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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, ]ij:>O@{$  
    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, |+=ctpx9&  
    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? 3>^B%qg6  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ul"Z% 1]  
    Ge24Lp;Y 6  
    s3~6[T?8  
    `2'*E\   
    :k~ p=ko  
    function z = zernfun(n,m,r,theta,nflag) W#^p%?8pR  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. l%`~aVGJ  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ">nFzg?Y  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 3>z+3!I z  
    %   unit circle.  N is a vector of positive integers (including 0), and 0%3T'N%  
    %   M is a vector with the same number of elements as N.  Each element H$&P=\8n  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) OW8TiM mK  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, l_2YPon  
    %   and THETA is a vector of angles.  R and THETA must have the same ~SEIIq  
    %   length.  The output Z is a matrix with one column for every (N,M) |G)bnmi7  
    %   pair, and one row for every (R,THETA) pair. [U{RDX  
    % 1 EHNg<J(  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike B_%O6  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), o7g6*hJz  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral tgu fU  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, eBW]hwhKzM  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized BFn}~\wzK  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. u?8e>a  
    % o5NrDDH  
    %   The Zernike functions are an orthogonal basis on the unit circle. "C+Fl /v  
    %   They are used in disciplines such as astronomy, optics, and D&8*4>  
    %   optometry to describe functions on a circular domain. \0l"9 B.  
    % ~I%JVX%  
    %   The following table lists the first 15 Zernike functions. l,Ixz1S3e  
    % _\FA}d@N  
    %       n    m    Zernike function           Normalization oc&yz>%q  
    %       -------------------------------------------------- 6EG`0h6  
    %       0    0    1                                 1 J_XbtCmt  
    %       1    1    r * cos(theta)                    2 JZcW?Or  
    %       1   -1    r * sin(theta)                    2 iS28p  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) OH~I+=}.  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Q__1QUu  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) wW^3/  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 65pC#$F<x  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) p5=VGKp  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) f@IL2DL}\  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) D5 ^WiQ<  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ]F,v#6qi  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) YZ+<+`Mz<  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) .{k^ tf4  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) h&?tF~h  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) HoKN<w  
    %       -------------------------------------------------- -ID!kZx  
    % m2Q#ATLW  
    %   Example 1: ]i:O+t/U  
    % Yy8%vDdJO  
    %       % Display the Zernike function Z(n=5,m=1) A*hc w  
    %       x = -1:0.01:1; zDof e*  
    %       [X,Y] = meshgrid(x,x); _6Fj&mw(u  
    %       [theta,r] = cart2pol(X,Y); YQ<O .E  
    %       idx = r<=1; ?gOZY\[ma  
    %       z = nan(size(X)); 1)wzSEV@  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); D|!^8jHj  
    %       figure 2qUC@d<K  
    %       pcolor(x,x,z), shading interp K)t+lJ  
    %       axis square, colorbar B (dq$+4  
    %       title('Zernike function Z_5^1(r,\theta)') p[-bu B]  
    % }c*6|B@f  
    %   Example 2: 0sKY;(  
    % -|xyj2M  
    %       % Display the first 10 Zernike functions nA^UF_rD-  
    %       x = -1:0.01:1; Yv jRJ  
    %       [X,Y] = meshgrid(x,x); nXqZkZE\  
    %       [theta,r] = cart2pol(X,Y); &/A?*2  
    %       idx = r<=1; 'y.'Xj:l  
    %       z = nan(size(X)); O hcPlr  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ^+ +ec>  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; co!#.  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; j:{d'OV  
    %       y = zernfun(n,m,r(idx),theta(idx)); 9rsty{J8  
    %       figure('Units','normalized') g&"__~dS-F  
    %       for k = 1:10 NI136P  
    %           z(idx) = y(:,k); 3YF*TxKx  
    %           subplot(4,7,Nplot(k)) /xRPQ|  
    %           pcolor(x,x,z), shading interp y2eeE CS]  
    %           set(gca,'XTick',[],'YTick',[]) -?WhJ.U  
    %           axis square #b4Pn`[   
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) nAJ<@a  
    %       end JY tM1d  
    % fu95-)M  
    %   See also ZERNPOL, ZERNFUN2. e'uI~%$NJL  
    \f_YJit  
    X} v]iX  
    %   Paul Fricker 11/13/2006 %Ot^G%34  
    ~Xg@,?Zr  
    S:GX!6>  
    +;Jb)8  
    I)Dd"I  
    % Check and prepare the inputs: tc'` 4O]c8  
    % ----------------------------- rSVU|O3m;  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) f|1GlUA{t  
        error('zernfun:NMvectors','N and M must be vectors.') W=Ru?sG=  
    end GJY7vS^#  
    J34lu{'if  
    \P_1@sH=  
    if length(n)~=length(m) ;$\d^i{N  
        error('zernfun:NMlength','N and M must be the same length.') T&=1IoOg  
    end FU|c[u|z  
    FXPw 5  
    n^;:V8k  
    n = n(:); W|@/<K$V  
    m = m(:); el*C8TWlw  
    if any(mod(n-m,2)) Q2)z1'Wv  
        error('zernfun:NMmultiplesof2', ... d aIt `}s  
              'All N and M must differ by multiples of 2 (including 0).') 47xJ(yO  
    end ruLi "d  
    ^t=Hl  
    Oi=>Usd  
    if any(m>n) MeqW/!72$L  
        error('zernfun:MlessthanN', ... Kd _tjWS  
              'Each M must be less than or equal to its corresponding N.') s:UQ~p}"S  
    end !tT$}?Ano  
    (ROurq"  
    >uuP@j  
    if any( r>1 | r<0 ) "|S \J5-%  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 0.-2FHc9L  
    end I% 43rdoPe  
    VrA9}"1x~*  
    9Kw4K#IqQ  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) _W4i?Bde  
        error('zernfun:RTHvector','R and THETA must be vectors.') 8]Xwj].^C  
    end O1Gd_wDC/i  
    *BYSfcX6  
    ~\c]!%)o  
    r = r(:); K4i#:7r'b  
    theta = theta(:); MX 2UYZ&  
    length_r = length(r); ANy=f-V  
    if length_r~=length(theta) UDHk@M  
        error('zernfun:RTHlength', ... g_}r)CgG|  
              'The number of R- and THETA-values must be equal.') CE>RAerY  
    end ~l%Dcp  
    !Re/W ykY  
    3VbQDPG  
    % Check normalization: -Rhxib|<  
    % -------------------- \. A~>=:  
    if nargin==5 && ischar(nflag) g83!il\  
        isnorm = strcmpi(nflag,'norm'); (u-i{<   
        if ~isnorm e*e}X&|(g  
            error('zernfun:normalization','Unrecognized normalization flag.') MPMJkL$F^  
        end <L@0w8i`  
    else >A|6 kzC  
        isnorm = false; 8@|_];9#.  
    end 9}Tf9>qP>M  
    4`G":nE?We  
    lcij}-z:%e  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% '+NmHu:q  
    % Compute the Zernike Polynomials +I#5?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% e =Vu;  
    G6xdGUM  
    J4h7] qt  
    % Determine the required powers of r: ho2o/>Ef3  
    % -----------------------------------  &(\z  
    m_abs = abs(m); !'Xk=+  
    rpowers = []; dRyK'Xr  
    for j = 1:length(n) 9 kzytx  
        rpowers = [rpowers m_abs(j):2:n(j)]; !SIGzj  
    end A"k6n\!n;  
    rpowers = unique(rpowers); q[Hx y  
    8zGe5Dn9  
    j|9;") 1  
    % Pre-compute the values of r raised to the required powers, }%[TJ@R;  
    % and compile them in a matrix: y@'8vOh`  
    % ----------------------------- za'Eom-<u  
    if rpowers(1)==0 "[(_C&Ot4  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); }-k<>~FA  
        rpowern = cat(2,rpowern{:}); ]v@#3,BV  
        rpowern = [ones(length_r,1) rpowern]; * I`, L/  
    else 4aGV1u+4  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 0]{h,W3]@[  
        rpowern = cat(2,rpowern{:}); 7am._K  
    end F'W{\4  
    fQlR;4QX]  
    xA#B1qbw  
    % Compute the values of the polynomials: BV$lMLD{r  
    % -------------------------------------- m>$+sMZE  
    y = zeros(length_r,length(n)); KP[ax2!x  
    for j = 1:length(n) ~qLbyzHaB  
        s = 0:(n(j)-m_abs(j))/2; vL{~?vq6  
        pows = n(j):-2:m_abs(j); vY<(3[pp  
        for k = length(s):-1:1 V{@<Z8sW#  
            p = (1-2*mod(s(k),2))* ... R{5Qb?&wOp  
                       prod(2:(n(j)-s(k)))/              ... U^?/nRZ  
                       prod(2:s(k))/                     ... mKtZ@r)u  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... AYd7qx:~  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); S& 8gZ~B  
            idx = (pows(k)==rpowers); Q1IN@Db}y  
            y(:,j) = y(:,j) + p*rpowern(:,idx); B%Dy;zdWd/  
        end }]N7CWy  
         G6_Kid}"q  
        if isnorm 3/>McZ@OH  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 7w_`<b6  
        end K!"[,=u_  
    end FJKt5}`8  
    % END: Compute the Zernike Polynomials c~b[_J)  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~ d^+yR-  
    WZ'8{XY8  
    bKYLBu:  
    % Compute the Zernike functions: "X g@X5BG  
    % ------------------------------ NQ !t`  
    idx_pos = m>0; FAJ\9  
    idx_neg = m<0; C;}~C:aJ  
    THWT\3~,  
    U_m<W$"HF  
    z = y; ~4'e)g.hG  
    if any(idx_pos) IrjKI.PR  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 7gfNe kr~W  
    end `MlQPLH  
    if any(idx_neg) 'ADt<m_$  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); &Hb6  
    end BJ<hP9 #  
    `QXO+'j4  
    JGX E{FT  
    % EOF zernfun 2PE|4zG  
     
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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)  |y9(qcKn$  
    @~k4,dJ  
    DDE还是手动输入的呢? kvcDa+#  
    K&ZN!VN/p  
    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