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    [讨论]如何从zernike矩中提取出zernike系数啊 [复制链接]

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    离线jssylttc
     
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    只看楼主 倒序阅读 楼主  发表于: 2012-04-23
    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, WR=e$ ;  
    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, ? &ew$%  
    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? w+bQpIP M  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? uYAPGs#k  
    ]%m0PU#  
    I~EQuQ>=  
    LbDhPG`u  
    #L.fGTb  
    function z = zernfun(n,m,r,theta,nflag) f_X]2in  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 6|3$43J,F  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N "; tl>Ot  
    %   and angular frequency M, evaluated at positions (R,THETA) on the NvWwj%6]  
    %   unit circle.  N is a vector of positive integers (including 0), and k2l(!0o|;  
    %   M is a vector with the same number of elements as N.  Each element =NwmhV  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) vRYQ4B4o  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 4lH$BIAW  
    %   and THETA is a vector of angles.  R and THETA must have the same K:fK! /  
    %   length.  The output Z is a matrix with one column for every (N,M) >I Aw Nr  
    %   pair, and one row for every (R,THETA) pair. $QmP' <  
    % :^FOh*H  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ipnvw4+  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), -Y%#z'^-  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral O.CRF-` t  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Ia$&SS)K  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized )Ac+5bs  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. MjNCn&c  
    % Ce}wgKzr  
    %   The Zernike functions are an orthogonal basis on the unit circle. h=um t<&D  
    %   They are used in disciplines such as astronomy, optics, and ~hPp)- A  
    %   optometry to describe functions on a circular domain. h|"98PI  
    % .P.TqT@)r  
    %   The following table lists the first 15 Zernike functions. 4;W eB   
    % 'WkDp a  
    %       n    m    Zernike function           Normalization :\x53-&hO4  
    %       -------------------------------------------------- d9h"Q  
    %       0    0    1                                 1 Gd1%6}<~  
    %       1    1    r * cos(theta)                    2 qlmz@kTb  
    %       1   -1    r * sin(theta)                    2 8;/`uB:zV  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 7!.%HhU0  
    %       2    0    (2*r^2 - 1)                    sqrt(3) @$z/=gsy  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) "knSc0 ,u  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) lG,/tMy  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 'CsD[<  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) r 11:T3  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) B5pM cw  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 6?Ul)'  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) n}PK0  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) )vO;=% GQ  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ~` v 7  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) .g_B KeU  
    %       -------------------------------------------------- z|[#6X6tT  
    % fRC(Yyx  
    %   Example 1: EU.vw0}u8  
    % Z W` Ur>  
    %       % Display the Zernike function Z(n=5,m=1)  `W< 7.  
    %       x = -1:0.01:1; #=UEx  
    %       [X,Y] = meshgrid(x,x); p"f=[awp  
    %       [theta,r] = cart2pol(X,Y); 3/mVdU?U  
    %       idx = r<=1; mz;S*ONlV  
    %       z = nan(size(X)); uhvmh  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); \dSMF,E  
    %       figure rMAH YH9  
    %       pcolor(x,x,z), shading interp [,)yc/{*  
    %       axis square, colorbar |xy r6gY  
    %       title('Zernike function Z_5^1(r,\theta)') | iEhe  
    % mz@`*^7?  
    %   Example 2: XH&Fn+  
    % ysD @yM,  
    %       % Display the first 10 Zernike functions 6z@OGExmd#  
    %       x = -1:0.01:1; rRyBGEj  
    %       [X,Y] = meshgrid(x,x); h"/FqO  
    %       [theta,r] = cart2pol(X,Y); pvM;2  
    %       idx = r<=1; `'9Kj9}   
    %       z = nan(size(X)); w_|R.T\7  
    %       n = [0  1  1  2  2  2  3  3  3  3]; yaV=e1W  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; g9(zJ  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; c0jdZ#H  
    %       y = zernfun(n,m,r(idx),theta(idx)); xevG)m  
    %       figure('Units','normalized') -Qx:-,.a  
    %       for k = 1:10 j|gv0SI_ w  
    %           z(idx) = y(:,k); }r^@Xh  
    %           subplot(4,7,Nplot(k)) 'bp*hqG[  
    %           pcolor(x,x,z), shading interp Vzf{gr?  
    %           set(gca,'XTick',[],'YTick',[]) CZyOAoc<  
    %           axis square ;V]EF  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) &P{  
    %       end %\#s@8=2u  
    % ;m$F~!Y  
    %   See also ZERNPOL, ZERNFUN2. ]z`Y'wSxd  
    Q/r0p>  
    ?T-6|vZA  
    %   Paul Fricker 11/13/2006 6dQa|ACX_  
     {n}6  
    -C.x;@!k  
    Okm&b g  
    R)?b\VK2$  
    % Check and prepare the inputs: f2Frb  
    % ----------------------------- INSI$tA~  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) |VM c,_D  
        error('zernfun:NMvectors','N and M must be vectors.') NfcY30}:  
    end A3ad9?LR[R  
    `C"Slz::  
    1 Y_e1tgmm  
    if length(n)~=length(m) <y5V],-U  
        error('zernfun:NMlength','N and M must be the same length.') iK{q_f\"  
    end 0-cqux2U  
    X1G[&  
    ! {lcF%  
    n = n(:); ZxkX\gl91  
    m = m(:); *9)7.} uY  
    if any(mod(n-m,2)) AH`D&V  
        error('zernfun:NMmultiplesof2', ... ]4SnOSV?S  
              'All N and M must differ by multiples of 2 (including 0).') p'1n'|$e  
    end -'+|r]  
    `HU`=a&d  
    8[5%l7's  
    if any(m>n) }CZ,WJz=  
        error('zernfun:MlessthanN', ... EB jiSQw  
              'Each M must be less than or equal to its corresponding N.') @<Au|l`  
    end 1) V,>)Ak  
     =Run  
    @OAX#iQl  
    if any( r>1 | r<0 ) FV^CSaN[R  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 6"Q/Y[y  
    end fEc}c.!5  
    -H~g+i*J  
    {LTb-CB  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) SPW @TF1  
        error('zernfun:RTHvector','R and THETA must be vectors.') `Yp\.K z  
    end %lNWaA  
    jzV*V<  
    g(<02t!OT=  
    r = r(:); GyJp! xFB  
    theta = theta(:); :/ns/~5xa:  
    length_r = length(r); ~jAOGo/&6  
    if length_r~=length(theta) b6_*ljM  
        error('zernfun:RTHlength', ... C3-l(N1O{  
              'The number of R- and THETA-values must be equal.') 65AXUTg  
    end [YP8z~  
    ,R0@`t1 p  
    W ]5kM~Q@  
    % Check normalization: 8 W8ahG}  
    % -------------------- gVCkj!{  
    if nargin==5 && ischar(nflag) q:#,b0|bv  
        isnorm = strcmpi(nflag,'norm'); ;5#P?   
        if ~isnorm *{tn/ro6a  
            error('zernfun:normalization','Unrecognized normalization flag.') FOpOS?Cr'  
        end  S]ZO*+  
    else &Th/Qv}[  
        isnorm = false; @;_r `AT7  
    end lJoMJS;S]}  
    F0:Fv;  
    kM]:~b2  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |tz1'YOB  
    % Compute the Zernike Polynomials ',8]vWsl  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  1JgnuBX"  
    UV)[a%/SB&  
    W5}.WFu  
    % Determine the required powers of r: m}6GVQ'Q  
    % ----------------------------------- C]*9:lK  
    m_abs = abs(m); %^^2  
    rpowers = []; xuO5|{h  
    for j = 1:length(n) {.SN  
        rpowers = [rpowers m_abs(j):2:n(j)]; hU5[k/ q  
    end hF+YZU]rT  
    rpowers = unique(rpowers); tc@v9`^_  
    jD0^,aiG  
     \A:m<::  
    % Pre-compute the values of r raised to the required powers, VJD$nh #M5  
    % and compile them in a matrix: L':;Vv~-  
    % ----------------------------- /fA:Fnv  
    if rpowers(1)==0 BMU~1[r  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); e`4OlM]  
        rpowern = cat(2,rpowern{:}); jnt0,y A  
        rpowern = [ones(length_r,1) rpowern]; 9C[3w[G~C  
    else Cst\_j  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ^Ot+,l)  
        rpowern = cat(2,rpowern{:}); *Au4q<   
    end 82Nh;5T r  
    JoKD6Q1D  
    Z4}Yw{=f  
    % Compute the values of the polynomials: B9iH+ ]W  
    % -------------------------------------- wED~^[]f  
    y = zeros(length_r,length(n)); W>dS@;E  
    for j = 1:length(n) Slq=;TDp  
        s = 0:(n(j)-m_abs(j))/2; }CaL:kY8  
        pows = n(j):-2:m_abs(j); y&lj+j  
        for k = length(s):-1:1 B^U5= L[:p  
            p = (1-2*mod(s(k),2))* ... EU ThH.  
                       prod(2:(n(j)-s(k)))/              ... !fwLC"QC  
                       prod(2:s(k))/                     ... +%eMm.(  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Cv{rd##Y8  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); IyOujdKa  
            idx = (pows(k)==rpowers);  gsc/IUk  
            y(:,j) = y(:,j) + p*rpowern(:,idx); U?>P6p  
        end ZNFn^iuQ  
         l+kI4B7--  
        if isnorm qOZe\<.V<  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); "6 dC  
        end +;`Cm.Iu  
    end ub}t3#  
    % END: Compute the Zernike Polynomials p(Y'fd}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mY(~94{d  
    @s2z/ h0H  
    |?V6__9  
    % Compute the Zernike functions: ],>Z' W  
    % ------------------------------ eXnMS!g%Z  
    idx_pos = m>0; cliP+#  
    idx_neg = m<0; \M=" R-&b  
    J.?6a:#bU/  
    *M/3 1qI  
    z = y; }_3<Q\j  
    if any(idx_pos) zjM+F{P8  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 5Tb93Q@c  
    end `P)atQ  
    if any(idx_neg) m]=|%a6  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); &Dqg<U  
    end 3tS~/o+]  
    B2 Tp;)  
    ?t'O\n)M  
    % EOF zernfun fseHuL=~  
     
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    离线phoenixzqy
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    只看该作者 1楼 发表于: 2012-04-23
    慢慢研究,这个专业性很强的。用的人又少。
    2024年6月28-30日于上海组织线下成像光学设计培训,欢迎报名参加。请关注子在川上光学公众号。详细内容请咨询13661915143(同微信号)
    离线sansummer
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    只看该作者 2楼 发表于: 2012-04-27
    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
    离线jssylttc
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    只看该作者 3楼 发表于: 2012-05-14
    回 sansummer 的帖子
    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,cV(  
    Nz'fMdaX,  
    DDE还是手动输入的呢? %d c=Q SL  
    _U?   
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    离线jssylttc
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    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    离线18257342135
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    只看该作者 5楼 发表于: 2016-12-13
    支持一下,慢慢研究