[讨论]如何从zernike矩中提取出zernike系数啊 [复制链接]

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， \}jMC  
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， h$cm:uks  
这个，跟我们用zygo干涉仪直接拟合出的36项zernike系数，有何关系呢？ %@$UIO,(  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ 3h:j.8Z  
FpoHm%+  
%!aU{E|@_  
.sMs_ 5D  
Z\&f"z?L  
function z = zernfun(n,m,r,theta,nflag) >)><u4}  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. h2l;xt
 
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N X{9^$/XsJ  
%   and angular frequency M, evaluated at positions (R,THETA) on the {#,<)wFV\  
%   unit circle.  N is a vector of positive integers (including 0), and /{M<FVXK+|  
%   M is a vector with the same number of elements as N.  Each element ! 'zd(kv<  
%   k of M must be a positive integer, with possible values M(k) = -N(k) c-LzluWi  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ?gH[la  
%   and THETA is a vector of angles.  R and THETA must have the same hor7~u+  
%   length.  The output Z is a matrix with one column for every (N,M) fFQ|dE;cF  
%   pair, and one row for every (R,THETA) pair. pYr"3BwG  
% qJey&_  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike &L o TO+  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), `lf_wB+I  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral kA:Y^2X'  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, SzULy >e  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized 1.hWgWDP  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. #-{<d%qk  
% xtV+Le%  
%   The Zernike functions are an orthogonal basis on the unit circle. FX:`7c]:9  
%   They are used in disciplines such as astronomy, optics, and UwN Vvo  
%   optometry to describe functions on a circular domain. W4^L_p>Tm^  
% i'tMpS3  
%   The following table lists the first 15 Zernike functions. k"wQ9=HP7  
% [W[{ 4 Xu  
%       n    m    Zernike function           Normalization KK|w30\f  
%       -------------------------------------------------- spK8^sh  
%       0    0    1                                 1 Sp`l>BL  
%       1    1    r * cos(theta)                    2 {X{R]  
%       1   -1    r * sin(theta)                    2 st'T._  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) hmy%X`%j  
%       2    0    (2*r^2 - 1)                    sqrt(3) $8EEtr,!  
%       2    2    r^2 * sin(2*theta)             sqrt(6) P.~UUS  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) -D^I;[j_  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) sXLW';Fz  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ' jciX]g  
%       3    3    r^3 * sin(3*theta)             sqrt(8) _nGx[1G( 5  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) F72#vS j  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /:|vJ|dJ  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Im]@#X  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 8R~<$xz  
%       4    4    r^4 * sin(4*theta)             sqrt(10) XF`2*:7  
%       -------------------------------------------------- ,p2UshOmd
 
% \;;M")$  
%   Example 1: 2+]5}'M  
% !R{IEray  
%       % Display the Zernike function Z(n=5,m=1) DE13x*2  
%       x = -1:0.01:1; B|`?hw@g+  
%       [X,Y] = meshgrid(x,x); ns[/M~_r  
%       [theta,r] = cart2pol(X,Y); B-I4(w($  
%       idx = r<=1; n Ja!&G&  
%       z = nan(size(X)); 7?lz$.*Avp  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); S"bN9?;#u  
%       figure vu0Ql1  
%       pcolor(x,x,z), shading interp i4D(8;  
%       axis square, colorbar *CN *G"  
%       title('Zernike function Z_5^1(r,\theta)') 1(' wg!  
% c[@_t.%)  
%   Example 2: "M%R{pGA7  
% #*A'<Zm  
%       % Display the first 10 Zernike functions 79DNNj~  
%       x = -1:0.01:1; VZ]iep  
%       [X,Y] = meshgrid(x,x); Z[O hZ 9  
%       [theta,r] = cart2pol(X,Y); HZrA}|:h  
%       idx = r<=1; F
`=p/IAJK  
%       z = nan(size(X)); uYW4$6S3  
%       n = [0  1  1  2  2  2  3  3  3  3]; Omd;  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 3Tr,waV  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; ]2zM~  
%       y = zernfun(n,m,r(idx),theta(idx)); A;cA|`b  
%       figure('Units','normalized') }G4I9Py  
%       for k = 1:10 KGt:  
%           z(idx) = y(:,k); }i9:k kfq2  
%           subplot(4,7,Nplot(k)) N2:Hdu:  
%           pcolor(x,x,z), shading interp y_PA9#v7  
%           set(gca,'XTick',[],'YTick',[]) cXXZ'y>FP  
%           axis square G1|1Z5r  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ?XKX&ws  
%       end ^[hAj>7_8$  
% ^^q&VL  
%   See also ZERNPOL, ZERNFUN2. @ZEBtM%.O  
'Oa3 6@  
@&T' h}|:  
%   Paul Fricker 11/13/2006 wd:Yy  
nDi^s{  
zC50 @S3|  
, ['}9:f9  
hcVu`Bn  
% Check and prepare the inputs: 2V~E <K-  
% ----------------------------- fY]"_P  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) # epP~J_f  
    error('zernfun:NMvectors','N and M must be vectors.') fW= N  
end he|Q(?  
%/dOV[/  
3ynkf77cn  
if length(n)~=length(m) K6{wM  
    error('zernfun:NMlength','N and M must be the same length.') ?NBae\6r  
end 6R :hsC$  
%9YY \a {  
XPhP1 ^>\  
n = n(:); Jm!,=}oP'  
m = m(:); Kebr>t8^  
if any(mod(n-m,2)) Q{~g<G  
    error('zernfun:NMmultiplesof2', ... 9]Jv >_W*  
          'All N and M must differ by multiples of 2 (including 0).') ?}`-?JB1  
end ^%!{qAp}Z  
8K4^05*S
 
l8~(bq1  
if any(m>n) >/ _#+,  
    error('zernfun:MlessthanN', ... (iKJ~bJ  
          'Each M must be less than or equal to its corresponding N.') xLed];2G  
end
S(@kdL  
|GMo"[  
iM
!Ya!  
if any( r>1 | r<0 ) ")KqPD6k  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') _DxHJl  
end -k + jMH  
hh4R  
?22U0UF  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) cr;:5D%_  
    error('zernfun:RTHvector','R and THETA must be vectors.') aEdA'>  
end K/9Jx(I,qL  
:]:)c8!6  
x[mz`0  
r = r(:); ;PaU"z+Je~  
theta = theta(:); qu^g~"s  
length_r = length(r); `h'+4  
if length_r~=length(theta) RB4n>&Y  
    error('zernfun:RTHlength', ... ;6@sC[  
          'The number of R- and THETA-values must be equal.') brp3xgQ`]  
end he(K  
S ,F[74K  
z5gVP8*z5  
% Check normalization: Uha.8  
% -------------------- 7:B/?E  
if nargin==5 && ischar(nflag) ~!ooIwNNz  
    isnorm = strcmpi(nflag,'norm'); YE@yts  
    if ~isnorm \k5"&]I3  
        error('zernfun:normalization','Unrecognized normalization flag.') A6[FH\f  
    end n*"r
!&Dg  
else dC,C[7\  
    isnorm = false; NC
h-BinK@  
end N!ihj:,  
eL~xS: VT  
W,EIBgR(R5  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~AjPa}@ f  
% Compute the Zernike Polynomials umns*U%T;  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% GXxI=,L8F  
x^@oY5}cr  
QM8Ic,QFvo  
% Determine the required powers of r: c2NB@T9'v  
% ----------------------------------- {C&Uq#V  
m_abs = abs(m); lrZ]c:%k  
rpowers = []; XB7*S*"!  
for j = 1:length(n) h
Zfj$|<  
    rpowers = [rpowers m_abs(j):2:n(j)]; g"748LY>=p  
end \dCGu~bT  
rpowers = unique(rpowers); vyDxX  
keC'/\e  
{
@CQ (  
% Pre-compute the values of r raised to the required powers, MrzD ah9UG  
% and compile them in a matrix: |kK5:\H  
% ----------------------------- sJKr%2nVV  
if rpowers(1)==0 "a].v 8l!  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); t
x7 zG.,  
    rpowern = cat(2,rpowern{:}); M?YNK]   
    rpowern = [ones(length_r,1) rpowern];  >%;i@"  
else W:
8MqVm34  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ]=t}8H  
    rpowern = cat(2,rpowern{:}); ,
r*Kxy  
end 27 XM&ZrZ  
lIS`_H}  
3F]Dh^IR9  
% Compute the values of the polynomials: 8!|vp7/  
% -------------------------------------- IQU1 JVkZ  
y = zeros(length_r,length(n)); .O"a:^i  
for j = 1:length(n) CIMI?  
    s = 0:(n(j)-m_abs(j))/2; ;&<N1  
    pows = n(j):-2:m_abs(j); W6T4Zsg  
    for k = length(s):-1:1 Jy/< {7j  
        p = (1-2*mod(s(k),2))* ... x?o#}:S  
                   prod(2:(n(j)-s(k)))/              ... iO?AY  
                   prod(2:s(k))/                     ... 7YD+zd:  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... o)XrC   
                   prod(2:((n(j)+m_abs(j))/2-s(k))); nEu:& 4  
        idx = (pows(k)==rpowers); qK7:[\T|?T  
        y(:,j) = y(:,j) + p*rpowern(:,idx); %d];h  
    end - (WH+  
     ('J@GTe@xj  
    if isnorm -_nQn  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); f$QkzWvr  
    end <&Xl b0  
end _!1LV[x!s  
% END: Compute the Zernike Polynomials 0F-{YQr>  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,V,mz?d^9  
?Fx~_
GT  
lXTE#,XVf  
% Compute the Zernike functions: %B\x %e;P  
% ------------------------------ Qu[QcB{ro-  
idx_pos = m>0; .F8[;+  
idx_neg = m<0; ^Zz^h@+  
B?i#m^S  
KGM__ZO.  
z = y; 0zNbux_  
if any(idx_pos) 2|^@=.4\  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); :.ZWYze  
end ,B'=$PO%  
if any(idx_neg) iH4LZ  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); H2BRId  
end #dae^UjM  
#?w07/~L  
[TOo9W  
% EOF zernfun NH|I>vyN  

支持一下，慢慢研究

顶顶·········

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)  0J'^<GTL  

0x~+=GUN  
DDE还是手动输入的呢？  it H  
|u%;"N'p)  
zygo和zemax的zernike系数，类型对应好就没问题了吧

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

慢慢研究，这个专业性很强的。用的人又少。