切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 9264阅读
    • 5回复

    [讨论]如何从zernike矩中提取出zernike系数啊 [复制链接]

    上一主题 下一主题
    离线jssylttc
     
    发帖
    25
    光币
    13
    光券
    0
    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, WAd5,RZ?  
    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, )9oF?l^q  
    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? C2l=7+X#W  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? . 5cL+G1k#  
    p }p@])}8  
    [ ;/4'  
    he/WqCZg  
    D9hV`fA  
    function z = zernfun(n,m,r,theta,nflag) Bf)}g4nYn  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. *wvd[q h  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ]2Vu+AP  
    %   and angular frequency M, evaluated at positions (R,THETA) on the &oU) ,H  
    %   unit circle.  N is a vector of positive integers (including 0), and RB,`I#z1f  
    %   M is a vector with the same number of elements as N.  Each element //x^[fkNq)  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) eUY/H1  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, %S'gDCwq  
    %   and THETA is a vector of angles.  R and THETA must have the same qdss(LZ  
    %   length.  The output Z is a matrix with one column for every (N,M) v--Qbu  
    %   pair, and one row for every (R,THETA) pair. ,sa%u Fm  
    % Wqy\yS [  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike PG51+#  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), }fS`jq;  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 4@qHS0$  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, e1Ne{zg~  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized :!'!V>#g  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. [WfigqY`b*  
    % 9a$\l2  
    %   The Zernike functions are an orthogonal basis on the unit circle. ?QJS6i'k  
    %   They are used in disciplines such as astronomy, optics, and `FJ2 ?  
    %   optometry to describe functions on a circular domain. uPbGQ:%}  
    % 6 h?v/\  
    %   The following table lists the first 15 Zernike functions. >e'Hz(~'/  
    % y TbOBl  
    %       n    m    Zernike function           Normalization NZ|(#` X  
    %       -------------------------------------------------- t)p . $  
    %       0    0    1                                 1 o(gEyK  
    %       1    1    r * cos(theta)                    2 qcmf*Yl:v  
    %       1   -1    r * sin(theta)                    2 x>ZnQ6x~m]  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) (=jztIZ C  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 4\#b@1]}  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) # $N)  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) )R+26wZ|n*  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) GR%h3HO2&  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) *v}3So  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ],W/IDv  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 0gIJ&h6*f  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ]Yw/}GKB  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) :j<ij]rsI  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ]%Db%A  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) KUE}^/%z  
    %       -------------------------------------------------- iXgy/>qgT  
    % \nzaF4+$  
    %   Example 1: i&di}x  
    % MEI.wJZ  
    %       % Display the Zernike function Z(n=5,m=1) aioN)V  
    %       x = -1:0.01:1; Vm"{m/K0  
    %       [X,Y] = meshgrid(x,x); =O.%)|  
    %       [theta,r] = cart2pol(X,Y); K(: _52rt  
    %       idx = r<=1; xY=%+o.?*  
    %       z = nan(size(X)); -W\1n#J  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); vl"{ovoC  
    %       figure N!Q~?/!d  
    %       pcolor(x,x,z), shading interp 4nz$J a)  
    %       axis square, colorbar Vlf=gP  
    %       title('Zernike function Z_5^1(r,\theta)') _!K@( dl  
    % *a[iq`499  
    %   Example 2: @p\te7(P%  
    % Rf4}4ixkj  
    %       % Display the first 10 Zernike functions &OXWD]5$6  
    %       x = -1:0.01:1; c]x'}K c  
    %       [X,Y] = meshgrid(x,x); Kqn{q4L  
    %       [theta,r] = cart2pol(X,Y); 3 {OZdl|  
    %       idx = r<=1; z0F'zN 3J  
    %       z = nan(size(X)); tsWzM9Yf  
    %       n = [0  1  1  2  2  2  3  3  3  3]; !xRboPg  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; jTh^#Q  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; T1_qAz+  
    %       y = zernfun(n,m,r(idx),theta(idx)); +gh*n,:|  
    %       figure('Units','normalized') -]-?>gkN5  
    %       for k = 1:10 R)Y*<Na  
    %           z(idx) = y(:,k); ? 3t]9z  
    %           subplot(4,7,Nplot(k)) kKHGcm^r  
    %           pcolor(x,x,z), shading interp |%tI!RN):  
    %           set(gca,'XTick',[],'YTick',[]) g-NfZj?  
    %           axis square Y2 oN.{IH  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) |EpL~ G_  
    %       end RHj<t");  
    % ([Da*Tk*  
    %   See also ZERNPOL, ZERNFUN2. OGGuVY  
    CW . O"_  
    hAvX{]  
    %   Paul Fricker 11/13/2006 k0>]7t$L  
    wQR0R~|M  
    )2Dm{T  
    {{+woL'C  
    T/YvCbo  
    % Check and prepare the inputs: (q+EP(Q  
    % ----------------------------- UPr8Q^wm  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) PpWn+''M  
        error('zernfun:NMvectors','N and M must be vectors.') +}Q@{@5w  
    end vbMt}bM(GD  
     cq,8^o&  
    e< E]8GAF  
    if length(n)~=length(m) sR*.i?lN  
        error('zernfun:NMlength','N and M must be the same length.') l6y*SW5+  
    end ,nnVHBN  
    r)/nx@x  
    4)OM58e}  
    n = n(:); ]*\m@lWu  
    m = m(:); 9i`sSi8   
    if any(mod(n-m,2)) lE 09Y  
        error('zernfun:NMmultiplesof2', ... C0#"U f  
              'All N and M must differ by multiples of 2 (including 0).') j{: >"6  
    end 5.o{A#/NTl  
    kM o7mkV  
    r_EuLFMA  
    if any(m>n) TQiDbgFo  
        error('zernfun:MlessthanN', ... |h{#r7H0  
              'Each M must be less than or equal to its corresponding N.') fd&=\~1_$  
    end A DW>  
    +^tw@b  
    ^Ss4<  
    if any( r>1 | r<0 ) #->#mshd4  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') -'F? |  
    end E2xcd#ZD  
    =0gfGwD{  
     `GQ'yv  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) k2t#O%_f  
        error('zernfun:RTHvector','R and THETA must be vectors.') Kulh:d:w  
    end =j$!N# L  
    DAHQ7#qfQC  
    -'wFaW0%I  
    r = r(:); B(l8&  
    theta = theta(:); %yJ $R2%*y  
    length_r = length(r); <-%OXEG  
    if length_r~=length(theta) #nS[]UbwZ  
        error('zernfun:RTHlength', ... xZpGSlA  
              'The number of R- and THETA-values must be equal.') I6B4S"Q5<  
    end p#6V|5~8  
    *LZ^0c:r  
    mok%TK  
    % Check normalization: ;+W9EbY2  
    % -------------------- @ApX43U(  
    if nargin==5 && ischar(nflag) {%cm;o[7o  
        isnorm = strcmpi(nflag,'norm'); JAA{5@ST  
        if ~isnorm Qk_` IlSd  
            error('zernfun:normalization','Unrecognized normalization flag.') @w]z"UCwV@  
        end w\f>.N  
    else @*}?4wU^k  
        isnorm = false; ^+)q@{\8Y  
    end Zv8I`/4?  
    3.vQ~Fvl  
    `E4OgO  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jh3X G  
    % Compute the Zernike Polynomials UC{Tmf  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sM0o,l(5  
    QTKN6P  
    $GcVI ;a  
    % Determine the required powers of r: 0*-nVC1  
    % ----------------------------------- 7Rix=*  
    m_abs = abs(m); 1E'/!|  
    rpowers = []; w\PCBY=  
    for j = 1:length(n) u>U4w68  
        rpowers = [rpowers m_abs(j):2:n(j)]; |DZ3=eWZ  
    end .gS x`|!  
    rpowers = unique(rpowers); gY=Ry=w9  
    Er]lObfQo  
    X8Ld\vZYn  
    % Pre-compute the values of r raised to the required powers, (K>=!&tlp=  
    % and compile them in a matrix: S7 _^E  
    % ----------------------------- vxrRkOU1  
    if rpowers(1)==0 FJj #  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); LtDQgel"  
        rpowern = cat(2,rpowern{:}); Edi`x5"l  
        rpowern = [ones(length_r,1) rpowern]; >*"6zR2 o  
    else :>t^B+  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); *w[\(d'T  
        rpowern = cat(2,rpowern{:}); zLa3Q\T  
    end XA%a7Xtni  
    y?1<7>L5~  
    y_Tc$g~  
    % Compute the values of the polynomials: aTx*6;-PH  
    % -------------------------------------- qauZ-Qoc9  
    y = zeros(length_r,length(n)); +#|):aF  
    for j = 1:length(n) :y!%GJW  
        s = 0:(n(j)-m_abs(j))/2; AvNU\$B4aG  
        pows = n(j):-2:m_abs(j); ZJ7<!?6  
        for k = length(s):-1:1 %}*0l8y  
            p = (1-2*mod(s(k),2))* ... G L> u3K  
                       prod(2:(n(j)-s(k)))/              ... xWa96U[  
                       prod(2:s(k))/                     ... hDf|9}/UQd  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... l`}Ag8Q  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); cIIt ;q[  
            idx = (pows(k)==rpowers); k;?Oi?]  
            y(:,j) = y(:,j) + p*rpowern(:,idx); dT9ekNQB  
        end 0B;cQSH!q  
         H"g$qSx  
        if isnorm q:9#Vcw  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); clwJ+kku@  
        end YsHZFF  
    end i(k]}Di:  
    % END: Compute the Zernike Polynomials c T!L+z g  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RRBokj)]  
    v FL\O  
    i{$h]D_fD  
    % Compute the Zernike functions: Po: )b  
    % ------------------------------ JvZNr?_w%  
    idx_pos = m>0; rkW2_UTZE  
    idx_neg = m<0; q Pc"A!-i  
    4&+;n[D  
    aB(6yBBoxj  
    z = y; > WsRCBA  
    if any(idx_pos) E|aPkq]  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); /<Doe SDJ|  
    end 8>}^W  
    if any(idx_neg) ;BR`}~m  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); N~%F/`Z<+  
    end gDmwJr  
    Z !qHL$  
    t1I` n(]n  
    % EOF zernfun ET&Q}UOE  
     
    分享到
    离线phoenixzqy
    发帖
    4352
    光币
    5478
    光券
    1
    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
    发帖
    959
    光币
    1087
    光券
    1
    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
    发帖
    25
    光币
    13
    光券
    0
    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)  T?W`g> yM  
    @x">e][B  
    DDE还是手动输入的呢? ^'=J'Q  
    J7BfH,o  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    离线jssylttc
    发帖
    25
    光币
    13
    光券
    0
    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    离线18257342135
    发帖
    51
    光币
    1518
    光券
    0
    只看该作者 5楼 发表于: 2016-12-13
    支持一下,慢慢研究