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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 正序阅读 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, X62h7?'Pd  
    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, iCCY222:  
    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? 4A:@+n%3m  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ++-\^'&1  
    r*<)QP^B~  
    O:3pp8  
    ;JMd(\+-  
    KFBo1^9N  
    function z = zernfun(n,m,r,theta,nflag) Af5O;v\  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. PA;RUe  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Esw#D90q  
    %   and angular frequency M, evaluated at positions (R,THETA) on the #r; ' AG  
    %   unit circle.  N is a vector of positive integers (including 0), and Fxy-_%a  
    %   M is a vector with the same number of elements as N.  Each element Bo8+ uRF|  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) A.m#wY8  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, vRYQ4B4o  
    %   and THETA is a vector of angles.  R and THETA must have the same SlI0p&2,  
    %   length.  The output Z is a matrix with one column for every (N,M) Wq8Uq}~_g  
    %   pair, and one row for every (R,THETA) pair. zr%lBHuW  
    % w1EYXe  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike MCU{@ \?Xf  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Lz2 AWqR  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 9VdVom|e  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, l@nkR&4[  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized TLzg*  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. KHKf+^uu  
    % Z3Os9X9p  
    %   The Zernike functions are an orthogonal basis on the unit circle. 8SK}#44Xz  
    %   They are used in disciplines such as astronomy, optics, and O`U&0lKi'  
    %   optometry to describe functions on a circular domain. @47MJzC  
    % o0^'x Vv  
    %   The following table lists the first 15 Zernike functions. 'x BBQP  
    % ;|e{J$  
    %       n    m    Zernike function           Normalization H[ocIw  
    %       -------------------------------------------------- JzMPLmgG/  
    %       0    0    1                                 1 :<4:h.gO8  
    %       1    1    r * cos(theta)                    2 Q^4j  
    %       1   -1    r * sin(theta)                    2 Ks:~Z9r}  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) g2.%x \d  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 8P.UB{QNe  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) x;89lHy@e  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) DbFTNoVR  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Xjc{={@p3  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) c%w@-n`  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) O{rgx~lLJt  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) _In[Z?P}  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) '`$a l7D  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) B) J.(k`p  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) My0h9'K  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) SC)4u l%  
    %       -------------------------------------------------- P|YBCH  
    % i X qB-4"  
    %   Example 1: J Sz'oA5  
    % f~-81ctu  
    %       % Display the Zernike function Z(n=5,m=1) tJo,^fdfv  
    %       x = -1:0.01:1; 8v"tOa4D7  
    %       [X,Y] = meshgrid(x,x); |^Nz/PN  
    %       [theta,r] = cart2pol(X,Y); w~@.&  
    %       idx = r<=1; $>1 'pV  
    %       z = nan(size(X)); p*)RP2  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ]YYjXg}%  
    %       figure :D6"h[7  
    %       pcolor(x,x,z), shading interp _,(]T&j #2  
    %       axis square, colorbar ^l;nBD#nJ  
    %       title('Zernike function Z_5^1(r,\theta)') U;o[>{L   
    % iD,iv  
    %   Example 2: cMOvM0f  
    % 3>qUYxG8  
    %       % Display the first 10 Zernike functions R?!xO-^t  
    %       x = -1:0.01:1; FU/yJy  
    %       [X,Y] = meshgrid(x,x); \)859x&(  
    %       [theta,r] = cart2pol(X,Y); 9H:5XR  
    %       idx = r<=1; Bi2be$nV  
    %       z = nan(size(X)); =SPuOy8  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 8`}(N^=}  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Tyt:Abym=  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 'jWd7w~(  
    %       y = zernfun(n,m,r(idx),theta(idx)); jXq~ x"(  
    %       figure('Units','normalized') }7YDe'5V  
    %       for k = 1:10 e_s9E{(  
    %           z(idx) = y(:,k); |E$Jt-'  
    %           subplot(4,7,Nplot(k)) =0 W`tx  
    %           pcolor(x,x,z), shading interp , "w`,c>!  
    %           set(gca,'XTick',[],'YTick',[]) 5\1Z"?  
    %           axis square g{w IdV  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) <r]7xsr  
    %       end CL%?K<um  
    % MVHj?  
    %   See also ZERNPOL, ZERNFUN2. |g]TWKc*  
    +RS>#zd/=  
    un0t zz  
    %   Paul Fricker 11/13/2006 Dgh|,LqUB  
    Q#P=t83  
    %\PnsnJ9Q  
    rhY>aj  
    Gb+cT  
    % Check and prepare the inputs: GczGW4\P'  
    % ----------------------------- Ai\"w0  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 2Cn^<(F^4I  
        error('zernfun:NMvectors','N and M must be vectors.') 33x3zEUt6  
    end %||}WT-wv  
    B%!z7AT  
    Z0T{1YEJ  
    if length(n)~=length(m) |,M&ks  
        error('zernfun:NMlength','N and M must be the same length.') RbX!^v<0f6  
    end h+F@apUS  
    ;;'b;,/  
    w#[Ul9=?6  
    n = n(:); Knsb`1"E^6  
    m = m(:); k+S+ : 5  
    if any(mod(n-m,2)) +4^XFPq~  
        error('zernfun:NMmultiplesof2', ... `EVTlq@<  
              'All N and M must differ by multiples of 2 (including 0).') <K!5N&vh  
    end M iIH&z  
    r4caIV  
    P{mV  
    if any(m>n) E 5}T_~-{  
        error('zernfun:MlessthanN', ... eCdx(4(\a  
              'Each M must be less than or equal to its corresponding N.') 0 z{S@  
    end *9e T#dH  
    UN_f2  
    =BJ/ZM  
    if any( r>1 | r<0 ) Ls#pe  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 8}h ^Frh  
    end ;SkC[;`J  
    )%%RI_J T  
    ;`g\Tu  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) , RfU1R  
        error('zernfun:RTHvector','R and THETA must be vectors.') KTxdZt  
    end vai.",b=n6  
    #EtS9D'd+  
    ;>[).fX>/  
    r = r(:); M`\c'|i/  
    theta = theta(:); XPXC7_fV  
    length_r = length(r); 8,2l >S  
    if length_r~=length(theta) \lHi=}0  
        error('zernfun:RTHlength', ... ^T"9ZBkb  
              'The number of R- and THETA-values must be equal.') V[,/Hw~d%  
    end T:x5 ,vpM  
    %Bmi3 =Rr  
    AC3K*)`E  
    % Check normalization: R[ S*ON  
    % -------------------- _v4TyJ  
    if nargin==5 && ischar(nflag)  A$ %5l  
        isnorm = strcmpi(nflag,'norm'); a*&P>Lwe7&  
        if ~isnorm XG<J'3  
            error('zernfun:normalization','Unrecognized normalization flag.') d+~c$(M)  
        end udB:ys  
    else $1oU^V Y  
        isnorm = false; Y{Kpopst  
    end o*97Nbjn  
    eJ)Bs20Q  
    Vi`+2%4  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !JUXq  
    % Compute the Zernike Polynomials &w:"e'FG`  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^ef:cS$;  
    mn\e(WoX  
    n|NI]Qi*  
    % Determine the required powers of r: z;1tJ  
    % ----------------------------------- k#`.!yI,  
    m_abs = abs(m); W-=~Afy  
    rpowers = []; liFNJd`|o+  
    for j = 1:length(n) aW %ulZ  
        rpowers = [rpowers m_abs(j):2:n(j)]; ~$#DB@b  
    end 8<3J!X+  
    rpowers = unique(rpowers); K]zBPfx  
    y% uUA]c*m  
    lE08UEk1i  
    % Pre-compute the values of r raised to the required powers, J/w?Fa<  
    % and compile them in a matrix: )z3mS2  
    % ----------------------------- ~CldqXeI  
    if rpowers(1)==0 ~b5aT;ObR  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); wQb")3dw  
        rpowern = cat(2,rpowern{:}); eJE?H]  
        rpowern = [ones(length_r,1) rpowern]; !l~tBJr*sB  
    else GB\.msls  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ?nrd$,  
        rpowern = cat(2,rpowern{:}); /YH Bhoat  
    end n?&G>`u*  
    8^p/?R^bu  
    zF<*h~  
    % Compute the values of the polynomials: dTyTj|"x{  
    % -------------------------------------- e{Om W  
    y = zeros(length_r,length(n)); cg7NtY  
    for j = 1:length(n) W5$jIQ}Bw  
        s = 0:(n(j)-m_abs(j))/2; \%&QIe;:k  
        pows = n(j):-2:m_abs(j); $ePAsJ  
        for k = length(s):-1:1 Mp?Ev.  
            p = (1-2*mod(s(k),2))* ... /-E>5wU  
                       prod(2:(n(j)-s(k)))/              ... RoM'+1nP:#  
                       prod(2:s(k))/                     ... 5'\/gvxIC  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ho#] ?Z#  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); R[wy{4<y  
            idx = (pows(k)==rpowers); Qz{:m  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Y1{6lhxgE  
        end f|?i6.N> f  
         zdyS"H}  
        if isnorm xex/L%!Rj  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ^O#,%>1J  
        end J\_tigd   
    end VyCBJK  
    % END: Compute the Zernike Polynomials >~TLgq*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |GL#E"[&'  
    h\C  
    4xT(Uj  
    % Compute the Zernike functions: p}R)qz-=5U  
    % ------------------------------ e.\d7_T+  
    idx_pos = m>0; 4&K~EX"^T  
    idx_neg = m<0; .pu]21m=  
    {qx}f^WV  
    93)&  
    z = y; !s\-i6S>  
    if any(idx_pos) vwZ2kk!|i  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ;. !AX|v  
    end qQ/j+  
    if any(idx_neg) $4>K2  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); + ?*,J=/  
    end zjM+F{P8  
    5Tb93Q@c  
    `P)atQ  
    % EOF zernfun 8NPt[*  
     
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    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
    离线jssylttc
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    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    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)  Z7V 1e<E  
    <2fZYt vt  
    DDE还是手动输入的呢? -L NJ*?b  
    Ev,>_1#Xm  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    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)