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    [求助]ansys分析后面型数据如何进行zernike多项式拟合? [复制链接]

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    离线niuhelen
     
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    只看楼主 倒序阅读 楼主  发表于: 2011-03-12
    小弟不是学光学的,所以想请各位大侠指点啊!谢谢啦 1oW]O@R  
    就是我用ansys计算出了镜面的面型的数据,怎样可以得到zernike多项式系数,然后用zemax各阶得到像差!谢谢啦! +sbacMfq  
     
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    离线phility
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    只看该作者 1楼 发表于: 2011-03-12
    可以用matlab编程,用zernike多项式进行波面拟合,求出zernike多项式的系数,拟合的算法有很多种,最简单的是最小二乘法,你可以查下相关资料,挺简单的
    离线phility
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    只看该作者 2楼 发表于: 2011-03-12
    泽尼克多项式的前9项对应象差的
    离线niuhelen
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    只看该作者 3楼 发表于: 2011-03-12
    回 2楼(phility) 的帖子
    非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 f2e$BA  
    function z = zernfun(n,m,r,theta,nflag) _^s SI<&m  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. $zA[5}{ZtQ  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N \yizIo.Y`  
    %   and angular frequency M, evaluated at positions (R,THETA) on the _~&v s<  
    %   unit circle.  N is a vector of positive integers (including 0), and ;HwJw\fo  
    %   M is a vector with the same number of elements as N.  Each element ;Wm)e~`,  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) \D k^\-  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Fm~}A4  
    %   and THETA is a vector of angles.  R and THETA must have the same 5{f/H] P  
    %   length.  The output Z is a matrix with one column for every (N,M) Bq =](<>>  
    %   pair, and one row for every (R,THETA) pair. DQXx}%Px  
    % U1tPw`0h  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike t7%Bv+Uo  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), j|8{Vyqd  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral X"59`Yh  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, @!HMd{r  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ptL}F~  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. BnY|t2r  
    % znpZ0O\!  
    %   The Zernike functions are an orthogonal basis on the unit circle. FOyfk$  
    %   They are used in disciplines such as astronomy, optics, and v" TH[}C9D  
    %   optometry to describe functions on a circular domain. xH-k~#  
    % 6>7LFV1tvy  
    %   The following table lists the first 15 Zernike functions. -mdPqVIJn:  
    % j-E>*N}-_  
    %       n    m    Zernike function           Normalization e' ;c8WF3E  
    %       -------------------------------------------------- UsKn4Kh  
    %       0    0    1                                 1 5 : >  
    %       1    1    r * cos(theta)                    2 *3oQS"8  
    %       1   -1    r * sin(theta)                    2 wpMQ 7:j  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 8j +;Xlh  
    %       2    0    (2*r^2 - 1)                    sqrt(3) +/8?+1E ^  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 3ZZI1_j  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) =v"{EmT[$  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) OtqLigt&l  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) g{{SY5qDj  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 0 1w/,r  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) +@v} (  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) $\H46Ji  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) jH/%Z5iu  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Mi-9sW  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) #>NZN1  
    %       -------------------------------------------------- YH$`r6\S  
    % l'R`XGT  
    %   Example 1: nXW1:  
    % i<![i5uAI  
    %       % Display the Zernike function Z(n=5,m=1) lK@r?w|<M  
    %       x = -1:0.01:1; Kwau:_B  
    %       [X,Y] = meshgrid(x,x); (acRYv(  
    %       [theta,r] = cart2pol(X,Y); M" \y2   
    %       idx = r<=1; 7:<>#  
    %       z = nan(size(X)); .6(i5K  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); g}h0J%s  
    %       figure -p~B -,  
    %       pcolor(x,x,z), shading interp }RK9Onh3G  
    %       axis square, colorbar aa!c>"g6  
    %       title('Zernike function Z_5^1(r,\theta)') _Y~?.hs^  
    % G _o4A:2  
    %   Example 2: >H! 2Wflm  
    % |a3b2x,  
    %       % Display the first 10 Zernike functions Dne&YVF9V  
    %       x = -1:0.01:1; pc>R|~J{2  
    %       [X,Y] = meshgrid(x,x); =^}2 /vA  
    %       [theta,r] = cart2pol(X,Y); 3<lDsb(}0A  
    %       idx = r<=1; RmCR"~   
    %       z = nan(size(X)); Ric$Xmu  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ;T(^riAEl  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 3EdPKM j&  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; AS ul  
    %       y = zernfun(n,m,r(idx),theta(idx)); ? 'nMZ  
    %       figure('Units','normalized') {[dqXG$v `  
    %       for k = 1:10 yK;I<8+>_  
    %           z(idx) = y(:,k); c Ix(;[U  
    %           subplot(4,7,Nplot(k)) ]|(?i ,p  
    %           pcolor(x,x,z), shading interp Nrh`DyF0D!  
    %           set(gca,'XTick',[],'YTick',[]) _l<"Qqt  
    %           axis square ~a Rq\fx{  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) dY^~^<{Lj  
    %       end a WC sLH  
    % mZ%\`H+  
    %   See also ZERNPOL, ZERNFUN2. `^x^= og'  
    Pd?YS!+S  
    %   Paul Fricker 11/13/2006 4|UIyDt8  
    #/6X44 *u  
    48VsHqG  
    % Check and prepare the inputs: v4Gkf  
    % ----------------------------- >@o*v*25  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) c{0?gt.  
        error('zernfun:NMvectors','N and M must be vectors.') ~<3yTl>  
    end ~Fh(4'  
    O jmz/W  
    if length(n)~=length(m) x(Z@ R\C-a  
        error('zernfun:NMlength','N and M must be the same length.') Ig2VJs;  
    end EWi@1PAZK  
    ah.Kb(d:  
    n = n(:); J/ ~]A1fP6  
    m = m(:); BH1To&ol  
    if any(mod(n-m,2)) {zcjTJ=Zt8  
        error('zernfun:NMmultiplesof2', ... #;)7~69  
              'All N and M must differ by multiples of 2 (including 0).')  Qy%/+9L  
    end bE{`g]C5  
    Gy5W;,$q  
    if any(m>n) 'lF|F+8   
        error('zernfun:MlessthanN', ... PC5FfX  
              'Each M must be less than or equal to its corresponding N.') mCo5 Gdt  
    end +( d2hSIF  
    !~#31kL&  
    if any( r>1 | r<0 ) l%O-c}X  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') {_JLmyaerZ  
    end &DV'%h>i=  
    4KKNw9L)  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 6r`g+Js/  
        error('zernfun:RTHvector','R and THETA must be vectors.') ~*qGH  
    end Vl%k:  
    C%&7,F7  
    r = r(:); J&?kezs  
    theta = theta(:); iT5%X   
    length_r = length(r); pJI H_H  
    if length_r~=length(theta) @ NF8?>!  
        error('zernfun:RTHlength', ... FWj~bn  
              'The number of R- and THETA-values must be equal.') =W6P>r_  
    end YY9q'x,w  
    w;:,W@K  
    % Check normalization: b({2|R  
    % -------------------- -p 1arA  
    if nargin==5 && ischar(nflag) #'[ f^xgJ  
        isnorm = strcmpi(nflag,'norm'); =[$*PTe  
        if ~isnorm BBDOjhik  
            error('zernfun:normalization','Unrecognized normalization flag.') 5D#*lMSP"'  
        end >3JOQ;:d8  
    else Q'N<jX[  
        isnorm = false; W$&Q.Z  
    end 1VeCAx[e  
    s}.nh>Q  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (]JJ?aAF  
    % Compute the Zernike Polynomials er_aol e  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% cb+!H>+  
    @1pdyKK  
    % Determine the required powers of r: ^ZsME,  
    % ----------------------------------- CNwhH)*  
    m_abs = abs(m); FR&RIFy  
    rpowers = []; `4o;Lz~  
    for j = 1:length(n) Vo\d&}Q  
        rpowers = [rpowers m_abs(j):2:n(j)]; * PZ=$>r  
    end ZE9*i}r  
    rpowers = unique(rpowers); Zqao4  
    E,;nx^`!l  
    % Pre-compute the values of r raised to the required powers, *6h.#$\  
    % and compile them in a matrix: mb#)w`<  
    % ----------------------------- D -jew&B  
    if rpowers(1)==0 )z aMycW  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); \6N\6=t!A  
        rpowern = cat(2,rpowern{:}); q/[)mr|~  
        rpowern = [ones(length_r,1) rpowern]; Deam%)bXM]  
    else 6Hz=VhQrN  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); S SzOz-&GA  
        rpowern = cat(2,rpowern{:}); ELm#  
    end k_ skn3,u  
    Z d%*,\`S  
    % Compute the values of the polynomials: 33; yt d  
    % -------------------------------------- 27MgwX NQ  
    y = zeros(length_r,length(n)); R_^:<F0  
    for j = 1:length(n) XdB8Oj~~  
        s = 0:(n(j)-m_abs(j))/2; {\%x{  
        pows = n(j):-2:m_abs(j); i,~{{XS<  
        for k = length(s):-1:1 m$4Gm(Up  
            p = (1-2*mod(s(k),2))* ... FGZOn5U6'  
                       prod(2:(n(j)-s(k)))/              ... !:>y.^O  
                       prod(2:s(k))/                     ... 29E^]IL?  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... &W ~,q(  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); NZl0sX.:  
            idx = (pows(k)==rpowers); rlds-j''  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ^PD a  
        end J sH9IK:  
         A_[65'*b  
        if isnorm 6Us#4 v,  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ^v,^.>P  
        end ci$o~b6V  
    end \Wo,^qR  
    % END: Compute the Zernike Polynomials L.8-nTg"y  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &BQ`4j~.  
    `'g%z: ~  
    % Compute the Zernike functions: E)`+1j  
    % ------------------------------ WUHijHo5(8  
    idx_pos = m>0; I|p(8 R!  
    idx_neg = m<0; /JvNJ f  
    [1s B  
    z = y; 0iwx$u 7[  
    if any(idx_pos) O*30|[  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); \}cEHLq  
    end /{Nx%PqL  
    if any(idx_neg) IQR?n}ce  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); En[cg  
    end FzNs >*  
    P2lj#aQLS  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) 'sJYt^  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ^'#vUj:"  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 1{_;`V  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive x%0Q W  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, d?'q(6&H  
    %   and THETA is a vector of angles.  R and THETA must have the same INi(G-!g  
    %   length.  The output Z is a matrix with one column for every P-value, ?&"-y)FG  
    %   and one row for every (R,THETA) pair. 0*x  
    % RHeql*`  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ]x?`&f8i  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) NKh8'=S  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) gLU #\d]  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 &_G^=Nc,H  
    %   for all p. 5ILce%#zL  
    % !@5B:n*  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 *GD?d2.6j  
    %   Zernike functions (order N<=7).  In some disciplines it is v, 9MAZ,  
    %   traditional to label the first 36 functions using a single mode aNw8][  
    %   number P instead of separate numbers for the order N and azimuthal NZCPmst  
    %   frequency M. j#zUO&Q@  
    % QF Vy2 q  
    %   Example:  {|a=  
    % Wu?4oF  
    %       % Display the first 16 Zernike functions 6o!+E@V b  
    %       x = -1:0.01:1; 8Y_wS&eB  
    %       [X,Y] = meshgrid(x,x); =UT*1-yh R  
    %       [theta,r] = cart2pol(X,Y); n}}$-xl  
    %       idx = r<=1; 7:<co  
    %       p = 0:15; +<7`Gn(n3  
    %       z = nan(size(X)); z q _*)V  
    %       y = zernfun2(p,r(idx),theta(idx)); ~{/"fTif  
    %       figure('Units','normalized') oYI7 .w  
    %       for k = 1:length(p) rK7m(  
    %           z(idx) = y(:,k); 6O>NDTd%  
    %           subplot(4,4,k) bC&*U|de  
    %           pcolor(x,x,z), shading interp *;5P65:u$>  
    %           set(gca,'XTick',[],'YTick',[]) XcD$xFDZ  
    %           axis square 4'_PLOgnX  
    %           title(['Z_{' num2str(p(k)) '}']) x(ue |UG  
    %       end s8Bbe t  
    % tUaDwIu#  
    %   See also ZERNPOL, ZERNFUN. 5R"iF+p4  
    ^Cs?FF@P  
    %   Paul Fricker 11/13/2006 ;Hk{bz(  
    f_I6g uDPz  
    __O@w.  
    % Check and prepare the inputs: DSf  
    % ----------------------------- P;G Rk6  
    if min(size(p))~=1 8A}cxk  
        error('zernfun2:Pvector','Input P must be vector.') A 0~uv4MC  
    end xy;u"JY*  
    qp;eBa  
    if any(p)>35 Y)lYEhF  
        error('zernfun2:P36', ... MROe"Xj  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... XA PqRJ*Z  
               '(P = 0 to 35).']) 3g ep_ aC  
    end wA$ JDf)Vg  
    G6@XRib3  
    % Get the order and frequency corresonding to the function number: R+}7]tva6C  
    % ---------------------------------------------------------------- F5s Pd  
    p = p(:); 0&wbGbg(W  
    n = ceil((-3+sqrt(9+8*p))/2); \k&2nYVHf  
    m = 2*p - n.*(n+2); }JPLhr|d^  
    Ps<;DE\$f4  
    % Pass the inputs to the function ZERNFUN: /zV&ebN]  
    % ---------------------------------------- W w\M3Q`h  
    switch nargin ~*NG~Kn"s  
        case 3 >JVdL\3  
            z = zernfun(n,m,r,theta); x)GpNkx:  
        case 4 .0 }eg$d  
            z = zernfun(n,m,r,theta,nflag); [C@ |q Ah  
        otherwise $DS|jnpV  
            error('zernfun2:nargin','Incorrect number of inputs.') *,az`U  
    end lW6$v* s9  
    ,y5,+:Y ~  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) yX7P5c.   
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 6>Dm cG:.  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 1buVV]*~  
    %   order N and frequency M, evaluated at R.  N is a vector of !94qF,#1  
    %   positive integers (including 0), and M is a vector with the a*2JLK  
    %   same number of elements as N.  Each element k of M must be a -_[ZRf?^  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 0Ba*"/U]t~  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 0^('hS&  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix &) qs0  
    %   with one column for every (N,M) pair, and one row for every y <] x  
    %   element in R. z ?L]5m` H  
    % K6Z/  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- fug F k  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 8.WZC1N  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to _<^mi!Y  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 Wd>gOE  
    %   for all [n,m]. X+7@8)1(  
    % >S}^0vNZX  
    %   The radial Zernike polynomials are the radial portion of the IoKN.#;^  
    %   Zernike functions, which are an orthogonal basis on the unit 3Z_\.Z1R@  
    %   circle.  The series representation of the radial Zernike a1dkB"Zp.p  
    %   polynomials is *e,GXU@  
    % O_ 4 j"0  
    %          (n-m)/2 /0 2-0mNv  
    %            __ .dPy<6E  
    %    m      \       s                                          n-2s 5}Z_A?gy  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r [cso$Tv  
    %    n      s=0 X+KQ%Efo  
    % q=x1:^rVH  
    %   The following table shows the first 12 polynomials. 2A&Y})D  
    % #Y<QEGb(  
    %       n    m    Zernike polynomial    Normalization B`w@Xk'D  
    %       --------------------------------------------- 4Ai#$SHLm  
    %       0    0    1                        sqrt(2) ~5:-;ZbZ  
    %       1    1    r                           2 hqc)Ydg_%  
    %       2    0    2*r^2 - 1                sqrt(6) b wqd` C  
    %       2    2    r^2                      sqrt(6) wOV}<.W  
    %       3    1    3*r^3 - 2*r              sqrt(8) A}W}H;8x  
    %       3    3    r^3                      sqrt(8) }AG dWt@  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) R>B4v+b  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) WH lvd  
    %       4    4    r^4                      sqrt(10) ]I: h4hgw  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) z8JdA%YBM  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) hQ_g OI  
    %       5    5    r^5                      sqrt(12) FA$1&Fu3Y  
    %       --------------------------------------------- G[lNgVbU@  
    % qr'P0+|~5  
    %   Example: l<-0@(x)  
    % ,M0#?j>  
    %       % Display three example Zernike radial polynomials d>hLnz1O  
    %       r = 0:0.01:1; cyXnZs ?|  
    %       n = [3 2 5]; /SKgN{tWe  
    %       m = [1 2 1]; wS;hC&~2  
    %       z = zernpol(n,m,r); ><w=  
    %       figure d9pZg=$8  
    %       plot(r,z) v]@ n'!  
    %       grid on }%Vx2Q  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') eb=#{  
    % u&Cu"-%=M  
    %   See also ZERNFUN, ZERNFUN2. &Xp<%[:  
    nO;t5d  
    % A note on the algorithm. `KHP?lX  
    % ------------------------ M ]uO%2  
    % The radial Zernike polynomials are computed using the series b |JM4jgK  
    % representation shown in the Help section above. For many special LWIPq"  
    % functions, direct evaluation using the series representation can xC YL3hl  
    % produce poor numerical results (floating point errors), because cIOM}/gqv  
    % the summation often involves computing small differences between HOb0\X  
    % large successive terms in the series. (In such cases, the functions rssn'h  
    % are often evaluated using alternative methods such as recurrence WmTg`[  
    % relations: see the Legendre functions, for example). For the Zernike l g43  
    % polynomials, however, this problem does not arise, because the L9^h .Y7  
    % polynomials are evaluated over the finite domain r = (0,1), and aqoxj[V^3L  
    % because the coefficients for a given polynomial are generally all BkJNu_{m?  
    % of similar magnitude. @R s3i;"W  
    % g3kF&+2i  
    % ZERNPOL has been written using a vectorized implementation: multiple XHYVcwmDz-  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] I3" GGp3L  
    % values can be passed as inputs) for a vector of points R.  To achieve [*z`p;n2D  
    % this vectorization most efficiently, the algorithm in ZERNPOL W?2Z31;7  
    % involves pre-determining all the powers p of R that are required to ,+o*>fD  
    % compute the outputs, and then compiling the {R^p} into a single BiI`oCX  
    % matrix.  This avoids any redundant computation of the R^p, and ,%U'>F?  
    % minimizes the sizes of certain intermediate variables. b$/ 'dnx  
    % by@}T@^\  
    %   Paul Fricker 11/13/2006 :GN7JxD#  
    >?)Df(n(9  
    FWl'='5L  
    % Check and prepare the inputs: :uQ~?amM  
    % ----------------------------- t`eUD>\  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 9xM7X?  
        error('zernpol:NMvectors','N and M must be vectors.') &'A8R;b}-?  
    end N3?@CM^hHw  
    +5oK91o[y  
    if length(n)~=length(m) oa:30@HSb  
        error('zernpol:NMlength','N and M must be the same length.') Qv/Kbw N{  
    end 6\GL|#G  
    [RFF&uy  
    n = n(:); qb?9i-(  
    m = m(:); !)+8:8H'  
    length_n = length(n);  KSB{Z TE  
    $Y&rci]  
    if any(mod(n-m,2)) >'e(|P4  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') =.yKl*WV{  
    end  "?(N  
    g!.k>  
    if any(m<0) uBqZ62{G  
        error('zernpol:Mpositive','All M must be positive.') sEm064  
    end I+g[ p  
    E'wJ+X9 +  
    if any(m>n) e{fm7Cc)D  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 1PnWgu  
    end fLR\@f  
    :Miri_l  
    if any( r>1 | r<0 ) J={R@}u  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 18];fC  
    end PA<<{\dp  
    59Lmv &s  
    if ~any(size(r)==1) Y!nxHRE  
        error('zernpol:Rvector','R must be a vector.') (OT&:WwW  
    end -3T~+  
     k.("<)  
    r = r(:); C,#FH}  
    length_r = length(r); i! DO  
    c]!Yb-  
    if nargin==4 N;.}g*_+}  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ZA Xw=O5  
        if ~isnorm M73d^z  
            error('zernpol:normalization','Unrecognized normalization flag.') > nOU 8  
        end UqEpeLK  
    else )MlT=k6S  
        isnorm = false; b%"Lwqdr7  
    end NvU~?WN  
    ,sln0  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6|AD]/t^K  
    % Compute the Zernike Polynomials KOHYeiry~A  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2c4x=%  
    eV)'@ 8p  
    % Determine the required powers of r: QfHO3Y6h[  
    % ----------------------------------- [mJmT->  
    rpowers = []; JOvRU DZ  
    for j = 1:length(n) afNqK~  
        rpowers = [rpowers m(j):2:n(j)]; *D6X&Hg&5  
    end (M,IgSn9  
    rpowers = unique(rpowers); oGXndfd"  
    Hd9vS"TN]  
    % Pre-compute the values of r raised to the required powers, ]> 36{k]&  
    % and compile them in a matrix: P}RewMJ$L  
    % ----------------------------- qTD^Vz V  
    if rpowers(1)==0 xhmrep6+<  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); TE!+G\@  
        rpowern = cat(2,rpowern{:}); eg$y,Tx  
        rpowern = [ones(length_r,1) rpowern]; d9kN @W  
    else 3HI- G.]hC  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); {'e%Hx  
        rpowern = cat(2,rpowern{:}); /;rPzP4K6  
    end W`2Xn?g  
    Obb"#W@3  
    % Compute the values of the polynomials: 8BgHoQ*  
    % -------------------------------------- ;%_s4  
    z = zeros(length_r,length_n); &)y$XsSMW  
    for j = 1:length_n m_pqU(sP  
        s = 0:(n(j)-m(j))/2; qPI1\!z6  
        pows = n(j):-2:m(j); }aC@ov]2  
        for k = length(s):-1:1 ,2C{X+t  
            p = (1-2*mod(s(k),2))* ... (yB)rBh>n  
                       prod(2:(n(j)-s(k)))/          ... 3j2#'Jf|:  
                       prod(2:s(k))/                 ... U'K{>"~1a  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... quGv q"Y>  
                       prod(2:((n(j)+m(j))/2-s(k))); -xk.wWpV  
            idx = (pows(k)==rpowers); iF#|Z$g-(  
            z(:,j) = z(:,j) + p*rpowern(:,idx); .\6q\7Ej  
        end 6+s10?  
         d$}z,~sN  
        if isnorm F\G-. 1  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); znxP.=GB   
        end @V>BG8Y  
    end jqeR{yo&0b  
    &?)? w-$p  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  q22@ZRw  
    h3(B7n7  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 :v%iF!+.P  
    $xK(bc'{  
    07年就写过这方面的计算程序了。
    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)