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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, C&K%Q3V  
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, qdvGBdF  
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? H\ 8.T:>  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? U/wY;7{)#  
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function z = zernfun(n,m,r,theta,nflag) qS+'#Sn  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle.  FxD\F  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ?^5W.`Y2i  
%   and angular frequency M, evaluated at positions (R,THETA) on the Y -7x**I  
%   unit circle.  N is a vector of positive integers (including 0), and N{L]H _=  
%   M is a vector with the same number of elements as N.  Each element %[&cy'  
%   k of M must be a positive integer, with possible values M(k) = -N(k) ^ D?;K8a-l  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Uw<Lt"ls.  
%   and THETA is a vector of angles.  R and THETA must have the same 82efqzT  
%   length.  The output Z is a matrix with one column for every (N,M)  2gb49y~  
%   pair, and one row for every (R,THETA) pair. "JbFbcj  
% 6D/5vM1  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike IeZ}`$[H  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ;{j:5+'  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral K;gm^  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, c|hKo[r)  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized laKMQLtv  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. wC..LdSR  
% ,K8PumM_  
%   The Zernike functions are an orthogonal basis on the unit circle. VCkhK9(N  
%   They are used in disciplines such as astronomy, optics, and (gs"2  
%   optometry to describe functions on a circular domain. z2wR]G5!  
% nYTI\f/8v  
%   The following table lists the first 15 Zernike functions. |o|0qG@g  
% +SZ#s :#SE  
%       n    m    Zernike function           Normalization /M\S^ !g@  
%       -------------------------------------------------- 3,S5>~R=  
%       0    0    1                                 1 v=iz*2+X  
%       1    1    r * cos(theta)                    2 M@ ! {m  
%       1   -1    r * sin(theta)                    2 akrEZ7A  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) e8--qV#<  
%       2    0    (2*r^2 - 1)                    sqrt(3) ?v8B;="#w  
%       2    2    r^2 * sin(2*theta)             sqrt(6) YmNBtGhT  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) =y[eQS$  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) j;J4]]R;o  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) `b# w3 2  
%       3    3    r^3 * sin(3*theta)             sqrt(8) Gk'J'9*  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) H:a(&Zb  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) P;mmK&&  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) +w%MwPC7`  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) htkn#s~=  
%       4    4    r^4 * sin(4*theta)             sqrt(10) V~NS<!+q  
%       -------------------------------------------------- *~:4&$  
% |!flR? OU  
%   Example 1: rR^VW^|f  
% "a>%tsl$K  
%       % Display the Zernike function Z(n=5,m=1) Cf@WjgR  
%       x = -1:0.01:1; oT_k"]~Q~2  
%       [X,Y] = meshgrid(x,x); e nDjP  
%       [theta,r] = cart2pol(X,Y); 57%:0loW  
%       idx = r<=1; rF>:pS,`&  
%       z = nan(size(X)); $l0^2o=  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); h8 $lDFo  
%       figure ){5  $8  
%       pcolor(x,x,z), shading interp . m@Sk`s  
%       axis square, colorbar kYmkKl_  
%       title('Zernike function Z_5^1(r,\theta)') mWfzL'*  
% .y#@~H($  
%   Example 2: '!b1~+PV  
% 7z5AI!s_  
%       % Display the first 10 Zernike functions Ym2![FC1  
%       x = -1:0.01:1; yLipuMNV  
%       [X,Y] = meshgrid(x,x); ]* ':  
%       [theta,r] = cart2pol(X,Y); AL3zE=BL  
%       idx = r<=1; &5~bJ]P   
%       z = nan(size(X)); +p>tO\mo  
%       n = [0  1  1  2  2  2  3  3  3  3]; G%Wjtrpj  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Z_ak4C  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; z;2kKQZm  
%       y = zernfun(n,m,r(idx),theta(idx)); GbBcC#0  
%       figure('Units','normalized') q: ?6  
%       for k = 1:10 nH/V2> Lm  
%           z(idx) = y(:,k); cQA;Y!Q #  
%           subplot(4,7,Nplot(k)) Ro$l/lXl8t  
%           pcolor(x,x,z), shading interp u01x}Ff~6  
%           set(gca,'XTick',[],'YTick',[]) `G@]\)-!  
%           axis square ?2aglj*"v,  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) at/besW  
%       end rB< UOe  
% @m<xpe l  
%   See also ZERNPOL, ZERNFUN2. C Rw.UC\  
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QZYD;&iY&  
%   Paul Fricker 11/13/2006 wS hsu_(i  
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i%:oO KI  
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% Check and prepare the inputs: Kj;Q;Ii  
% ----------------------------- DrC4oxS 1  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) MOJKz!%  
    error('zernfun:NMvectors','N and M must be vectors.') =NQDxt}  
end S)>L 0^M1  
?|w>."F  
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if length(n)~=length(m) Z[[q W f  
    error('zernfun:NMlength','N and M must be the same length.') x32hO;  
end 5.q2<a :  
4703\ HK  
;1'X_tp  
n = n(:); 9!NL<}]{  
m = m(:); h'|{@X  
if any(mod(n-m,2)) p$`71w)'[  
    error('zernfun:NMmultiplesof2', ... (1^AzE%U+Z  
          'All N and M must differ by multiples of 2 (including 0).') wzwEYZN(q  
end m6a`OkP  
'-N `u$3Y  
zn@<>o8hU  
if any(m>n) }~DlOvsq  
    error('zernfun:MlessthanN', ... Pv|g.hH9m  
          'Each M must be less than or equal to its corresponding N.') mCb(B48]%X  
end a`  s2 z  
| (JxtQqQg  
G3 rTzMO  
if any( r>1 | r<0 ) !ow:P8K?  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') >B!E 6ah  
end  z"Miy  
HIsIW%B  
jhgS@g=@ZC  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) >E;kM B  
    error('zernfun:RTHvector','R and THETA must be vectors.') 'y[74?1  
end #>i Bu:\J  
|r>+\" X  
_~/F-  
r = r(:); aMe]6cWHV>  
theta = theta(:); r'/&{?Je/  
length_r = length(r); Kkcb' aDR  
if length_r~=length(theta) ?<)4_  
    error('zernfun:RTHlength', ... Tsch:r S  
          'The number of R- and THETA-values must be equal.') ' ^E7T'v%  
end b"Hc==`  
&&T\PspM  
Z}#'.y\ f  
% Check normalization: CT1@J-np  
% -------------------- 1y'8bt~7Pf  
if nargin==5 && ischar(nflag) VwvL  
    isnorm = strcmpi(nflag,'norm'); o&0fvCpW  
    if ~isnorm o@:${> jw  
        error('zernfun:normalization','Unrecognized normalization flag.') _N)/X|=~s  
    end 5.! OC5tO  
else gR1vUad7  
    isnorm = false; |>|f?^  
end ?1w{lz(P  
{*mf Is  
(K xI*  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G5*"P!@6  
% Compute the Zernike Polynomials ]8@s+ N  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jV{?.0/h|  
D+#OB|&Dn  
I]Ev6>=;  
% Determine the required powers of r: xB-\yWDZe  
% ----------------------------------- :j^IXZW  
m_abs = abs(m); M^IEu }  
rpowers = []; K|L&mL&8  
for j = 1:length(n) PWciD '!  
    rpowers = [rpowers m_abs(j):2:n(j)]; qlSI|@CO  
end c"KN;9c,  
rpowers = unique(rpowers); |BGB60}]f  
7[=\bL  
lCafsIB  
% Pre-compute the values of r raised to the required powers, +pUG6.j%  
% and compile them in a matrix: ]31>0yj[Q  
% ----------------------------- Z9wKjxu+  
if rpowers(1)==0 9K!kU6Gh  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); )7]la/0  
    rpowern = cat(2,rpowern{:}); (A(j.[4a  
    rpowern = [ones(length_r,1) rpowern]; s>J\h  
else D/[;Y<X#V  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); \-Vja{J]  
    rpowern = cat(2,rpowern{:}); CP0;<}k  
end :j2?v(jT_l  
M$u.lI  
H 48YX(HI  
% Compute the values of the polynomials: ^4/   
% -------------------------------------- 0<i8 ;2KD  
y = zeros(length_r,length(n)); |j}D2q=  
for j = 1:length(n) '4KN  
    s = 0:(n(j)-m_abs(j))/2; /a,"b8  
    pows = n(j):-2:m_abs(j); lA{JpH_Y8s  
    for k = length(s):-1:1 jOUM+QO  
        p = (1-2*mod(s(k),2))* ... dNu?O>=  
                   prod(2:(n(j)-s(k)))/              ... X9 N4  
                   prod(2:s(k))/                     ... o$QC:%[#  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ^[x6p}$  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); *@I/TX'\rY  
        idx = (pows(k)==rpowers); 7Pe<0K)s(  
        y(:,j) = y(:,j) + p*rpowern(:,idx); 5GK> ~2c(  
    end ^(kmFUV,Z  
     @.&KRAZ  
    if isnorm ?B+]Ex(\B,  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ^HhV ?Iqg  
    end o9rZ&Q<  
end oRo[WQla  
% END: Compute the Zernike Polynomials bvW3[ V  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LpK? C<?x  
BOfl hoUX  
>,x&L[3  
% Compute the Zernike functions: l{I.l  
% ------------------------------ Sx:JuK@  
idx_pos = m>0; P5KpFL`B  
idx_neg = m<0; 6t\0Ui  
r>#4Sr  
A^c  (  
z = y; 9!_JV;2  
if any(idx_pos) +iqzj-e&e[  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); HV&i! M@T  
end B/*\Ih9y  
if any(idx_neg) ^ Paf-/  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 00B,1Q HP  
end b*(, W  
x) jc  
>*/:"!u  
% EOF zernfun {[4.<|26  
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)  >;]S+^dXY  
i%GiWanG  
DDE还是手动输入的呢? EL%Pv1  
S}VN(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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