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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, Dn1aaN6  
    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, -pvF~P?8U  
    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? J4EQhuQ  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? +G)L8{FY(  
    i|2CZ  
    hV_bm@f/y  
    $DBJ"8n2  
    ei%L[>N  
    function z = zernfun(n,m,r,theta,nflag)   
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. A%(t'z  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N /x\{cHAt8J  
    %   and angular frequency M, evaluated at positions (R,THETA) on the KPTp91  
    %   unit circle.  N is a vector of positive integers (including 0), and CEzwI _  
    %   M is a vector with the same number of elements as N.  Each element 9}G.Fr  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) A0JlQE&U  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, dz_~_|  
    %   and THETA is a vector of angles.  R and THETA must have the same u)J&3Ah%  
    %   length.  The output Z is a matrix with one column for every (N,M) W~b->F  
    %   pair, and one row for every (R,THETA) pair. kbu.KU+  
    % VEFUj&t;xW  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike "h58I)O  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), s 1~&PH^  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral |}$ZOwc  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 7 G37V"''  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized J?DJA2o  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. , !0-;H.Y  
    % ;l4 epN  
    %   The Zernike functions are an orthogonal basis on the unit circle. xQ~}9Kt\  
    %   They are used in disciplines such as astronomy, optics, and )/Z% HBn  
    %   optometry to describe functions on a circular domain. pQ2'0u5w5  
    % D6z*J?3^#&  
    %   The following table lists the first 15 Zernike functions. BeFCt;  
    % T3H\KRe6  
    %       n    m    Zernike function           Normalization V[#eeH)/  
    %       -------------------------------------------------- B\*"rSP\  
    %       0    0    1                                 1 xWR<>Og.  
    %       1    1    r * cos(theta)                    2 9IfeaoZZ4q  
    %       1   -1    r * sin(theta)                    2 T*pcS'?'  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) \+,%RN.  
    %       2    0    (2*r^2 - 1)                    sqrt(3) T'8d|$X  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ZF@T,i9  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Ynxzkm S  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) J A!?vs  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Ah#bj8}  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ;cpQ[+$nKp  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 7:Cq[u fl  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^VL",Nt  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ip)gI&kN`z  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) J)I|Xot  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) L>@:Xo@  
    %       -------------------------------------------------- o!$O+%4  
    % 7gxC xfL$  
    %   Example 1: /u&{=nU  
    % n=_jmR1  
    %       % Display the Zernike function Z(n=5,m=1) yUY* l@v]  
    %       x = -1:0.01:1; CQ;.}=j ,  
    %       [X,Y] = meshgrid(x,x); x b6X8:  
    %       [theta,r] = cart2pol(X,Y); HE BKRpt  
    %       idx = r<=1; { VK   
    %       z = nan(size(X)); `514HgR  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); :n0czO6 E  
    %       figure /k_?S?  
    %       pcolor(x,x,z), shading interp zqJ0pDS  
    %       axis square, colorbar ~[[(_C3  
    %       title('Zernike function Z_5^1(r,\theta)') B QxU~s  
    % E!rgR5Bd  
    %   Example 2: <<vT"2Q]  
    % P,RdY M06  
    %       % Display the first 10 Zernike functions a Byetc88/  
    %       x = -1:0.01:1; _]aA58,j  
    %       [X,Y] = meshgrid(x,x); =wcqCW,]  
    %       [theta,r] = cart2pol(X,Y); P&kjtl68 Y  
    %       idx = r<=1; N0mP EF2  
    %       z = nan(size(X)); wbImE;-Z  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ?yNg5z  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; $C.;GUEQ  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; qvHRP@  
    %       y = zernfun(n,m,r(idx),theta(idx)); '.$va<  
    %       figure('Units','normalized') T*3>LY+bb  
    %       for k = 1:10 n-)Xs;`2  
    %           z(idx) = y(:,k); ] -}Zd\Rs  
    %           subplot(4,7,Nplot(k)) ~tM+!  
    %           pcolor(x,x,z), shading interp qZ=%r u  
    %           set(gca,'XTick',[],'YTick',[]) Gm1[PAj  
    %           axis square a9%^Jvm"  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) {];8jdg/?  
    %       end aK+jpi4?  
    % 0x1#^dII  
    %   See also ZERNPOL, ZERNFUN2. I&Dp~aEM]  
    -ufO,tJRLL  
    yRdME>_L  
    %   Paul Fricker 11/13/2006 L `6 R  
    aMq|xHZ  
    "54t7  
    k. @OFkX.  
    7Z7e}| \W  
    % Check and prepare the inputs: |XV@/ZGl~  
    % ----------------------------- z]d2 rzV(_  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) &ZR}Z7E*=  
        error('zernfun:NMvectors','N and M must be vectors.') Bsc&#  
    end ~k[mowz0  
    kKlcK_b;  
    u|eV'-R)s  
    if length(n)~=length(m) p*ic@n*G  
        error('zernfun:NMlength','N and M must be the same length.') E9]\ I> v  
    end 1;FtQnvH  
    fBw"<J{  
    8p0ZIrD%  
    n = n(:); kuI%0) iZn  
    m = m(:); {wq~+O  
    if any(mod(n-m,2)) B{6wf)[O  
        error('zernfun:NMmultiplesof2', ... WJA0 `<~  
              'All N and M must differ by multiples of 2 (including 0).') xZc].l6  
    end sCrOdJ6|  
    $!q(-+(  
    knb 9s`wR  
    if any(m>n) 1RM@~I$0  
        error('zernfun:MlessthanN', ... M[1!#Q><!  
              'Each M must be less than or equal to its corresponding N.') 9o<5Z=  
    end \#%1t  
    O*dtVX  
    kS)azV  
    if any( r>1 | r<0 ) 0*{ 2^\  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') j8[RDiJ  
    end 8?z7!k]  
    HCIS4}lQ  
    X:kqX[\>  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) +5xVgIk#  
        error('zernfun:RTHvector','R and THETA must be vectors.') *%p`Jk-U  
    end 1Ax{Y#<  
    \YMe&[C:o  
    d:&=|kKw  
    r = r(:); U5!~ @XjG>  
    theta = theta(:); kh5VuXpe  
    length_r = length(r); wRsh@I<  
    if length_r~=length(theta) ra]lC7<H  
        error('zernfun:RTHlength', ... y c:y}"  
              'The number of R- and THETA-values must be equal.') (5\VOCT>4%  
    end }Y`D^z~  
    MIx,#]C&  
    P g.j]  
    % Check normalization: ~[ZRE @  
    % -------------------- .tQeOZW'  
    if nargin==5 && ischar(nflag) 4mM?RGWv  
        isnorm = strcmpi(nflag,'norm'); lFT` WO  
        if ~isnorm H$4 4,8,m  
            error('zernfun:normalization','Unrecognized normalization flag.') W^8MsdM  
        end zNRR('B?  
    else /OtLIM+7~{  
        isnorm = false; efUa[XO  
    end [#mRlL0yk  
    'fS&WVR?  
    + rN&@}Jt.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% n~Qo@%Jr  
    % Compute the Zernike Polynomials {$P')> /  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fMl uVND  
    +DwE~l  
    kPvR ,  
    % Determine the required powers of r: /]>8V'e\  
    % ----------------------------------- ,C;%AS/  
    m_abs = abs(m); #C#*yE  
    rpowers = []; ?Jy /]j5fI  
    for j = 1:length(n) ,We'A R3X  
        rpowers = [rpowers m_abs(j):2:n(j)]; @ CNe)&U  
    end 0/TP`3$X#"  
    rpowers = unique(rpowers); j[Z<|Da  
    `&w{-om\  
    `x:8m?q05  
    % Pre-compute the values of r raised to the required powers, Rn9e#_Az  
    % and compile them in a matrix: :c}"a(|  
    % ----------------------------- Tg_#z  
    if rpowers(1)==0 155vY  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); UB2Ft=  
        rpowern = cat(2,rpowern{:}); pSKw Xx  
        rpowern = [ones(length_r,1) rpowern]; -J]j=  
    else }-N4D"d4o  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); '4e, e|r  
        rpowern = cat(2,rpowern{:}); H{U(Rt]K  
    end kkU#0p?7  
    5KgAY;|  
    z{wZLqG  
    % Compute the values of the polynomials: q#_<J1)z  
    % -------------------------------------- uWDWf5@  
    y = zeros(length_r,length(n)); (U([T-H  
    for j = 1:length(n) # ~(lY}  
        s = 0:(n(j)-m_abs(j))/2; 8{DW$Z tR  
        pows = n(j):-2:m_abs(j); mPJ@hr%3  
        for k = length(s):-1:1 lEXI<b'2  
            p = (1-2*mod(s(k),2))* ... K)N'~jCG  
                       prod(2:(n(j)-s(k)))/              ... B1 Y   
                       prod(2:s(k))/                     ... :zp9L/eh  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... rk8Cea  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); <pIel   
            idx = (pows(k)==rpowers); ZO8r8 [  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ap wA  
        end `;)op3A'  
         )~be<G( a  
        if isnorm L2> )HG  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 7RCVqc"  
        end qlm7eS"sy  
    end THy{r_dx  
    % END: Compute the Zernike Polynomials 0lLg uBW@  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  N~vK8j@  
    'b:UafV  
    ;MH_pE/m  
    % Compute the Zernike functions: ]FEsN6  
    % ------------------------------ fRK=y+gl@  
    idx_pos = m>0; KMP[Ledr  
    idx_neg = m<0; zn#lFPj12  
    *hlinQKs  
    9S/X,|i  
    z = y; D!rD-e  
    if any(idx_pos) 1 &-%<o  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); kJ"}JRA<  
    end Z)!#+m83>-  
    if any(idx_neg) ODCv^4}9  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 3B5 `Y  
    end 0) Q*u  
    UL0n>Wa5  
    1xjw=  
    % EOF zernfun f{+X0Oj  
     
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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)
    在线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)  Y1L7sH 9  
    I] 0 D*z  
    DDE还是手动输入的呢? 1n EW'F  
    rPF2IS(5  
    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