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, - :z5m+
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, G2{ M#H
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? r tmt 3
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? m{dyVE
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function z = zernfun(n,m,r,theta,nflag) aX'g9E
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. |abst&yp
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ;=\5$J9
% and angular frequency M, evaluated at positions (R,THETA) on the 'qF3,Rw
% unit circle. N is a vector of positive integers (including 0), and 3]OP9!\6
% M is a vector with the same number of elements as N. Each element tDHHQ
% k of M must be a positive integer, with possible values M(k) = -N(k) }>X\"
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ^~<Rz q!
% and THETA is a vector of angles. R and THETA must have the same [^}>AC*im
% length. The output Z is a matrix with one column for every (N,M) Bx : So6:
% pair, and one row for every (R,THETA) pair. pkN:D+gS
% u$=ogp=0
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike >{qK]xj
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), lH@E %
% with delta(m,0) the Kronecker delta, is chosen so that the integral _Z66[T+M
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, kbp(
a+5
% 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]. QJIItx4hE
% ;.Oh88|k
% The Zernike functions are an orthogonal basis on the unit circle. Tb0;Mbr
% They are used in disciplines such as astronomy, optics, and H(G^O&ppdB
% optometry to describe functions on a circular domain. n &\'Hm
% +fP/|A8P
% The following table lists the first 15 Zernike functions. @Gn?8Ur%
% 1'v !9
% n m Zernike function Normalization ZG/8 Ds
% -------------------------------------------------- [X">vaa
% 0 0 1 1 ')u5 l
% 1 1 r * cos(theta) 2 ]O7.ss/2
% 1 -1 r * sin(theta) 2 AXh3LA
% 2 -2 r^2 * cos(2*theta) sqrt(6) (4/]dTb
% 2 0 (2*r^2 - 1) sqrt(3) yg+IkQDf4U
% 2 2 r^2 * sin(2*theta) sqrt(6) }EedHS
% 3 -3 r^3 * cos(3*theta) sqrt(8) NB
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% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) =yTa,PY
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) X=p3KzzX
% 3 3 r^3 * sin(3*theta) sqrt(8) XHZ:
mLf
% 4 -4 r^4 * cos(4*theta) sqrt(10) a?,[w'7FU
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) yXTK(<'
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) S\3AW,c]w
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4Ay`rG
% 4 4 r^4 * sin(4*theta) sqrt(10) ;]&~D
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% -------------------------------------------------- u3*NO
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% "0'*q<8
% Example 1: eN]>l
% Hw?2XDv j
% % Display the Zernike function Z(n=5,m=1) Cl t5
% x = -1:0.01:1; Jny)uo8
% [X,Y] = meshgrid(x,x); M<Wn]}7!
% [theta,r] = cart2pol(X,Y); 5w,Z 7I8
% idx = r<=1; #6N+5Yx_[
% z = nan(size(X)); {C/L5cZ]J
% z(idx) = zernfun(5,1,r(idx),theta(idx)); i+)}aA
% figure [*9YIjn
% pcolor(x,x,z), shading interp !]rETP_
% axis square, colorbar :>P4L,Da]
% title('Zernike function Z_5^1(r,\theta)') UR1JbyT
% hg?j)jl|
% Example 2: 9|N"@0<B
% fou_/Nrue
% % Display the first 10 Zernike functions <Qcex3
% x = -1:0.01:1; f2O*8^^Y{Q
% [X,Y] = meshgrid(x,x); Y^f94s:2S
% [theta,r] = cart2pol(X,Y); ePq13!FC/
% idx = r<=1; -t@y\vZF,
% z = nan(size(X)); 7b&JX'`Mb
% n = [0 1 1 2 2 2 3 3 3 3]; <G~}N
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; +}7Ea:K
% Nplot = [4 10 12 16 18 20 22 24 26 28]; %NT`C9][
% y = zernfun(n,m,r(idx),theta(idx)); M&qh]v gC
% figure('Units','normalized') n5Nan
% for k = 1:10 8_a$kJJ2
% z(idx) = y(:,k); sK`~Csb
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% subplot(4,7,Nplot(k)) 4<G?
% pcolor(x,x,z), shading interp *xE"8pN/
% set(gca,'XTick',[],'YTick',[]) <%d51~@={I
% axis square O{k89{
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) -?< Ww{
% end w4e%-Ln
% t&GA6ML#s
% See also ZERNPOL, ZERNFUN2. 0?lp/|K
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% Paul Fricker 11/13/2006 Em e'Gk
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% Check and prepare the inputs: uY^v"cw/F
% ----------------------------- xS6(K
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) \Fj5v$J-
error('zernfun:NMvectors','N and M must be vectors.') "?apgx 6
end 9=t#5J#O
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if length(n)~=length(m) 9$7&URwSDI
error('zernfun:NMlength','N and M must be the same length.') `]*%:NZP@
end J=I:T2bV&s
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n = n(:); *#3*;dya]
m = m(:); C=fsJ=a5;
if any(mod(n-m,2)) 9YP*f
error('zernfun:NMmultiplesof2', ... `J72+ RA
'All N and M must differ by multiples of 2 (including 0).') ?h/xAl
end 8 YNu<
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if any(m>n) <,\ `Psa)N
error('zernfun:MlessthanN', ... uxWFM
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'Each M must be less than or equal to its corresponding N.') OE_QInb<
end tbtI1"$
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if any( r>1 | r<0 ) |#{- .r6Y]
error('zernfun:Rlessthan1','All R must be between 0 and 1.') {jvOHu
end x&'o ]Y
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ^NXcLEaP*<
error('zernfun:RTHvector','R and THETA must be vectors.') ujU=JlJ7dl
end !RS9%ES_?
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r = r(:); N;uUx#z
theta = theta(:); KkEv#2n
length_r = length(r); :z]}ZZ
if length_r~=length(theta) CdY8#+"
error('zernfun:RTHlength', ...
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'The number of R- and THETA-values must be equal.') }.p<wCPy6
end _2b9QP p
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% Check normalization: iZaeoy
% -------------------- S='
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if nargin==5 && ischar(nflag) :-?Ct
isnorm = strcmpi(nflag,'norm'); ] /+D^6
if ~isnorm u_PuqRcs
error('zernfun:normalization','Unrecognized normalization flag.') 2R]&v;A
end !YiuwFt
else f;gZ|a
isnorm = false;
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end h35Hu_c&
@9Q2$
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Compute the Zernike Polynomials %a];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {XgnZ`*
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% Determine the required powers of r: YzAFC11,
% ----------------------------------- p~2UUmV
m_abs = abs(m); ;#TaZN
rpowers = []; @b2`R3}9R
for j = 1:length(n) q]\X~
9#
rpowers = [rpowers m_abs(j):2:n(j)]; (DDyK[t+VX
end Q/ZkW
rpowers = unique(rpowers); =oX>Ph+ P
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|9Yi7.
% Pre-compute the values of r raised to the required powers, QV qK
% and compile them in a matrix: (vc|7DX M
% ----------------------------- M\oTZ@
if rpowers(1)==0 09S6#; N&
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);
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rpowern = cat(2,rpowern{:}); aE|OTm+@9;
rpowern = [ones(length_r,1) rpowern]; vMla'5|l
else Ue*C>F
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )zq.4
rpowern = cat(2,rpowern{:}); K=?VDN
end ar.AL'
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% Compute the values of the polynomials: Jq
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% -------------------------------------- 9b;A1gu
y = zeros(length_r,length(n)); Xf
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for j = 1:length(n) ?":'O#E
s = 0:(n(j)-m_abs(j))/2; U7iuY~L
pows = n(j):-2:m_abs(j); ]XA4;7
for k = length(s):-1:1 %UZVb V
p = (1-2*mod(s(k),2))* ... ir16
prod(2:(n(j)-s(k)))/ ... Y[Ltrk{
prod(2:s(k))/ ... ZH ,4oF
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... &v!WVa?
prod(2:((n(j)+m_abs(j))/2-s(k))); FP^{=0
idx = (pows(k)==rpowers); Nt:9 MG>1
y(:,j) = y(:,j) + p*rpowern(:,idx); nkDy!"K
end HKO739&n}
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if isnorm 2;`=P5V
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); %7hB&[ 5
end 2Y!S_Hw8
end Bi3+)k>u7
% END: Compute the Zernike Polynomials a j\nrD1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2F`cv1 M
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% Compute the Zernike functions: w}R~C
% ------------------------------ 5 BtX63
idx_pos = m>0; Jb["4X;h
idx_neg = m<0; SP]IUdE\
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z = y; zTbVp8\pI
if any(idx_pos) ,Gk}"w
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); C1EtoOv K
end HO)/dZNU
if any(idx_neg) Rli:x
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); qU6nJi+-I
end _c$9eAe
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% EOF zernfun ;a{ :%t