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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, h\|T(597.  
    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, Upc_"mkI.  
    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? $F@ ,,*  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? AZ. j>+0xx  
    {afIr1j/m  
    Dd:48sN:Jq  
    K{iC'^wP  
    gS!zaD7Nr  
    function z = zernfun(n,m,r,theta,nflag) xE6hE'rh.O  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. rDLgQ{Sea  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N riSgb=7q9  
    %   and angular frequency M, evaluated at positions (R,THETA) on the N3\vd_D(  
    %   unit circle.  N is a vector of positive integers (including 0), and 5C5OLAl v  
    %   M is a vector with the same number of elements as N.  Each element C!|Yz=e  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) g7v(g?  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, gq.l=xS  
    %   and THETA is a vector of angles.  R and THETA must have the same kq> I?wg  
    %   length.  The output Z is a matrix with one column for every (N,M) \| 'Yuh  
    %   pair, and one row for every (R,THETA) pair. *dx E (dP  
    % Z1U@xQj  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike To,*H OP  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), R-Gg= l5  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral YN7JJJ/~T  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ~Vf A  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized |0VZ1{=*  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. $AdBX}{  
    % d*LW32B@  
    %   The Zernike functions are an orthogonal basis on the unit circle. !{b4+!@p  
    %   They are used in disciplines such as astronomy, optics, and ;esOe\z jE  
    %   optometry to describe functions on a circular domain. (J.k\d   
    % Pk`3sfz  
    %   The following table lists the first 15 Zernike functions. 6P0 2=  
    % B|r'  
    %       n    m    Zernike function           Normalization #1p\\Av  
    %       -------------------------------------------------- xk s M e  
    %       0    0    1                                 1 3]pHc)p!.  
    %       1    1    r * cos(theta)                    2 D/Py?<n-B  
    %       1   -1    r * sin(theta)                    2 5Rae?* XH  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) JD]uDuE  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ,k}(]{ -  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) aqv'c j>  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 9<5S!?JL  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) V}Ce3wgvA  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) _>4)q=  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) I M G^L  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 8Y/1+-  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) YVPLHwh/5  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) &BN#"- J  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -]Q\G  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) dQy K4T  
    %       -------------------------------------------------- JgBC:t^\pV  
    % -[[( Zx  
    %   Example 1: ~re~Ys  
    % $t0JfDd6Ky  
    %       % Display the Zernike function Z(n=5,m=1) k&17 (Tv$  
    %       x = -1:0.01:1; sEi9<$~R@0  
    %       [X,Y] = meshgrid(x,x); QSOG(}w  
    %       [theta,r] = cart2pol(X,Y); H^M>(kT#&  
    %       idx = r<=1; jW>K#vj  
    %       z = nan(size(X)); #Sg"/Cc  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); bbT$$b-  
    %       figure iWIq~t*,H]  
    %       pcolor(x,x,z), shading interp kq@~QI?9  
    %       axis square, colorbar Pk;YM}  
    %       title('Zernike function Z_5^1(r,\theta)') pk0{*Z?@  
    % PJ^qE| X  
    %   Example 2: @4n>I+6*&  
    % WWATG=  
    %       % Display the first 10 Zernike functions pj7v{H+  
    %       x = -1:0.01:1; J M`[|"R%  
    %       [X,Y] = meshgrid(x,x); ^,aI2vC  
    %       [theta,r] = cart2pol(X,Y); )W&{OMr  
    %       idx = r<=1; "<LWz&e^^  
    %       z = nan(size(X)); ri6KD  
    %       n = [0  1  1  2  2  2  3  3  3  3]; jwP5pu  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; %*#+(A"V  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; >T-4!ZvS\j  
    %       y = zernfun(n,m,r(idx),theta(idx)); nkf7Fq}  
    %       figure('Units','normalized') ?hViOh$.  
    %       for k = 1:10 |#'n VN.;  
    %           z(idx) = y(:,k); mv^X{T  
    %           subplot(4,7,Nplot(k)) Eihn%Esa  
    %           pcolor(x,x,z), shading interp }_5R9w]"  
    %           set(gca,'XTick',[],'YTick',[]) n]i#&[*A(  
    %           axis square D}Sww5ZmP  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) h}kJ,n  
    %       end mhB2l/  
    % QW tDZ>  
    %   See also ZERNPOL, ZERNFUN2. ^b.#4i (v  
    aemi;61T\  
    ck\W'Y*Q7  
    %   Paul Fricker 11/13/2006 `evF?t11X  
    m.<u !MI  
    Xj~%kPe  
    A0:rn\$l3  
    :Qh rh(i  
    % Check and prepare the inputs: hCS}  
    % ----------------------------- qG ? :Q  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) !#=3>\np+X  
        error('zernfun:NMvectors','N and M must be vectors.') *"OUwEl a  
    end !F.h+&^D;  
    #'0Yzh]qc  
    n 4y]h  
    if length(n)~=length(m) `.J17mQe"  
        error('zernfun:NMlength','N and M must be the same length.') ,Vw>3|C  
    end ~9E_L?TW*  
    YV!hlYOBi  
    @\o"zU  
    n = n(:); =1@LMIi5x  
    m = m(:); C511 hbF  
    if any(mod(n-m,2)) s^K2,D]P  
        error('zernfun:NMmultiplesof2', ... ^3 9lUKL  
              'All N and M must differ by multiples of 2 (including 0).') cv G*p||  
    end H2+b3y-1a]  
    cqSXX++CS,  
    4QTHBT+2`  
    if any(m>n) gKQV99  
        error('zernfun:MlessthanN', ... {M5t)-  
              'Each M must be less than or equal to its corresponding N.') b?c/J {me  
    end qR_>41JU"  
    *3r s+0  
    O1S7t)ag  
    if any( r>1 | r<0 ) ts9wSx~[+  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') lo(C3o'  
    end oeB'{bG  
    S_/S2(V"  
    _DH^ K 9,9  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) R TpNxr{[  
        error('zernfun:RTHvector','R and THETA must be vectors.') U3Z=X TB  
    end QS5t~rb  
    XbB(<\0+  
    =$fz</S=J  
    r = r(:); \~"Ub"~I  
    theta = theta(:); &9fQW?Czs  
    length_r = length(r); zX4RqI  
    if length_r~=length(theta) e6Y>Bk   
        error('zernfun:RTHlength', ... 5af0- hj  
              'The number of R- and THETA-values must be equal.') ,(pp+hNq  
    end \yC/OLXq  
    &i(Ip'r  
    YrB-n  
    % Check normalization: | @$I<  
    % -------------------- V`1{*PrI@L  
    if nargin==5 && ischar(nflag) j~G(7t  
        isnorm = strcmpi(nflag,'norm'); )38%E;T{X  
        if ~isnorm e-`.Ht  
            error('zernfun:normalization','Unrecognized normalization flag.') {;u,04OVK  
        end UZ2_FP  
    else W>C?a=r~  
        isnorm = false; jr?/wtw  
    end X<$Tn60,  
    @$ Zh^+x!  
    $_E.D>5^%7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8c+V$rH_  
    % Compute the Zernike Polynomials nOvR, 6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% } &B6  
    rB$~,q&.V  
    1o%#kf  
    % Determine the required powers of r: G rp{ .  
    % ----------------------------------- jDpA>{O[  
    m_abs = abs(m); 9hfg/3t('  
    rpowers = []; 8O9^g4?  
    for j = 1:length(n) 3NLn}  
        rpowers = [rpowers m_abs(j):2:n(j)]; Z%]K,9K  
    end -smN}*3[  
    rpowers = unique(rpowers); )>:~XA|?  
    jRU: un4  
    `\62 iUN  
    % Pre-compute the values of r raised to the required powers, W~;Jsd=f  
    % and compile them in a matrix: !SW0iq[7j  
    % ----------------------------- 1vj@ qw3  
    if rpowers(1)==0 -je} PwT  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); XNWtX-[ ^@  
        rpowern = cat(2,rpowern{:}); OW4j!W  
        rpowern = [ones(length_r,1) rpowern]; $G9LaD#;M  
    else /DHgwpJ  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); {MUiK 5:  
        rpowern = cat(2,rpowern{:}); CF{b Yf^%  
    end @)6b  
    k773h`;  
    kg]6q T;Y  
    % Compute the values of the polynomials: ly17FLJ].  
    % -------------------------------------- +9b{Y^^~T  
    y = zeros(length_r,length(n)); I]v2-rB&-  
    for j = 1:length(n) z/ 1$G"  
        s = 0:(n(j)-m_abs(j))/2; :}zyd;Rc  
        pows = n(j):-2:m_abs(j); >,{s Fc  
        for k = length(s):-1:1 hi1Ial\Y  
            p = (1-2*mod(s(k),2))* ... U]sAYp^$  
                       prod(2:(n(j)-s(k)))/              ... QPDh!A3T  
                       prod(2:s(k))/                     ... hLLSmW (  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... f[k#Znr  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); =#V^t$  
            idx = (pows(k)==rpowers); uGMzU&+  
            y(:,j) = y(:,j) + p*rpowern(:,idx); .P)lQk\  
        end -<s Gu9  
         1n8[fgz  
        if isnorm Kd5'2"DI  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); >o?v[:u*  
        end 4|`>}Nu  
    end <u!cdYo@  
    % END: Compute the Zernike Polynomials 1y'Y+1.<  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z v@nK%#J  
    :snO*Zg  
    (SBhU:^h  
    % Compute the Zernike functions: aV'bI  
    % ------------------------------ *H QcI-  
    idx_pos = m>0; q*&R&K;q  
    idx_neg = m<0; ]$@a.#}  
    Food<(!.>  
    \;X7DK2  
    z = y; JI5o~; }m  
    if any(idx_pos) bqcCA9 1  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); kXSX<b<%  
    end .T'@P7Hdx  
    if any(idx_neg) }<04\t?  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ODxZO3  
    end ' k,2*.A  
    rm[C{Pn  
    Z>9@)wo  
    % EOF zernfun  'o-4'  
     
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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)
    在线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)  _#8hgwf>  
    pjeNBSu6  
    DDE还是手动输入的呢? Y ;Ym=n'  
    >*-%:ub  
    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