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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 \pjRv  
    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! 4JX`>a{<  
     
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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 `jD8(}_  
    function z = zernfun(n,m,r,theta,nflag) OqfhCNAY  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 4kW 30Ma  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N N0y;PVAGu  
    %   and angular frequency M, evaluated at positions (R,THETA) on the -XS+Uv  
    %   unit circle.  N is a vector of positive integers (including 0), and nUI63?  
    %   M is a vector with the same number of elements as N.  Each element Uv @!i0W  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) e.)yV'%L  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, J8sJ~FnUj  
    %   and THETA is a vector of angles.  R and THETA must have the same d1srV`  
    %   length.  The output Z is a matrix with one column for every (N,M) iQ]T+}nn_  
    %   pair, and one row for every (R,THETA) pair. 9TYw@o5V  
    % IqvqvHxLX  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike C?GvTc  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), B)j`}7O 06  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral [?|l X$<  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, tJ?qcT?  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 2 pM  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?V&Ld$db  
    % A&NC0K}G!  
    %   The Zernike functions are an orthogonal basis on the unit circle. R`Ys;g/!  
    %   They are used in disciplines such as astronomy, optics, and >cwJl@wx-  
    %   optometry to describe functions on a circular domain. ue6/EN;}  
    % rE1np^z7  
    %   The following table lists the first 15 Zernike functions. !uj!  
    % W,9k0t  
    %       n    m    Zernike function           Normalization X7XCZSh#A  
    %       -------------------------------------------------- [M7iJcwt  
    %       0    0    1                                 1 pz*/4  
    %       1    1    r * cos(theta)                    2 N3XVT{ yo  
    %       1   -1    r * sin(theta)                    2 ct2_N  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) mr{k>Un\  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ++J Bbuzj!  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) XhlI|h-j  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ZXssvjWQV}  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 7':5  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) *@bg/S K%  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) "xvV'&lQ  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) CI~hmL0  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) bGMeBj"R  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) C,OB3y  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) A:8FJ3'  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) SHXa{-  
    %       -------------------------------------------------- 7(A G]  
    % =FtM;(\  
    %   Example 1: ;3.T* ?|o  
    % V',m $   
    %       % Display the Zernike function Z(n=5,m=1) 4 BE:&A  
    %       x = -1:0.01:1; {Gk}3u/  
    %       [X,Y] = meshgrid(x,x); 8^P2GG'+-  
    %       [theta,r] = cart2pol(X,Y); ;*>QG6Fh  
    %       idx = r<=1; _-|yCo  
    %       z = nan(size(X)); xVHQ[I%  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ?vht~5'  
    %       figure +hgaBJy  
    %       pcolor(x,x,z), shading interp Pq{YZMr  
    %       axis square, colorbar 9AVK_   
    %       title('Zernike function Z_5^1(r,\theta)') DiGUxnP  
    % m &3HFf  
    %   Example 2: Sq?6R}q%  
    % 6?<`wGs(  
    %       % Display the first 10 Zernike functions }OX>(  
    %       x = -1:0.01:1; $X.'W\o|  
    %       [X,Y] = meshgrid(x,x); .=b +O~  
    %       [theta,r] = cart2pol(X,Y); XqE55Jclp  
    %       idx = r<=1; QRg"/62WCD  
    %       z = nan(size(X)); Y>dg10=  
    %       n = [0  1  1  2  2  2  3  3  3  3]; %CsTB0Y7n,  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; N) V7yo?  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 2t]! {L  
    %       y = zernfun(n,m,r(idx),theta(idx)); 9|G=KN)P:  
    %       figure('Units','normalized') 8,H#t@+MT  
    %       for k = 1:10 RBv=  
    %           z(idx) = y(:,k); 9sO{1rF  
    %           subplot(4,7,Nplot(k)) 0-t4+T  
    %           pcolor(x,x,z), shading interp R+ #.bQg  
    %           set(gca,'XTick',[],'YTick',[]) )K\k6HC.  
    %           axis square QX.F1T 2e?  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Be14$7r  
    %       end x%:> Ol  
    % HhQPgjZ/  
    %   See also ZERNPOL, ZERNFUN2. A\PV@w%A i  
    vU \w3  
    %   Paul Fricker 11/13/2006 !Lg}q!*%>V  
    g*w-"%"O  
    ]Gd]KP@S  
    % Check and prepare the inputs: V)?x*R*T)  
    % ----------------------------- 9TXm Z  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) d'g{K]=tF  
        error('zernfun:NMvectors','N and M must be vectors.') @=<TA0;LL  
    end $CQwBsYb=  
    `X.=uG+m  
    if length(n)~=length(m) d=+Lv<  
        error('zernfun:NMlength','N and M must be the same length.') rY_C3;B  
    end rfZj8R&  
    S}xDB  
    n = n(:); )Ido|!]0d  
    m = m(:); 1o6J9kCq^3  
    if any(mod(n-m,2)) 5f`XFe$8  
        error('zernfun:NMmultiplesof2', ... lA^Kh  
              'All N and M must differ by multiples of 2 (including 0).') HU'`kimWb  
    end 1Sc~Vb|>  
    ]BS{,sI  
    if any(m>n) {</$ObK  
        error('zernfun:MlessthanN', ... $RFu m'`5  
              'Each M must be less than or equal to its corresponding N.') dXK~ Z:  
    end PEQvEruZ}  
    nO.+&kA  
    if any( r>1 | r<0 ) Ci#5@Q9#w  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') \%4+mgiD  
    end C;:1CK  
    ~3-YxCn%  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) H R!>g  
        error('zernfun:RTHvector','R and THETA must be vectors.') 9:Z~}yX  
    end kV(DnZ#jq  
    sp |y/r#  
    r = r(:); k s`  
    theta = theta(:); r0$9c  
    length_r = length(r); @okm@6J*X  
    if length_r~=length(theta) g7Q*KA+  
        error('zernfun:RTHlength', ... "y ,(9_#  
              'The number of R- and THETA-values must be equal.') :;#}9g9  
    end hr}R,BR|  
    1oW]O@R  
    % Check normalization: kA :;c}p  
    % -------------------- zl8\jP  
    if nargin==5 && ischar(nflag) Y  X{  
        isnorm = strcmpi(nflag,'norm'); .LTFa.jxA  
        if ~isnorm KZ >"L  
            error('zernfun:normalization','Unrecognized normalization flag.') 0@/E% T1c"  
        end H2_>Av{m  
    else )I UWM  
        isnorm = false; au}0PnA;  
    end Hr,lA(  
    E#V-F-@2  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^l2d?v8  
    % Compute the Zernike Polynomials Qs[EA_  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% N{z(|2{A#  
    FEi,^V  
    % Determine the required powers of r: Y&Vbf>Hi+  
    % ----------------------------------- 7Hlh (k  
    m_abs = abs(m); K[;,/:Y  
    rpowers = []; VKfHN_m*  
    for j = 1:length(n) 3LnyQ  
        rpowers = [rpowers m_abs(j):2:n(j)]; Mw7UU1 ei  
    end j<-o{6r  
    rpowers = unique(rpowers); Jz8#88cY  
    ZC-evy  
    % Pre-compute the values of r raised to the required powers, o>rlrqr?_  
    % and compile them in a matrix: 8uD%]k=#!  
    % ----------------------------- oW1olmpp=  
    if rpowers(1)==0 eS%6 h U b  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); (>lqp%G~  
        rpowern = cat(2,rpowern{:}); CpdY)SMSL  
        rpowern = [ones(length_r,1) rpowern]; *8eh%3_$h  
    else v&,VC~RN-J  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); mb6?$1j  
        rpowern = cat(2,rpowern{:}); K>JU/(  
    end ,ui'^8{gK  
    ?-v?SN#  
    % Compute the values of the polynomials: ?B:wV?-`  
    % -------------------------------------- krY.Cc]  
    y = zeros(length_r,length(n)); =` >Nfa+,  
    for j = 1:length(n) bD[W~ku  
        s = 0:(n(j)-m_abs(j))/2; (=B7_jrl  
        pows = n(j):-2:m_abs(j); ?Lb7~XKt\  
        for k = length(s):-1:1 c@{^3V##T  
            p = (1-2*mod(s(k),2))* ... KFG^vmrn  
                       prod(2:(n(j)-s(k)))/              ... Vx8.FNJh  
                       prod(2:s(k))/                     ... TK?N^ly  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... `X03Q[:q"[  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); *jSc&{s~  
            idx = (pows(k)==rpowers); S5 vMP N  
            y(:,j) = y(:,j) + p*rpowern(:,idx); I{UB!0H  
        end I,Y^_(JW  
         QN5N h s  
        if isnorm RwHXn]1  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); g[)hm`{?  
        end u<r('IW0  
    end XE%6c3s  
    % END: Compute the Zernike Polynomials Z+Zh;Ms  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% rxA)&  
    ^Iq.0E9_  
    % Compute the Zernike functions: aV#;o9H{  
    % ------------------------------ "Z?":|%7  
    idx_pos = m>0; itMc!bUQ  
    idx_neg = m<0; } +Z;zm@/6  
    QZP;k!"w  
    z = y; 56aJE .?<  
    if any(idx_pos) [NDYJ'VGe  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ,fL e%RP  
    end G?(:Z=  
    if any(idx_neg) {D.0_=y~2  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); pMrf i}esx  
    end e+aQ$1^t  
    #?| z&9  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) {(Z1JoSl  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ^jC0S[csw2  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated |Q+v6r(<zZ  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive w-/Tb~#E  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, J#nEGl|a  
    %   and THETA is a vector of angles.  R and THETA must have the same Jc6 D^=  
    %   length.  The output Z is a matrix with one column for every P-value, |9JYg7<  
    %   and one row for every (R,THETA) pair. Xb;`WE gC  
    % L2tmo-]nw  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike IC42O_^  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) !qq@F%tv  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) SS-   
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 7MfvU|D[d/  
    %   for all p. ?+_"2XY  
    % )E|Bb=%  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 }])f^  
    %   Zernike functions (order N<=7).  In some disciplines it is 9bvd1bKEW  
    %   traditional to label the first 36 functions using a single mode :oQaN[3>_  
    %   number P instead of separate numbers for the order N and azimuthal bZJiubBRI  
    %   frequency M. 5$w1[}UUd  
    % JJa?"82FXZ  
    %   Example: $S/ 8T  
    % BC+qeocg  
    %       % Display the first 16 Zernike functions y3GIR f;>  
    %       x = -1:0.01:1; {^iV<>J  
    %       [X,Y] = meshgrid(x,x); bSzb! hT`  
    %       [theta,r] = cart2pol(X,Y); nwYeOa/t  
    %       idx = r<=1; Yzz8:n  
    %       p = 0:15; lnUy ? 0(  
    %       z = nan(size(X)); Z m>69gl  
    %       y = zernfun2(p,r(idx),theta(idx)); M,P_xkLp  
    %       figure('Units','normalized') }qg&2M%\  
    %       for k = 1:length(p) Pr"ESd>Y  
    %           z(idx) = y(:,k); <Do89  
    %           subplot(4,4,k) t@v8>J%K  
    %           pcolor(x,x,z), shading interp ([A;~ p;n  
    %           set(gca,'XTick',[],'YTick',[]) _\zf XHp  
    %           axis square !LA#c'  
    %           title(['Z_{' num2str(p(k)) '}']) lRq!|.C  
    %       end yDrJn* r^  
    % K(Nk|gQ  
    %   See also ZERNPOL, ZERNFUN. M~4!gKs  
    [;bLlS,  
    %   Paul Fricker 11/13/2006 OduTg^R  
    WJWrLu92\U  
    }I0^nv1  
    % Check and prepare the inputs: 'aJ?Syn  
    % ----------------------------- S3r\)5%;  
    if min(size(p))~=1 :A[/;|&  
        error('zernfun2:Pvector','Input P must be vector.') 1['A1 ,  
    end 0%GWc}o  
    6 s/O\A  
    if any(p)>35 6>Fw,$  
        error('zernfun2:P36', ... 6Xa2A 6  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... rv[\2@}  
               '(P = 0 to 35).']) R_&>iu'[  
    end t&0p@xLQ  
    0J" 3RTt  
    % Get the order and frequency corresonding to the function number: Ra5cfkH;  
    % ---------------------------------------------------------------- hG U &C]  
    p = p(:); JqO( ]*"Hi  
    n = ceil((-3+sqrt(9+8*p))/2); 2t'&7>Ys{  
    m = 2*p - n.*(n+2); w>e OERZa  
    ;-F#a+2]!  
    % Pass the inputs to the function ZERNFUN: , /pE*Yk  
    % ---------------------------------------- &N#)(rQ1  
    switch nargin @ NF8?>!  
        case 3 FWj~bn  
            z = zernfun(n,m,r,theta); =W6P>r_  
        case 4 YY9q'x,w  
            z = zernfun(n,m,r,theta,nflag); w;:,W@K  
        otherwise b({2|R  
            error('zernfun2:nargin','Incorrect number of inputs.') S70ERRk  
    end Jg:'gF]jt  
    [O3R(`<e5  
    % 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) REJ}T:  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. nD_g84us  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of biJU r^n  
    %   order N and frequency M, evaluated at R.  N is a vector of o8" [6Ys  
    %   positive integers (including 0), and M is a vector with the HTC7fS  
    %   same number of elements as N.  Each element k of M must be a .C1^QY-wL  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) myYe~f4=HQ  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is $?GF]BT  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix o)$sZ{` ="  
    %   with one column for every (N,M) pair, and one row for every i|<*EXB"  
    %   element in R. i-Z@6\/a5  
    % Vq*p?cF .  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- YC$pT  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is @cx!m   
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to b~|B(lL6Xm  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 -*WD.|k  
    %   for all [n,m]. 6 @d( <Z  
    % hZpFI?lqc\  
    %   The radial Zernike polynomials are the radial portion of the A4# m&o  
    %   Zernike functions, which are an orthogonal basis on the unit NzEuiI}  
    %   circle.  The series representation of the radial Zernike Nb$)YMbA  
    %   polynomials is %VdJ<=@  
    % :( `Q4D~l  
    %          (n-m)/2 d#(xP2  
    %            __ fhC=MJ @  
    %    m      \       s                                          n-2s f_ ::?  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r FnCHbPlb  
    %    n      s=0 *33Zt+  
    % w<'mV^S  
    %   The following table shows the first 12 polynomials. XW19hG  
    % q3;HfZ  
    %       n    m    Zernike polynomial    Normalization $FAl9  
    %       --------------------------------------------- 0$UE|yDs>  
    %       0    0    1                        sqrt(2) JeO(sj$e  
    %       1    1    r                           2 eVy,7goh  
    %       2    0    2*r^2 - 1                sqrt(6) v(af aN  
    %       2    2    r^2                      sqrt(6) rR7}SEa  
    %       3    1    3*r^3 - 2*r              sqrt(8) <mpkkCl,  
    %       3    3    r^3                      sqrt(8) 8\[6z0+;  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) s^ 6S{XJ  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ON$u581 y  
    %       4    4    r^4                      sqrt(10) 5r.{vQ  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) NZ Xmrc{S  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) $,R|$0B7  
    %       5    5    r^5                      sqrt(12) kY*D s;  
    %       --------------------------------------------- Y+D#Dv |  
    % iR_X,&p   
    %   Example: GI/g@RV  
    % ?&N JN/+%  
    %       % Display three example Zernike radial polynomials SL*B `P~{  
    %       r = 0:0.01:1; gHTo|2 Q{  
    %       n = [3 2 5]; lc*<UZR  
    %       m = [1 2 1]; f#[Fqkmj  
    %       z = zernpol(n,m,r); /N~.,vf  
    %       figure E")82I  
    %       plot(r,z) Fd3V5h  
    %       grid on VPf=LSxJe  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') or0f%wAF  
    % {| Tl3  
    %   See also ZERNFUN, ZERNFUN2. R7vO,kZ6Q  
    O7E0{8  
    % A note on the algorithm. * c xYB  
    % ------------------------ HogT#BMs  
    % The radial Zernike polynomials are computed using the series kMK-E<g  
    % representation shown in the Help section above. For many special / S]<MS  
    % functions, direct evaluation using the series representation can :]:q=1;c  
    % produce poor numerical results (floating point errors), because ,%Dn}mWu  
    % the summation often involves computing small differences between ]81P<Y(7  
    % large successive terms in the series. (In such cases, the functions @q|I$'K]x  
    % are often evaluated using alternative methods such as recurrence D;m>9{=  
    % relations: see the Legendre functions, for example). For the Zernike F(mm0:lT  
    % polynomials, however, this problem does not arise, because the I>:M1Yc0  
    % polynomials are evaluated over the finite domain r = (0,1), and q&7J1  
    % because the coefficients for a given polynomial are generally all Yf<6[(6 O  
    % of similar magnitude. |LWG7 ZE  
    % !}<Y^="  
    % ZERNPOL has been written using a vectorized implementation: multiple Ioj F/  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] IE,xiV  
    % values can be passed as inputs) for a vector of points R.  To achieve E7ixl~  
    % this vectorization most efficiently, the algorithm in ZERNPOL HPT$)NeNc  
    % involves pre-determining all the powers p of R that are required to ]H%y7kH8  
    % compute the outputs, and then compiling the {R^p} into a single -FdhV%5]  
    % matrix.  This avoids any redundant computation of the R^p, and 8eQ 4[wJY  
    % minimizes the sizes of certain intermediate variables. tKu'Q;J  
    % Y=\;$:L[  
    %   Paul Fricker 11/13/2006 bfhap(F~(e  
    P6@(nGgK<  
    r,aV11{  
    % Check and prepare the inputs: .r$d 8J  
    % ----------------------------- 9*U3uyPi  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) m&cVda/  
        error('zernpol:NMvectors','N and M must be vectors.') HvLvSy1U  
    end d%8hWlffz  
    rISg`-  
    if length(n)~=length(m) 6]1cy&SG  
        error('zernpol:NMlength','N and M must be the same length.') U TC|8  
    end  1ti+ Q0~  
    [HLXWu3  
    n = n(:); *\L\Bzm  
    m = m(:); 3%p^>D\  
    length_n = length(n); h`;w/+/Zr  
    OLg=kF[[  
    if any(mod(n-m,2)) #+>8gq^5  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') +a0q?$\  
    end TldqF BX  
    unY+/p $  
    if any(m<0) oF7o"NHaWa  
        error('zernpol:Mpositive','All M must be positive.') Db3# ;  
    end fq-e2MCX5  
    Yi:@>A<#  
    if any(m>n) jv_z%`  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') Xt& rYv  
    end Wo+fMn(O  
    8A}cxk  
    if any( r>1 | r<0 ) A 0~uv4MC  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') xy;u"JY*  
    end Y- esD'MD  
    > PHin%#  
    if ~any(size(r)==1) ^--kcTiR%  
        error('zernpol:Rvector','R must be a vector.') RzgA;ZC'  
    end H!PMb{e  
    Vz[tgb]-  
    r = r(:); :QGgtTEV""  
    length_r = length(r); -q'G]}  
    J$"3w,O6+U  
    if nargin==4 ny'?Hl'Q  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); AYb-BaIc  
        if ~isnorm l=4lhFG,Mk  
            error('zernpol:normalization','Unrecognized normalization flag.') +J [<zxh\  
        end $z[FL=h)?+  
    else JiH^N!  
        isnorm = false; p`N+9t&I4  
    end H;D 5)eJ90  
    IqD;*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% l4: B(  
    % Compute the Zernike Polynomials CvkZ<i){  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ' m^nKG$"  
    jTHgh>n  
    % Determine the required powers of r: X3mHg5zt  
    % ----------------------------------- T%q@jv{c  
    rpowers = []; wjEyU:  
    for j = 1:length(n) bSJ@ 5qS  
        rpowers = [rpowers m(j):2:n(j)]; v_G1YC7TU  
    end Fw.df<  
    rpowers = unique(rpowers); `|&#=hl~  
    V)<Jj  
    % Pre-compute the values of r raised to the required powers, I.dS-)Y  
    % and compile them in a matrix: Q7#Yw"#G!  
    % ----------------------------- }o,-@R~  
    if rpowers(1)==0 j3=%J5<  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); sH_B*cr3  
        rpowern = cat(2,rpowern{:}); 6~b)Hc/  
        rpowern = [ones(length_r,1) rpowern]; Rq`d I~5!b  
    else Nl$b;~ u  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); W *.j=?)\[  
        rpowern = cat(2,rpowern{:}); 6>Dm cG:.  
    end 1buVV]*~  
    !94qF,#1  
    % Compute the values of the polynomials: a*2JLK  
    % -------------------------------------- $Sls9H+.  
    z = zeros(length_r,length_n); 0Ba*"/U]t~  
    for j = 1:length_n 0^('hS&  
        s = 0:(n(j)-m(j))/2; aWS_z6[t#6  
        pows = n(j):-2:m(j); ,::f? Gc7j  
        for k = length(s):-1:1 z ?L]5m` H  
            p = (1-2*mod(s(k),2))* ... [%LIW%t|  
                       prod(2:(n(j)-s(k)))/          ... BZP{{  
                       prod(2:s(k))/                 ... $ VTk0J-W  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... JfLoGl;p m  
                       prod(2:((n(j)+m(j))/2-s(k))); ~8 S2BV3@  
            idx = (pows(k)==rpowers); K3dg.>O  
            z(:,j) = z(:,j) + p*rpowern(:,idx); P1G;JK  
        end &iI5^b-P  
         )=TS)C4  
        if isnorm *e,GXU@  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); O_ 4 j"0  
        end 89Ch'D  
    end \%/Y(YVm  
    T/$hN hQK  
    % 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)  WJF#+)P:Y  
    V7q-Pfh!y  
    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。 0zrZrl  
    #b5V/)K  
    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)