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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, JStEOQF4  
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, mxu!$wx  
这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? j,SZJ{ebXg  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? E$ &bl  
AX'-}5T=  
3.g4X?=zd  
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function z = zernfun(n,m,r,theta,nflag) w~jm0jK]  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. C rl:v8  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N {t.S_|IE  
%   and angular frequency M, evaluated at positions (R,THETA) on the ori[[~OyB  
%   unit circle.  N is a vector of positive integers (including 0), and _ b</ ::Tp  
%   M is a vector with the same number of elements as N.  Each element P}>>$$b\Yi  
%   k of M must be a positive integer, with possible values M(k) = -N(k) sTep2W.9  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 'H4?V  
%   and THETA is a vector of angles.  R and THETA must have the same +EqL|  
%   length.  The output Z is a matrix with one column for every (N,M) gjFQDrz(  
%   pair, and one row for every (R,THETA) pair. R3LIN-g(  
% 1_]%,  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike )O$S3ojZ  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), GXNkl?#  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral y+V>,W)r7  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Y7 K2@257  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized `s3:Vsv4  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. YfMs~}h,  
% U!K#g_}  
%   The Zernike functions are an orthogonal basis on the unit circle. z]LVq k  
%   They are used in disciplines such as astronomy, optics, and g!r) yzK  
%   optometry to describe functions on a circular domain. DRTT3;,N  
% VVpJ +  
%   The following table lists the first 15 Zernike functions. OECVExb@eH  
% =vriraV"  
%       n    m    Zernike function           Normalization \:'6_K  
%       -------------------------------------------------- -V[!qI  
%       0    0    1                                 1 .I$+ E  
%       1    1    r * cos(theta)                    2 1CM 8P3  
%       1   -1    r * sin(theta)                    2 hd[t&?{=  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) wlslG^^(!  
%       2    0    (2*r^2 - 1)                    sqrt(3) Dkh=(+> <  
%       2    2    r^2 * sin(2*theta)             sqrt(6) w>}n1Nc$G  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) ~r'ApeI9  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) qPJSVo  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ;B(16&l=q  
%       3    3    r^3 * sin(3*theta)             sqrt(8) 86dz Jh  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) v6E5#pse8  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) zy8+~\a+Y&  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) =NnG[#n%  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,_D@ggL-  
%       4    4    r^4 * sin(4*theta)             sqrt(10) /F''4%S?E  
%       -------------------------------------------------- hx/A215L  
% (?lT @RY/  
%   Example 1: r>PKl'IbE  
% CyB4apJ  
%       % Display the Zernike function Z(n=5,m=1) 5B8fz;l= B  
%       x = -1:0.01:1; 8]O#L}"  
%       [X,Y] = meshgrid(x,x); #e[r0f?U  
%       [theta,r] = cart2pol(X,Y); 7 s2*VKr  
%       idx = r<=1; _F^NX%  
%       z = nan(size(X)); a5d_= :S ;  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); :<0lCj  
%       figure cS@p`A7Tpo  
%       pcolor(x,x,z), shading interp  Bs>S2]  
%       axis square, colorbar ~DB:/VSmu  
%       title('Zernike function Z_5^1(r,\theta)') ]@}hyM[D;  
% huR ^l  
%   Example 2: se}$/Y}t  
% X &G]ci  
%       % Display the first 10 Zernike functions XaoVv2=G~  
%       x = -1:0.01:1; D5].^*AbZ  
%       [X,Y] = meshgrid(x,x); ymnK`/J!Q  
%       [theta,r] = cart2pol(X,Y); A2\3.3  
%       idx = r<=1; Y`6<:8[?  
%       z = nan(size(X)); A_2lG!! 6  
%       n = [0  1  1  2  2  2  3  3  3  3]; g0s4ZI+T  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; h gwS_L  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; ?[WUix;  
%       y = zernfun(n,m,r(idx),theta(idx)); Nd@/U c  
%       figure('Units','normalized') vkM_a}%<  
%       for k = 1:10 6{g&9~V  
%           z(idx) = y(:,k); wsc=6/#u  
%           subplot(4,7,Nplot(k)) U^DR'X=  
%           pcolor(x,x,z), shading interp i1]}Q$  
%           set(gca,'XTick',[],'YTick',[]) Z[,,(M  
%           axis square AXnKhYlu  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) - ku8n%u  
%       end *TCV}=V G  
% hQNUA|Q=%  
%   See also ZERNPOL, ZERNFUN2. o>m*e7l,  
) :Px`] 5  
f3h]t0M  
%   Paul Fricker 11/13/2006 Y;dqrA>@  
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y`/:E<fVk  
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^?A>)?Sq  
% Check and prepare the inputs: [ p(0g;bx  
% ----------------------------- W*n|T{n  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) UF}Ji#fqn  
    error('zernfun:NMvectors','N and M must be vectors.') <Skf n`).  
end &0d5".|s  
&b-&0 rTqz  
tZ*>S]qD  
if length(n)~=length(m) (#qQ;ch  
    error('zernfun:NMlength','N and M must be the same length.') vo~Qo;m  
end $`lGPi(Jc  
wARd^Iw  
d*@K5?O.  
n = n(:); %.fwNS  
m = m(:); TIF  =fQ  
if any(mod(n-m,2)) Q]dKyMSSA  
    error('zernfun:NMmultiplesof2', ... y"K[#&,0  
          'All N and M must differ by multiples of 2 (including 0).') hxw6^EA  
end J$`5KbT3  
o+- 0`!yj  
8\PI1U  
if any(m>n) tCu.Fc@  
    error('zernfun:MlessthanN', ... bcAk$tA2  
          'Each M must be less than or equal to its corresponding N.') -f?,%6(1  
end ItZ*$I1<  
9w1`_r[J  
U_UN& /f  
if any( r>1 | r<0 ) qmNG|U&  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') R#rfnP >  
end !?K#f?x<?  
Tv|i CYB?  
I-Am9\   
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) e Dpt1  
    error('zernfun:RTHvector','R and THETA must be vectors.') {rygIl{V  
end gTd r  
3wPUP+)c7  
2cZgG^  
r = r(:); i7&ay\+@  
theta = theta(:); [LV>z  
length_r = length(r); "DX 2Mu=  
if length_r~=length(theta) iRV=I,  
    error('zernfun:RTHlength', ... w 5t|C>  
          'The number of R- and THETA-values must be equal.') jm'^>p,9G  
end {GGP8  
tK 6=F63e  
AMK(-=  
% Check normalization: vVjk9_Ul  
% -------------------- I:;umyRH  
if nargin==5 && ischar(nflag) |>wGl  
    isnorm = strcmpi(nflag,'norm'); 5d-rF:#  
    if ~isnorm XXXQAY-,C  
        error('zernfun:normalization','Unrecognized normalization flag.') B!4~A{  
    end g]d0B!Ar~  
else !';;q  
    isnorm = false; ,=: -&~?  
end H6lZ<R{=  
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e^N}(Kpy  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% y<l(F?_  
% Compute the Zernike Polynomials m@",Zr `f=  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% U[Lr+nKo\  
k/)h@K8@  
8KsPAK_  
% Determine the required powers of r: a/[)A _-  
% ----------------------------------- $M$-c{>s  
m_abs = abs(m); fGWXUJ  
rpowers = []; =}Yz[-I  
for j = 1:length(n) q RRvZhf  
    rpowers = [rpowers m_abs(j):2:n(j)]; :*YnH&  
end W< $!H V$  
rpowers = unique(rpowers); fT YlIT9  
bKEiS8x  
jVqpokWH  
% Pre-compute the values of r raised to the required powers, F(Je$c/J|~  
% and compile them in a matrix: B#3Q4c$  
% ----------------------------- UtR wZ(09  
if rpowers(1)==0 eYevj[c;  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); bL5u;iy)  
    rpowern = cat(2,rpowern{:}); 3u< ntx ><  
    rpowern = [ones(length_r,1) rpowern]; S F da?>  
else >Sb3]$$  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); pm[+xM9PB  
    rpowern = cat(2,rpowern{:}); h05<1>?|  
end x0lAJaG  
PZI6{KOis  
?P/73p  
% Compute the values of the polynomials: 5kojh _\  
% -------------------------------------- 8Y:x+v5  
y = zeros(length_r,length(n)); )jh~jU?c@  
for j = 1:length(n) 3PlIn0+LX  
    s = 0:(n(j)-m_abs(j))/2; lNTbd"}$:  
    pows = n(j):-2:m_abs(j); *;U<b  
    for k = length(s):-1:1 i1*0'x  
        p = (1-2*mod(s(k),2))* ... rbl^ aik  
                   prod(2:(n(j)-s(k)))/              ... x~K79Mya  
                   prod(2:s(k))/                     ... AJ)&+H  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... <,X=M6$0n  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); 7y_<BCx h  
        idx = (pows(k)==rpowers); D0>Pc9  
        y(:,j) = y(:,j) + p*rpowern(:,idx); }'K-1:  
    end  GInw7  
     5Vai0Qfcu:  
    if isnorm _(I)C`8m  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); vin3 i&k  
    end unKgOvtj  
end e0j4t-lL  
% END: Compute the Zernike Polynomials dnh~An 9  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0 ZSn r+  
=ADOf_n}  
YOUB%N9+  
% Compute the Zernike functions: G,<l}(tEG  
% ------------------------------ lQy-&d|=#^  
idx_pos = m>0; wuM'M<J@  
idx_neg = m<0; {|B[[W\TN  
l]gW_wUQd  
Ey=}bBx  
z = y; |sEuhP\A3  
if any(idx_pos) y|zIu I-p  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); LF#[$ so{i  
end d8U<V<H<  
if any(idx_neg) 5"X@<;H%  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ;>S|?M4GZ  
end >(.Y%$9"E  
G6+6u Wvl  
s+z5"3'n  
% EOF zernfun {s@ 0<!  
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)  `@v;QLD"d<  
f-ceDn  
DDE还是手动输入的呢? 57 Bx-  
|0?v4%g  
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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