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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, Y3Oz'%B  
    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, d ~`V7B2Y  
    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? </[: 9Cl  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? hH>``gK  
    HC*?DJ,  
    C&<~f#lB  
    )d`mvZBn1  
    !<<AzLVL  
    function z = zernfun(n,m,r,theta,nflag) #_JYh?  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. r.yK,  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N @Jn!0Y1_3  
    %   and angular frequency M, evaluated at positions (R,THETA) on the F #`=oM $5  
    %   unit circle.  N is a vector of positive integers (including 0), and <RXwM6G2  
    %   M is a vector with the same number of elements as N.  Each element &7>zURv  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 91Z'  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, [k<1`z3  
    %   and THETA is a vector of angles.  R and THETA must have the same 2C=Q8ayvX  
    %   length.  The output Z is a matrix with one column for every (N,M) 8sOQ9  
    %   pair, and one row for every (R,THETA) pair. *O~e T  
    % =9wy/c$  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 6'vbT~S!  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), |~0UM$OB^3  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral F3M aqr y  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, j;0vAf  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized EGVM)ur  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. A8r^)QJP{  
    % H t(n%;<  
    %   The Zernike functions are an orthogonal basis on the unit circle. 3Q^fVn$tk  
    %   They are used in disciplines such as astronomy, optics, and GVGlVAo|@  
    %   optometry to describe functions on a circular domain. N2C7[z+l`  
    % ino:N5&;;  
    %   The following table lists the first 15 Zernike functions. QzvHm1,@  
    % 8\.b4FNJ  
    %       n    m    Zernike function           Normalization S \i@s_  
    %       -------------------------------------------------- soRv1)el  
    %       0    0    1                                 1 \ 0W!4D  
    %       1    1    r * cos(theta)                    2 SmwQET<H  
    %       1   -1    r * sin(theta)                    2 > L2HET  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Q\ppfc{,  
    %       2    0    (2*r^2 - 1)                    sqrt(3) /]^#b  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) L{-LX= G^  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) saf&dd  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) KLWn?`  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) PNs~[  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) W-Hoyn>?2  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) j=RRfFg)  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) NoE*/!Sr  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) kYzKU2T\W  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) : rMM4  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) FzQTDu9  
    %       -------------------------------------------------- W,5Hx1z R  
    % 8,P- 7^  
    %   Example 1: l7H qo)  
    % b?X.U}62_  
    %       % Display the Zernike function Z(n=5,m=1) HBS\<}  
    %       x = -1:0.01:1; }@ Z56  
    %       [X,Y] = meshgrid(x,x); t_^X$pL  
    %       [theta,r] = cart2pol(X,Y); aT!'}GjL  
    %       idx = r<=1; OJ|r6  
    %       z = nan(size(X)); 1RcSTg  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); H %JaZ?(  
    %       figure ^&@w$  
    %       pcolor(x,x,z), shading interp ?'86d_8  
    %       axis square, colorbar K_)eWf0a  
    %       title('Zernike function Z_5^1(r,\theta)') Q/uwQ o/  
    % e}/Lk5q!  
    %   Example 2: TxjYrzC  
    % a7zcIwk '{  
    %       % Display the first 10 Zernike functions )c n+1R  
    %       x = -1:0.01:1; 7w,FX.=;cv  
    %       [X,Y] = meshgrid(x,x); 3s\.cG?`r  
    %       [theta,r] = cart2pol(X,Y); 9{k97D/  
    %       idx = r<=1; ]^':Bmq  
    %       z = nan(size(X)); *_a@z1  
    %       n = [0  1  1  2  2  2  3  3  3  3]; N{ Z  H  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; $vC1 K5sLk  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; wO ?+Nh  
    %       y = zernfun(n,m,r(idx),theta(idx)); _v Sn`  
    %       figure('Units','normalized') k.("3R6v:  
    %       for k = 1:10 \%PaceH  
    %           z(idx) = y(:,k); NI#X @  
    %           subplot(4,7,Nplot(k)) p&+;w  
    %           pcolor(x,x,z), shading interp }bY; q-  
    %           set(gca,'XTick',[],'YTick',[]) pyLRgD0 g  
    %           axis square (py]LBZ  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) &%})wZ+Dj  
    %       end mxb(<9O  
    % H 0+dV3  
    %   See also ZERNPOL, ZERNFUN2. $o$ maA0  
    j83p)ido  
    =mA: ctu~v  
    %   Paul Fricker 11/13/2006 z<H~ItX,n  
    )* 4fzo  
    /}Jj  
    *K> l*l(f]  
    a=M\MZK>  
    % Check and prepare the inputs: 'F>'(XWWQ  
    % ----------------------------- XGP6L0j  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) q_-7i  
        error('zernfun:NMvectors','N and M must be vectors.') _g1b{$  
    end TXe$<4"  
    3nZo{p:E  
    J7r|atSk  
    if length(n)~=length(m) !9Aaj<yxm  
        error('zernfun:NMlength','N and M must be the same length.') ^Z~;4il_F  
    end 9Xx's%U  
    !\&;h  
    h/~n\0,J/  
    n = n(:); LpR3BP@At  
    m = m(:); PO 6&bIr  
    if any(mod(n-m,2)) xg)v0y~  
        error('zernfun:NMmultiplesof2', ... dtpoU&?6s  
              'All N and M must differ by multiples of 2 (including 0).') 7o8{mp'_  
    end ZDbc  
    2WU@*%sk"  
    /_ `lz^  
    if any(m>n) }Ho Qwy|&  
        error('zernfun:MlessthanN', ... 4 :U?u  
              'Each M must be less than or equal to its corresponding N.') Pp )3(T:  
    end Mbjvh2z  
    X7s `U5'l  
    #dM9pc jh  
    if any( r>1 | r<0 ) '` pDngX  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 27;ci:5  
    end aH^RoG}  
    6`f2-f9%iq  
    lsJnI|  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Z)jw|T'X  
        error('zernfun:RTHvector','R and THETA must be vectors.') 33!oS&L  
    end lbpq_=  
    } \ZaE~  
    F&&$Qn_+  
    r = r(:); \hB5@e4i2  
    theta = theta(:); TY1I=8  
    length_r = length(r); OoWyPdC+P  
    if length_r~=length(theta) iezO9`  
        error('zernfun:RTHlength', ... Q=]w !I\  
              'The number of R- and THETA-values must be equal.') ih7/}   
    end l5"OIq  
    Hla0 5N' 4  
    `QXErw  
    % Check normalization: JU4q zi  
    % -------------------- nw-I|PVTNa  
    if nargin==5 && ischar(nflag) ]MxC_V+P`  
        isnorm = strcmpi(nflag,'norm'); #5f-`~^C{  
        if ~isnorm t^w"w`v\u  
            error('zernfun:normalization','Unrecognized normalization flag.') B[KJR?>  
        end ONe# rKJ_  
    else Nqu>6^-z0  
        isnorm = false; 5O\*h;U 6  
    end y~FV2$  
    lBaR  
    \_lod kf  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% m}X`> aD/  
    % Compute the Zernike Polynomials W VI{oso#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% NUp<e%zB  
    L31HG H2l  
    ]wtb-PC  
    % Determine the required powers of r: 6q{HU]N+  
    % ----------------------------------- 3? HhG  
    m_abs = abs(m); Vv ?-"\Z>  
    rpowers = []; <TP=oq?I/  
    for j = 1:length(n) V>b\[(=s  
        rpowers = [rpowers m_abs(j):2:n(j)]; 5=Di<!a;  
    end ~P@Q7T*  
    rpowers = unique(rpowers); Z /9>  
    Nd;K u6  
    % ClHCoyA  
    % Pre-compute the values of r raised to the required powers, #Y4=J 6  
    % and compile them in a matrix: /K(o]J0F  
    % ----------------------------- G%s 2P.cd  
    if rpowers(1)==0 "T|PS 6R~  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); |6$p;Aar  
        rpowern = cat(2,rpowern{:}); #*KNPh  
        rpowern = [ones(length_r,1) rpowern]; 8p:j&F  
    else o}$ EG  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); H XmS|PX  
        rpowern = cat(2,rpowern{:}); 6*3.SGUY  
    end bLwAXW2K+  
    R.yC(r  
    43@{JK9G  
    % Compute the values of the polynomials: 2'W<h)m)z  
    % -------------------------------------- aXoVy&x=  
    y = zeros(length_r,length(n)); 7eiV{tYF  
    for j = 1:length(n) 7D;cw\ |  
        s = 0:(n(j)-m_abs(j))/2; 4B^ZnFJ%m  
        pows = n(j):-2:m_abs(j); WUh$^5W  
        for k = length(s):-1:1 lb3]$Da  
            p = (1-2*mod(s(k),2))* ... FGoy8+nB1M  
                       prod(2:(n(j)-s(k)))/              ... @9lUSk^9  
                       prod(2:s(k))/                     ... N9v1[~ bv_  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... BDjn !3  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); z& fwE$Nm  
            idx = (pows(k)==rpowers); =,UWX3`f  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ,d{"m)r<  
        end [_Qa9e  
         v uoQz\  
        if isnorm J{k79v  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ;oy-#p>N%  
        end L{8xlx`  
    end Mw;sLsu  
    % END: Compute the Zernike Polynomials BBtzs^C|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <=>=.kmGt  
    %|^fi8!:|  
    <psZQdH  
    % Compute the Zernike functions: Ro9tZ'N!S  
    % ------------------------------ =fO5cA6Z  
    idx_pos = m>0; L2[f]J%  
    idx_neg = m<0; 0Nnsjh  
    #6CC3TJ'k  
    <y?r!l=Am  
    z = y; 9R7 A8  
    if any(idx_pos) 7eV di*  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); pP*a  
    end pLLGus+W  
    if any(idx_neg) b)e *$)  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); d2H|LMhJ  
    end 2(#7[mgPI  
    %3ICI  
    f PM8f  
    % EOF zernfun *q-['"f  
     
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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)  f=mZu1(FZ  
    Q^e}?v%=%3  
    DDE还是手动输入的呢? |qmu _x\  
    BC&9fr  
    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