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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>"cc@8&~  
    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, e=n{f*KG`  
    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? m~j\?mb{+  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? FH`'1iVH  
    o&XMgY~  
    Neo^C_[vN  
    r0g/:lJi  
    bDFCZH-:'O  
    function z = zernfun(n,m,r,theta,nflag) PZ!dn%4jy  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. >xZhK63C/  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N aa0`y  
    %   and angular frequency M, evaluated at positions (R,THETA) on the (XG[_  
    %   unit circle.  N is a vector of positive integers (including 0), and ueE?"Hk  
    %   M is a vector with the same number of elements as N.  Each element Y7:Y{7E7  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) +{C9uY)$vf  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, C>:/(O  
    %   and THETA is a vector of angles.  R and THETA must have the same }rY?=I  
    %   length.  The output Z is a matrix with one column for every (N,M) eb.cq"C  
    %   pair, and one row for every (R,THETA) pair. 3?*M{Y|  
    % Y0 X"Zw  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike =(|xU?OL  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), CmJ?_>  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ?lc[ hH  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, N,/BudF o  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized I>kiah*  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. EOBs}M;  
    % $['7vcB^  
    %   The Zernike functions are an orthogonal basis on the unit circle. mP)3cc5T  
    %   They are used in disciplines such as astronomy, optics, and KCJN<  
    %   optometry to describe functions on a circular domain. ,\S pjE  
    % _Vo)<--+I  
    %   The following table lists the first 15 Zernike functions. pVV}1RDa  
    % uK;K{  
    %       n    m    Zernike function           Normalization (! 0j4'  
    %       -------------------------------------------------- Tbi]oB#  
    %       0    0    1                                 1 >St. &#c  
    %       1    1    r * cos(theta)                    2 )H;pGM:  
    %       1   -1    r * sin(theta)                    2 XJ:>UNf5;  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Y3P.|  
    %       2    0    (2*r^2 - 1)                    sqrt(3) t":W.q<  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) T}n}.JwU  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) zmB31' _  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 7>'uj7r]=  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) %qS]NC  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ^zaKO'KcV  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ':!3jZP"m  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) A[^qq UL'  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) z29qARiX  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Sg.+`xww3  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) d1~_?V'r]  
    %       -------------------------------------------------- VDByj "%  
    % |RR%bQ^{  
    %   Example 1: *%T)\\H2  
    % T|o`a+?  
    %       % Display the Zernike function Z(n=5,m=1) I!$jYY2  
    %       x = -1:0.01:1; gf68iR.Gs  
    %       [X,Y] = meshgrid(x,x); 0^GbpSW{  
    %       [theta,r] = cart2pol(X,Y); :M22P`:  
    %       idx = r<=1; J+)'-OFt0  
    %       z = nan(size(X)); > $w^%I  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 0T9@,scY  
    %       figure a>wfhmr  
    %       pcolor(x,x,z), shading interp %s$rP  
    %       axis square, colorbar /OQK/ t63  
    %       title('Zernike function Z_5^1(r,\theta)') \!+-4,CbZY  
    % vix&E`0yD  
    %   Example 2: 5l41Q  
    % I#|ocz  
    %       % Display the first 10 Zernike functions 4GG1E. z}  
    %       x = -1:0.01:1; uQGz;F x  
    %       [X,Y] = meshgrid(x,x); Q'Jv} 'eK_  
    %       [theta,r] = cart2pol(X,Y); La"o)L +m_  
    %       idx = r<=1; V I6\   
    %       z = nan(size(X)); <u/a`E?  
    %       n = [0  1  1  2  2  2  3  3  3  3]; [_y9"MMwn  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; s<A*[  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; H-eEhI(;O  
    %       y = zernfun(n,m,r(idx),theta(idx)); ! jbEm8bt  
    %       figure('Units','normalized') uy/y wm/?=  
    %       for k = 1:10 `%-4>jI9-  
    %           z(idx) = y(:,k); m"lE&AM64p  
    %           subplot(4,7,Nplot(k)) h [nH<m  
    %           pcolor(x,x,z), shading interp Vh"MKJ'R^  
    %           set(gca,'XTick',[],'YTick',[]) 79)A%@YHQQ  
    %           axis square OSp?okV  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) cCM j\H@  
    %       end cu[!D}tVU  
    % NTqo`VWe  
    %   See also ZERNPOL, ZERNFUN2. W8f`J2^"M  
    2HcsQ*H] G  
    ^C!mCTL1N  
    %   Paul Fricker 11/13/2006 \ ,>_c  
    <"* "1(wN  
    3c c1EQ9  
    {mNdL J  
    Q]< (bD.7  
    % Check and prepare the inputs: 12idM*  
    % ----------------------------- C&=x3Cz  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ecn}iN  
        error('zernfun:NMvectors','N and M must be vectors.') O$a#2p&  
    end Xo2^N2I  
    bfFmTI$,  
    S8\+XJ  
    if length(n)~=length(m) |<sf:#YzY&  
        error('zernfun:NMlength','N and M must be the same length.') m"n.Dz/S  
    end [}z?1Gj;W(  
    e#tIk;9Xz  
    m7JPH7P@BM  
    n = n(:); *5 e<\{!  
    m = m(:); f%c06Un=  
    if any(mod(n-m,2)) 3 h#s([uL  
        error('zernfun:NMmultiplesof2', ... F&xv z2G  
              'All N and M must differ by multiples of 2 (including 0).') Hw Z^D= A  
    end >A3LA3( c  
    4<u;a46Z#M  
    |VK:2p^ u  
    if any(m>n) 0f1H8zV  
        error('zernfun:MlessthanN', ... z;J  
              'Each M must be less than or equal to its corresponding N.') \I;cZ>{u"}  
    end lqF>=15  
    im=5{PbJ^  
    XJUEwX  
    if any( r>1 | r<0 ) cST\~SUm  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') I-,>DLG  
    end ?FN9rhAC  
    iAK/d)bq  
    [eyb7\#   
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) @: Z#E[N H  
        error('zernfun:RTHvector','R and THETA must be vectors.') !}ilN 1>  
    end ) !i!3  
    Jz0K}^Dj[  
    0C]4~F x~  
    r = r(:);  =^Th[B  
    theta = theta(:); r&SO:#rOSM  
    length_r = length(r); QP:9%f>=  
    if length_r~=length(theta) Lx%:t YZ  
        error('zernfun:RTHlength', ... bhYU5I 9  
              'The number of R- and THETA-values must be equal.') }wfI4?}j}  
    end WHP;Neb6  
    AuAT]`  
    y1iX!m~)  
    % Check normalization: *<r%aeG$em  
    % -------------------- `NQ{)N0!  
    if nargin==5 && ischar(nflag) bo1I&I  
        isnorm = strcmpi(nflag,'norm'); 6GzzG P^  
        if ~isnorm -,^WaB7u\  
            error('zernfun:normalization','Unrecognized normalization flag.') 45) D+  
        end \#++s&06  
    else "qS!B.rt:  
        isnorm = false; VG)="g[%)  
    end +#~O'r]%GG  
    #&V5H{  
    .a,(pq Jg  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !R74J=#(  
    % Compute the Zernike Polynomials ^!}F%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _s*! t  
    %K?iNe  
    wu2:'y>n  
    % Determine the required powers of r: _IxamWpX$  
    % ----------------------------------- wWTQ6~Y%d  
    m_abs = abs(m); EjSD4  
    rpowers = []; IcFK,y%1  
    for j = 1:length(n) K6hfauWd[  
        rpowers = [rpowers m_abs(j):2:n(j)]; [/OQyb4F<  
    end p![&8i@ym  
    rpowers = unique(rpowers); ~ M*gsW$  
    3u_oRs  
    Vv7PCaq  
    % Pre-compute the values of r raised to the required powers, vTd- x>n  
    % and compile them in a matrix: .E$q&7@/j  
    % -----------------------------  2:'lZQ  
    if rpowers(1)==0 | ]# +v@  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); C8.W5P[U  
        rpowern = cat(2,rpowern{:}); G#0,CLGN^  
        rpowern = [ones(length_r,1) rpowern]; =Z`0>R`  
    else )b92yP{  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); BI.V0@qZ  
        rpowern = cat(2,rpowern{:}); ;&kn"b}G;  
    end Pbe7SRdr^  
    ?E7=:h(@t  
    9|=nV|R'6  
    % Compute the values of the polynomials: {y6C0A*  
    % -------------------------------------- U:n*<l-k}  
    y = zeros(length_r,length(n)); h<Wg3o  
    for j = 1:length(n) v459},!P  
        s = 0:(n(j)-m_abs(j))/2; k 4B_W  
        pows = n(j):-2:m_abs(j); ~<,Sh~Ana.  
        for k = length(s):-1:1 U5<@<j(@  
            p = (1-2*mod(s(k),2))* ... W-XpJ\_  
                       prod(2:(n(j)-s(k)))/              ... P}@*Z>j:#  
                       prod(2:s(k))/                     ... &@6 GI<  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... r6t&E%b  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ~ziexZ=N  
            idx = (pows(k)==rpowers); e+@xs n3  
            y(:,j) = y(:,j) + p*rpowern(:,idx); )6{P8k4Zr  
        end GV8)Kor%  
         M&yqfb[  
        if isnorm oZ:{@ =  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); e$|VG* d  
        end Wc|z7P~',%  
    end 5UO k)rOf  
    % END: Compute the Zernike Polynomials VR4%v9[1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G),db%,X2  
    B 8{ uR  
    dy:d=Z  
    % Compute the Zernike functions: /{X_ .fv<v  
    % ------------------------------ w$>3pQ8d  
    idx_pos = m>0; H$tb;:  
    idx_neg = m<0; KlU qoJ;"  
    Rla4L`X;  
    O]qPmEj  
    z = y; bulboyA&#  
    if any(idx_pos)  $Nu)E  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); u D(t`W"  
    end L~eAQR  
    if any(idx_neg) ?N>pZR  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); m r4b  
    end ~/|zlu*jpc  
    r1Z<:}ZwK  
    =i6:puf  
    % EOF zernfun C).2gQ G  
     
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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)  g6euXI  
    oI@ 9}*  
    DDE还是手动输入的呢? "!q?P" @C  
    L[C*@ uK  
    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