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, q\U4n[Zk
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, b=_{/F*b?
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? lO_c/o$
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? xDLMPo&
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function z = zernfun(n,m,r,theta,nflag) s@z{dmL
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. YJc%h@ _=]
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N v\'rXy
% and angular frequency M, evaluated at positions (R,THETA) on the Y.9~Bo<<r
% unit circle. N is a vector of positive integers (including 0), and yP%o0n/"x
% M is a vector with the same number of elements as N. Each element 9iK&f\#5H
% k of M must be a positive integer, with possible values M(k) = -N(k) Lb^(E-
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, mw Z'=H
% and THETA is a vector of angles. R and THETA must have the same [NZ-WU&&LP
% length. The output Z is a matrix with one column for every (N,M) a!?.F_T9A
% pair, and one row for every (R,THETA) pair.
Db,= 2e
% ]DU61Z"v?b
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike t5n2eOy~T
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), PC[cHgSYU
% with delta(m,0) the Kronecker delta, is chosen so that the integral +/w(K,
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, <g*.p@o
% and theta=0 to theta=2*pi) is unity. For the non-normalized Fj,(_^
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Li ij{ahm
% n3*UgNg%fK
% The Zernike functions are an orthogonal basis on the unit circle. ) (+)Q'*
% They are used in disciplines such as astronomy, optics, and ;*.(.
% optometry to describe functions on a circular domain. >"O1`xdG
% ZXh~79
% The following table lists the first 15 Zernike functions. l3BD
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% |@+8]dy:l
% n m Zernike function Normalization 0FTRm2(
% -------------------------------------------------- Y=3X9%v9g
% 0 0 1 1 0Ux<16#
% 1 1 r * cos(theta) 2 _ r~+p
% 1 -1 r * sin(theta) 2 %
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% 2 -2 r^2 * cos(2*theta) sqrt(6) Tt`L(oF
% 2 0 (2*r^2 - 1) sqrt(3) v&e-`.xR
% 2 2 r^2 * sin(2*theta) sqrt(6) L)1C'8).
% 3 -3 r^3 * cos(3*theta) sqrt(8) =zz+<!!
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) K q/~T7Ru
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) O1|B3M[P
% 3 3 r^3 * sin(3*theta) sqrt(8) I'xC+nL@
% 4 -4 r^4 * cos(4*theta) sqrt(10) xJN |w\&
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) L>0!B8X2
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) y{YXf!AS
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #>@<n3rq
% 4 4 r^4 * sin(4*theta) sqrt(10) I Jqv w
% -------------------------------------------------- gH5CB%)
% Xm%iPrl D
% Example 1: B'<!k7Ewy
% )\D2\1e(c
% % Display the Zernike function Z(n=5,m=1) O<4Q$|=&?
% x = -1:0.01:1; yLjV[qP
% [X,Y] = meshgrid(x,x); Y+!Ouc!$
% [theta,r] = cart2pol(X,Y); 4}+xeGA$
% idx = r<=1; `i=JjgG@
% z = nan(size(X)); Z+r%_|kZ
% z(idx) = zernfun(5,1,r(idx),theta(idx)); bd,Uz%o_
% figure +:fqL
% pcolor(x,x,z), shading interp <"hb#Tn
% axis square, colorbar YW'{|9KnI
% title('Zernike function Z_5^1(r,\theta)') 4[2=L9MIo~
% ?]s%(R,B5
% Example 2: eVZa6la"
% 3%_
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% % Display the first 10 Zernike functions uE"5 cq'B/
% x = -1:0.01:1; Po'-z<}wS
% [X,Y] = meshgrid(x,x); Sjw2 j#Q
% [theta,r] = cart2pol(X,Y); 8mk}nex
% idx = r<=1; j?Cr31
% z = nan(size(X)); d&NCFx
% n = [0 1 1 2 2 2 3 3 3 3]; AGl|>f)
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ;,<r|.6U
% Nplot = [4 10 12 16 18 20 22 24 26 28]; I/mvQxp
% y = zernfun(n,m,r(idx),theta(idx)); j#7wyi5q
% figure('Units','normalized') m$7x#8gF
% for k = 1:10 kuWK/6l4
% z(idx) = y(:,k); c:3@[nF~
% subplot(4,7,Nplot(k)) wy,Jw3
% pcolor(x,x,z), shading interp K~`n}_:
% set(gca,'XTick',[],'YTick',[]) l. XknF
% axis square \R6;Fef
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) _Wm(/ +G_|
% end p.@0=)
% n33JTqX
% See also ZERNPOL, ZERNFUN2. 8FB\0LA!g
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% Paul Fricker 11/13/2006 _7r qXkp%
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q:TZ=bs^
% Check and prepare the inputs: X*TuQ\T
% ----------------------------- QN)/,=#
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) J!=](s5|
error('zernfun:NMvectors','N and M must be vectors.') `
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end (iHf9*i CV
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if length(n)~=length(m) jf$JaY
error('zernfun:NMlength','N and M must be the same length.') P3+)pOE-SI
end <{$ev&bQ
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n = n(:); RnMB Gxa
m = m(:); a/`c ef
if any(mod(n-m,2)) 6Y;Y}E
error('zernfun:NMmultiplesof2', ... 4a(g<5wfI
'All N and M must differ by multiples of 2 (including 0).') Vpug"aR&_
end F3kC"H
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if any(m>n) z1Bj_u{
error('zernfun:MlessthanN', ... Gl?P.BCW.&
'Each M must be less than or equal to its corresponding N.') X@6zI-Y%
end {toyQ)C7
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if any( r>1 | r<0 ) Ft3N#!ubl
error('zernfun:Rlessthan1','All R must be between 0 and 1.') tb-OKZq
end Q3B'-BZe
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 5?V?
error('zernfun:RTHvector','R and THETA must be vectors.') Nb^zkg
end c[wQJc
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r = r(:); 7,V!Iv^X
theta = theta(:); &[?u1qQ%o
length_r = length(r); L Q I: ]d
if length_r~=length(theta) eh({K;>
error('zernfun:RTHlength', ... Z$OF|ZZQ
'The number of R- and THETA-values must be equal.') q|47;bK'
end Gt\K Ln
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% Check normalization: XEf&Yd
% -------------------- 4b3 F9
if nargin==5 && ischar(nflag) s
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:tFK\
isnorm = strcmpi(nflag,'norm'); :$SRG^7md
if ~isnorm %nDPM? aO
error('zernfun:normalization','Unrecognized normalization flag.') H6%!v1 u
end F:*[
else RE`J"&
isnorm = false; j61BP8E
end }5o~R~H
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -!C
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% Compute the Zernike Polynomials GvZac
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [6,]9|~
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% Determine the required powers of r: C0t+Q
% ----------------------------------- ?BHWzo!
m_abs = abs(m); 1c<CEq:?e%
rpowers = []; bMqu5G_q
for j = 1:length(n) h30QCk
rpowers = [rpowers m_abs(j):2:n(j)]; 4i[v
ew
end O]Ry3j
rpowers = unique(rpowers); F(KH-
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% Pre-compute the values of r raised to the required powers, [}9XHhY1O=
% and compile them in a matrix: YmO"EWb
% ----------------------------- 6yu*a_
if rpowers(1)==0 PxP?hk
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); DSDl[;3O{s
rpowern = cat(2,rpowern{:}); UALg!M#
rpowern = [ones(length_r,1) rpowern]; fncwe ';?
else d}wa[WRv
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); [/+dHW|
rpowern = cat(2,rpowern{:}); X>6~{3
end sO{0hZkc
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% Compute the values of the polynomials: h87L8qh9
% -------------------------------------- aV?}+Y{#
y = zeros(length_r,length(n)); 2#n$x*CY
for j = 1:length(n) q5I4'6NF
s = 0:(n(j)-m_abs(j))/2; ~/|unV
pows = n(j):-2:m_abs(j); `G ;Lz^
for k = length(s):-1:1 w}U5dM`
p = (1-2*mod(s(k),2))* ... o%4&1^ Vg
prod(2:(n(j)-s(k)))/ ... ohc/.5Kl
prod(2:s(k))/ ... wCq)w=,
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... TN |{P
prod(2:((n(j)+m_abs(j))/2-s(k))); YA;8uMqh;
idx = (pows(k)==rpowers); WnJLX ^;
y(:,j) = y(:,j) + p*rpowern(:,idx); $@u^Jt, ?
end j quSR=
zNsL^;uT
if isnorm DX%8.@
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Ghq'k:K,
end +3o)L?:g
end St3(1mApl
% END: Compute the Zernike Polynomials *(\;}JF-
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .~A"Wyu\
*nsnX/e(-
2LxVt@_R!%
% Compute the Zernike functions: ~kj(s>xP
% ------------------------------ %8}ksl07
idx_pos = m>0; LG&Q>pt.
idx_neg = m<0; ,
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Z'>eT)
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z = y; K&D}!.~/
if any(idx_pos) [BZ(p
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); l6`d48U
end -4^@)~Y
if any(idx_neg) C>\!'^u1
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); p=`x
end vZ nO
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% EOF zernfun vn%U;}