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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, c[dzO .~  
    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, !LI<%P)  
    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? *Y m? gCig  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? =^nb+}Nz(  
    $RPW/Lyiq  
    F1t(P 8  
    BPba3G9H  
    )\fY1WD  
    function z = zernfun(n,m,r,theta,nflag) &r5q,l&@n  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. h4? x_"V"  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N DmiBM6t3N  
    %   and angular frequency M, evaluated at positions (R,THETA) on the w )R5P[b  
    %   unit circle.  N is a vector of positive integers (including 0), and &7aWVKon  
    %   M is a vector with the same number of elements as N.  Each element wSTul o:9  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) @p+;iS1}  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ~7P)$[  
    %   and THETA is a vector of angles.  R and THETA must have the same ?['!0PF  
    %   length.  The output Z is a matrix with one column for every (N,M) K9lgDk"i  
    %   pair, and one row for every (R,THETA) pair. 4>hHUz[_  
    % i--t ?@#  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike cj/`m$  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), \c=I!<9  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Lx^ eaP5  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, gb ga"WO  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized T # \  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. X*8y"~X|vq  
    % qLN^9PdEE  
    %   The Zernike functions are an orthogonal basis on the unit circle. %@/^UE:  
    %   They are used in disciplines such as astronomy, optics, and m~ tvuz I  
    %   optometry to describe functions on a circular domain. "F<CGSo  
    % ~Iu!B Y  
    %   The following table lists the first 15 Zernike functions. z$32rt8{`v  
    % ~C.*Vc?|  
    %       n    m    Zernike function           Normalization @;Ttdwg#J  
    %       -------------------------------------------------- 'rD6MY  
    %       0    0    1                                 1 CLb6XnkcA\  
    %       1    1    r * cos(theta)                    2 '|C3t!H`  
    %       1   -1    r * sin(theta)                    2 kI{DxuTad  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) tZrc4$D-  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 3FEJ 9ZyG  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) Zp_(vOc  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) nV;'UpQw  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) &|>+LP@8  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) U2oCSo5:3N  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) *sho/[~_  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) _"DS?`z6  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) I5$P9UE+^9  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Nk`UQ~g$  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) DX>a0-Xj  
    %       4    4    r^4 * sin(4*theta)             sqrt(10)  `zwz  
    %       -------------------------------------------------- KhCP9(A=Qo  
    % OG,P"sv  
    %   Example 1: Lpchla$  
    % S2~cAhR|M  
    %       % Display the Zernike function Z(n=5,m=1) c8qr-x1HG  
    %       x = -1:0.01:1; ^rkKE dd  
    %       [X,Y] = meshgrid(x,x); j]a$RC#  
    %       [theta,r] = cart2pol(X,Y); TOYK'|lwM  
    %       idx = r<=1; ]Z JoC!u  
    %       z = nan(size(X)); P:qmg"i@3  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); qfkHGW?1/j  
    %       figure S)rZE*~2  
    %       pcolor(x,x,z), shading interp =BsV`p7rU  
    %       axis square, colorbar c PGlT"  
    %       title('Zernike function Z_5^1(r,\theta)') +8=$-E=  
    % p|4qkJK8  
    %   Example 2: Y4T")  
    % ,w }Po  
    %       % Display the first 10 Zernike functions g|=_@ pL  
    %       x = -1:0.01:1; _B4&Fb.  
    %       [X,Y] = meshgrid(x,x); T:|/ux3  
    %       [theta,r] = cart2pol(X,Y); .b :!qUE^  
    %       idx = r<=1; 7\u+%i;YZ  
    %       z = nan(size(X)); SGd]o"VF  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 0bNvmZ$  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 6Z/`p~e  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ]`E+HLEQ'  
    %       y = zernfun(n,m,r(idx),theta(idx)); Nz{dnV{&x;  
    %       figure('Units','normalized') )n/%P4l  
    %       for k = 1:10 ;?6vKpj;  
    %           z(idx) = y(:,k); WKf<% E$  
    %           subplot(4,7,Nplot(k)) od;-D~  
    %           pcolor(x,x,z), shading interp K,f:X g!:  
    %           set(gca,'XTick',[],'YTick',[]) mgxIxusR  
    %           axis square JFq wC=-  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) `h}eP[jA  
    %       end ? @V R%z  
    % $o6/dEKQ  
    %   See also ZERNPOL, ZERNFUN2. Iw1Y?Qia  
    @WJ;T= L  
    I8F+Z  
    %   Paul Fricker 11/13/2006 NGra/s,9 |  
    TyxIlI4"  
    yr]ja-Y  
    }ze+ tf  
    U%{GLO   
    % Check and prepare the inputs: \?bV\/GBR  
    % ----------------------------- (Guzj*12  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 2FcL-?  
        error('zernfun:NMvectors','N and M must be vectors.') p< R:[rz  
    end Hg+<GML  
    mDD.D3RS  
    ~KK 9aV{  
    if length(n)~=length(m) V>$( N/1  
        error('zernfun:NMlength','N and M must be the same length.') [f6uwp  
    end <+8'H:wz  
    ,'NasL8?We  
     >DL  
    n = n(:); 2:+8]b3i  
    m = m(:); |@ mz@  
    if any(mod(n-m,2)) npP C;KD  
        error('zernfun:NMmultiplesof2', ... *0WVrM06?  
              'All N and M must differ by multiples of 2 (including 0).') Z:b?^u4.  
    end OhF55,[  
    3CUQQ_  
    Z[vx0[av&  
    if any(m>n) M,Gy.ivz  
        error('zernfun:MlessthanN', ... gv!8' DKn  
              'Each M must be less than or equal to its corresponding N.') S :HOlJze  
    end Ht`fC|E  
    5zuwqOD*  
    2Gyq40  
    if any( r>1 | r<0 ) NW|B|kc  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') l,ny=Q$[1'  
    end l\U Q2i  
    1- RY5R}VR  
    j*=!M# D  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) dQX-s=XJ  
        error('zernfun:RTHvector','R and THETA must be vectors.') ^[ae )}  
    end ktu?-?#0,  
    u#05`i:Z  
    (qcFGM22U  
    r = r(:); zI88IM7/  
    theta = theta(:); J_s`G  
    length_r = length(r); E4#{&sRT  
    if length_r~=length(theta) aRd~T6I  
        error('zernfun:RTHlength', ... bC&A@.g{  
              'The number of R- and THETA-values must be equal.') glAS$<  
    end [i.@q}c~E  
    Po>6I0y  
    S)CsH1Q  
    % Check normalization: "+DA)K  
    % -------------------- B=Hd:P|  
    if nargin==5 && ischar(nflag) O[X*F2LC4  
        isnorm = strcmpi(nflag,'norm'); dT`nR"  
        if ~isnorm Z bRRDXk!  
            error('zernfun:normalization','Unrecognized normalization flag.') F`}'^>  
        end YSjc=  
    else &9jJ\+:7  
        isnorm = false; wGHft`Z  
    end o)Q4+njT@  
    2"0VXtv6  
    2OG/0cP  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0'&C5v'  
    % Compute the Zernike Polynomials ^&3vGu9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *0U#Z]t  
    gx\V)8Zr  
    }OkzP)(  
    % Determine the required powers of r: YznL+TD  
    % ----------------------------------- 32GI+NN  
    m_abs = abs(m); %p7 ?\>  
    rpowers = []; mR}8}K]L  
    for j = 1:length(n) ,>|tQ'  
        rpowers = [rpowers m_abs(j):2:n(j)]; 1q}32^>+o  
    end a[ULSYEi  
    rpowers = unique(rpowers); R P{pEd  
    QPy h.9:N  
    E2hsSqsu=  
    % Pre-compute the values of r raised to the required powers, vbJ<|#|r-  
    % and compile them in a matrix: a-5UG#o  
    % ----------------------------- eI-fH  
    if rpowers(1)==0 zJ`u>:*$  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Uo3  
        rpowern = cat(2,rpowern{:}); LcKc#)'EE  
        rpowern = [ones(length_r,1) rpowern]; s'O%@/;J  
    else 5{H)r   
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); V .Kjcy  
        rpowern = cat(2,rpowern{:}); y)r`<B  
    end <XL%*  
    F"Dr(V  
    Ust +g4  
    % Compute the values of the polynomials: AB=%yM7V*  
    % -------------------------------------- COi15( G2  
    y = zeros(length_r,length(n)); h]zok}$  
    for j = 1:length(n) l6zAMyau5  
        s = 0:(n(j)-m_abs(j))/2; 3P_.SF  
        pows = n(j):-2:m_abs(j); s:<y\1Ay  
        for k = length(s):-1:1 ?M90K)&g{  
            p = (1-2*mod(s(k),2))* ... 4Q+,_iP  
                       prod(2:(n(j)-s(k)))/              ... eKP >} `  
                       prod(2:s(k))/                     ... za>%hZf\  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Y] 1U1 08  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 1dD%a91  
            idx = (pows(k)==rpowers); +5fB?0D;  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 1D%P;eUDp  
        end /G5KNSi  
         Z%#e* O0  
        if isnorm FC 8<D  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); fwpp qIM  
        end tFcQ.1  
    end W{Qb*{9  
    % END: Compute the Zernike Polynomials X&Oo[Z  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 03?ADjO  
    :M6|V_Yp  
    h`Jc%6o  
    % Compute the Zernike functions: ' !huU   
    % ------------------------------ "'B DVxp'w  
    idx_pos = m>0; R14&V1 tZ  
    idx_neg = m<0; j1Ys8k%$l  
    3 EAr=E]  
    LBio$67F  
    z = y; $%U}k=-  
    if any(idx_pos)  7]@M  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 3SM'vV0[  
    end %n]jsdE^|  
    if any(idx_neg) ]:ca=&>  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 9f['TG,"  
    end u!xgLf'`  
    ,T;sWl  
    dLQp"vs$  
    % EOF zernfun r E1ouz!D  
     
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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)  }>Lz\.Z/+[  
    "%t !+E>nr  
    DDE还是手动输入的呢? 4p&SlJ  
    RG_)<U/B  
    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。
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