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, i 61k
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, /"u37f?[^
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? nM b@
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那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? hEsiAbTyF
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function z = zernfun(n,m,r,theta,nflag) L*A-&9.p3
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Z
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% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N *`Vm ncv3
% and angular frequency M, evaluated at positions (R,THETA) on the A 0k?$ko
% unit circle. N is a vector of positive integers (including 0), and b7Z o~Z
% M is a vector with the same number of elements as N. Each element vI5lp5( -3
% k of M must be a positive integer, with possible values M(k) = -N(k) DmLx"%H3
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, zB`woI28
% and THETA is a vector of angles. R and THETA must have the same uXh:/KO
% length. The output Z is a matrix with one column for every (N,M) pxd=a!(
% pair, and one row for every (R,THETA) pair. d,JDfG)
% )(-;H|]?
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ` K{k0_{
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), >5L_t
% with delta(m,0) the Kronecker delta, is chosen so that the integral [{BY$"b#:
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, SFDTHvXu#_
% and theta=0 to theta=2*pi) is unity. For the non-normalized {o)pwM"@(
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Q^rR }Ws
% Y`bTf@EP>
% The Zernike functions are an orthogonal basis on the unit circle. rHX^bcYK
% They are used in disciplines such as astronomy, optics, and %|D)%|Z
% optometry to describe functions on a circular domain. S\&3t}_
% %sr- xE
% The following table lists the first 15 Zernike functions. IyyBW2
% yivu|q
% n m Zernike function Normalization L8PX SJ
% -------------------------------------------------- tULGfvp
% 0 0 1 1 4#0 3x:/<\
% 1 1 r * cos(theta) 2 y-1e(:GF
% 1 -1 r * sin(theta) 2 o"
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% 2 -2 r^2 * cos(2*theta) sqrt(6) >dt*^}*
% 2 0 (2*r^2 - 1) sqrt(3) M[YFyM(
% 2 2 r^2 * sin(2*theta) sqrt(6) \{lv~I
% 3 -3 r^3 * cos(3*theta) sqrt(8) J}X{8Ds9
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) HN{c)DIm]
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) _f^KP@^j
% 3 3 r^3 * sin(3*theta) sqrt(8) ?";SUku
% 4 -4 r^4 * cos(4*theta) sqrt(10)
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% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ob{'Z]-V
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) D'#Q`H
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #lLUBJ#:
% 4 4 r^4 * sin(4*theta) sqrt(10) ([u|j
% -------------------------------------------------- "[|b,fxR
% x [FLV8`b|
% Example 1: 'Be'!9K*d
% n_e'n|T
% % Display the Zernike function Z(n=5,m=1) UUJQc~=
% x = -1:0.01:1; L9D`hefz
% [X,Y] = meshgrid(x,x); k k3^m1
% [theta,r] = cart2pol(X,Y); sV
% idx = r<=1; MCT1ZZpPr
% z = nan(size(X)); M`Er&nQs
% z(idx) = zernfun(5,1,r(idx),theta(idx));
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% figure XWp8[Cxs
% pcolor(x,x,z), shading interp y~
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% axis square, colorbar J/?Nf2L4
% title('Zernike function Z_5^1(r,\theta)') ~y!'\d>q<
% $>XeC}"x68
% Example 2: i/ilG3m>
% c~Ka) dF|
% % Display the first 10 Zernike functions } p'ZMj&
% x = -1:0.01:1; &[.`xZ(|
% [X,Y] = meshgrid(x,x); !.]JiT'o
% [theta,r] = cart2pol(X,Y); d!y*z
% idx = r<=1; =^nb+}Nz(
% z = nan(size(X)); +J X;T(T
% n = [0 1 1 2 2 2 3 3 3 3]; M<fhQJ
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; H$(bSw$
% Nplot = [4 10 12 16 18 20 22 24 26 28]; \n)',4mY
% y = zernfun(n,m,r(idx),theta(idx)); 1PkCWRpR
% figure('Units','normalized') u]J@65~'b
% for k = 1:10 []>'Dw_r
% z(idx) = y(:,k); '-;[8:y.
% subplot(4,7,Nplot(k)) qos7u91z
% pcolor(x,x,z), shading interp JbMTULA
% set(gca,'XTick',[],'YTick',[]) e`D}[G#
% axis square B[t^u\Fk
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) %iN>4;T8
% end Dm>"c;2
% 5AYOM=O]t
% See also ZERNPOL, ZERNFUN2. W(s4R,j
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% Paul Fricker 11/13/2006 {*ak>Wud
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% Check and prepare the inputs: c3A\~tHW
% ----------------------------- P#,u9EIJ
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) SUncQJJ0S*
error('zernfun:NMvectors','N and M must be vectors.') n|SV)92o1
end (yOkf-e2y
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if length(n)~=length(m)
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error('zernfun:NMlength','N and M must be the same length.') ~GaGDS\V
end ly[LF1t
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n = n(:); l>?c AB[
m = m(:); |?` 5 ~f
if any(mod(n-m,2)) [4Z 31v>
error('zernfun:NMmultiplesof2', ... "/#JC}]
'All N and M must differ by multiples of 2 (including 0).') H"C'<(4*\
end u2V-V#jS
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if any(m>n) qS!r<'F3dP
error('zernfun:MlessthanN', ... n/H
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'Each M must be less than or equal to its corresponding N.') 5gszAvOO
end XH:*J+$O
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if any( r>1 | r<0 ) [|u^:&az
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ] $ew 5%
end j]a$RC#
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) |D3u"Y!:^
error('zernfun:RTHvector','R and THETA must be vectors.') gt&|T
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end |.IH4
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r = r(:); i%glQT
theta = theta(:); [&Xp]:M'D
length_r = length(r); TBhM^\z
if length_r~=length(theta) Tt[zSlIMx
error('zernfun:RTHlength', ... h$>F}n
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'The number of R- and THETA-values must be equal.') )^h6'h`
end ?HZp@&
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ZO}Og&%
% Check normalization: _`$LdqgE
% -------------------- q!c(~UVw
if nargin==5 && ischar(nflag) 0bNvmZ$
isnorm = strcmpi(nflag,'norm'); 6 Z/`p~e
if ~isnorm ]`E+HLEQ'
error('zernfun:normalization','Unrecognized normalization flag.') Nz{dnV{&x;
end Ycm)PU ["
else 4DXbeQs:
isnorm = false; FoefBo?g65
end bIKg>U'5d
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% gjF5~
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% Compute the Zernike Polynomials sE9FT#iE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% XGlt^<`
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