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jssylttc 2012-04-23 19:23

如何从zernike矩中提取出zernike系数啊

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, <<d#  
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, ZXQ5fBx  
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? _oc6=Z  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? bDWL Hdu a  
&{8:XJe*,%  
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function z = zernfun(n,m,r,theta,nflag) :} =lE"2  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. )Q`Ycz-  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ASy?^Jrs5  
%   and angular frequency M, evaluated at positions (R,THETA) on the Gf.ywqE$Y$  
%   unit circle.  N is a vector of positive integers (including 0), and )KFxtM-  
%   M is a vector with the same number of elements as N.  Each element e: Sd#H!  
%   k of M must be a positive integer, with possible values M(k) = -N(k) ~2rQ80_  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, l3b=8yn.  
%   and THETA is a vector of angles.  R and THETA must have the same [6l0|Y  
%   length.  The output Z is a matrix with one column for every (N,M) > .NLmzUX  
%   pair, and one row for every (R,THETA) pair. [G8EX3  
% $BehU  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike H3$py|}lL  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), #w|v.35%?  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral )=GPhC/sw  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, b(N\R_IQ~  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized 7 w,D2T  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. i=<;$+tW  
% V $I8iVGL  
%   The Zernike functions are an orthogonal basis on the unit circle. e]1'D  
%   They are used in disciplines such as astronomy, optics, and 1]''@oh{6U  
%   optometry to describe functions on a circular domain. L3\#ufytb  
% \12G,tBH  
%   The following table lists the first 15 Zernike functions. u4FD}nV  
% ')q4d0B`"  
%       n    m    Zernike function           Normalization \ejHM}w3,  
%       -------------------------------------------------- 3\}u#/Vb  
%       0    0    1                                 1 $`GlXiV  
%       1    1    r * cos(theta)                    2 H'#06zP>5  
%       1   -1    r * sin(theta)                    2 =M-=94  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) &u&WP  
%       2    0    (2*r^2 - 1)                    sqrt(3) K!\v ?WbF  
%       2    2    r^2 * sin(2*theta)             sqrt(6) [(Z(8{3i  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) }y*D(`  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) HUjX[w8  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) C)j/!+nh  
%       3    3    r^3 * sin(3*theta)             sqrt(8) w.58=Pr  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) [ S  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) u3qx G3  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ?`e@ o?  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) stK}K-=`  
%       4    4    r^4 * sin(4*theta)             sqrt(10) ?l%4 P5  
%       -------------------------------------------------- BhDg\oxZ  
% &G_#=t&  
%   Example 1: U%bm{oVn  
% tgG 8pL  
%       % Display the Zernike function Z(n=5,m=1) m2^vH+wD  
%       x = -1:0.01:1; s i2@k  
%       [X,Y] = meshgrid(x,x); 'i$. _Tx  
%       [theta,r] = cart2pol(X,Y); roc DO8f  
%       idx = r<=1; (<>??(VM  
%       z = nan(size(X)); ZMlBd}H  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); v|~=rvXFC  
%       figure ^cNuEF9  
%       pcolor(x,x,z), shading interp OF`J{`{r  
%       axis square, colorbar N9|J\;fzT  
%       title('Zernike function Z_5^1(r,\theta)') ^z!=,M<+{  
% (U# ,;  
%   Example 2: (bv{1 7K  
% wG MhKZE  
%       % Display the first 10 Zernike functions })<u ~r  
%       x = -1:0.01:1;  'V^M+ng  
%       [X,Y] = meshgrid(x,x); mW!n%f  
%       [theta,r] = cart2pol(X,Y); ~vt*%GN3  
%       idx = r<=1; rfVQX<95=/  
%       z = nan(size(X)); }Fu1Y@M%  
%       n = [0  1  1  2  2  2  3  3  3  3]; x&DqTX?b,  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 8y6dT  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; _+ 9i  
%       y = zernfun(n,m,r(idx),theta(idx)); @2. :fK  
%       figure('Units','normalized') :|kO}NGM  
%       for k = 1:10 w;}5B~).  
%           z(idx) = y(:,k); bP-(N14x+  
%           subplot(4,7,Nplot(k)) @!oN]0`F;  
%           pcolor(x,x,z), shading interp _!zc <&~I  
%           set(gca,'XTick',[],'YTick',[]) ZKrK >X  
%           axis square M2ex 3m  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) >b"@{MZ@t  
%       end I~F&@  
% bC+Z R{M  
%   See also ZERNPOL, ZERNFUN2. iCpm^XT  
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> "rM\ Q  
%   Paul Fricker 11/13/2006 9Br+]F _i  
 kSEA  
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% Check and prepare the inputs: *JX$5bZsI  
% ----------------------------- `^{G`es  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Z?xaXFm_  
    error('zernfun:NMvectors','N and M must be vectors.') tH;9"z# ~  
end MkFWZ9c3  
9;XbyA]  
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if length(n)~=length(m) 7Z:3xb&>   
    error('zernfun:NMlength','N and M must be the same length.') KB~1]cYMp  
end KO8vUR*2R  
6!0NFP~b  
j%3 $ytf|p  
n = n(:); 5!9y nIC+>  
m = m(:); !-T#dU  
if any(mod(n-m,2)) 32+N?[9 *  
    error('zernfun:NMmultiplesof2', ... >CKa?N;  
          'All N and M must differ by multiples of 2 (including 0).') XelFGTE  
end @=w)a  
f 5bX,e)!  
A4l"^dZc  
if any(m>n) ,9 ^ 5  
    error('zernfun:MlessthanN', ... l+6@,TY1U  
          'Each M must be less than or equal to its corresponding N.') i/ o  
end =CFg~8W  
Eyf17  
fXnTqKAfu6  
if any( r>1 | r<0 ) Ke@Bf  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') NM9ViYm>P  
end "Vc|D (g  
}mp`!7?>O  
n5xG4.#G  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) EhO\N\p(Q=  
    error('zernfun:RTHvector','R and THETA must be vectors.') Kc r)W  
end #q34>}O< O  
6C.!+km  
(*eX'^Q)d  
r = r(:); oU3gy[wF;b  
theta = theta(:); 3<FqK\P  
length_r = length(r); }T902RL0  
if length_r~=length(theta) G2[2y-Rv  
    error('zernfun:RTHlength', ... eWYet2!Q  
          'The number of R- and THETA-values must be equal.') &flcJ`  
end hHw1<! M  
)I<VH +6  
A+Je?3/.  
% Check normalization: X+emJ&Z$@  
% -------------------- -$s1k~o  
if nargin==5 && ischar(nflag) zXGI{P0O  
    isnorm = strcmpi(nflag,'norm'); 0=`aXb-  
    if ~isnorm rf$[8d  
        error('zernfun:normalization','Unrecognized normalization flag.') %E, -dw  
    end D0f7I:i1  
else *sfz+8Y  
    isnorm = false; Obc,    
end Q4ii25]*  
]6=opvm  
<9=RLENmY"  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Jl"DMUy[kW  
% Compute the Zernike Polynomials e!i.u'z  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% N9)ERW2`*  
Z#%77!3  
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% Determine the required powers of r: K1<l/ s  
% ----------------------------------- z9#jXC#OdN  
m_abs = abs(m); DML0paOm5  
rpowers = []; |@-y+vbA*  
for j = 1:length(n) gJOswN;([  
    rpowers = [rpowers m_abs(j):2:n(j)]; jT QN(a9Y  
end b[;3y/X  
rpowers = unique(rpowers); n |,}   
E\}Q9, Z$  
'?fn} V  
% Pre-compute the values of r raised to the required powers, 9*|An  
% and compile them in a matrix: @qJv  
% ----------------------------- m} =<@b:l  
if rpowers(1)==0 A,ao2)  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Y50$ 2%kM  
    rpowern = cat(2,rpowern{:}); 8>:2li  
    rpowern = [ones(length_r,1) rpowern]; IZ4jFgpR  
else M[T!AO-S$  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); DNM~/Oo  
    rpowern = cat(2,rpowern{:}); 0Zl1(;hx@  
end s{j3F  
:\1&5Pm]  
gwF@'Uu  
% Compute the values of the polynomials: z!j`Qoh?V9  
% -------------------------------------- %^ E>~  
y = zeros(length_r,length(n)); Ipmr@%~  
for j = 1:length(n) 0$49X  
    s = 0:(n(j)-m_abs(j))/2; a-DE-V Uls  
    pows = n(j):-2:m_abs(j); Y!s/uvRI  
    for k = length(s):-1:1 BQ&h&57K  
        p = (1-2*mod(s(k),2))* ... 55|$Imnf  
                   prod(2:(n(j)-s(k)))/              ... 2Z..~1r  
                   prod(2:s(k))/                     ... 1$2Rs-J  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... B "}GAk}V  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); ~JT{!wcE}o  
        idx = (pows(k)==rpowers); oq<#  
        y(:,j) = y(:,j) + p*rpowern(:,idx); vqxTf)ys  
    end TT&!WbA-Hk  
     m :^,qC  
    if isnorm uTJ?@ ^nq  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Wj*6}N/  
    end &k{@:z  
end If#7SF)n'  
% END: Compute the Zernike Polynomials I2l'y8)d  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -s!PO;qm  
1Q;` <=  
& ='uAw  
% Compute the Zernike functions: "Ii!)n,  
% ------------------------------ V+K.' J ^@  
idx_pos = m>0; yq,5M1vR  
idx_neg = m<0; XAFTLNV>  
XKK*RVs#  
^e 6(#SqR  
z = y; #5I "M WA  
if any(idx_pos) ,^,J[F  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); XZT( :(  
end 1Q$ M/}  
if any(idx_neg) dCyQCA[  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ff hD+-gTU  
end / q!&I  
J8D-a!  
t-Fl"@s  
% EOF zernfun \%VoX` B  
phoenixzqy 2012-04-23 20:38
慢慢研究,这个专业性很强的。用的人又少。
sansummer 2012-04-27 10:22
这个太牛了,我目前只能把zygo中的zernike的36项参数带入到zemax中,但是我目前对其结果的可信性表示质疑,以后多交流啊
jssylttc 2012-05-14 11:28
sansummer:这个太牛了,我目前只能把zygo中的zernike的36项参数带入到zemax中,但是我目前对其结果的可信性表示质疑,以后多交流啊 (2012-04-27 10:22)  YGO@X(ej,  
73.b9mF  
DDE还是手动输入的呢? a*8}~p,  
{ 1+H\ (v  
zygo和zemax的zernike系数,类型对应好就没问题了吧
jssylttc 2012-05-14 11:37
顶顶·········
18257342135 2016-12-13 10:03
支持一下,慢慢研究
查看本帖完整版本: [-- 如何从zernike矩中提取出zernike系数啊 --] [-- top --]

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