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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 正序阅读 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, =55V<VI  
    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, G6pR?K+  
    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? iz3Hoj  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? :eFyd`Syw  
    %J+k.UrM  
    BFZ\\rN`  
    E&>;a!0b]  
    emIF{oP  
    function z = zernfun(n,m,r,theta,nflag) {Zo*FZcaX  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. %lGT |XrY  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N L'O=;C"f  
    %   and angular frequency M, evaluated at positions (R,THETA) on the MUUhg  
    %   unit circle.  N is a vector of positive integers (including 0), and A`1-c   
    %   M is a vector with the same number of elements as N.  Each element ;i!$rL  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) R0e!b+MZ.  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, )}@Z*.HZL  
    %   and THETA is a vector of angles.  R and THETA must have the same )i[K1$x2  
    %   length.  The output Z is a matrix with one column for every (N,M) X0]Se(  
    %   pair, and one row for every (R,THETA) pair. Ls'8  
    % )3^#CD  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike &/?OP)N,}  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), )kIjZ  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral MbeK{8~E%l  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, `KUL 4) g~  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized HpS1(%d"  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. j43i:c;F  
    % Awe'MGp%  
    %   The Zernike functions are an orthogonal basis on the unit circle. -qG7,t  
    %   They are used in disciplines such as astronomy, optics, and WnhH]WY  
    %   optometry to describe functions on a circular domain. |nY+Nen7  
    % 5hfx2 O)  
    %   The following table lists the first 15 Zernike functions. (zw.?ADPCT  
    % H[N~)3x  
    %       n    m    Zernike function           Normalization m5 l&  
    %       -------------------------------------------------- q#`;G,rs  
    %       0    0    1                                 1 = Q"(9[Az  
    %       1    1    r * cos(theta)                    2 at(gem  
    %       1   -1    r * sin(theta)                    2 J]|S0JC`  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) kfq<M7y  
    %       2    0    (2*r^2 - 1)                    sqrt(3) o<rbC < U  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) =z'533C  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) orhze Oi\  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 1-Q>[Uz,  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) RQ,X0 pS  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) JC9OL.Ob  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) $YK~7!!  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) j#${L6  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) aZ}z/.b]  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1~vv<`-  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) =cxG4R1x  
    %       -------------------------------------------------- xLw[ aYy4  
    % -l{ wB"  
    %   Example 1: ZK8DziO  
    % 9g7Ok9dF  
    %       % Display the Zernike function Z(n=5,m=1) 5D>cbzP@  
    %       x = -1:0.01:1; 0$|wj^?U  
    %       [X,Y] = meshgrid(x,x); i8.OM*[f  
    %       [theta,r] = cart2pol(X,Y); M] W5 %3do  
    %       idx = r<=1; xI8v'[3  
    %       z = nan(size(X)); d4o_/[  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); sNJ?Z"5k1h  
    %       figure JB HnJm  
    %       pcolor(x,x,z), shading interp [yVcH3GcjI  
    %       axis square, colorbar E#n: d9WA:  
    %       title('Zernike function Z_5^1(r,\theta)') u HXb=U  
    % Co`:D  
    %   Example 2: kv`5"pa7M  
    % vr$z6m ^  
    %       % Display the first 10 Zernike functions uR82},r$m  
    %       x = -1:0.01:1; dq3"L!0u  
    %       [X,Y] = meshgrid(x,x); z_a7HCG2  
    %       [theta,r] = cart2pol(X,Y); >2tosxH M  
    %       idx = r<=1; y|Y hDO  
    %       z = nan(size(X)); rm,h\  
    %       n = [0  1  1  2  2  2  3  3  3  3]; =%wBC;  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 6H:EBj54?  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; /!-ypIY  
    %       y = zernfun(n,m,r(idx),theta(idx)); 7/BA!V(na  
    %       figure('Units','normalized') I# |ib  
    %       for k = 1:10 {>l`P{{y  
    %           z(idx) = y(:,k); LsNJ3oy  
    %           subplot(4,7,Nplot(k)) i(kr#XsU  
    %           pcolor(x,x,z), shading interp DkBVk+  
    %           set(gca,'XTick',[],'YTick',[]) l%7^'nDn  
    %           axis square Wl3fR[@3Q  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) #4!6pMW(&7  
    %       end kYkck]|  
    % UbSD?Ew@35  
    %   See also ZERNPOL, ZERNFUN2. G_?qY#"(  
    *JVJKqed  
    6dYUMqQ  
    %   Paul Fricker 11/13/2006 <(Ktf0'__  
    I)4NCjcCw  
    Zy > W2(<  
    ]BBjFs4#  
    x[2eA!NC  
    % Check and prepare the inputs: &r V  
    % ----------------------------- C-ipxL"r  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 2LH.If  
        error('zernfun:NMvectors','N and M must be vectors.') YR$d\,#R  
    end 5VW*h  
    ) 2Hl\"F  
    xiQ;lE   
    if length(n)~=length(m) |HK/*B  
        error('zernfun:NMlength','N and M must be the same length.') LzkwgcR  
    end 3(La)|k  
    oOK&+r7  
    vm"dE4W=  
    n = n(:); 1 *$-.  
    m = m(:); 0G/_"} @  
    if any(mod(n-m,2)) q=cH ^`<.  
        error('zernfun:NMmultiplesof2', ... JU0|pstf  
              'All N and M must differ by multiples of 2 (including 0).') !u|s| 6{\  
    end TzK[:o  
    4gK_' b6"  
    <4P"1#nHQ+  
    if any(m>n) ;923^*\:F{  
        error('zernfun:MlessthanN', ... =%oKYQ  
              'Each M must be less than or equal to its corresponding N.') 9$P*fx&m  
    end X.!|#FWb+  
    XxE>KeP  
    O PVc T  
    if any( r>1 | r<0 ) /h73'"SpDy  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') @60/IE{-v  
    end a]_eSU@  
    8&9'1X5)8_  
    FBJ Lkg0  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) BA|*V[HBE  
        error('zernfun:RTHvector','R and THETA must be vectors.') j4.deQ,  
    end !RwOU Ck  
    pwO U6A!  
    {F+iL&e)  
    r = r(:); ' fP`ET5  
    theta = theta(:); :i:M7}r  
    length_r = length(r); j /=4f�  
    if length_r~=length(theta) ^{Y9!R*9U*  
        error('zernfun:RTHlength', ... QAh6!<.;@  
              'The number of R- and THETA-values must be equal.') 2s:$4]K D  
    end 5A=FEg  
    Qape DU;  
    Acib<Mi2!-  
    % Check normalization: )s4a<S c]  
    % -------------------- I<ta2<h  
    if nargin==5 && ischar(nflag) iSxuor ^;  
        isnorm = strcmpi(nflag,'norm'); Rc k k  
        if ~isnorm ThSB\  
            error('zernfun:normalization','Unrecognized normalization flag.') _ -/<bO  
        end ^J Y]w^u  
    else x>=8~wIK  
        isnorm = false; 9n[ovX 7n!  
    end H '(Ky  
    /NFcIU  
    2k$~Mv@L  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s>^$: wzu  
    % Compute the Zernike Polynomials ==pGRauq  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A[O'e  
    D/-$~u_o  
    Lcpz(W ^  
    % Determine the required powers of r: B`SHr"k!V[  
    % ----------------------------------- KcQe1mT!+  
    m_abs = abs(m); dLn Md0  
    rpowers = []; 5?n@.hcL  
    for j = 1:length(n) NYcF]K}[  
        rpowers = [rpowers m_abs(j):2:n(j)]; Fb{kql=  
    end MKN],l N  
    rpowers = unique(rpowers); =^LX,!2zp{  
    eDPmUlC+-  
    )2jBhT  
    % Pre-compute the values of r raised to the required powers, {g(-C&  
    % and compile them in a matrix: %VD>S  
    % ----------------------------- oH|<(8efD  
    if rpowers(1)==0 UI>?"b6 L  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); >1n[Y- r  
        rpowern = cat(2,rpowern{:}); ]TmxCTVL  
        rpowern = [ones(length_r,1) rpowern]; NJ{M-K%>  
    else \.%GgTF  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); B:Xmc,|,  
        rpowern = cat(2,rpowern{:}); V]$Tbxg  
    end qOk=:1`3  
    7pY7iR_  
    4/tp-dBip  
    % Compute the values of the polynomials: Tn7(A^h'  
    % -------------------------------------- (;@\gRL  
    y = zeros(length_r,length(n)); a5AD$bP  
    for j = 1:length(n) a!US:^}lu  
        s = 0:(n(j)-m_abs(j))/2; 'sCj|=y2Qc  
        pows = n(j):-2:m_abs(j); TE.O@:7Z  
        for k = length(s):-1:1 (wRJ"Nwu  
            p = (1-2*mod(s(k),2))* ... CF:s@Z+  
                       prod(2:(n(j)-s(k)))/              ... eQRY xx{  
                       prod(2:s(k))/                     ... )Q%hd|R  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 71# ipZ  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); DVWqrK}q  
            idx = (pows(k)==rpowers); *uq}jlD`!  
            y(:,j) = y(:,j) + p*rpowern(:,idx); @m=xCg.Z  
        end 0cwb^ffN  
         #&cNR_"w  
        if isnorm fv",4L  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); %fyah}=  
        end *"pf3x6  
    end XOe8(cXa9  
    % END: Compute the Zernike Polynomials 8sG0HI$f+  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% };:+0k/  
    AGe\PCn-  
    ;m+*R/  
    % Compute the Zernike functions: E?z~)0z2`  
    % ------------------------------ l?N|Gj;ZFZ  
    idx_pos = m>0; w<ol$2&B  
    idx_neg = m<0; \MA 4>  
    J}9 I5O  
    wewYlm5@  
    z = y; bH-QF\>  
    if any(idx_pos) #0WGSIht<  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); vThK@P!s  
    end QD}'2{M!  
    if any(idx_neg) Whd2mKwiO  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 4[ 7) $  
    end 'w8k*@cQ  
    [a<u cJ  
    #xMl<  
    % EOF zernfun SGd[cA Ko  
     
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    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
    离线jssylttc
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    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    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)  Smt&/~7D%  
    ezq<)gJc  
    DDE还是手动输入的呢? U_&v|2o#3  
    IO@Ti(,  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    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。
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