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, .VfBwTh7q8
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, ?Y#x`DMh
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? $tFmp)
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? lG!We'?
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function z = zernfun(n,m,r,theta,nflag) Nh/B8:035
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. *^-~J/
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Q GQ}I
% and angular frequency M, evaluated at positions (R,THETA) on the K\vyfYi
% unit circle. N is a vector of positive integers (including 0), and DAt Zp%
% M is a vector with the same number of elements as N. Each element C%\.
% k of M must be a positive integer, with possible values M(k) = -N(k) Wk&g!FR
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, zz~AoX7V6
% and THETA is a vector of angles. R and THETA must have the same BjyGk+A
% length. The output Z is a matrix with one column for every (N,M) Hwm]l`E]
% pair, and one row for every (R,THETA) pair. ~xaPq=AH
% Y)]x1I
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike f-/zR %s{
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), lZ` CFZR0
% with delta(m,0) the Kronecker delta, is chosen so that the integral d~-Cr-s4
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, @u}1 S1
% and theta=0 to theta=2*pi) is unity. For the non-normalized ag\xwS#i5H
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 6YeEr!zt%
% c$[cDf~
% The Zernike functions are an orthogonal basis on the unit circle. vpl>
5 %
% They are used in disciplines such as astronomy, optics, and &>&UqWL
% optometry to describe functions on a circular domain. c O[Hr
% .q^+llM
% The following table lists the first 15 Zernike functions. Pn[R.u(l
% /MUa
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% n m Zernike function Normalization MTxe5ob`$Q
% -------------------------------------------------- 2En^su$
% 0 0 1 1 2PrUI;J$
% 1 1 r * cos(theta) 2 +)eI8o0#
% 1 -1 r * sin(theta) 2 ]NrA2i?
% 2 -2 r^2 * cos(2*theta) sqrt(6) bF
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% 2 0 (2*r^2 - 1) sqrt(3) bzt(;>_8
% 2 2 r^2 * sin(2*theta) sqrt(6) I "<ACM
% 3 -3 r^3 * cos(3*theta) sqrt(8)
*[^[!'kT&
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) [Q5>4WY
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) p%+uv\Ix
% 3 3 r^3 * sin(3*theta) sqrt(8) `78:TU~5S
% 4 -4 r^4 * cos(4*theta) sqrt(10) #nOS7Q#uW
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) {BA1C
(
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) &n)=OConge
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) L)`SNN\ipR
% 4 4 r^4 * sin(4*theta) sqrt(10) ;m[-yqX
% -------------------------------------------------- [9S?
% ,J3s1 ]~^
% Example 1: !jeoB
% e
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% % Display the Zernike function Z(n=5,m=1) Nhnw'9
% x = -1:0.01:1; wgb
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% [X,Y] = meshgrid(x,x); *$eMM*4
% [theta,r] = cart2pol(X,Y); O-D${==
% idx = r<=1; !b0ANIp
% z = nan(size(X)); D|`I"N[<
% z(idx) = zernfun(5,1,r(idx),theta(idx)); dO{a!Ca
% figure np#RBy
% pcolor(x,x,z), shading interp "DniDA
% axis square, colorbar SQ_w~'(
% title('Zernike function Z_5^1(r,\theta)') d/fg
% cn~M:LW23
% Example 2: ?!4xtOA
% HoIK^t~VT#
% % Display the first 10 Zernike functions l,pI~A`w_
% x = -1:0.01:1; ]N\J~Gm
% [X,Y] = meshgrid(x,x); )S;pYVVAl
% [theta,r] = cart2pol(X,Y); &r)i6{w81
% idx = r<=1; dP0%<Q|
% z = nan(size(X)); ,a&&y0,
% n = [0 1 1 2 2 2 3 3 3 3]; :Rq>a@Rp
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; {|;5P.,l
% Nplot = [4 10 12 16 18 20 22 24 26 28]; j6NK7Li
% y = zernfun(n,m,r(idx),theta(idx)); $Z!$E,@c
% figure('Units','normalized') =68CR[H
% for k = 1:10 F"k.1.
% z(idx) = y(:,k); #@*;Y(9Ol
% subplot(4,7,Nplot(k)) q
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% pcolor(x,x,z), shading interp 8hK\Ya:mP
% set(gca,'XTick',[],'YTick',[]) HX(Z(rcI
% axis square &ZmHR^Flz
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) V@QWJZ"
% end yMQZulCWE
% m,,FNYW
% See also ZERNPOL, ZERNFUN2. /Lf+*u>"
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% Paul Fricker 11/13/2006 2Op\`Ht&
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% Check and prepare the inputs: ww_gG5Fc$
% ----------------------------- ]7*Z'E
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) zSpL^:~
error('zernfun:NMvectors','N and M must be vectors.') vbDSNm#Yv
end _x.<Zc\x
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if length(n)~=length(m) J^Dkx"1GD
error('zernfun:NMlength','N and M must be the same length.') ,}("es\b
end 7lo`)3mB
@+9x8*~S'
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n = n(:); Ac0^`
m = m(:); i|@lUXBp
if any(mod(n-m,2)) Qj?qWVapA
error('zernfun:NMmultiplesof2', ... `W3;LTPEb
'All N and M must differ by multiples of 2 (including 0).') Yt 9{:+[RK
end }\9elVt'2
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if any(m>n) 8IAf9
error('zernfun:MlessthanN', ... u x[h\Tp
'Each M must be less than or equal to its corresponding N.') ^`W8>czi
end +w(sDH~kd
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if any( r>1 | r<0 ) FD}hw9VyF@
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 4`x.d
end KxEy
N (n
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) %s+H& vfQs
error('zernfun:RTHvector','R and THETA must be vectors.') igoXMsifT+
end Ya#,\;dTT
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r = r(:); {^r8uKo:~
theta = theta(:); 8{m5P8w'
length_r = length(r); d)G'y
if length_r~=length(theta) 4K_ fN
error('zernfun:RTHlength', ... _I("k:E7
'The number of R- and THETA-values must be equal.') cJ6n@\
end `}b#O}z)^
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% Check normalization: Orb('Z,-3
% -------------------- u?OyvvpH
if nargin==5 && ischar(nflag) 7J
0=HbH
isnorm = strcmpi(nflag,'norm'); : ryE`EhB
if ~isnorm kRCuc}:SB
error('zernfun:normalization','Unrecognized normalization flag.')
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end u7rA8u|TO
else cULASS`,
isnorm = false; }U)g<Kzh
end ?s4-2g
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9#:b+Amzz
% Compute the Zernike Polynomials y7K&@Y
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% y"|QY!fK
d?j_L`?+
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% Determine the required powers of r: Pj4WWK X
% ----------------------------------- 0P(U^rkR~
m_abs = abs(m); =j%B`cJ66_
rpowers = []; 8hx4s(1!
for j = 1:length(n) TM|M#hMS
rpowers = [rpowers m_abs(j):2:n(j)]; 0JQ0lzk1
end
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rpowers = unique(rpowers); 'baew8Q#
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o80"ZU|=
% Pre-compute the values of r raised to the required powers, +*dG'U6
% and compile them in a matrix: fS08q9,S /
% ----------------------------- -ZTe#@J
if rpowers(1)==0 d$>TC(E=t
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); EXlmIY4
rpowern = cat(2,rpowern{:}); }b9"&io
rpowern = [ones(length_r,1) rpowern]; UL81x72O
else m5O;aj* i
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); e:SBX/\j
rpowern = cat(2,rpowern{:}); KeU|E<|!
end 7
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% Compute the values of the polynomials: 7Ao9MF-
% -------------------------------------- 4)L(41h
y = zeros(length_r,length(n)); ff.(X!
for j = 1:length(n) &PHejG_#
s = 0:(n(j)-m_abs(j))/2; / S32)=(
pows = n(j):-2:m_abs(j); 72hN%l
for k = length(s):-1:1 I{8fTod
p = (1-2*mod(s(k),2))* ... \)\uAI-
prod(2:(n(j)-s(k)))/ ... 3;M7^DM
prod(2:s(k))/ ... _ZM$&6EC
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... %2 A-u
prod(2:((n(j)+m_abs(j))/2-s(k))); ;x 9_
idx = (pows(k)==rpowers); \;al@yC=T
y(:,j) = y(:,j) + p*rpowern(:,idx); !N\<QRb\q
end bSOxM/N
m3Mo2};?
if isnorm T&M*sydA
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); j]-0m4QF
end 8>T#sO?+
end 3[R<JrO
% END: Compute the Zernike Polynomials I
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Y7WxV>E
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% Compute the Zernike functions: .YV{w L@cB
% ------------------------------ xvP=i/SO
idx_pos = m>0; !Zowe*`
idx_neg = m<0; m:kXr^!D
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z = y; G`n|fuv
if any(idx_pos) #[|~m;K(w
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); nkI+"$Rz0
end p~Tp=d)/
if any(idx_neg) kF%EJuu
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 9Fo00"q
end r]e1a\)r
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% EOF zernfun [NG~FwpRf