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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, wUcp_)aE|  
    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, _q3|Ddm2LN  
    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? 9\KMU@Ne  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? b]Z>P{ j  
    t BKra  
    'uU{.bq  
    ;'4 HR+E"  
    =SLCG.  
    function z = zernfun(n,m,r,theta,nflag) "D?:8!\!  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. K#4Toc#=V  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N d2 (3 ,  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Rg~F[j$N  
    %   unit circle.  N is a vector of positive integers (including 0), and rxQ&N[r2  
    %   M is a vector with the same number of elements as N.  Each element R>dd#`r"  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) `u#N  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, o 6A1;e  
    %   and THETA is a vector of angles.  R and THETA must have the same Bf{c4YiF  
    %   length.  The output Z is a matrix with one column for every (N,M) ZCz#B2Sf8  
    %   pair, and one row for every (R,THETA) pair. &M*f4PeXb  
    % eD?f|bif  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike :XeRc"m<  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), (I\qTfN4  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral hLF;MH@  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, jC_m0Iwc  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized klSAY  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?"L ^ 0%  
    % *g!7PzJ'  
    %   The Zernike functions are an orthogonal basis on the unit circle. )l[bu6bM  
    %   They are used in disciplines such as astronomy, optics, and 5Za%EaW%G  
    %   optometry to describe functions on a circular domain. .l +yK-BZ  
    % .+$ox-EK8  
    %   The following table lists the first 15 Zernike functions. p@iU9K\,  
    % c!dc`R  
    %       n    m    Zernike function           Normalization JpC_au7CX  
    %       -------------------------------------------------- 2tI,`pSU  
    %       0    0    1                                 1 jCp`woV  
    %       1    1    r * cos(theta)                    2 S0mzDLgE  
    %       1   -1    r * sin(theta)                    2 0=Mu|G|Z  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) IHcR/\mz  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ,#Mt10e{  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) OS sYmF  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) sglH=0MP  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 9N V.<&~  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) #9CLIYJAd  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 2i)vT)~  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) #8@o%%F d  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^j]_MiA4  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 'ocPG.PaU  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) d34BJ<  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) tzrvIVD  
    %       -------------------------------------------------- ]oxi~TwY^  
    % xASH- 9  
    %   Example 1: &AP`k  
    % MZ"|Jn  
    %       % Display the Zernike function Z(n=5,m=1) ,v_NrX=f?  
    %       x = -1:0.01:1; Aqo90(jffx  
    %       [X,Y] = meshgrid(x,x); e"&QQ-q  
    %       [theta,r] = cart2pol(X,Y); 3o BR  
    %       idx = r<=1; 1"UHe*2  
    %       z = nan(size(X)); ;bRyk#  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); :s>x~t8g#n  
    %       figure oMHTB!A=2  
    %       pcolor(x,x,z), shading interp =Hx]K8N)  
    %       axis square, colorbar P$5K[Y4f  
    %       title('Zernike function Z_5^1(r,\theta)') '^%kTNn  
    % aM YtWj  
    %   Example 2: ;"|QW?>$D  
    % ~}RfepM  
    %       % Display the first 10 Zernike functions RAj>{/E#W  
    %       x = -1:0.01:1; 9nS fFGu  
    %       [X,Y] = meshgrid(x,x); fs0EbVDF  
    %       [theta,r] = cart2pol(X,Y); %uDH_J|^  
    %       idx = r<=1; +F+M[ef<ws  
    %       z = nan(size(X)); <h%I-e6  
    %       n = [0  1  1  2  2  2  3  3  3  3]; {BzE  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ;Q,, i  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; <.hutU*1  
    %       y = zernfun(n,m,r(idx),theta(idx)); _ o.j({S  
    %       figure('Units','normalized') |dhKeg_  
    %       for k = 1:10 9J$-E4G.M  
    %           z(idx) = y(:,k); 2]=`^rC*  
    %           subplot(4,7,Nplot(k)) bX>R9i$  
    %           pcolor(x,x,z), shading interp ym_p49  
    %           set(gca,'XTick',[],'YTick',[]) H{hzw&dZ<P  
    %           axis square *USG p<iH  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) {r'+icvLX  
    %       end ^09-SUl^  
    % `IT]ZAem`/  
    %   See also ZERNPOL, ZERNFUN2. 5GbC}y>  
    !cW!zP-B*p  
    ($-m}UF\/  
    %   Paul Fricker 11/13/2006 dozC[4mF  
    )6(|A$~C+  
    .F G%QFF~  
    1Eb2X}XC  
    y/+ IPR  
    % Check and prepare the inputs: bvS6xU- J  
    % ----------------------------- \,pObWm  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }$i/4?dYsQ  
        error('zernfun:NMvectors','N and M must be vectors.') O L 9(~p  
    end _!,Ees=b  
    */2nh%>$  
    p>B-Ubu  
    if length(n)~=length(m) 9{ #5~WP  
        error('zernfun:NMlength','N and M must be the same length.') 54=*vokX_  
    end -e"A)Bpl(  
    <~P!yLr  
    pQ>|d H+.  
    n = n(:); b0Dco0U(  
    m = m(:); [iZH[7&j  
    if any(mod(n-m,2)) RL3*fRlb  
        error('zernfun:NMmultiplesof2', ... 4w)>}  
              'All N and M must differ by multiples of 2 (including 0).')  1D_&n@  
    end Cz &3=),G  
    E^A S65%bL  
    +lb&_eD  
    if any(m>n) B<i(Y1n[  
        error('zernfun:MlessthanN', ... LI].*n/v  
              'Each M must be less than or equal to its corresponding N.') v3]5`&3~  
    end W^)mz,%x  
    `QtkC>[  
    \*[DR R0  
    if any( r>1 | r<0 ) qsQ{`E0  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 7hTpjox2  
    end +abb[  
    7Mk>`4D'c  
    V~p01f"J  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 4XAs^>N+  
        error('zernfun:RTHvector','R and THETA must be vectors.') ]6M,s0  
    end c g)> A  
    ==Xy'n9'  
    JOJuGB-d  
    r = r(:); 3dlY_z=0  
    theta = theta(:); 3!$+N\ #w  
    length_r = length(r); .]s? 01Z  
    if length_r~=length(theta) ZZ  Hjv  
        error('zernfun:RTHlength', ... -+Ot' ^  
              'The number of R- and THETA-values must be equal.') e^oGiL ~  
    end I=:"Fqj'N  
    6VVxpDAi:  
    r}es_9*~Z  
    % Check normalization: FSm.o?>  
    % -------------------- 3n)$\aBE  
    if nargin==5 && ischar(nflag) P;o  {t  
        isnorm = strcmpi(nflag,'norm'); ^RO<r}B u  
        if ~isnorm 6<T:B[a-  
            error('zernfun:normalization','Unrecognized normalization flag.') @HPr;m!  
        end Cf9{lhE8  
    else Arm'0)B>  
        isnorm = false; 0|.jIix;  
    end oyr b.lu/  
    3E^qh03(  
    l5.k2{'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _ xTpW  
    % Compute the Zernike Polynomials }X?#"JFX?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% y*ZA{  
    ox%j_P9@:  
    3}!u8,P  
    % Determine the required powers of r: R?{xs  
    % ----------------------------------- !+A%`m  
    m_abs = abs(m); |9=A"092{  
    rpowers = []; \pfa\, rW  
    for j = 1:length(n) q&J5(9]O|L  
        rpowers = [rpowers m_abs(j):2:n(j)]; #>("(euXMF  
    end yvj/u c  
    rpowers = unique(rpowers); ]J'TebP=L5  
    IdN3Ea]  
    rJkJ/9s  
    % Pre-compute the values of r raised to the required powers, z=) m6\  
    % and compile them in a matrix: Ak,JPz T  
    % ----------------------------- (Hj[9[=  
    if rpowers(1)==0 A&)2m  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); +Wg/ O -  
        rpowern = cat(2,rpowern{:}); M:GpyE%  
        rpowern = [ones(length_r,1) rpowern]; ]95VM yN  
    else pB\:.?.pd  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); '/NpmNY:L  
        rpowern = cat(2,rpowern{:}); bj}Lxc],  
    end X!K>.r_Dg  
    ""jW'%wR  
    h?p_jI  
    % Compute the values of the polynomials: v}N\z2A  
    % -------------------------------------- ` PQQU~^  
    y = zeros(length_r,length(n)); oe]* Q  
    for j = 1:length(n) mjWU0.  
        s = 0:(n(j)-m_abs(j))/2; NI#]#yM+  
        pows = n(j):-2:m_abs(j); _%=CW' B  
        for k = length(s):-1:1 OPDT:e86Y=  
            p = (1-2*mod(s(k),2))* ... 'I&0$<  
                       prod(2:(n(j)-s(k)))/              ... ,c|MB  
                       prod(2:s(k))/                     ... 8 5X}CCQ  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... w(&EZDe  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); R%RxF=@  
            idx = (pows(k)==rpowers); Ao8ua|:  
            y(:,j) = y(:,j) + p*rpowern(:,idx); >fzyD(>  
        end c>K]$;}  
         l;0([_>*j  
        if isnorm $uDgBZA\  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); X':FFD4h  
        end Z::I3 Q  
    end eZAMV/]jH  
    % END: Compute the Zernike Polynomials ,\iHgsZ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TLVsTM8 P  
    QF/_?Tm4  
    G |KA!q  
    % Compute the Zernike functions: i,r:R g~  
    % ------------------------------ ` = O  
    idx_pos = m>0; =yZq]g6Q  
    idx_neg = m<0; fV|uKs(W  
     x)Bbo9J  
    0>Snps3*Z  
    z = y; > v%.q]E6n  
    if any(idx_pos) kEnGr6e  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); dEtjcId  
    end H?];8wq$G  
    if any(idx_neg) jeWv~JA%L|  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); (T#$0RFq  
    end Cjr]l!  
    ;,[0bmL  
    {WrEe7dLy  
    % EOF zernfun qx5`lm~L  
     
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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)  q%4X1 W  
    i&:SWH=  
    DDE还是手动输入的呢? 0zH-g  
     =1Sny7G  
    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