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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, 5sj$XA?5  
    我输入10阶的n、m,r,theta为38025*1向量,最后得到的z是29525*10阶的矩阵, _3NH"o d  
    这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? [@B!N+P5;  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? `Q/\w1-Q  
    .JJ50p  
    [0]J 2  
    Vg :''!4t2  
    kY6_n4  
    function z = zernfun(n,m,r,theta,nflag) Zz]/4 4t  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. G:wO1f6  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N  =zDvZ(5  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ?A24h !7  
    %   unit circle.  N is a vector of positive integers (including 0), and "q!*RO'a  
    %   M is a vector with the same number of elements as N.  Each element ZR"qrCSw`  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) sY?wQ:  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, (d* | |"  
    %   and THETA is a vector of angles.  R and THETA must have the same Sfp-ns32%A  
    %   length.  The output Z is a matrix with one column for every (N,M) 5*Qzw[[=  
    %   pair, and one row for every (R,THETA) pair. ts("(zI1E  
    % (ip3{d{CT]  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ,U+>Q!$`\^  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), U!K#g_}  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral z]LVq k  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, g!r) yzK  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized `*`ZgTV  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. N3a ]!4Y\  
    % \3%3=:  
    %   The Zernike functions are an orthogonal basis on the unit circle. 4x?I,cAN  
    %   They are used in disciplines such as astronomy, optics, and :S7[<SwL  
    %   optometry to describe functions on a circular domain. I)0_0JXs  
    % Tj\hAcD  
    %   The following table lists the first 15 Zernike functions. h?} S|>9  
    % l Ft&cy2  
    %       n    m    Zernike function           Normalization + Okw+v  
    %       -------------------------------------------------- TDWD8??e  
    %       0    0    1                                 1 ,^ dpn  
    %       1    1    r * cos(theta)                    2 :f7vGO"t  
    %       1   -1    r * sin(theta)                    2 Ke]'RfO\  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) {yEL$8MC  
    %       2    0    (2*r^2 - 1)                    sqrt(3) %M`zkA2]J  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 0ia-D`^me  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) V?`|Ha}  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) \%%M>4c  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) tK'9%yA\  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) :Z_abKt  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) *,*XOd:3TL  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 5Z"N2D)."  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) klY, @  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Jw^my4  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ,JTyOBB<I  
    %       -------------------------------------------------- FL&Y/5  
    % 8]O#L}"  
    %   Example 1: #e[r0f?U  
    % aSJD'u4w.a  
    %       % Display the Zernike function Z(n=5,m=1) 78<fbN5}r  
    %       x = -1:0.01:1; 5lM 3In@  
    %       [X,Y] = meshgrid(x,x); jHA(mU)b  
    %       [theta,r] = cart2pol(X,Y); O'.{6H;t  
    %       idx = r<=1; H`Zg-j`  
    %       z = nan(size(X)); PlgpH'z4$  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); c? GV  
    %       figure TC@F*B;  
    %       pcolor(x,x,z), shading interp N+H[Y4c?F&  
    %       axis square, colorbar 6Bexwf<u  
    %       title('Zernike function Z_5^1(r,\theta)') De>,i%`Q,D  
    % ]=/?Ooh  
    %   Example 2: IlI5xkJ(  
    % 'P4V_VMK  
    %       % Display the first 10 Zernike functions /oGaA@#+  
    %       x = -1:0.01:1; hw)z]  
    %       [X,Y] = meshgrid(x,x); g?Rq .py]!  
    %       [theta,r] = cart2pol(X,Y); jYBiC DD  
    %       idx = r<=1; LcNI$g;}Yf  
    %       z = nan(size(X)); EQM[!g^a  
    %       n = [0  1  1  2  2  2  3  3  3  3]; rg 0u#-  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Yfs eX;VX  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 1:./f|m  
    %       y = zernfun(n,m,r(idx),theta(idx)); |%3>i"Y@AK  
    %       figure('Units','normalized') l <Z7bo  
    %       for k = 1:10 !ZCxi  
    %           z(idx) = y(:,k); |S]fs9  
    %           subplot(4,7,Nplot(k)) /#L4ec-'  
    %           pcolor(x,x,z), shading interp J*ZcZ FbWN  
    %           set(gca,'XTick',[],'YTick',[]) o"A?Aq  
    %           axis square <A`SC;k\u  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) <$#^)]Ts  
    %       end *7#5pT~  
    % f3h]t0M  
    %   See also ZERNPOL, ZERNFUN2. Y;dqrA>@  
    uBC#4cX`D*  
    tn(6T^u  
    %   Paul Fricker 11/13/2006 - &)  
    "avG#rsH  
    E5*pD*#  
    1,we: rwX  
    fl4'dv  
    % Check and prepare the inputs: e<~bDFH  
    % ----------------------------- 1:u~T@;" `  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) gh `_{l  
        error('zernfun:NMvectors','N and M must be vectors.') ,Hp7`I>/  
    end hVJ}EF 0  
    ^(BE_<~  
    r $YEq5  
    if length(n)~=length(m) ?f!&M  
        error('zernfun:NMlength','N and M must be the same length.') >{Xyl):  
    end H6KBXMYO  
    fN9uSnu  
    ^.*zBrFx  
    n = n(:); "1p, r&}  
    m = m(:); OL@$RTh  
    if any(mod(n-m,2)) 9tmnx')_  
        error('zernfun:NMmultiplesof2', ... 4ZYywDwn  
              'All N and M must differ by multiples of 2 (including 0).') ZK<c(,oZ^  
    end 8zjJshE/  
    L/5th}m  
    bcAk$tA2  
    if any(m>n) -f?,%6(1  
        error('zernfun:MlessthanN', ... 7$*x&We  
              'Each M must be less than or equal to its corresponding N.') rV*Ri~Vx  
    end 6.|[;>Km  
    EQ"+G[j~x  
    R[QBFL<  
    if any( r>1 | r<0 ) =t|,6Vp  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') P#rS.CIh  
    end vJX0c\e  
    w.+G+ r=  
    SI=7$8T5=5  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) '+*'sQvH[  
        error('zernfun:RTHvector','R and THETA must be vectors.') ]L3MIaO2T  
    end &,\my-4c>  
    {qs>yQ6a:-  
    xlc2,L;i  
    r = r(:); ws$kwSHq  
    theta = theta(:); fOP3`G^\  
    length_r = length(r); y3P4]sq  
    if length_r~=length(theta) w ,0OO f  
        error('zernfun:RTHlength', ... {CX06BP  
              'The number of R- and THETA-values must be equal.') \J-D@b;  
    end _Y)Wi[  
    bH%d*  
    E0u&hBd3_  
    % Check normalization: I(z16wQ  
    % -------------------- #f_.  
    if nargin==5 && ischar(nflag) 3A.lS+P1  
        isnorm = strcmpi(nflag,'norm'); \9}DAM_  
        if ~isnorm [&lH[:Y#  
            error('zernfun:normalization','Unrecognized normalization flag.') uu/2C \n}  
        end AH:0h X6+  
    else m<J:6^H@  
        isnorm = false; \]3[Xw-$  
    end E+$D$a  
    ~CHVU3  
    0u +_D8G  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &@w0c>Y  
    % Compute the Zernike Polynomials s'BlFB n  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RxVZn""  
    tCv}+7)   
    hzA+,  
    % Determine the required powers of r: RP k'1nD  
    % ----------------------------------- I2,AT+O<  
    m_abs = abs(m); ~{pds  
    rpowers = []; VDiW9]  
    for j = 1:length(n) O-3aU!L  
        rpowers = [rpowers m_abs(j):2:n(j)]; 3KtJT&RuL  
    end 1I40N[PE)  
    rpowers = unique(rpowers); U&#`5u6'j  
    {T DZDH  
    Zb:Z,O(vn  
    % Pre-compute the values of r raised to the required powers, ryb81.|  
    % and compile them in a matrix: |<MSV KW  
    % ----------------------------- /. >%IcK  
    if rpowers(1)==0 {+EnJ"  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ?}(B8^  
        rpowern = cat(2,rpowern{:}); RNt9Qdr4y  
        rpowern = [ones(length_r,1) rpowern]; {HFx+<JG  
    else S F da?>  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 8d&%H,  
        rpowern = cat(2,rpowern{:}); D&qJ@PR  
    end \m=k~Cf:f  
    vhDtjf/*  
    }]=@Y/p  
    % Compute the values of the polynomials: N*)O_Ki  
    % -------------------------------------- OP\L  
    y = zeros(length_r,length(n)); wVX2.D'n<  
    for j = 1:length(n) }T}xVd0  
        s = 0:(n(j)-m_abs(j))/2; AS'+p%(  
        pows = n(j):-2:m_abs(j); ?%n"{k?#  
        for k = length(s):-1:1 Fh/sD?  
            p = (1-2*mod(s(k),2))* ... yD@1H(yM  
                       prod(2:(n(j)-s(k)))/              ... *Rxn3tR7  
                       prod(2:s(k))/                     ... Mh {>#Gs  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... X8wtdd]64  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); `$q0fTz  
            idx = (pows(k)==rpowers); tq51;L  
            y(:,j) = y(:,j) + p*rpowern(:,idx); I+31:#d  
        end s'bTP(wl9  
         p1W6s0L  
        if isnorm Y~?Z'uR  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); $EzWUt  
        end PKQ.gPu6*@  
    end <(H<*Xf9  
    % END: Compute the Zernike Polynomials <~S]jtL.j:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /U`p|M;  
    hD4>mpk  
    n~0MhE0H  
    % Compute the Zernike functions: KLG29G  
    % ------------------------------ d]MpE9@'v  
    idx_pos = m>0; C>SO d]  
    idx_neg = m<0; P'DcNMdw  
    wuM'M<J@  
    {|B[[W\TN  
    z = y; l]gW_wUQd  
    if any(idx_pos) Xz9[0;Q  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); &9"Y:),  
    end :Gew8G  
    if any(idx_neg) >]o>iOz;]  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); B#cN'1c  
    end @4]{ZUV  
    d24_,o\_  
    iio-RT?!  
    % EOF zernfun ?7J::}R  
     
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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)  XWQ `]m)  
    R! On  
    DDE还是手动输入的呢? u' Q82l&Y  
    ,\cV,$  
    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