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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, 4[ *G  
    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, Y_FQB K U  
    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? _oE 7<  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? }a"koL  
    _)Ad%LPsd7  
    `$Y%c1;  
    mM2DZ^"j(  
    "!R*f $  
    function z = zernfun(n,m,r,theta,nflag) 8wLGmv^  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 8 +mW  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N = G>Y9Sc  
    %   and angular frequency M, evaluated at positions (R,THETA) on the f%/6kz  
    %   unit circle.  N is a vector of positive integers (including 0), and 7?ILmYBw  
    %   M is a vector with the same number of elements as N.  Each element qV)hCc/ ~  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ) S-Fuq4i4  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, L>n^Q:M  
    %   and THETA is a vector of angles.  R and THETA must have the same zmhAeblA  
    %   length.  The output Z is a matrix with one column for every (N,M) ;qs^+  
    %   pair, and one row for every (R,THETA) pair. ~IFafAO&  
    % 4xF}rm  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike [M2xF<r6t  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), OyQ[}w3o|  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral }\QXPU{UVd  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 6Z5$cR_vC7  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized `0`#Uf_/$  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. v)aV(Oa  
    % ' L-h2  
    %   The Zernike functions are an orthogonal basis on the unit circle. r2\ }_pIj  
    %   They are used in disciplines such as astronomy, optics, and xD9ZL  
    %   optometry to describe functions on a circular domain. y/>Nx7C0=2  
    % J4Ca0Ag  
    %   The following table lists the first 15 Zernike functions. +4F; m_G6  
    % 5R6QZVc  
    %       n    m    Zernike function           Normalization 5& _R+g  
    %       -------------------------------------------------- `( 'NH]^  
    %       0    0    1                                 1 L>pSE'}  
    %       1    1    r * cos(theta)                    2 TVVu_ib  
    %       1   -1    r * sin(theta)                    2 ,x utI  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) #n+sbx5~7  
    %       2    0    (2*r^2 - 1)                    sqrt(3) a1x].{  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) {<zE}7/2-  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) _6->D[dB  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) g&\;62lV%  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) | Pqs)Mb]  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) r-Oz k$  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Ky*xAx:  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .uB[zJc  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ]dT]25V  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) y!x-R !3  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Hp@cBj_@P2  
    %       -------------------------------------------------- Ch]q:o4  
    % Uv(}x 7e)  
    %   Example 1: PiLLUyQx  
    % G+WCE*  
    %       % Display the Zernike function Z(n=5,m=1) t&-c?&FO\;  
    %       x = -1:0.01:1; tPDB'S:&3  
    %       [X,Y] = meshgrid(x,x); '.e 5Ku  
    %       [theta,r] = cart2pol(X,Y); PPh1y;D  
    %       idx = r<=1; Xy9'JVV6  
    %       z = nan(size(X)); (kx>\FIK*  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); !v*#E{r"g=  
    %       figure ~]BR(n  
    %       pcolor(x,x,z), shading interp KF7d`bRe  
    %       axis square, colorbar Cyud)BZvm  
    %       title('Zernike function Z_5^1(r,\theta)') xzRC %  
    % eTt{wn;6  
    %   Example 2: =|d5V%mK  
    %  <JZa  
    %       % Display the first 10 Zernike functions w$749jGx  
    %       x = -1:0.01:1; Y3xEFqMU  
    %       [X,Y] = meshgrid(x,x); V{{UsEVO  
    %       [theta,r] = cart2pol(X,Y); z]sQ3"cmX  
    %       idx = r<=1; k,y#|bf,Y  
    %       z = nan(size(X)); .>'J ^^  
    %       n = [0  1  1  2  2  2  3  3  3  3]; hG3RZN#ejq  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; R~bLEo  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; (; Zl  
    %       y = zernfun(n,m,r(idx),theta(idx)); 2Mu(GUe;  
    %       figure('Units','normalized') U27ja|W^  
    %       for k = 1:10 _K~?{".  
    %           z(idx) = y(:,k); 'YEiT#+/  
    %           subplot(4,7,Nplot(k)) ;e~K<vMm;y  
    %           pcolor(x,x,z), shading interp %;`3I$  
    %           set(gca,'XTick',[],'YTick',[]) 5JZZvc$au  
    %           axis square 94XRf"^  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Kw>gg  
    %       end t;[Q&Jl  
    % p-/}@r3Z+  
    %   See also ZERNPOL, ZERNFUN2. 7p18;Z+6>X  
    ^N~Jm&I  
    *c@]c~hY,  
    %   Paul Fricker 11/13/2006 _[ `"E'  
    .sUL5`  
    NO#^_N`#\  
    wJF$<f7P  
    <7X+-%yb;  
    % Check and prepare the inputs: D7$xY\0r  
    % ----------------------------- yNQ 9~P2  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 8\Eq(o}7  
        error('zernfun:NMvectors','N and M must be vectors.') L^nS%lm  
    end m$$98N  
    CY9`HQ1  
    J~G"D-l<9/  
    if length(n)~=length(m) p|w;StLy  
        error('zernfun:NMlength','N and M must be the same length.') dk2o>jI4;  
    end o6 [i0S  
    yM34GS=,J  
    /XW,H0pR  
    n = n(:); ;D<rGkry  
    m = m(:); vGPaWYV  
    if any(mod(n-m,2)) z~a]dMs"(P  
        error('zernfun:NMmultiplesof2', ... ]%%cc  
              'All N and M must differ by multiples of 2 (including 0).') 9$'Edi=6  
    end g:c @  
    kC[nY  
    m;I;{+"u  
    if any(m>n) 'w7{8^Z2  
        error('zernfun:MlessthanN', ... ~ .Eln+N  
              'Each M must be less than or equal to its corresponding N.') >:P3j<xTv  
    end gM3gc;  
    }~5xlg$B<<  
    DSHpM/7  
    if any( r>1 | r<0 ) ("BFI  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Yui:=GgUrr  
    end Wkv **X}  
    ]j:Ikb}  
     yQ8H-a.  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) IA;KEGJ  
        error('zernfun:RTHvector','R and THETA must be vectors.') *)d|:q3  
    end z*>CP  
    ^q$vyY   
    ss 3fq}  
    r = r(:); Z_Ma|V?6  
    theta = theta(:); {1YT a:evl  
    length_r = length(r); D2Go,1  
    if length_r~=length(theta) z:R2Wksg  
        error('zernfun:RTHlength', ... &f qmO>M  
              'The number of R- and THETA-values must be equal.') _.06^5o  
    end _?_Svx2  
    RN:#+S(8  
    U>x2'B v  
    % Check normalization: z_l3=7R  
    % -------------------- 0QIocha  
    if nargin==5 && ischar(nflag) .^.UJo;4G  
        isnorm = strcmpi(nflag,'norm'); T[q-$8U  
        if ~isnorm @4B2O"z`  
            error('zernfun:normalization','Unrecognized normalization flag.') {Q(6 .0R  
        end a\m10Ih:  
    else gkk< -j'  
        isnorm = false; /9w}[y*E  
    end 1I^Sv  
    6l vx  
    p go\(K0  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q%:Jmi>  
    % Compute the Zernike Polynomials |PJW2PN  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )Y&De)=  
    sqZHk+<%  
    *u{.K:.I  
    % Determine the required powers of r: JN KZ'9  
    % ----------------------------------- kyo ,yD  
    m_abs = abs(m); Z%OSW  
    rpowers = []; C aJD*  
    for j = 1:length(n) 2aje$w-  
        rpowers = [rpowers m_abs(j):2:n(j)]; xf]4!zE  
    end !d0@^JbM"  
    rpowers = unique(rpowers); "^D6%I#T  
    cT0g, ^&  
    T&23Pf1  
    % Pre-compute the values of r raised to the required powers, ( L6`_)  
    % and compile them in a matrix: %-'U9e KN  
    % ----------------------------- d|NNIf  
    if rpowers(1)==0 N8{>M,  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); P*T)/A%4  
        rpowern = cat(2,rpowern{:}); BVNh>^W5B  
        rpowern = [ones(length_r,1) rpowern]; anwn!Eqk"  
    else |B`tRq  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); %ej"ZeM  
        rpowern = cat(2,rpowern{:}); |}|;OG  
    end 5#F+-9r  
    Q8~pIv  
    NR[mzJv  
    % Compute the values of the polynomials: LGMFv  
    % -------------------------------------- mDmWTq\  
    y = zeros(length_r,length(n)); 7f$Lb,\y  
    for j = 1:length(n) l&A`  
        s = 0:(n(j)-m_abs(j))/2; mHMej@  
        pows = n(j):-2:m_abs(j); 09?<K)_G  
        for k = length(s):-1:1 f\^QV  
            p = (1-2*mod(s(k),2))* ... rh l5r"%  
                       prod(2:(n(j)-s(k)))/              ... IyuT=A~Ki  
                       prod(2:s(k))/                     ... Q}T9NzOH%  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... *j* WE\  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ~GeYB6F  
            idx = (pows(k)==rpowers); ]xG4T>S  
            y(:,j) = y(:,j) + p*rpowern(:,idx); T7Ac4LA  
        end \nyFN  
         ({9!P30:  
        if isnorm Y"jDZG?  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ;~bn@T-  
        end S_CtE M  
    end W<L6,  
    % END: Compute the Zernike Polynomials M Sj0D2H  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PS22$_}   
    !T~d5^l!  
    y {]%,  
    % Compute the Zernike functions: A!kyga6F5  
    % ------------------------------ |Q;o538  
    idx_pos = m>0; ]>L]?Rm  
    idx_neg = m<0; jb2:O,+!  
    [s2V-'2  
    {ybuHC  
    z = y; !q/lgpEi  
    if any(idx_pos) YeLOd  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); KIFx &A  
    end [VW;L l  
    if any(idx_neg) 0)]1)z(P  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 2~DPq p[  
    end (r4VIlap  
    kU/=Du  
    J":9  
    % EOF zernfun H=#Jg;_w  
     
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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)  :-u-hO5*8  
    Vv]$\`d#  
    DDE还是手动输入的呢? WiNr866nB  
    2rO)qjiH  
    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