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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, O_=2{k~s0  
    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, (j<FS>##  
    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? Ub[SUeBGH  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? <46> v<  
    K|^PHe  
    j'L/eps?S  
    U'4j+vUc  
    1,Ams  
    function z = zernfun(n,m,r,theta,nflag) a ]~Rp  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. >- S?rXO  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N jGm`Qg{<  
    %   and angular frequency M, evaluated at positions (R,THETA) on the QjT$.pU d  
    %   unit circle.  N is a vector of positive integers (including 0), and P}"=67$  
    %   M is a vector with the same number of elements as N.  Each element zEM  c)  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) d `MTc  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, |h^[/  
    %   and THETA is a vector of angles.  R and THETA must have the same D;?cf+6$  
    %   length.  The output Z is a matrix with one column for every (N,M) '%Fg+cZN\  
    %   pair, and one row for every (R,THETA) pair. \NZ(Xk  
    % # <?igtUO  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike OdKfU^  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), :zA/~/Wo  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral L i g7Ac,  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 5r2A^<)  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized y  J|/^qs  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. L<D<3g|4  
    % pVm]<jO  
    %   The Zernike functions are an orthogonal basis on the unit circle. `GDWy^-Q+!  
    %   They are used in disciplines such as astronomy, optics, and srbES6  
    %   optometry to describe functions on a circular domain. z:Y Z]   
    % w]@H]>sHd  
    %   The following table lists the first 15 Zernike functions. ^U q%-a  
    % I3I1<}>]Z  
    %       n    m    Zernike function           Normalization gDN7ly]6M  
    %       -------------------------------------------------- #}xw *)3  
    %       0    0    1                                 1 o:wI{?%-3  
    %       1    1    r * cos(theta)                    2 QG1+*J76b@  
    %       1   -1    r * sin(theta)                    2 gPE` mE  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 6y+_x'  
    %       2    0    (2*r^2 - 1)                    sqrt(3) {<}9r6k;f  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ! V^wq]D2  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 42oW]b%P{;  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) XJZ\ss  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) M&[bb $00j  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) !{1;wC(b  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) #}p@+rkg2  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) | V: 9 ][\  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) v:F_! Q  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) W5*Kq^6Pd  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 1a gNwFd~  
    %       -------------------------------------------------- 11^.oa+`  
    % 8P?p  
    %   Example 1: oBS m>V  
    % ]qd$rX   
    %       % Display the Zernike function Z(n=5,m=1) A+=K<e  
    %       x = -1:0.01:1; ?S<`*O +  
    %       [X,Y] = meshgrid(x,x); h}y]Pt?  
    %       [theta,r] = cart2pol(X,Y); Q]{ `m  
    %       idx = r<=1; wi/qI(O!  
    %       z = nan(size(X)); 3<x1s2U  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ;7>k[?'e  
    %       figure x%'5 rnm|  
    %       pcolor(x,x,z), shading interp |1pD n7  
    %       axis square, colorbar `nCVO;B  
    %       title('Zernike function Z_5^1(r,\theta)') f6,?Yex8B  
    % =OeLF  
    %   Example 2: gs"w 0[$  
    % p:NIRs  
    %       % Display the first 10 Zernike functions OQ&'3hv{  
    %       x = -1:0.01:1; "h5.^5E6  
    %       [X,Y] = meshgrid(x,x); e?7Oom  
    %       [theta,r] = cart2pol(X,Y); ^)E# c  
    %       idx = r<=1; 60R]Q  
    %       z = nan(size(X)); %a:>3! +  
    %       n = [0  1  1  2  2  2  3  3  3  3]; X \BxRgl},  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; *e25!#o1  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 8|{d1dy  
    %       y = zernfun(n,m,r(idx),theta(idx)); vlqL  
    %       figure('Units','normalized') l3xI\{jn  
    %       for k = 1:10 :+_  
    %           z(idx) = y(:,k); ~f:"Q(f+  
    %           subplot(4,7,Nplot(k))  y 2C Jk~  
    %           pcolor(x,x,z), shading interp hLr\;Swyp  
    %           set(gca,'XTick',[],'YTick',[]) udOdXz6K?  
    %           axis square FEO /RMh  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) /E-s g, k  
    %       end ?J@P0(M#  
    % f+lPQIB  
    %   See also ZERNPOL, ZERNFUN2. cN:dy#  
    u[HamGxx$u  
    w|1O-k`  
    %   Paul Fricker 11/13/2006 mpNS}n6  
    *zwo="WA\t  
    W1&"dT@  
    6~-,.{Y  
    #}lWM%9Dy  
    % Check and prepare the inputs: h0?w V5H  
    % ----------------------------- 4" pU\g  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) {TZV^gT4  
        error('zernfun:NMvectors','N and M must be vectors.') jp7cPpk:LG  
    end s6QD^[  
    >qVSepK3  
    =gB8(1g8  
    if length(n)~=length(m) ffMk.SqI  
        error('zernfun:NMlength','N and M must be the same length.') vSy[lB|)24  
    end mqtYny'  
    ?=im  ~  
    w6h*dh$w  
    n = n(:); SZUo RWx  
    m = m(:); ZfXgVTJ`  
    if any(mod(n-m,2)) V KxuK0{  
        error('zernfun:NMmultiplesof2', ... q8!]x-5$6j  
              'All N and M must differ by multiples of 2 (including 0).') Ae%AG@L  
    end [1mEdtqf*  
    [tRb{JsUd  
    ME66BWg{  
    if any(m>n) $*:g~#bh  
        error('zernfun:MlessthanN', ... q+} \ (|  
              'Each M must be less than or equal to its corresponding N.') !X9^ L^v}  
    end n]6-`fpD  
    4peRbm  
    qLPuKIF  
    if any( r>1 | r<0 ) 6OB3%R'p  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') dQz#&&s-  
    end IL1iTR H  
    lD{*Z spz  
    _'4S1  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) K $WMrp  
        error('zernfun:RTHvector','R and THETA must be vectors.') G^';9 UK  
    end OIIA^QyV  
    <'QI_mP*  
    :cf#Tpq"  
    r = r(:); !L@<?0x LW  
    theta = theta(:); W4bN']?  
    length_r = length(r); "lrQC`?  
    if length_r~=length(theta) 0cDP:EzR;  
        error('zernfun:RTHlength', ... da i+"  
              'The number of R- and THETA-values must be equal.') NTEN  
    end 7xFZJ#  
    Cg|\UKfy$  
    S>)[n]f  
    % Check normalization: +&dkJ 4g[  
    % -------------------- ddN G :  
    if nargin==5 && ischar(nflag) do*aE  
        isnorm = strcmpi(nflag,'norm'); :[CEHRc7x  
        if ~isnorm h#c7v !g  
            error('zernfun:normalization','Unrecognized normalization flag.') Uu52uR  
        end 'tDUPm38  
    else f7|Tp m  
        isnorm = false; . :>e"D  
    end &po!X )  
    Pf/8tXs}  
    1w,34*-}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% IT)3Et@Y  
    % Compute the Zernike Polynomials 1J72*`4OK  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I~6 o<HO  
    !{{gL=_@  
    6`vW4]zu  
    % Determine the required powers of r: pp@B]We  
    % ----------------------------------- yn"4qC#Z  
    m_abs = abs(m); AW E ab  
    rpowers = []; ;4(}e{  
    for j = 1:length(n) bdLi _k  
        rpowers = [rpowers m_abs(j):2:n(j)]; L`e19I$  
    end d S'J@e=#  
    rpowers = unique(rpowers); Nu OxEyC  
    U82mO+}  
    )0]U"Nf ho  
    % Pre-compute the values of r raised to the required powers, #vhN$H:&q  
    % and compile them in a matrix: N'-[>w7vK2  
    % ----------------------------- znPh7{|<  
    if rpowers(1)==0 u>G9r#~`k  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); =Xg/[J%  
        rpowern = cat(2,rpowern{:}); h5pfmN\-5  
        rpowern = [ones(length_r,1) rpowern]; @g4o8nH}  
    else hF$qH^-c*A  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); N>,`TsUwW  
        rpowern = cat(2,rpowern{:}); zsd1n`r  
    end A)V*faD  
    9(nq 4 HvI  
    &oq 0XV.M^  
    % Compute the values of the polynomials: @K+gh#  
    % -------------------------------------- tXH;4K@  
    y = zeros(length_r,length(n)); D/~1?p  
    for j = 1:length(n) xb<|m2<)H  
        s = 0:(n(j)-m_abs(j))/2; ,[t? $Cy ;  
        pows = n(j):-2:m_abs(j); $B_%MfI  
        for k = length(s):-1:1 ajtH 1Z#  
            p = (1-2*mod(s(k),2))* ... 9cUa@;*1  
                       prod(2:(n(j)-s(k)))/              ... =*jFaj  
                       prod(2:s(k))/                     ... #{{p4/:  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... zL9~gJ  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); eBs.RR ]O  
            idx = (pows(k)==rpowers); y(MB _B7j  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Eu:/U*j  
        end 80_w_i+  
         DyCzRkH  
        if isnorm gwQMy$  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); <n06(9BF  
        end fZ5 UFq_~s  
    end Su"Z3gm5Kw  
    % END: Compute the Zernike Polynomials c9fz x  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% u~j H  
    op5 `#{  
     \20} /&  
    % Compute the Zernike functions: Zfcf?&><  
    % ------------------------------ A8{ xZsH  
    idx_pos = m>0; !CcDA/0  
    idx_neg = m<0; MO0NNVVi%U  
    WV.hQX9P  
    %" 7UYLX  
    z = y; bTmhz  
    if any(idx_pos) #` gu<xlW  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); L;u5  
    end W h9L!5  
    if any(idx_neg) ~{tO8 ]  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Vjv6d&Q  
    end q%e'WMG~n  
    _^#eO`4"  
    *2->>"kh  
    % EOF zernfun JJ ?'<)EF  
     
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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)  9!X3Cv|+L  
    G8+&fn6  
    DDE还是手动输入的呢? tcXXo&ZS  
    o!+%|V8Y  
    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