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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 5PCqYN(:B  
    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! >IafUy  
     
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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 2|y"!JqE1  
    function z = zernfun(n,m,r,theta,nflag) u#fM_>ML  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. :G=fl)!fE  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N TqQB@-!  
    %   and angular frequency M, evaluated at positions (R,THETA) on the K3&qq[8.e  
    %   unit circle.  N is a vector of positive integers (including 0), and c]<5zyl"j1  
    %   M is a vector with the same number of elements as N.  Each element wu6;.xTLl  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) DK~xrU'  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, p>N(Typ0b  
    %   and THETA is a vector of angles.  R and THETA must have the same j_[tu!~  
    %   length.  The output Z is a matrix with one column for every (N,M) 7+cO_3AB  
    %   pair, and one row for every (R,THETA) pair. bs&43Ae  
    % ]cvwIc">  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 3%|&I:tI  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), aK~8B_5k8  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ]A `n( "%  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, @bLy,Xr&  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized }#+^{P3;  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. e"cXun4nS=  
    % 59L\|OR  
    %   The Zernike functions are an orthogonal basis on the unit circle. rXq.DvQ  
    %   They are used in disciplines such as astronomy, optics, and FxY}m  
    %   optometry to describe functions on a circular domain. Hio0HL-  
    % 7z,C}-q  
    %   The following table lists the first 15 Zernike functions. Y-z(zS^1  
    % B mb0cF Q  
    %       n    m    Zernike function           Normalization est9M*Fn  
    %       -------------------------------------------------- /s?`&1v|r  
    %       0    0    1                                 1 W i.& e  
    %       1    1    r * cos(theta)                    2 Lb-OsKU  
    %       1   -1    r * sin(theta)                    2 Oo~; L,  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Uc>lGo1j  
    %       2    0    (2*r^2 - 1)                    sqrt(3) $wa{~'  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) {Mk6T1Bkq  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) SulY1,  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 6|=f$a  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Rv>-4@fMJ  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Ne!lH@ql  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) RP|`HkP-2  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ue"~9JK.  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Nx;~@  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) IPpN@  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) {Xy5pfW Q  
    %       -------------------------------------------------- M3y NAN  
    % 372rbY  
    %   Example 1: N~gzDQ3  
    % :OZrH<SW  
    %       % Display the Zernike function Z(n=5,m=1) t?gic9 q  
    %       x = -1:0.01:1; .{^5X)  
    %       [X,Y] = meshgrid(x,x); 0mVNQxHI  
    %       [theta,r] = cart2pol(X,Y); WU` rh^  
    %       idx = r<=1; wlvgg  
    %       z = nan(size(X)); ~?}Emn;t  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); %1L,Y  
    %       figure @mBQ?; qlK  
    %       pcolor(x,x,z), shading interp 'LC1(V!_j  
    %       axis square, colorbar uW{l(}0N  
    %       title('Zernike function Z_5^1(r,\theta)') Q&;9 x?e  
    % o|:b;\)b  
    %   Example 2: pv&sO~!iC  
    % rlLMT6r.8  
    %       % Display the first 10 Zernike functions 6 "sSoj  
    %       x = -1:0.01:1; *fxG?}YT  
    %       [X,Y] = meshgrid(x,x); J@'wf8Ub  
    %       [theta,r] = cart2pol(X,Y); ITBE|b  
    %       idx = r<=1; e T{ 4{  
    %       z = nan(size(X)); 'H!Uh]!  
    %       n = [0  1  1  2  2  2  3  3  3  3]; m0SlOgRsk  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; \\qZl)P_  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; X_h}J=33Q  
    %       y = zernfun(n,m,r(idx),theta(idx)); cI*;k.KU  
    %       figure('Units','normalized') 7}>EJ  
    %       for k = 1:10 {\5  
    %           z(idx) = y(:,k); [q -h|m  
    %           subplot(4,7,Nplot(k)) SnfYT)Ph  
    %           pcolor(x,x,z), shading interp ]ieeP4*  
    %           set(gca,'XTick',[],'YTick',[]) \b x$i*  
    %           axis square "+s++@ z  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) oc`H}Wvn  
    %       end Otuf] B^s  
    % D@.6>:;il  
    %   See also ZERNPOL, ZERNFUN2. ?a5!H*,  
    ##*3bDf$-5  
    %   Paul Fricker 11/13/2006 Y3b *a".X  
    `;C  V=,M  
    Z9|P'R(l  
    % Check and prepare the inputs: 0,")C5j  
    % ----------------------------- QWYJ *  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~>|ziHx  
        error('zernfun:NMvectors','N and M must be vectors.') }}~|!8  
    end }7Q%6&IR  
    '=pU^Oz<}  
    if length(n)~=length(m) L,!?Nt\  
        error('zernfun:NMlength','N and M must be the same length.') L8B! u9%  
    end 0(HU}I  
    (<9u-HF#  
    n = n(:); fHFE){  
    m = m(:); ]a`$LW}  
    if any(mod(n-m,2)) Zy/_ E@C}u  
        error('zernfun:NMmultiplesof2', ... 7@Qcc t4A  
              'All N and M must differ by multiples of 2 (including 0).') g 7H(PF?  
    end ktIFI`@ w)  
    z03K=aZ  
    if any(m>n) })%{AfDRF  
        error('zernfun:MlessthanN', ... `c$V$/IT  
              'Each M must be less than or equal to its corresponding N.') 2^7`mES  
    end @yYkti;4-  
    !a\^Sk /  
    if any( r>1 | r<0 ) ? J0y|  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') !nnC3y{G  
    end [/r(__.  
    {Sh ;(.u^  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Pm7}"D'/  
        error('zernfun:RTHvector','R and THETA must be vectors.') E1 2uZ$X  
    end 9(Xn>G'iT  
    wCBplaojJ  
    r = r(:); TWTb?HP  
    theta = theta(:); [a(#1  
    length_r = length(r); ~} ~4  
    if length_r~=length(theta) * ;FdD{+  
        error('zernfun:RTHlength', ... pb,d'z\S  
              'The number of R- and THETA-values must be equal.') -~w'Xo#  
    end vY3h3o  
    .%-8 t{dt  
    % Check normalization: ueNS='+m  
    % -------------------- ?Bmb' 3  
    if nargin==5 && ischar(nflag) :`sUt1Fw.  
        isnorm = strcmpi(nflag,'norm'); -{vD: Il=6  
        if ~isnorm Y]a@j !  
            error('zernfun:normalization','Unrecognized normalization flag.') -Y8B~@]P?  
        end |w=zOC;v  
    else Z\sDUJ  
        isnorm = false; P+}h$ _x  
    end *4 n)  
    |s_GlJV.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ALHIGJW:6$  
    % Compute the Zernike Polynomials =_^X3z0  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i.#:zU%o  
    *qq+jsA6wH  
    % Determine the required powers of r: LP=)~K<  
    % ----------------------------------- i XN1I  
    m_abs = abs(m); Hn:Crl y#  
    rpowers = []; ]M3yLYK/P  
    for j = 1:length(n) %so]L+r2!  
        rpowers = [rpowers m_abs(j):2:n(j)]; %iB,IEw  
    end j<$2hiI/?&  
    rpowers = unique(rpowers); jEwIn1  
    <VE@DBWyl~  
    % Pre-compute the values of r raised to the required powers, !R$`+wZ62  
    % and compile them in a matrix: F0# 'WfM#  
    % ----------------------------- w-jVC^C]  
    if rpowers(1)==0 ~LC-[&$  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Ys7]B9/1O  
        rpowern = cat(2,rpowern{:}); p ll)Y  
        rpowern = [ones(length_r,1) rpowern]; $cg cX  
    else "N#Y gSr  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); H?w6C):]  
        rpowern = cat(2,rpowern{:}); dr"1s-D4IQ  
    end |j|rS5  
    D_MmW  
    % Compute the values of the polynomials: '%;m?t% q  
    % -------------------------------------- naNghGQ  
    y = zeros(length_r,length(n)); HOi`$vX }N  
    for j = 1:length(n) gM]:Ma  
        s = 0:(n(j)-m_abs(j))/2; !x)R=Z/C  
        pows = n(j):-2:m_abs(j); $~kA B8z  
        for k = length(s):-1:1 TqQ[_RKg2  
            p = (1-2*mod(s(k),2))* ... +`15le`R  
                       prod(2:(n(j)-s(k)))/              ... OrW  
                       prod(2:s(k))/                     ... \7_y%HR  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... E{@[k%,_  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); SX#&5Ka/  
            idx = (pows(k)==rpowers); Ul# r  
            y(:,j) = y(:,j) + p*rpowern(:,idx); $VR{q6[0S?  
        end >mkFV@`  
         ,: ^u-b|  
        if isnorm VN.Je: Ju  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ?A0)L27UE&  
        end x~sBzTa  
    end u@444Vzg  
    % END: Compute the Zernike Polynomials $Kd>:f=A  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% AFn7uW!9Gw  
    mZBo~(}  
    % Compute the Zernike functions: @+DX.9  
    % ------------------------------ 3$/IC@+  
    idx_pos = m>0; 1Ws9WU  
    idx_neg = m<0; T>>c2$ x  
    4z)]@:`}z  
    z = y; afk>+4q  
    if any(idx_pos) !~Z"9(v'C  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ;a3}~s  
    end 1*7@BP5  
    if any(idx_neg) L.IlBjD  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 1x^GWtRp  
    end |uDdHX8T  
    ULW~90  
    % 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) %)wjR/o  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Pc9H0\+Xk  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated e(yh[7p=  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 0$njMnB2l  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, F[0]/  
    %   and THETA is a vector of angles.  R and THETA must have the same OJxl<Q=z  
    %   length.  The output Z is a matrix with one column for every P-value, 9FX-1,Jx  
    %   and one row for every (R,THETA) pair. <vP=zk  
    % $8FUfJ1@  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike /O9EQPm(  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) @XVTU  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) cnLro  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 Wjc'*QCPl  
    %   for all p. %$mA03[MQ  
    % ;Qq\DFe.w  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 y)*RV;^  
    %   Zernike functions (order N<=7).  In some disciplines it is YK\X+"lB  
    %   traditional to label the first 36 functions using a single mode qWw=8Bq  
    %   number P instead of separate numbers for the order N and azimuthal wS*E(IAl  
    %   frequency M. )X!,3Ca{43  
    % (#'>(t(4  
    %   Example: / j^  
    % K%d&EYoW]  
    %       % Display the first 16 Zernike functions =QsYXK7Mn4  
    %       x = -1:0.01:1; :pUtSs7p}  
    %       [X,Y] = meshgrid(x,x); h$*!8=M  
    %       [theta,r] = cart2pol(X,Y); [gB+C84%%  
    %       idx = r<=1; =#\:}@J5I  
    %       p = 0:15; +qoRP2  
    %       z = nan(size(X)); 7Ix973^  
    %       y = zernfun2(p,r(idx),theta(idx));  )*[3Vq  
    %       figure('Units','normalized') @.C2LIb  
    %       for k = 1:length(p) {8OCXus3m  
    %           z(idx) = y(:,k); Lv%x81]K  
    %           subplot(4,4,k) 7 3m1  
    %           pcolor(x,x,z), shading interp ZW}_DT0  
    %           set(gca,'XTick',[],'YTick',[]) #F#%`Rv1  
    %           axis square PM+[,H  
    %           title(['Z_{' num2str(p(k)) '}']) :r[`.`  
    %       end  `]X>V,  
    % grYe&(`X  
    %   See also ZERNPOL, ZERNFUN. ;fJ.8C  
    /\Ef%@  
    %   Paul Fricker 11/13/2006 7"mc+QOp  
    P%6~&woF  
    z}@7'_iJ  
    % Check and prepare the inputs: liZxBs :%i  
    % ----------------------------- [~ fraK,)  
    if min(size(p))~=1 H.c7Nle  
        error('zernfun2:Pvector','Input P must be vector.') sRW<me;  
    end rZF*q2?  
    OPi0~s  
    if any(p)>35 `gJ(0#ac  
        error('zernfun2:P36', ... S:Hl/:iV  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... \8 ":]EU  
               '(P = 0 to 35).']) nEfK53i_  
    end (ZGbh MK  
    U(Zq= M  
    % Get the order and frequency corresonding to the function number: ]yu:i-SfP  
    % ---------------------------------------------------------------- j [a(#V{  
    p = p(:); VQs5"K"  
    n = ceil((-3+sqrt(9+8*p))/2); nNm`Hfi  
    m = 2*p - n.*(n+2); J05e#-)<K  
    C+]I@Go'Tk  
    % Pass the inputs to the function ZERNFUN: /{[o ~:'p  
    % ---------------------------------------- lk!@?  
    switch nargin .6> w'F{>  
        case 3 Fs{*XKv&lH  
            z = zernfun(n,m,r,theta); FlQGg VN  
        case 4 D@KlOU{<  
            z = zernfun(n,m,r,theta,nflag); \GBuWY3B  
        otherwise ==B6qX8T  
            error('zernfun2:nargin','Incorrect number of inputs.') 5s XXM  
    end 7nSxi+6e  
    No$3"4wk  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag)  \*da6Am  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 3^ClAE"8  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of l}h!B_P'  
    %   order N and frequency M, evaluated at R.  N is a vector of dQvcXl]  
    %   positive integers (including 0), and M is a vector with the [Pp'Ye~K@c  
    %   same number of elements as N.  Each element k of M must be a =D(j)<9$A  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ?M2J wAK5  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is LD?sh"?b  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix "4Nt\WQ  
    %   with one column for every (N,M) pair, and one row for every pCDmXB  
    %   element in R. VUc%4U{Cti  
    % RCrCs  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- iscz}E,Y  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is u +hX  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to o-\[,}T)M  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 Ef\ -VKh  
    %   for all [n,m]. \['Cj*ek  
    % c L]1f  
    %   The radial Zernike polynomials are the radial portion of the +|v90ed  
    %   Zernike functions, which are an orthogonal basis on the unit (:_$5&i7  
    %   circle.  The series representation of the radial Zernike 1 zZlC#V  
    %   polynomials is 9$t( &z=  
    % hgmCRC  
    %          (n-m)/2 Xvv6~  
    %            __ -=="<0c  
    %    m      \       s                                          n-2s |pK !S  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 1oS/`)  
    %    n      s=0 '91/md5  
    % 1\Xw3prH  
    %   The following table shows the first 12 polynomials. 0sqFF[i  
    % oDR%\VY6T  
    %       n    m    Zernike polynomial    Normalization \zY!qpX<  
    %       --------------------------------------------- 9x8fhAy}4  
    %       0    0    1                        sqrt(2) ,}PgOJZ  
    %       1    1    r                           2 XX@ZQcN  
    %       2    0    2*r^2 - 1                sqrt(6) Y73C5.dNcE  
    %       2    2    r^2                      sqrt(6) [GR; ?R5  
    %       3    1    3*r^3 - 2*r              sqrt(8) eRYK3W  
    %       3    3    r^3                      sqrt(8) ok[i<zl; '  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 1x)J[fyId  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) +0&/g&a\R  
    %       4    4    r^4                      sqrt(10) p?!/+  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) zda 3 ,U2o  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) \G[$:nS  
    %       5    5    r^5                      sqrt(12) F847pyOJnf  
    %       --------------------------------------------- @- xjfC\d  
    % %4H%?4  
    %   Example: !Ee:o"jG{  
    % x4 yR8n(  
    %       % Display three example Zernike radial polynomials r" y.KD^  
    %       r = 0:0.01:1; *g%yRU{N  
    %       n = [3 2 5]; >j/w@Fj  
    %       m = [1 2 1]; NJ<F>3  
    %       z = zernpol(n,m,r); o4X{L`m  
    %       figure `Oa WGZ[  
    %       plot(r,z) $]d^-{|  
    %       grid on qna8|3eP  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') NOva'qk  
    % "x-j~u?  
    %   See also ZERNFUN, ZERNFUN2. +rd+0 `}C  
    29Ki uP  
    % A note on the algorithm. 0;k# *#w  
    % ------------------------ cr3^6HB  
    % The radial Zernike polynomials are computed using the series py4 h(04u  
    % representation shown in the Help section above. For many special WcAkCH!L  
    % functions, direct evaluation using the series representation can b;n[mk  
    % produce poor numerical results (floating point errors), because xp t:BBo  
    % the summation often involves computing small differences between CrLrw T  
    % large successive terms in the series. (In such cases, the functions HtFDlvdy]  
    % are often evaluated using alternative methods such as recurrence .]^?<bG  
    % relations: see the Legendre functions, for example). For the Zernike ;+%rw2Z,B  
    % polynomials, however, this problem does not arise, because the d-qUtgqV86  
    % polynomials are evaluated over the finite domain r = (0,1), and b=vkiO`2  
    % because the coefficients for a given polynomial are generally all z_HdISy0  
    % of similar magnitude. HfVZ~PP  
    % CTb%(<r  
    % ZERNPOL has been written using a vectorized implementation: multiple L,\Iasv  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] KoT\pY^7\  
    % values can be passed as inputs) for a vector of points R.  To achieve ^!d3=}:0  
    % this vectorization most efficiently, the algorithm in ZERNPOL kmW4:EA%  
    % involves pre-determining all the powers p of R that are required to 7I}uZ/N  
    % compute the outputs, and then compiling the {R^p} into a single eFgA 8kY)  
    % matrix.  This avoids any redundant computation of the R^p, and s!J9|]o  
    % minimizes the sizes of certain intermediate variables. 9w"*y#_  
    % j%kncGS  
    %   Paul Fricker 11/13/2006 %EH)&k  
    h{Y",7] !  
    ZVBXx\{s  
    % Check and prepare the inputs: Vr}'.\$  
    % ----------------------------- tw;}jh  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) *@5@,=d  
        error('zernpol:NMvectors','N and M must be vectors.') =bOW~0Z1  
    end dd;~K&_Q/i  
    fC`&g~yK'  
    if length(n)~=length(m) 4x34u}l  
        error('zernpol:NMlength','N and M must be the same length.') 4s- !7  
    end e6*8K@LHB  
    dPlV>IM$z  
    n = n(:); jA1 +x:Wq  
    m = m(:); fhiM U8(&  
    length_n = length(n); Ui~>SN>s  
    kP:!/g  
    if any(mod(n-m,2)) N8jIMb'<  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') (QEG4&9  
    end [y(MCf19  
    pBHRa?Y5  
    if any(m<0) 01]f2.5  
        error('zernpol:Mpositive','All M must be positive.') Et$2Y-L.  
    end B\~}3!j  
    Lbgi7|&  
    if any(m>n) ah"o~Cbj  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') VA%J\T|G2\  
    end yWK)vju"  
    (PL UFT  
    if any( r>1 | r<0 ) BGSw~6  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') )lkjqFQ(  
    end C%u28|  
    {7[Ox<Ho  
    if ~any(size(r)==1) x2xRBkRg=  
        error('zernpol:Rvector','R must be a vector.') C|bET  
    end 6nn *]|7  
    K(4_a``05  
    r = r(:); sHj/;  
    length_r = length(r); "oyo#-5z  
    /ZX }Nc g  
    if nargin==4 =X}J6|>X  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); OUnA;_  
        if ~isnorm 4W75T2q#  
            error('zernpol:normalization','Unrecognized normalization flag.') F9^S"qv$  
        end E .h*g8bXe  
    else F,kZU$  
        isnorm = false; a?1Wq  
    end KNl$3nX  
    w0. u\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xT8?&Bx  
    % Compute the Zernike Polynomials @7 }W=HB  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PCA4k.,T  
    mpyt5#f  
    % Determine the required powers of r: h[ ZN+M  
    % ----------------------------------- 4eu O1=  
    rpowers = []; gGYKEq{j(  
    for j = 1:length(n) JF]JOI6.e  
        rpowers = [rpowers m(j):2:n(j)];  *CMx-_  
    end bA 2pbjg=  
    rpowers = unique(rpowers); i b m4fa  
    %b0*H_ok7  
    % Pre-compute the values of r raised to the required powers, BtZyn7a  
    % and compile them in a matrix: _1^'(5f$  
    % ----------------------------- YSMAd-Ef-  
    if rpowers(1)==0 #yen8SskB  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); !D6]JPX  
        rpowern = cat(2,rpowern{:}); lZ0 =;I  
        rpowern = [ones(length_r,1) rpowern]; $G>.\t  
    else 4i bc  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); [ ~,AfY  
        rpowern = cat(2,rpowern{:}); <@}9Bid!o  
    end bt *k.=p  
    N`i/mP  
    % Compute the values of the polynomials: nN;u,}e  
    % -------------------------------------- =N@t'fOr  
    z = zeros(length_r,length_n); :k"]5>(^  
    for j = 1:length_n *Ex|9FCt$  
        s = 0:(n(j)-m(j))/2; =Qq+4F)MD  
        pows = n(j):-2:m(j); [aS*%Heu  
        for k = length(s):-1:1 %y@AA>x!  
            p = (1-2*mod(s(k),2))* ... :&Nbw  
                       prod(2:(n(j)-s(k)))/          ... 8L XHk l  
                       prod(2:s(k))/                 ... <3iMRe  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... E^PB)D(.  
                       prod(2:((n(j)+m(j))/2-s(k))); Z)!C'cb  
            idx = (pows(k)==rpowers); c> af  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 0x7'^Z>-oe  
        end dx]>(e@(t{  
         ^8tEach  
        if isnorm R]dg_Da  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); t) +310w  
        end K,]=6 Rj  
    end PFR:>^wK2  
    neh(<>  
    % 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)  7= DdrG<  
    n}V_,:Z  
    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。 S@Hf &hJ  
    CA#,THty  
    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)