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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， &v+8RY^F=  
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， xf8C$|,  
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？ cvpcadN[  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ c <[?Z7y  
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 X I\zEXO  
function z = zernfun(n,m,r,theta,nflag) 7FMg6z8~  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. j+:q:6=  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N mnM#NT5]  
%   and angular frequency M, evaluated at positions (R,THETA) on the 317Lv \[  
%   unit circle.  N is a vector of positive integers (including 0), and w!7f*  
%   M is a vector with the same number of elements as N.  Each element M0<gea\ =  
%   k of M must be a positive integer, with possible values M(k) = -N(k) F/[vg  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, o$>A;<  
%   and THETA is a vector of angles.  R and THETA must have the same ^$aj,*Aj~  
%   length.  The output Z is a matrix with one column for every (N,M) B*A{@)_  
%   pair, and one row for every (R,THETA) pair. !o2lB^e8  
% QDS=M]  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike NAjK0]SRY  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ]<mXf~zg  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral \?-`?QPux  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ~xqRCf{8  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized &[}T41  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. k9`Bi`wp  
% Bry\"V"'g  
%   The Zernike functions are an orthogonal basis on the unit circle. [ZS}P  
%   They are used in disciplines such as astronomy, optics, and c *(]pM  
%   optometry to describe functions on a circular domain. 8Letpygm  
% az~4sx$+}  
%   The following table lists the first 15 Zernike functions. [ "}0umt  
% UUy|/z%  
%       n    m    Zernike function           Normalization m]J Z@  
%       -------------------------------------------------- R_ojK&%  
%       0    0    1                                 1 lL~T@+J~  
%       1    1    r * cos(theta)                    2 dV<|ztv  
%       1   -1    r * sin(theta)                    2 eLcP.;Z  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) ~WK>+T,%  
%       2    0    (2*r^2 - 1)                    sqrt(3) MNNPBE  
%       2    2    r^2 * sin(2*theta)             sqrt(6) ? &ew$%  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) w+bQpIP M  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ygr[5Tl  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) _B)LRD+Hj  
%       3    3    r^3 * sin(3*theta)             sqrt(8) LD5n_W  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) [>+(zlK"  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `<2y [<y  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ,x}p1EZ  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) "; tl>Ot  
%       4    4    r^4 * sin(4*theta)             sqrt(10) S`TP#uzKu]  
%       -------------------------------------------------- MNOT<(  
% ?y!0QAIXK  
%   Example 1: Ub%+8 M  
% P& C,EE$  
%       % Display the Zernike function Z(n=5,m=1) t0p^0   
%       x = -1:0.01:1; #q40  >)]  
%       [X,Y] = meshgrid(x,x); GQoaBO.  
%       [theta,r] = cart2pol(X,Y); ?J,hv'L]  
%       idx = r<=1; -Y%#z'^-  
%       z = nan(size(X)); d paZ6g  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); sY!PXD0Q  
%       figure /o#!9H   
%       pcolor(x,x,z), shading interp D+d\<":  
%       axis square, colorbar $}r*WZ  
%       title('Zernike function Z_5^1(r,\theta)') Oz!#);v  
% n.p6+^ES  
%   Example 2: ZurQr}  
% 'WkDp a  
%       % Display the first 10 Zernike functions )e|Cd} 2  
%       x = -1:0.01:1; I{AteL  
%       [X,Y] = meshgrid(x,x); QN:gSS{30  
%       [theta,r] = cart2pol(X,Y); S#dkJu]]#  
%       idx = r<=1; g nJe!E  
%       z = nan(size(X)); J6/Mm7R  
%       n = [0  1  1  2  2  2  3  3  3  3]; tpj({   
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; $A,fO~  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; X*VHi  
%       y = zernfun(n,m,r(idx),theta(idx)); Q[`J=  
%       figure('Units','normalized') &Al9%W  
%       for k = 1:10 M@fUZh  
%           z(idx) = y(:,k); LGZ5py=xb  
%           subplot(4,7,Nplot(k)) *`[dC,+`.  
%           pcolor(x,x,z), shading interp .j:[R.  
%           set(gca,'XTick',[],'YTick',[]) cZT;VmC  
%           axis square #z 3tSnmp  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) |rkj$s,  
%       end RX:wt  
% !xyO  
%   See also ZERNPOL, ZERNFUN2. "&%: 9O  
~>zml1aJ6  
 GJW+'-f  
%   Paul Fricker 11/13/2006 W@v@|D@  
U.~, Bwb  
 mz;S*ONlV  
uhvmh  
 \dSMF,E  
% Check and prepare the inputs: rMAH YH9  
% ----------------------------- a(&!{Y1bt  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 1$oVcDLl  
    error('zernfun:NMvectors','N and M must be vectors.') |9ro&KA  
end b}4k-hZL  
evryk,x  
 =A&x d"  
if length(n)~=length(m) NKB,D$!~&  
    error('zernfun:NMlength','N and M must be the same length.') WV_y@H_  
end d)`XG cx{=  
mcAg,~"HB  
 ~Fv&z'R  
n = n(:); sL|lfc'bB  
m = m(:); 2P`QS@v0a=  
if any(mod(n-m,2)) -=,%9r  
    error('zernfun:NMmultiplesof2', ... eSf e s  
          'All N and M must differ by multiples of 2 (including 0).') I9P< !#q>  
end 2MwR jh_  
dk~h  
 l^4[;%*f#l  
if any(m>n) 'bp*hqG[  
    error('zernfun:MlessthanN', ... Vzf{gr?  
          'Each M must be less than or equal to its corresponding N.') CZyOAoc<  
end ;V]EF  
2f(5C*~  
 /'?Fz*b  
if any( r>1 | r<0 ) IQ[ ?ej3W  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') Q>f^*FyOw<  
end Q >[*Y/`I  
}Zu2GU$6  
 )iadu  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) qR0V\OtgY~  
    error('zernfun:RTHvector','R and THETA must be vectors.') 2xRb$QF  
end $+P9@Q$  
+F q`I2l|  
 <Ur(< WTV  
r = r(:); 2h0I1a,7  
theta = theta(:); H pXMPHd  
length_r = length(r); ?z0f5<dL  
if length_r~=length(theta) a*JM2^,HO  
    error('zernfun:RTHlength', ... d!/@+i  
          'The number of R- and THETA-values must be equal.') mN3}wJ}J  
end s mub> V  
[o8a(oC  
 ' 8`{u[:  
% Check normalization: {Pm^G^EP  
% -------------------- b9%}< w  
if nargin==5 && ischar(nflag) -a(f-  
    isnorm = strcmpi(nflag,'norm'); '8>h4s4  
    if ~isnorm Ti`<,TA54  
        error('zernfun:normalization','Unrecognized normalization flag.') > kOca  
    end q:sDNj)R\  
else e'aKI]>a  
    isnorm = false; |sz`w^#  
end ])h={gI  
U I|L;5  
 }CZ,WJz=  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% j{N;2#.u  
% Compute the Zernike Polynomials  !J!zi  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% p3O%|)yV  
\V`O-wcJ]S  
 =MO2M~e!  
% Determine the required powers of r: :7%JD.;W  
% ----------------------------------- *)"U5A/v)  
m_abs = abs(m); Yu=4j9e_mG  
rpowers = []; L^rtypkJ  
for j = 1:length(n) quk~z};R>\  
    rpowers = [rpowers m_abs(j):2:n(j)]; 6~GaFmW=  
end .E!7}O6  
rpowers = unique(rpowers); $+Ke$fq.>  
w=\Lw+X  
 "{;]T  
% Pre-compute the values of r raised to the required powers, x^_Wfkch]  
% and compile them in a matrix: 5P{dey!  
% ----------------------------- xjOy3_Js  
if rpowers(1)==0 3P Twpq1  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @8C^[fDL  
    rpowern = cat(2,rpowern{:}); DrbjqQL+.  
    rpowern = [ones(length_r,1) rpowern]; oQ~Q?o]Ri  
else k\_>/)g  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); G;615p1  
    rpowern = cat(2,rpowern{:}); 6"WR}S0o  
end C-]H+p  
Gdnk1_D>  
 'GQ1;9A57  
% Compute the values of the polynomials: [,Ts;Hy6Q  
% -------------------------------------- R0+v5E  
y = zeros(length_r,length(n)); eJ)Bs20Q  
for j = 1:length(n) LfyycC2E  
    s = 0:(n(j)-m_abs(j))/2; !JUXq  
    pows = n(j):-2:m_abs(j); \*6%o0c  
    for k = length(s):-1:1 |DfYH~@(  
        p = (1-2*mod(s(k),2))* ... rgILOtk[  
                   prod(2:(n(j)-s(k)))/              ... C.@R#a'  
                   prod(2:s(k))/                     ... N J:]jd  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Tz58@VYV  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); =Y|TShKk  
        idx = (pows(k)==rpowers); jEklf0Z  
        y(:,j) = y(:,j) + p*rpowern(:,idx); tr7FV1p  
    end l W'6rat  
     s2g}IZfo  
    if isnorm yXY8 o E  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); gqNd@tYI  
    end hF+YZU]rT  
end gj\r>~S  
% END: Compute the Zernike Polynomials 'mpY2|]\$  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $y\'j5nk3  
kxoJL6IC  
 !l~tBJr*sB  
% Compute the Zernike functions: td q;D  
% ------------------------------ ~FH''}3:3  
idx_pos = m>0; &GwBxJ  
idx_neg = m<0; 2|tZ xlt-  
_]1dm)%  
 ywmx6q4MFL  
z = y; !40{1U&@a`  
if any(idx_pos) xZtA) Bp  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ;M8N%  
end r$;DA<<|<c  
if any(idx_neg) sBS\S  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ckP&N:tC  
end )H S|pS:  
m^U\l9LE  
 Slq=;TDp  
% EOF zernfun }CaL:kY8  
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)  r@"Vbq%  
#`); UAf  
DDE还是手动输入的呢？ "1&C\}.7  
z]=A3!H/Y  
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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