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, N~kYT\$b#
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, [eN{Ft0x
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? )K6{_~Kc\
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? gc:>HX);)
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function z = zernfun(n,m,r,theta,nflag) srkOad
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. M:$nL
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ?C{N0?[P-
% and angular frequency M, evaluated at positions (R,THETA) on the q'r3a+
% unit circle. N is a vector of positive integers (including 0), and q<8HG_
% M is a vector with the same number of elements as N. Each element TK>}$.c%+
% k of M must be a positive integer, with possible values M(k) = -N(k) zK92:+^C
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, <coCu0
% and THETA is a vector of angles. R and THETA must have the same pp`U]Q5"gX
% length. The output Z is a matrix with one column for every (N,M) ;CZcY] ol
% pair, and one row for every (R,THETA) pair. HXQrtJ
% =R"tnjR
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike i5" q1dRQ
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), qsRh ihPX
% with delta(m,0) the Kronecker delta, is chosen so that the integral QMY4%uyY!
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 8(;i~f:bCW
% and theta=0 to theta=2*pi) is unity. For the non-normalized J#]yKgT
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. l :"*]m7o_
% _JDr?Kg
% The Zernike functions are an orthogonal basis on the unit circle. T1&H!
% They are used in disciplines such as astronomy, optics, and VLN3x.BY
% optometry to describe functions on a circular domain. 9="sx 8?
% do,X{\
% The following table lists the first 15 Zernike functions. nSiNSLv
% %R>S"
% n m Zernike function Normalization <hbbFL}|%
% -------------------------------------------------- WXU6J?tIm
% 0 0 1 1 k]iS3+nD
% 1 1 r * cos(theta) 2 h)vTu%J:
% 1 -1 r * sin(theta) 2 O2dgdtm
% 2 -2 r^2 * cos(2*theta) sqrt(6) gEsR-A!m
% 2 0 (2*r^2 - 1) sqrt(3) A~V\r<N
j
% 2 2 r^2 * sin(2*theta) sqrt(6) >6 #\1/RP
% 3 -3 r^3 * cos(3*theta) sqrt(8) !y?hn$w0
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 88j
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% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Gf\_WNrSE+
% 3 3 r^3 * sin(3*theta) sqrt(8) du,-]fF
% 4 -4 r^4 * cos(4*theta) sqrt(10) }0RFo96)v
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) &:*+p-!2<
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) f4_G[?9,
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) gj^]}6-P
% 4 4 r^4 * sin(4*theta) sqrt(10) auW]rwY
% -------------------------------------------------- |/;5|
z
% 6DW|O<k^j
% Example 1: C>dJ:.K%H
% ew$Z5N:
% % Display the Zernike function Z(n=5,m=1) Dys"|,F
% x = -1:0.01:1; X)OP316yx
% [X,Y] = meshgrid(x,x); Uc0'XPo3I
% [theta,r] = cart2pol(X,Y); #>B1$(@
% idx = r<=1; #UD
% z = nan(size(X)); ?/MXcI(
% z(idx) = zernfun(5,1,r(idx),theta(idx)); )du{ZWr
% figure );DIrA
% pcolor(x,x,z), shading interp B31-<w
% axis square, colorbar S(h*\we
% title('Zernike function Z_5^1(r,\theta)') oZ:F3 GQ4Q
% >L`mF_WG
% Example 2: pw yl,A
% .G~5F- 8'
% % Display the first 10 Zernike functions @I6 A9do
% x = -1:0.01:1; p|V1Gh<
% [X,Y] = meshgrid(x,x); {OrE1WHB
% [theta,r] = cart2pol(X,Y); F|`B2Gr
% idx = r<=1; \Pmk`^T
% z = nan(size(X)); ^X%4@,AE
% n = [0 1 1 2 2 2 3 3 3 3]; 'a?.X _t
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; > C&<dO#i
% Nplot = [4 10 12 16 18 20 22 24 26 28]; G3^]Wwu
% y = zernfun(n,m,r(idx),theta(idx)); mm<iT59
% figure('Units','normalized') u>6/_^iq
% for k = 1:10 1>x@1Mo+K
% z(idx) = y(:,k); -xIhN?r)
% subplot(4,7,Nplot(k)) kQl cT"R
% pcolor(x,x,z), shading interp _hL4@C
% set(gca,'XTick',[],'YTick',[]) ,nRwwFd.
% axis square XPo'iI-
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) L)Ar{*xC
% end v^_]W3K
% !>Y\&zA
% See also ZERNPOL, ZERNFUN2. %]$p ^m
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% Paul Fricker 11/13/2006 FXEfD"
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% Check and prepare the inputs: SH$cn,3F8
% ----------------------------- ":^
NLBm>5
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) cBR8HkP~
error('zernfun:NMvectors','N and M must be vectors.') 02#Iip3t
end ,W8Iabi^
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if length(n)~=length(m) #L=
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error('zernfun:NMlength','N and M must be the same length.') KM(9&1/
end )u )$ `a
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n = n(:); ,3--ERf
m = m(:); \ j X N*A
if any(mod(n-m,2)) ;(0$~O$3u
error('zernfun:NMmultiplesof2', ... 7O9hn2?e
'All N and M must differ by multiples of 2 (including 0).') #iU8hUbo
end bd
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if any(m>n) ^C)T M@+
error('zernfun:MlessthanN', ... =>ztB w\
'Each M must be less than or equal to its corresponding N.') >aC\_Mc
end !a&SB*%^I3
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if any( r>1 | r<0 ) fDEu%fUYZ
error('zernfun:Rlessthan1','All R must be between 0 and 1.') BS,5W]ervE
end , 64t
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Ao,!z
error('zernfun:RTHvector','R and THETA must be vectors.') [aM'
end -S%q!%}u
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r = r(:); 6^IqSNn-
theta = theta(:); X})Imk7&E
length_r = length(r); MjXE|3&
if length_r~=length(theta) jy(+
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error('zernfun:RTHlength', ... *zVLy^L_8
'The number of R- and THETA-values must be equal.') vuo'"^ =p0
end =e!l=d|/
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% Check normalization: {pb9UUP2
% -------------------- #;"D)C
if nargin==5 && ischar(nflag) ~@4ZV
isnorm = strcmpi(nflag,'norm'); ;64mf`
if ~isnorm jWK@NXMH
error('zernfun:normalization','Unrecognized normalization flag.') Z 5)_B,E:X
end 'LbeL1ca
else A6NxM8ybn+
isnorm = false; Gkv~e?Kc~^
end Gl8&FrR
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .-KtB(t
% Compute the Zernike Polynomials I& M36f
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% phgexAq
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@<OO
% Determine the required powers of r: /j' B\,
% ----------------------------------- IObx^N_K
m_abs = abs(m); MZ5Y\-nq\
rpowers = []; Cl6m$YUt
for j = 1:length(n) @1qdd~B}
rpowers = [rpowers m_abs(j):2:n(j)]; .5Knb c
end 7Y32p'
rpowers = unique(rpowers); (/SGT$#8
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{ eEC:[
% Pre-compute the values of r raised to the required powers, %-#
qO
% and compile them in a matrix: ZMoJ#p(
% ----------------------------- eB=v~I3
if rpowers(1)==0 os1?6z~
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); WDEe$k4.
rpowern = cat(2,rpowern{:}); !6zyJc@01
rpowern = [ones(length_r,1) rpowern]; Il{^
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else L\ }Pzxn
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);
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rpowern = cat(2,rpowern{:}); Re*|$r#
end kG)2%
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% Compute the values of the polynomials: F&QTL-pQW
% -------------------------------------- $s-9|Lbs`
y = zeros(length_r,length(n)); <t{?7_ 8
for j = 1:length(n) PMgQxM*h
s = 0:(n(j)-m_abs(j))/2; =n-z;/NL
pows = n(j):-2:m_abs(j); Q !9HA[Ly
for k = length(s):-1:1 g.x=pt
p = (1-2*mod(s(k),2))* ... 9<|m4
prod(2:(n(j)-s(k)))/ ... Ys-Keyg
prod(2:s(k))/ ... _+twqi
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ch@x]@-;A3
prod(2:((n(j)+m_abs(j))/2-s(k))); PSTu /^
idx = (pows(k)==rpowers); d/XlV]#2x\
y(:,j) = y(:,j) + p*rpowern(:,idx); ~ww?Emrw
end ^
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[N)#/6j
if isnorm x*.Ye5Jb
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 1GtOA3,~;-
end E:
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end F=XF]
% END: Compute the Zernike Polynomials ,>;!%Ui/p
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2B7h9P.N B
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% Compute the Zernike functions: mCM|&u
% ------------------------------ Kb}MF9?:e
idx_pos = m>0; q0&Wk"X%rr
idx_neg = m<0; *B0V<mV
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: *ERRSL)
z = y; f1A_`$>
if any(idx_pos) nV'3sUvR#
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); -#Np7/
end <^xfcYx\
if any(idx_neg) _=ugxL #eB
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); wGPotPdE2
end ],n%Xp
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W.^Ei\w/t
% EOF zernfun Vo%Yf9C