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, Jm1AJ4mw
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, XJ1nhE
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? =smY/q^3
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? N+@@EOmH
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function z = zernfun(n,m,r,theta,nflag) Ko1?jPE
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. :tDGNz*zG
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N /s0VyUV=
% and angular frequency M, evaluated at positions (R,THETA) on the kC#B7*[RM
% unit circle. N is a vector of positive integers (including 0), and bDh(;%=
% M is a vector with the same number of elements as N. Each element e$+? v2.
% k of M must be a positive integer, with possible values M(k) = -N(k) 5xV/&N
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, !I+u/f?TO7
% and THETA is a vector of angles. R and THETA must have the same j9fL0$+FI
% length. The output Z is a matrix with one column for every (N,M) ['YRY B
% pair, and one row for every (R,THETA) pair. `DY4d$!4
% uH;^>`DT
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike =&G|} M
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Vm8_
!$F
% with delta(m,0) the Kronecker delta, is chosen so that the integral op{(mn
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, l|QFNW[i
% and theta=0 to theta=2*pi) is unity. For the non-normalized LZbHK.G=
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. YG+Yb{^"
% 0`Qs=R`OM
% The Zernike functions are an orthogonal basis on the unit circle. aj-uk(r
% They are used in disciplines such as astronomy, optics, and ',ybHW%D%i
% optometry to describe functions on a circular domain. jQlK-U=oi
% u=i^F|
% The following table lists the first 15 Zernike functions. MZF ;k$R
% s OHAW*+
% n m Zernike function Normalization g
wiC ,
% -------------------------------------------------- 8l,hP .
% 0 0 1 1 2%%U)|39mB
% 1 1 r * cos(theta) 2 2Rp{]s$jo
% 1 -1 r * sin(theta) 2 8@#Y
<{
% 2 -2 r^2 * cos(2*theta) sqrt(6) lMf5F8
% 2 0 (2*r^2 - 1) sqrt(3) 0#nXxkw
% 2 2 r^2 * sin(2*theta) sqrt(6) o|im
% 3 -3 r^3 * cos(3*theta) sqrt(8) ]
:#IZ0#
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) H;te)km}
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 13@| {H CB
% 3 3 r^3 * sin(3*theta) sqrt(8) ;rdLYmmx^
% 4 -4 r^4 * cos(4*theta) sqrt(10) iiFKt(
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,Yt&PE
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) r?>Hg+
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (ZSSp1Rv
% 4 4 r^4 * sin(4*theta) sqrt(10) }Q(I&uz
% -------------------------------------------------- 4T^WRS
% laJ%fBWmbi
% Example 1: AlhiF\+ C
% wi
>ta
% % Display the Zernike function Z(n=5,m=1) };sm8P{M
% x = -1:0.01:1; TzXl ?N
% [X,Y] = meshgrid(x,x); _$lQK{@rY
% [theta,r] = cart2pol(X,Y); 3c6)
% idx = r<=1; W5;sps
% z = nan(size(X)); /;ITnG
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ![n`n(oN
% figure / /rWc,c
% pcolor(x,x,z), shading interp nuA!Jln_
% axis square, colorbar o~>go_Y
% title('Zernike function Z_5^1(r,\theta)') b=l}|)a
% wfzb:Aig`
% Example 2: ,DZLEsFM
% fs12<~+z
% % Display the first 10 Zernike functions g?M69~G$:x
% x = -1:0.01:1; Sw)ftC~d
% [X,Y] = meshgrid(x,x); >D aS*r
% [theta,r] = cart2pol(X,Y); xKux5u_
% idx = r<=1; #pFybk
% z = nan(size(X)); M 4?3l
% n = [0 1 1 2 2 2 3 3 3 3]; xI8*sTx
6
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; @jeV[N,0
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Br??Gdd
% y = zernfun(n,m,r(idx),theta(idx)); ITiw) M
% figure('Units','normalized') !7D S
% for k = 1:10 1OL~)X3
% z(idx) = y(:,k); 2kve?/
% subplot(4,7,Nplot(k)) 5gEK$7Vp
% pcolor(x,x,z), shading interp lEs/_f3;A
% set(gca,'XTick',[],'YTick',[]) M
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% axis square "de:plMofy
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) (*]Y<ve
% end Z;:-8 HPDY
% p,fin?nW c
% See also ZERNPOL, ZERNFUN2. ha5\T'
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% Paul Fricker 11/13/2006 nU)f]4q{Ec
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% Check and prepare the inputs: Q1yTDJ(2
% ----------------------------- {n'}S(
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) yfrgYA
error('zernfun:NMvectors','N and M must be vectors.') -9EbU7>!
end [u:_Jqf-
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if length(n)~=length(m) E*vi@aI
error('zernfun:NMlength','N and M must be the same length.') hZy*E [i
end |99eDgK,
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n = n(:); =Ju}{ bX
m = m(:); XJ+sm^`vOf
if any(mod(n-m,2)) teb(\% ,
error('zernfun:NMmultiplesof2', ...
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'All N and M must differ by multiples of 2 (including 0).') T5)?6i-N
end C{-pVuhK+
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if any(m>n) )U`
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error('zernfun:MlessthanN', ... UpbzH(?#
'Each M must be less than or equal to its corresponding N.') (WC<X Kf
end 7w|s8B
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if any( r>1 | r<0 ) H b.oKo$T
error('zernfun:Rlessthan1','All R must be between 0 and 1.') )XFMlSx)
end 5:wf"3%%
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) A}8U;<\Ig
error('zernfun:RTHvector','R and THETA must be vectors.') bc-"If Z&
end KH-.Z0
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r = r(:); ^*W3{eyi(L
theta = theta(:); Vufw:}i+^
length_r = length(r); !?96P|G
if length_r~=length(theta) 8eNGPuoL)
error('zernfun:RTHlength', ... Kmtr.]Nj
'The number of R- and THETA-values must be equal.') Dqki}k~{
end m(Oup=\%b}
!**q20-aP
H={,zZ11{
% Check normalization: U1Oq"Ij~
% -------------------- V+ Z22
if nargin==5 && ischar(nflag) kDrGl{U}
isnorm = strcmpi(nflag,'norm'); 1{*x+GC^/
if ~isnorm =vWnqF:
error('zernfun:normalization','Unrecognized normalization flag.') G} p~VLf
end wBf
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else *+G K?Ga
isnorm = false; /cg!Ap5
end A/MOY@%G
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% R$\ieNb
% Compute the Zernike Polynomials eWFlJ;=
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *oF{ R^
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% Determine the required powers of r: '{|87kI
% ----------------------------------- ,PB?pp8C}
m_abs = abs(m); ;J4_8N-
rpowers = []; 2iUF%>
for j = 1:length(n) |1neCP@ng
rpowers = [rpowers m_abs(j):2:n(j)]; (wTg aV1
end wL{Qni3A
rpowers = unique(rpowers); EV}%D9:
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% Pre-compute the values of r raised to the required powers, ALG +
% and compile them in a matrix: V/03m3!q
% ----------------------------- dCinbAQ
if rpowers(1)==0 _|F h^hq
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); =Vi+wH{xM
rpowern = cat(2,rpowern{:}); 4)`{ L$
rpowern = [ones(length_r,1) rpowern]; qRr;&M &t_
else {5,CW
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); -v]7}[
.[
rpowern = cat(2,rpowern{:}); y(%6?a @
end -1@kt<Es
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% Compute the values of the polynomials: u+ -}|
% -------------------------------------- J^u{7K,
y = zeros(length_r,length(n)); RW3&]l=
for j = 1:length(n) U+\\#5$
s = 0:(n(j)-m_abs(j))/2; J~~WV<6
pows = n(j):-2:m_abs(j); rTx]%{
for k = length(s):-1:1 oRCj]9I$
p = (1-2*mod(s(k),2))* ... ,i'>+Ix<
prod(2:(n(j)-s(k)))/ ... kw!! 5U;7
prod(2:s(k))/ ... j_k!9"bt
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... x]F:~(P
prod(2:((n(j)+m_abs(j))/2-s(k))); #
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idx = (pows(k)==rpowers); m~2PpO
y(:,j) = y(:,j) + p*rpowern(:,idx); gI[xOK#
end &L_(yJ~-
VLR W,lR9O
if isnorm d5h:py5
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); |_Vlw&qu+
end D&.+Dx^G
end 1B 2>8N
% END: Compute the Zernike Polynomials m'Ran3rp
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% O
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% Compute the Zernike functions: ^W;\faG
% ------------------------------ Lb(=:Z!{
idx_pos = m>0; @<h@d_8^k
idx_neg = m<0; o4U9jU4<"
f`T#=6C4|
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z = y; iOw'NxmY
if any(idx_pos) :Oxrw5`=
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 4v Ug:'DM
end ?8pR RzV$
if any(idx_neg) J#MUtpPdQ
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Oo$i,|$$
end G{)2f&<
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% EOF zernfun SIVzc Hm