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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, :vz_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, 'j6PL;~c  
    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? L1D{LzlBti  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? WBFG_])  
    rR@ t5  
    *A':^vgk  
    >:!TfuU^R  
    W'hE,  
    function z = zernfun(n,m,r,theta,nflag) /-TJtR4>  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. $`W .9  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N <i``#" /  
    %   and angular frequency M, evaluated at positions (R,THETA) on the x_CB'Rr6  
    %   unit circle.  N is a vector of positive integers (including 0), and ^} P|L  
    %   M is a vector with the same number of elements as N.  Each element qcEiJ}-  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) _I l/ i&  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ))^rk 6  
    %   and THETA is a vector of angles.  R and THETA must have the same Pou-AzEP$  
    %   length.  The output Z is a matrix with one column for every (N,M) .|}ogTEf  
    %   pair, and one row for every (R,THETA) pair. |FG t'  
    % `X'-4/Y  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike W|_ @ju  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), }\k"azQ`  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral F/sXr(7  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, R| [mp%Q  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized "z)dz,&T  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. *T' /5,rX2  
    % w'oP{=y[  
    %   The Zernike functions are an orthogonal basis on the unit circle. fEf ",{I  
    %   They are used in disciplines such as astronomy, optics, and h4N!zj[  
    %   optometry to describe functions on a circular domain. uF_gfjR[m  
    % rT9<_<  
    %   The following table lists the first 15 Zernike functions. )F4H'  
    % xa#0y   
    %       n    m    Zernike function           Normalization y Dg  
    %       -------------------------------------------------- ye=*m  
    %       0    0    1                                 1 Sr Z\]  
    %       1    1    r * cos(theta)                    2 3CK4a,]Dm  
    %       1   -1    r * sin(theta)                    2 N>!RKf:ir  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) >MZWm6M8  
    %       2    0    (2*r^2 - 1)                    sqrt(3) GzxtC  &  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) kKFmTo   
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) iD+Q\l;%  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) mJe;BU"y]  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 8gwJ%"-K  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) hn\<'|n  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) (yIl]ZN*  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) fYU/Jn#  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 4^vEMq8lB  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (oO*|\9u  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) U\'.rT[#  
    %       -------------------------------------------------- H'|b$rP0@  
    % M>^Ho2  
    %   Example 1: | Z'NMJU  
    % ?JO x9;`  
    %       % Display the Zernike function Z(n=5,m=1) }w .[ZeP  
    %       x = -1:0.01:1; gBfYm  
    %       [X,Y] = meshgrid(x,x); VcKufV'  
    %       [theta,r] = cart2pol(X,Y); m-9{@kgAM?  
    %       idx = r<=1; ZRN*.  
    %       z = nan(size(X)); !N:!x[5  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); b)RU+9x &  
    %       figure m`C c U`s  
    %       pcolor(x,x,z), shading interp +InAK>NZ'  
    %       axis square, colorbar l6Wa~E  
    %       title('Zernike function Z_5^1(r,\theta)') )\#w=P  
    % +M-x*;.  
    %   Example 2: |;3Ru vX?+  
    % ?Iy$'am]L  
    %       % Display the first 10 Zernike functions ; mnV)8:F  
    %       x = -1:0.01:1; 'X&sH/>r  
    %       [X,Y] = meshgrid(x,x); lj0"2@z3"E  
    %       [theta,r] = cart2pol(X,Y); (PAkKY}  
    %       idx = r<=1; dx}) 1%  
    %       z = nan(size(X)); !wy Qk  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ~Z-M?8:  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; rmXxid  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; )jk X&7x  
    %       y = zernfun(n,m,r(idx),theta(idx)); 1Q1NircJ  
    %       figure('Units','normalized') <?UbzT7X  
    %       for k = 1:10 Y/cnj n  
    %           z(idx) = y(:,k); G?$|aQ0j  
    %           subplot(4,7,Nplot(k)) (n:d {bKV  
    %           pcolor(x,x,z), shading interp <>JN&#3?  
    %           set(gca,'XTick',[],'YTick',[]) _)s<E9t2N  
    %           axis square Au Ib>@a  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 1|$V  
    %       end !]AM#LJ  
    % 7x` dEi<  
    %   See also ZERNPOL, ZERNFUN2. ArWMbT>Zqw  
    2z9\p%MX  
    |hBX"  
    %   Paul Fricker 11/13/2006 ~/Gx~P]  
    /RD@ [ 8  
    {(;dHF%{  
    lnuf_;0  
    $D{ KXkrd  
    % Check and prepare the inputs: 1OB,UU"S$  
    % ----------------------------- 8xs}neDg*  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) !$&k@#v:  
        error('zernfun:NMvectors','N and M must be vectors.')  1@Abs  
    end gz fs9e  
    xCU^4DO3p  
    ZC}'! $r7  
    if length(n)~=length(m) Y_m/? [:  
        error('zernfun:NMlength','N and M must be the same length.') wh4ik`S 1  
    end 48;6C g  
    }  IJ  
    xs+pCK|  
    n = n(:); %Jy0?WN  
    m = m(:); 533n z8&9@  
    if any(mod(n-m,2)) M- inlZNR  
        error('zernfun:NMmultiplesof2', ... t^eWFX  
              'All N and M must differ by multiples of 2 (including 0).') hBb&-/  
    end N-XOPwx'  
    G.v zz-yG  
    #[ZF'9x  
    if any(m>n) ZH'- >/  
        error('zernfun:MlessthanN', ... 9G njJ  
              'Each M must be less than or equal to its corresponding N.') &o{=  
    end ;',hwo_LBf  
    %`*`HU#X  
    6)<g%bH!  
    if any( r>1 | r<0 ) [O)(0  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') >6fc` 3*!  
    end p4l^b[p  
    OZ{YQ}t{^1  
    JjBG9Rp{  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) <dzfD;  
        error('zernfun:RTHvector','R and THETA must be vectors.') B~S"1EE[  
    end +?"N5%a%F  
    ;. jnRPo";  
    \HR<^xY  
    r = r(:); Xvy3D@o  
    theta = theta(:); c6 O1Z\M@\  
    length_r = length(r); IE/F =Wr  
    if length_r~=length(theta) Wh PwD6l>  
        error('zernfun:RTHlength', ... 7G,{BBB  
              'The number of R- and THETA-values must be equal.') {NmpTb  
    end uu08q<B5b)  
    b*C\0D  
    k^A17Nf`2  
    % Check normalization: S b0p?  
    % -------------------- "J51\8G@@  
    if nargin==5 && ischar(nflag) ]J<2a`IK!  
        isnorm = strcmpi(nflag,'norm'); 4sU*UePr  
        if ~isnorm [!^Q_O  
            error('zernfun:normalization','Unrecognized normalization flag.') }^T7S2_Qy  
        end w8MQA!=l  
    else 2|="!c8K  
        isnorm = false; 8:W," "  
    end 8GRp1'\Hi  
    82w;}(!  
    ~!PAs_O  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vTrjhTa\  
    % Compute the Zernike Polynomials M5$YFGGR  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Gk"o/]Sf  
    \*>r[6]*&5  
    R$[nYw  
    % Determine the required powers of r: +TA 'P$j  
    % ----------------------------------- ;rBd_  
    m_abs = abs(m); ].E89_|O  
    rpowers = []; 5U%J,W  
    for j = 1:length(n) G8]DK3#  
        rpowers = [rpowers m_abs(j):2:n(j)]; I` `S%`h  
    end &Z kY9XO  
    rpowers = unique(rpowers); OR{<)L  
    kNC.^8ryz[  
    h$F.(NIYe  
    % Pre-compute the values of r raised to the required powers, RQaB _bg7  
    % and compile them in a matrix: jO` b&]0  
    % ----------------------------- 2Fi ~GY_  
    if rpowers(1)==0 (#|CL/&  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); [73 \jT  
        rpowern = cat(2,rpowern{:}); )<J|kC\r6c  
        rpowern = [ones(length_r,1) rpowern]; 0F"W~OQ6  
    else yH\3*#+  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); E5^P*6c(  
        rpowern = cat(2,rpowern{:}); R`IFKmA EJ  
    end hW^*b:v{  
    QNH-b9u>8  
    79DzrLu  
    % Compute the values of the polynomials: DC&3=Nd  
    % -------------------------------------- (8Q0?SZN  
    y = zeros(length_r,length(n)); 4rcNBmA,  
    for j = 1:length(n) ~0;l\^  
        s = 0:(n(j)-m_abs(j))/2; W^a-K  
        pows = n(j):-2:m_abs(j); goE \C  
        for k = length(s):-1:1 s} I8:ufT  
            p = (1-2*mod(s(k),2))* ... GJu[af  
                       prod(2:(n(j)-s(k)))/              ... 7H$I9e  
                       prod(2:s(k))/                     ... |4$.mb.  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... M2pe*z  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); :i{Svb*_'  
            idx = (pows(k)==rpowers); Ri<7!Y?l  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 4AIo,{(  
        end 1Q5:Vo^B#  
         iMT[s b  
        if isnorm &dH[lB  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); jOkc'  
        end `Z#0kpXk_  
    end nrhzNW>]  
    % END: Compute the Zernike Polynomials )S:,q3gxJ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2HNAB4 E  
    n7|8`? R^  
    Z[ NO`!<  
    % Compute the Zernike functions: cuw 7P  
    % ------------------------------ Ipp#{'Do  
    idx_pos = m>0; '-,$@l#  
    idx_neg = m<0; F K7cDaI  
    6UAn# d9  
    As}e I!  
    z = y; Rudj"OGO  
    if any(idx_pos) 65HP9`5Tm  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); {h}0"5  
    end P&>!B,f  
    if any(idx_neg) <:n !qQS6  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); s~z~9#G(6  
    end gNWTzz<[f>  
    rexNsKRK_  
    r_x|2 A oO  
    % EOF zernfun Qm"&=<  
     
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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)  uE,g|51H/  
    w_P2\B^  
    DDE还是手动输入的呢? a];1)zVA6  
    \YPv pUg  
    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