:NMvectors','N and M must be vectors.') =u=Kw R
end |@RpWp>2
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if length(n)~=length(m) E!!
alc{
error('zernfun:NMlength','N and M must be the same length.') #!})3_Qc(y
end ubbnFE&PD
=K(JqSw+M
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n = n(:); I]R9HGJNlJ
m = m(:); %dW%o{
if any(mod(n-m,2)) q]FBl}nwl%
error('zernfun:NMmultiplesof2', ... >zngJ$
'All N and M must differ by multiples of 2 (including 0).') dJD(\a>r.u
end <a|@t@R
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if any(m>n) F!yV8XQ
error('zernfun:MlessthanN', ... %
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'Each M must be less than or equal to its corresponding N.') ->gZ)?Fqy
end ss
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if any( r>1 | r<0 ) *Y6xvib9*
error('zernfun:Rlessthan1','All R must be between 0 and 1.') {mTytT
end |+}G|hx@9
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Q(Gl{#b
error('zernfun:RTHvector','R and THETA must be vectors.') *%gF2@=r8F
end ,[!LCXp
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r = r(:); Um.qRZ?
theta = theta(:); $}o
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length_r = length(r); 1AD]v<M
if length_r~=length(theta) q(IQa@$SR
error('zernfun:RTHlength', ... ]!
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'The number of R- and THETA-values must be equal.') =^;P#kX
end
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% Check normalization: t18$x"\4k
% -------------------- jp2Q9Z
if nargin==5 && ischar(nflag) &[[K"aM1
isnorm = strcmpi(nflag,'norm'); (5Nv8H8|
if ~isnorm sW@krBxMv
error('zernfun:normalization','Unrecognized normalization flag.') *m+BuGt|
end {w6/[-^
else e^1uVN
isnorm = false; Nf41ZT~
end dt\jGD
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% KvFMs\o6p
% Compute the Zernike Polynomials ,E )|y4
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yR5XJ;Tct
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% Determine the required powers of r: _l`e#XbG
% ----------------------------------- B=f,QU
m_abs = abs(m); W!Gdf^Yy<
rpowers = []; WiL2
for j = 1:length(n) Io`P,l:
rpowers = [rpowers m_abs(j):2:n(j)]; +0wT!DZW\=
end Elj_,z
rpowers = unique(rpowers); J5Z%ImiT^O
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% Pre-compute the values of r raised to the required powers, p/Sbt/R
% and compile them in a matrix: y;cUl, :v
% ----------------------------- IA zZ1#/3
if rpowers(1)==0 .0 )Y
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); J@pb[O L,
rpowern = cat(2,rpowern{:}); /'2O.d0}.
rpowern = [ones(length_r,1) rpowern]; RrZM&lXY
else +yob)%
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); @:0ddb71
rpowern = cat(2,rpowern{:}); 3f Xv4R;!:
end 'nQVj
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% Compute the values of the polynomials: s&kQlQ=
% -------------------------------------- )5j;KI%t
y = zeros(length_r,length(n)); `O?TUQGR
for j = 1:length(n) qSqI7ptA\
s = 0:(n(j)-m_abs(j))/2; LTV{{Z+
pows = n(j):-2:m_abs(j); LH 3}d<{
for k = length(s):-1:1 )0vU
k
p = (1-2*mod(s(k),2))* ... Qp"y?S
prod(2:(n(j)-s(k)))/ ... 87%*+n:?*
prod(2:s(k))/ ... (6CN/A{qe
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... <^~FLjsfg
prod(2:((n(j)+m_abs(j))/2-s(k))); (bOpV>\Q7
idx = (pows(k)==rpowers); 'bGX-C
y(:,j) = y(:,j) + p*rpowern(:,idx); w;}@'GgL
end 9`jcC-;iv
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if isnorm v%k9M{
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); G2LK]
end
s cuHmY0
end F62V3 Xy
% END: Compute the Zernike Polynomials ri`R<l8
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Compute the Zernike functions: <PPNhf8
% ------------------------------ 2 oa#0`{
idx_pos = m>0;
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idx_neg = m<0; I$Qs;- (
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z = y; xD&^j$Em
if any(idx_pos) 2j(h+?N7k
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); mcz+P |
end 22kp l)vbU
if any(idx_neg) LG~S8u
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ^8 ' sib
end h8\
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% EOF zernfun T!wo2EzE