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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 *(Dmd$|0|  
    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! #; ?3k uq(  
     
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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 `R[Hxi  
    function z = zernfun(n,m,r,theta,nflag) Y?0/f[Ax,y  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. qLN\%}69/  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 9njl,Q:  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ;5Sdx5`_  
    %   unit circle.  N is a vector of positive integers (including 0), and 4%(Ji  
    %   M is a vector with the same number of elements as N.  Each element 6}4})B2  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) jj5S+ >4  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, gYKz,$  
    %   and THETA is a vector of angles.  R and THETA must have the same ScPVjqG2{  
    %   length.  The output Z is a matrix with one column for every (N,M) aB]0?C y9(  
    %   pair, and one row for every (R,THETA) pair. XjX  
    % F/>_PH57  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike / ;]5X  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 9j,g&G.K  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 3N7H7(IR  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, M;(,0dk  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized .|UQ)J?s  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. hx;f/E Px  
    % 5(1:^:LGK  
    %   The Zernike functions are an orthogonal basis on the unit circle. 6  63o  
    %   They are used in disciplines such as astronomy, optics, and * QgKo$IF  
    %   optometry to describe functions on a circular domain. d$dy6{/YD  
    % {1W:@6tl  
    %   The following table lists the first 15 Zernike functions. NxT"A)u  
    % XX6Z|Y5.  
    %       n    m    Zernike function           Normalization {@Mr7*u  
    %       -------------------------------------------------- |c_qq Bd  
    %       0    0    1                                 1 qQ&uU7,#  
    %       1    1    r * cos(theta)                    2 'rQ"Dc1D  
    %       1   -1    r * sin(theta)                    2 ?pDr"XH~  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Gr"CHz/  
    %       2    0    (2*r^2 - 1)                    sqrt(3) QLA.;`HIE  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) em$pU*`P  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) yV30x9i!2  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8)  C0rf  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ~E<2gMKjO  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ,Kit@`P%  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) eim+oms  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) F MfpjuHk  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 6% D9;-N)  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) wb-yAQ8  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) {h<D/:^v  
    %       -------------------------------------------------- [,%=\%5  
    % *6uccx7{  
    %   Example 1: D7Q+w  
    % Fe+ @;  
    %       % Display the Zernike function Z(n=5,m=1) +Y\:Q<eMFg  
    %       x = -1:0.01:1; b}J%4Lx%m  
    %       [X,Y] = meshgrid(x,x); boh?Xt-$  
    %       [theta,r] = cart2pol(X,Y); >.h:Y5  
    %       idx = r<=1; u8e_Lqx?  
    %       z = nan(size(X)); 8;PkuJR_]  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); K*6"c.D  
    %       figure 8~=*\ @^  
    %       pcolor(x,x,z), shading interp "Fz.# U  
    %       axis square, colorbar U"oNJ8&%|  
    %       title('Zernike function Z_5^1(r,\theta)') KKeMi@N  
    % MvVpp;bd  
    %   Example 2: 99m2aT()  
    % Zg;$vIhn  
    %       % Display the first 10 Zernike functions >rG>Bz^Pu  
    %       x = -1:0.01:1; %36x'Dn ?  
    %       [X,Y] = meshgrid(x,x); P rt} 01$  
    %       [theta,r] = cart2pol(X,Y); .nV2 n@SR  
    %       idx = r<=1; 8^mE<  
    %       z = nan(size(X)); j!;LN)s@?  
    %       n = [0  1  1  2  2  2  3  3  3  3]; [B0 BHJ~  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 8\"<t/_ W  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; OATdmHW  
    %       y = zernfun(n,m,r(idx),theta(idx)); 1/_g36\l$  
    %       figure('Units','normalized') [K{{P|(q  
    %       for k = 1:10 +X?ErQm  
    %           z(idx) = y(:,k); b0P3S!E  
    %           subplot(4,7,Nplot(k)) b F=MQ  
    %           pcolor(x,x,z), shading interp !oLn=  
    %           set(gca,'XTick',[],'YTick',[]) i,Ct AbMx  
    %           axis square pP<8zTLn  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) H R>Y?B{  
    %       end fv+]iK<{  
    % }$6L]   
    %   See also ZERNPOL, ZERNFUN2. 24sMX7Q,i  
     C=D*  
    %   Paul Fricker 11/13/2006 K;K0D@>]HR  
    ]4:QqdV  
    wmVmGa R  
    % Check and prepare the inputs: 9#8vPjXW}.  
    % ----------------------------- xATx2*@X2  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Y&2FH/(M  
        error('zernfun:NMvectors','N and M must be vectors.') ~id6^#&>  
    end `[ZswLE  
    m^X51,+<  
    if length(n)~=length(m) *>fr'jj1$  
        error('zernfun:NMlength','N and M must be the same length.') g_>&R58  
    end z?/_b  
    l4C{LZ  
    n = n(:); F^,:p.ihm<  
    m = m(:); AXyXK??  
    if any(mod(n-m,2)) >$Y/B=e  
        error('zernfun:NMmultiplesof2', ... wMS%/l0p1  
              'All N and M must differ by multiples of 2 (including 0).') y oW ~  
    end g`fG84  
    Y6{^cZ!=  
    if any(m>n) Vl.,e1)6  
        error('zernfun:MlessthanN', ... N0 {e7M  
              'Each M must be less than or equal to its corresponding N.') @SfQbM##%  
    end ,dC.|P' `  
    <4r8H-(%  
    if any( r>1 | r<0 ) s'HsLe0|  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') *)m:u:   
    end b.*LmSX#  
    \Vc[/Qp7Bb  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ~${~To8$CW  
        error('zernfun:RTHvector','R and THETA must be vectors.') , Ckcc  
    end U[C>Aoze  
    Zue3Z{31T  
    r = r(:); <83Ky;ry  
    theta = theta(:); EP:`l  
    length_r = length(r); 0ix(1`Z  
    if length_r~=length(theta) .W]k 8N E  
        error('zernfun:RTHlength', ... yr\ClIU  
              'The number of R- and THETA-values must be equal.') Jn+-G4h$  
    end @CNJpQ ujn  
    l";Yw]:^  
    % Check normalization: !*]i3 ,{7v  
    % -------------------- BU -;P  
    if nargin==5 && ischar(nflag) gIo\^ktW  
        isnorm = strcmpi(nflag,'norm'); n)yDep]$G  
        if ~isnorm bPVk5G*ruP  
            error('zernfun:normalization','Unrecognized normalization flag.') nJI2IPZ  
        end [,Ehu<mEK  
    else $+j1^  
        isnorm = false; >zJHvb)b\  
    end uV:R3#^  
    (d4btcg  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% * JK0X  
    % Compute the Zernike Polynomials @c^ Dl  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |]-Zz7N)  
    dWz?`B{'  
    % Determine the required powers of r: Pd7\Q]of  
    % ----------------------------------- DS?.'"n[u  
    m_abs = abs(m); bK:U:vpYm  
    rpowers = []; ^kfqw0!  
    for j = 1:length(n) $udhTI#,  
        rpowers = [rpowers m_abs(j):2:n(j)]; - l X4;  
    end J>(X0@eWz  
    rpowers = unique(rpowers); m-t: ' B  
    ?t YZ/  
    % Pre-compute the values of r raised to the required powers, %`F;i)Zz  
    % and compile them in a matrix: F! =l r  
    % ----------------------------- y,5qY}P+  
    if rpowers(1)==0 8>YF}\D V  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); JP8}+  
        rpowern = cat(2,rpowern{:}); xr'1CP  
        rpowern = [ones(length_r,1) rpowern]; l 5-[a  
    else 8m") )i-  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); m,)s8_a  
        rpowern = cat(2,rpowern{:}); IV,4BQ$  
    end x3ZF6)@  
    xFF!)k #  
    % Compute the values of the polynomials: H*0Y_H=  
    % -------------------------------------- Na91K4r#  
    y = zeros(length_r,length(n)); dk&e EDvfd  
    for j = 1:length(n) s70Z&3A  
        s = 0:(n(j)-m_abs(j))/2; os5$(  
        pows = n(j):-2:m_abs(j); LwB1~fF  
        for k = length(s):-1:1 -,")GA+[7  
            p = (1-2*mod(s(k),2))* ... *?<N3Rr*  
                       prod(2:(n(j)-s(k)))/              ... %}@iz(*}>  
                       prod(2:s(k))/                     ... -m@c{&r  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... M|blg!j;  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 2Wzx1_D "a  
            idx = (pows(k)==rpowers); mn@1&#c4y  
            y(:,j) = y(:,j) + p*rpowern(:,idx); l\Ozy  
        end g@#he95 }  
         ?TA7i b_  
        if isnorm PIH\*2\/  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); jC&fnt,O  
        end WX4sTxJK  
    end J2qsZ  
    % END: Compute the Zernike Polynomials eB~\~@  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [~N;d9H+*1  
    +;W%v7 %<  
    % Compute the Zernike functions: v_zt$bf{Y  
    % ------------------------------ h8(#\E  
    idx_pos = m>0; \d"\7SA  
    idx_neg = m<0; -`' |z+V  
    65aYH4"  
    z = y; oB 1Qw'J w  
    if any(idx_pos) 4 $R!)  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); -f|/#1  
    end 30HUY?'K  
    if any(idx_neg) 9(]_so24,  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); TFkG"ev  
    end br=e+]C Y)  
    |V5BL<4  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) *{?2M6Z  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 9-n]_AF`0  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated NATi)A"TZ  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive R}Zaz3( Hd  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ]/ !*^;cY(  
    %   and THETA is a vector of angles.  R and THETA must have the same s-*N_Dv  
    %   length.  The output Z is a matrix with one column for every P-value, `_vPElQXZ#  
    %   and one row for every (R,THETA) pair. q"Bd-?9  
    % =3,<(F5Y[  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ;2h"YU-b  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 6iQqOAG  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) Z2{$FN  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ruqE]Hx9(  
    %   for all p. toF6 Z  
    % lU|ltnU  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ~(m6dPm$}m  
    %   Zernike functions (order N<=7).  In some disciplines it is zX=%BL?  
    %   traditional to label the first 36 functions using a single mode ^ &KH|qRrO  
    %   number P instead of separate numbers for the order N and azimuthal jM*wm~4>@  
    %   frequency M. a w~a /T:  
    % KI<x`b  
    %   Example: dd=5`Bo9Yh  
    % 6:r1^q6A9L  
    %       % Display the first 16 Zernike functions (`n*d3  
    %       x = -1:0.01:1; -'W:P'BG  
    %       [X,Y] = meshgrid(x,x); 5}N O~Xd<  
    %       [theta,r] = cart2pol(X,Y); ~$a%& ]\  
    %       idx = r<=1; nkkGJV!  
    %       p = 0:15; }xrrHp  
    %       z = nan(size(X)); jP @t!=  
    %       y = zernfun2(p,r(idx),theta(idx)); DF#WQ8?$]  
    %       figure('Units','normalized') If9!S} wa  
    %       for k = 1:length(p) @7PE&3  
    %           z(idx) = y(:,k); XeBSHvO_  
    %           subplot(4,4,k) of<>M4/g4y  
    %           pcolor(x,x,z), shading interp qj|B #dU  
    %           set(gca,'XTick',[],'YTick',[]) qf] OSd  
    %           axis square ?yz}  
    %           title(['Z_{' num2str(p(k)) '}']) =TI|uD6T  
    %       end w_J`29uc  
    % h$zPQ""8  
    %   See also ZERNPOL, ZERNFUN. e$(i!G)  
    .Lu=16  
    %   Paul Fricker 11/13/2006 :S7yM8 b`  
    !c`Q?aGV)  
    ~(0Y`+gC  
    % Check and prepare the inputs: ZCJ8I  
    % ----------------------------- *&2#;mf3  
    if min(size(p))~=1 ?y45#Tk]  
        error('zernfun2:Pvector','Input P must be vector.') nt[0krG  
    end )J88gMk+  
    w)8@Tu:Q  
    if any(p)>35 "RX?"pB  
        error('zernfun2:P36', ... UZX)1?U  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... s* YFN#Wuc  
               '(P = 0 to 35).']) IKr7"`  
    end $xZk{ rK  
    &{=~)>h  
    % Get the order and frequency corresonding to the function number: ?RzT0HRd  
    % ---------------------------------------------------------------- fY"28#   
    p = p(:); tx1jBh:e=  
    n = ceil((-3+sqrt(9+8*p))/2); v!U#C[a^  
    m = 2*p - n.*(n+2); d=KOV;~);  
    c_a$g  
    % Pass the inputs to the function ZERNFUN: IP;@unBl  
    % ---------------------------------------- 7~Ga>BK  
    switch nargin .}<B*e=y  
        case 3 Q%xY/xH]  
            z = zernfun(n,m,r,theta); Mqtp}<*@-  
        case 4 #rr-4$w+  
            z = zernfun(n,m,r,theta,nflag); ,< icW &a  
        otherwise KJ)&(Yx  
            error('zernfun2:nargin','Incorrect number of inputs.')  ck~xj0  
    end ?^vZ{B)&0E  
    oX4uRc7wR  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) o {LFXNcg[  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 3p W MS&  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of !S-U8KI|  
    %   order N and frequency M, evaluated at R.  N is a vector of \W^+vuD8  
    %   positive integers (including 0), and M is a vector with the hFfaaB  
    %   same number of elements as N.  Each element k of M must be a |=T<WU1$  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) eBIR *TZ):  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is & }}WP:U  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix "'B%.a#k  
    %   with one column for every (N,M) pair, and one row for every a?~csP^?}  
    %   element in R. "\r~,S{:  
    % Jk`)`94 I  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- H`lD@q'S  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is i%4k5[f.:  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to :&)/vq  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 _lv:"/3R  
    %   for all [n,m]. ~"2@A F  
    % L_K\i?  
    %   The radial Zernike polynomials are the radial portion of the f\U&M,L\ '  
    %   Zernike functions, which are an orthogonal basis on the unit ]NV ]@*`tO  
    %   circle.  The series representation of the radial Zernike XuoEAu8]  
    %   polynomials is Wx)U<:^e  
    % &MZy;Sq  
    %          (n-m)/2 65#:2,s  
    %            __ Ca@=s  
    %    m      \       s                                          n-2s DE\bYxJ  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r i6#]$B  
    %    n      s=0 Df:7P>  
    % _<RR`  
    %   The following table shows the first 12 polynomials. #lshN,CPm  
    % h` h>H X  
    %       n    m    Zernike polynomial    Normalization q6JW@GT  
    %       --------------------------------------------- k;_KKvQ  
    %       0    0    1                        sqrt(2) 14n="-9  
    %       1    1    r                           2 sC5uA .?>9  
    %       2    0    2*r^2 - 1                sqrt(6) 3R%UPT0>  
    %       2    2    r^2                      sqrt(6) IgVo%)n  
    %       3    1    3*r^3 - 2*r              sqrt(8) w-H%B`/  
    %       3    3    r^3                      sqrt(8) %:w% o$  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) >[H&k8\7n  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) nz(q)"A  
    %       4    4    r^4                      sqrt(10) 1 73<x){  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Vv yrty  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) _@/C~  
    %       5    5    r^5                      sqrt(12) ^\[LrPq e  
    %       --------------------------------------------- wHOlj)CZ  
    % lQdnL.w$.4  
    %   Example: 88#qu.  
    % wAu[pWD'6;  
    %       % Display three example Zernike radial polynomials 2&gd"Ak(  
    %       r = 0:0.01:1; m +Q5vkW  
    %       n = [3 2 5]; dgco*TIGO  
    %       m = [1 2 1]; ^$=tcoQG  
    %       z = zernpol(n,m,r); uS,p|}Q&  
    %       figure 1C0Y0{6,  
    %       plot(r,z) ;V(H7 ZM  
    %       grid on #ox9&  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') JFG",09]  
    % kRN|TDx(  
    %   See also ZERNFUN, ZERNFUN2. C#Hcv*D  
    ?{ B[^  
    % A note on the algorithm. GSz @rDGY  
    % ------------------------ mW~i c  
    % The radial Zernike polynomials are computed using the series ptV4s=G2  
    % representation shown in the Help section above. For many special :cc[Jco@w  
    % functions, direct evaluation using the series representation can p-5P as  
    % produce poor numerical results (floating point errors), because 4? m/*VV  
    % the summation often involves computing small differences between (TEo_BW|+  
    % large successive terms in the series. (In such cases, the functions VLcyPM@"Q!  
    % are often evaluated using alternative methods such as recurrence ycgfZ 3K  
    % relations: see the Legendre functions, for example). For the Zernike -|m$YrzG  
    % polynomials, however, this problem does not arise, because the  d?:`n 9`  
    % polynomials are evaluated over the finite domain r = (0,1), and 3<c*v/L{C\  
    % because the coefficients for a given polynomial are generally all XnBm`vk?V!  
    % of similar magnitude. paW'R+Rck  
    % j#N(1}r=1  
    % ZERNPOL has been written using a vectorized implementation: multiple ;"1/#CY773  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] \Y*!f|=of  
    % values can be passed as inputs) for a vector of points R.  To achieve A)8rk_92Q  
    % this vectorization most efficiently, the algorithm in ZERNPOL o;3j:# 3 |  
    % involves pre-determining all the powers p of R that are required to ?E % +}P  
    % compute the outputs, and then compiling the {R^p} into a single K<c2PFo)Q  
    % matrix.  This avoids any redundant computation of the R^p, and 6 |QTS|!  
    % minimizes the sizes of certain intermediate variables. %fHH{60  
    % [ L  
    %   Paul Fricker 11/13/2006 }A-{6Qe  
    ,1,&b_  
    <BSSa`N`  
    % Check and prepare the inputs: u[)_^kIE(n  
    % ----------------------------- FtE90=$  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 4 bn t=5]  
        error('zernpol:NMvectors','N and M must be vectors.') ^[#=L4  
    end ?fV?|ZGZI  
    *m*`}9  
    if length(n)~=length(m) Q |%-9^  
        error('zernpol:NMlength','N and M must be the same length.') 1t.R+1[c  
    end "9w}dQ  
    mB~~_]M N  
    n = n(:);  7SaiS_{:  
    m = m(:); P/G>/MD/l  
    length_n = length(n); / $_M@>  
    (Izf L1  
    if any(mod(n-m,2)) H37Qg ApB  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') m^Glc?g<  
    end G{+2x N a(  
    Fwb5u!_,  
    if any(m<0) Tpb"uBiXoo  
        error('zernpol:Mpositive','All M must be positive.') -G-3q6A  
    end W!91tzs:  
    P3!Atnv2  
    if any(m>n) FIS "Z(  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') GWdSSr>  
    end |NJ}F@t/5  
    N'QqJe7Z  
    if any( r>1 | r<0 ) q_sEw~~@!  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') $Asr`Q1i   
    end {~'H  
    H:Y&OZ  
    if ~any(size(r)==1) :RiF3h(  
        error('zernpol:Rvector','R must be a vector.') e>AXXUEf  
    end bq NP#C  
    ~x#vZ=]8  
    r = r(:); <*wM=aq  
    length_r = length(r); \f%.n]>  
    $gvr -~  
    if nargin==4 .T8K-<R  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); v@n_F  
        if ~isnorm cU8xUpq  
            error('zernpol:normalization','Unrecognized normalization flag.') A gWPa.'3  
        end cG_Vc[  
    else q7_+}"i  
        isnorm = false; x)oRSsv!Tr  
    end 24B<[lSK  
    !MV@) (.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^B]@Lr E^  
    % Compute the Zernike Polynomials [Y/:@t"2y  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% kO_5|6  
    BC77<R!E)  
    % Determine the required powers of r: e fO jTA%  
    % ----------------------------------- +@:L|uFU  
    rpowers = []; {/ 2E*|W~I  
    for j = 1:length(n) \TbVS8e^  
        rpowers = [rpowers m(j):2:n(j)]; '*T]fND4  
    end /+[63=fl  
    rpowers = unique(rpowers); {_>em*Vb  
    SSA W52xC  
    % Pre-compute the values of r raised to the required powers, Tv$sqVe9  
    % and compile them in a matrix: i$uN4tVKT  
    % ----------------------------- qTmD '2  
    if rpowers(1)==0 1[PMDS_X  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ^}$O|t  
        rpowern = cat(2,rpowern{:}); 'EhBRU%  
        rpowern = [ones(length_r,1) rpowern]; VaLs`q&3>  
    else ]A+o>#n}x  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); JAjku6  
        rpowern = cat(2,rpowern{:}); ))T>jh   
    end j*R,m1e8  
    N4!<Xj  
    % Compute the values of the polynomials: MaY682}|y  
    % -------------------------------------- (< c7<_-H  
    z = zeros(length_r,length_n); '%;\YD9  
    for j = 1:length_n 5iX! lAFJ  
        s = 0:(n(j)-m(j))/2; ~r*P]*51x  
        pows = n(j):-2:m(j); LIpEQ7;  
        for k = length(s):-1:1 Dr#c)P~Wd  
            p = (1-2*mod(s(k),2))* ... |U' I/A  
                       prod(2:(n(j)-s(k)))/          ... ko\):DN  
                       prod(2:s(k))/                 ... [2c{k  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... AC- )BM';  
                       prod(2:((n(j)+m(j))/2-s(k))); X$n(-65  
            idx = (pows(k)==rpowers); i@P}{   
            z(:,j) = z(:,j) + p*rpowern(:,idx); ~Ci{3j :]  
        end !q"cpL'4  
         r6.d s^  
        if isnorm D-p.kA3MJ  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ~qP[eWe  
        end 5FeFN)  
    end o^p  
    Sv[5NZn0&  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  -_VG;$,jE  
    ewsKH\#  
    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。 vm23U^VJ  
    (LAXM x  
    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)