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, R>DaOH2K*
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, +iRq8aS_
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? 4h5g'!9-g
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? M02uO`Y9
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function z = zernfun(n,m,r,theta,nflag) -.g|l\
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. |mdi]TL
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ?$f)&O
% and angular frequency M, evaluated at positions (R,THETA) on the iXyO(w4D
% unit circle. N is a vector of positive integers (including 0), and 0sI1GhVR
% M is a vector with the same number of elements as N. Each element u0P)7~%
% k of M must be a positive integer, with possible values M(k) = -N(k) z0|&W&&D
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, GN KF&M
% and THETA is a vector of angles. R and THETA must have the same "ZTTg>r
% length. The output Z is a matrix with one column for every (N,M) N`)$[&NG]
% pair, and one row for every (R,THETA) pair. y5Tlpi`g
% +?p.?I
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike f|y:vpd%
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 'J,T{s1J
% with delta(m,0) the Kronecker delta, is chosen so that the integral {]"]uT#
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ; 7N
Z<k
% and theta=0 to theta=2*pi) is unity. For the non-normalized |_omr&[_
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. \~LQ%OM
% ix#epuN
% The Zernike functions are an orthogonal basis on the unit circle. Vi4~`;|&b+
% They are used in disciplines such as astronomy, optics, and ]f]<4HD=i
% optometry to describe functions on a circular domain. e/->_T(I
% `%09xMPu
% The following table lists the first 15 Zernike functions. )DYI
.
% W8lx~:v
% n m Zernike function Normalization DGevE~
% -------------------------------------------------- J9K3s_SN
% 0 0 1 1 AfG/JWSo}
% 1 1 r * cos(theta) 2 jy]JiQB
% 1 -1 r * sin(theta) 2 p{PE@KO:
% 2 -2 r^2 * cos(2*theta) sqrt(6) + >cBVx6
% 2 0 (2*r^2 - 1) sqrt(3) Rb(SBa
% 2 2 r^2 * sin(2*theta) sqrt(6) 9;?UvOI;
% 3 -3 r^3 * cos(3*theta) sqrt(8) /r12h|
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) e"
]2=5g
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) a>mm+L8y
% 3 3 r^3 * sin(3*theta) sqrt(8) S(\9T1DVe
% 4 -4 r^4 * cos(4*theta) sqrt(10) ='TE,et@d
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) z>w`ZD}XY
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) wH~kTU2br
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) %*#n d
% 4 4 r^4 * sin(4*theta) sqrt(10) w
'3#&k+
% -------------------------------------------------- xoOJauSX1
% V138d?Mm
% Example 1: ~EK'&Y"1
% WD'#5]#Y
% % Display the Zernike function Z(n=5,m=1) Isx#9C
% x = -1:0.01:1; ~tOAT;g}q
% [X,Y] = meshgrid(x,x); "zIFxDR#
% [theta,r] = cart2pol(X,Y); -{`@=U
% idx = r<=1; w`l{LHrR
% z = nan(size(X)); 1^i Pji/
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Fq9Q+RNMZL
% figure 8u!"#S#>a
% pcolor(x,x,z), shading interp o[E_Ge}g8
% axis square, colorbar D1nq2GwS
% title('Zernike function Z_5^1(r,\theta)') U35AX9/
% `GXkF:f=
% Example 2: !Ci~!)$z6
% N41 R
% % Display the first 10 Zernike functions pIbdN/z
% x = -1:0.01:1; nI0[;'Hn,
% [X,Y] = meshgrid(x,x); Py`N4y~
% [theta,r] = cart2pol(X,Y); pHoEa7:
% idx = r<=1; w,Ee>cV]a
% z = nan(size(X)); QM?#{%31
% n = [0 1 1 2 2 2 3 3 3 3]; $<ld3[l i
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; t'm;:J1
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ^[15&T5
% y = zernfun(n,m,r(idx),theta(idx)); nNXgW
% figure('Units','normalized') mqq;H}
% for k = 1:10 h5yzwj:C?
% z(idx) = y(:,k); %7O?JI[
% subplot(4,7,Nplot(k)) ." $
% pcolor(x,x,z), shading interp ':R,53tjl
% set(gca,'XTick',[],'YTick',[]) v`1,4,;,qs
% axis square cWajrLw
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) x1 1U@jd+1
% end t\$U`V)
% "`asFg
% See also ZERNPOL, ZERNFUN2. K!,<7[MBg
/t-fjB{=G
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% Paul Fricker 11/13/2006 0EJ(.8hwm
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% Check and prepare the inputs: -X+H2G
% ----------------------------- gl&5l1&
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) "`[!L z
error('zernfun:NMvectors','N and M must be vectors.') >hH0Q5aL
end Y?534l)j
e*j.
ly WwGR
if length(n)~=length(m) fqu}Le
error('zernfun:NMlength','N and M must be the same length.') [=%TnT+^9
end -7!&@wuQ
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n = n(:); t tXjn
m = m(:); s}j1"@
if any(mod(n-m,2)) .@-$5Jw
error('zernfun:NMmultiplesof2', ... KsrjdJx, '
'All N and M must differ by multiples of 2 (including 0).') jgS%1/&
end 0P>OJYFr'
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if any(m>n) w-@6|o,S
error('zernfun:MlessthanN', ... g/CxXSv@0
'Each M must be less than or equal to its corresponding N.') 8>/Q1(q0
end M/Pme&%
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if any( r>1 | r<0 ) h d~$WV0#
error('zernfun:Rlessthan1','All R must be between 0 and 1.') m5G \}8|
end kF7V.m/~o
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) _tr<}PnZ
error('zernfun:RTHvector','R and THETA must be vectors.') A8A~!2V
end y0~Ia:y
#"fJa:IYG7
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r = r(:); (Toq^+`c
theta = theta(:); *)]"27^
length_r = length(r); )6~1 ^tD
if length_r~=length(theta) ;@h0qRXW:h
error('zernfun:RTHlength', ... 7m#[!%D
'The number of R- and THETA-values must be equal.') }bU8G '
end b%f[p/no
/WPv\L
R_sC! -
% Check normalization: qz4^{
% -------------------- YC]L)eafo`
if nargin==5 && ischar(nflag) w<9>Q1(
isnorm = strcmpi(nflag,'norm'); yk2 !8
if ~isnorm @5)
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error('zernfun:normalization','Unrecognized normalization flag.') midsnG+jnf
end g/UaYCjM
else hC_Vts[v/
isnorm = false; fQ+VT|jzx
end VCy5JH
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V E2tq k%
% Compute the Zernike Polynomials avp;*G}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6I_Hd>4
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Ut]+k+ 4
% Determine the required powers of r: ,D6v4<jh
% ----------------------------------- {J/I-=CmML
m_abs = abs(m); Wl^R8w#Z$
rpowers = [];
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for j = 1:length(n) b{DiM098
rpowers = [rpowers m_abs(j):2:n(j)]; sM1RU
end h?\2_s
rpowers = unique(rpowers); `nR %Cav,U
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cD1o"bq
% Pre-compute the values of r raised to the required powers, &@"]+33
% and compile them in a matrix: O$`UCq
% ----------------------------- %[<Y9g,:Q
if rpowers(1)==0 5sde
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);
IGX:H)&*
rpowern = cat(2,rpowern{:}); bt+,0\Vg5
rpowern = [ones(length_r,1) rpowern]; 0h$GI"dR
else tNs~M4TVVH
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 1-I
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rpowern = cat(2,rpowern{:}); 7=4 A;Ybq
end O\;= V`z-
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% Compute the values of the polynomials: ]Pn!nSg
% -------------------------------------- cd;NpN
y = zeros(length_r,length(n)); o7&4G$FX~
for j = 1:length(n) RK9>dkW
s = 0:(n(j)-m_abs(j))/2; J3S&3+2G
pows = n(j):-2:m_abs(j); /7$mxtB5%L
for k = length(s):-1:1 z}}]jR\y?
p = (1-2*mod(s(k),2))* ... LU!1s@
prod(2:(n(j)-s(k)))/ ... ZeasYSo4P
prod(2:s(k))/ ... X_; *`,<T
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... |c-LSs'\
prod(2:((n(j)+m_abs(j))/2-s(k))); kR.wOJ7'
idx = (pows(k)==rpowers); ]0c Pml
y(:,j) = y(:,j) + p*rpowern(:,idx); #:3r4J%+~
end QL"gWr`R
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if isnorm zW{ 6Eg
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); P#GD?FUc
end )&W|QH=AI
end dGH_ z8
% END: Compute the Zernike Polynomials t\j!K2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a
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% Compute the Zernike functions: w`yx=i#
% ------------------------------ "2n;3ByR
idx_pos = m>0; j~ym<-[{a
idx_neg = m<0; &B-[oqC?
G=M] 8+h
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z = y; PNF4>)
if any(idx_pos) AfWl6a?T8:
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); [J\DB)V/
end <4F7@q,V
if any(idx_neg) 7{BnXN[
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); H$!-f>Rxa
end !Cj(A"uqY
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% EOF zernfun 5]"BRn1*