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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 G u-#wv5@  
    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! hIPU%  
     
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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 t3.;W/0_  
    function z = zernfun(n,m,r,theta,nflag) _ a|zvH  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. CfA^Xp@vc  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N R g7  O  
    %   and angular frequency M, evaluated at positions (R,THETA) on the {i)k#`  
    %   unit circle.  N is a vector of positive integers (including 0), and hTZaI*  
    %   M is a vector with the same number of elements as N.  Each element y_:i'Ri.  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) vlAYKtl3]  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, VQO6!ToKY  
    %   and THETA is a vector of angles.  R and THETA must have the same #`rvL6W q}  
    %   length.  The output Z is a matrix with one column for every (N,M) b/='M`D}#G  
    %   pair, and one row for every (R,THETA) pair. x8xSA*@k  
    % E=.4(J7K  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike qr[H0f]  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), z^to"j  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral pmR6(/B#  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, \e64Us>"x  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized o/bmS57  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ;H5PiSq;z  
    % Q<.84 7 )  
    %   The Zernike functions are an orthogonal basis on the unit circle. UGK4uK+I`  
    %   They are used in disciplines such as astronomy, optics, and V8w!yc  
    %   optometry to describe functions on a circular domain. 5"=qVmT)  
    % /(Se:jH$>  
    %   The following table lists the first 15 Zernike functions. pJ7M.C!  
    % 7KOM,FWKe  
    %       n    m    Zernike function           Normalization e$M \HPc  
    %       -------------------------------------------------- u/3 4E=  
    %       0    0    1                                 1 &)@|WLW  
    %       1    1    r * cos(theta)                    2 o;+$AU1f  
    %       1   -1    r * sin(theta)                    2 hiWfVz{~  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) E(F<shT#  
    %       2    0    (2*r^2 - 1)                    sqrt(3) V )CS,w  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) :!a'N3o>  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) C~IsYdln  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Zb<IZ)i#1  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) vs-%J 6}G  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ,C%fA>?UF8  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) <RfPd+</  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #;59THdtPk  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) pBV_'A}ioh  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) c|8[$_2  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) AvF:$ kG  
    %       -------------------------------------------------- M8 oCh  
    % dYdZt<6W<(  
    %   Example 1: `,XCD-R^  
    % d?G ~k[C!a  
    %       % Display the Zernike function Z(n=5,m=1) .}W#YN$  
    %       x = -1:0.01:1; m%Ah]x;  
    %       [X,Y] = meshgrid(x,x); 2JNO@  
    %       [theta,r] = cart2pol(X,Y); 9~8 A>  
    %       idx = r<=1; z DDvXz  
    %       z = nan(size(X)); Gzxq] Mg  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); bjvpYZC\5  
    %       figure v o vc,4}  
    %       pcolor(x,x,z), shading interp Uf#.b2]  
    %       axis square, colorbar R4+Gmx1  
    %       title('Zernike function Z_5^1(r,\theta)') 0F$;]zg  
    % 8zv=@`4@G  
    %   Example 2: cNX,%  
    % Ve,h]/G  
    %       % Display the first 10 Zernike functions >\=~2>FCD  
    %       x = -1:0.01:1; !;'#f xW[  
    %       [X,Y] = meshgrid(x,x); = WFn+#&^  
    %       [theta,r] = cart2pol(X,Y); q3a`Y)aVB  
    %       idx = r<=1; HAa2q=  
    %       z = nan(size(X)); _&!%yW@  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 6[g~p< 8n}  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 5ve4u  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 6(1xU\x  
    %       y = zernfun(n,m,r(idx),theta(idx)); f>$Ld1  
    %       figure('Units','normalized') [C)JI;\  
    %       for k = 1:10 ^MJTlRUb  
    %           z(idx) = y(:,k); u2=gG.  
    %           subplot(4,7,Nplot(k)) . C_\xb  
    %           pcolor(x,x,z), shading interp NHKIZx8sR  
    %           set(gca,'XTick',[],'YTick',[]) 7O6VnKl  
    %           axis square b'\a 4  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) sU>!sxW  
    %       end cR.[4rG'  
    % TG ,T>'   
    %   See also ZERNPOL, ZERNFUN2. |BrD:+  
    e_3KNQ`kA  
    %   Paul Fricker 11/13/2006 r?Y+TtF\e  
    NPjh2 AJm  
    &^WJ:BvA|^  
    % Check and prepare the inputs: |)'gQvDM  
    % ----------------------------- Z Z1s}TG  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 2w>l nJ-  
        error('zernfun:NMvectors','N and M must be vectors.') " jefB6k9h  
    end xi5/Wc6  
    6n9;t\'Gt  
    if length(n)~=length(m) P $4h_dw  
        error('zernfun:NMlength','N and M must be the same length.') pyPS5vWG  
    end qkX}pQkG)h  
    OE,uw2uaT  
    n = n(:); V&)lS Qw  
    m = m(:); XAN{uD^3\%  
    if any(mod(n-m,2)) v/%q*6@  
        error('zernfun:NMmultiplesof2', ... E8]PV,#xY  
              'All N and M must differ by multiples of 2 (including 0).') UPtWj8h  
    end y?BzZ16\bL  
    Jz(!eTVs  
    if any(m>n) Mv9q-SIc[  
        error('zernfun:MlessthanN', ... `V N $ S  
              'Each M must be less than or equal to its corresponding N.') (GnwK1f  
    end 7ky$9+~  
    rx^vh%/ Q!  
    if any( r>1 | r<0 ) IEb"tsel  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') }Ip"j]h  
    end **I9Nw!IH  
    fneg[K  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) XxT7YCi  
        error('zernfun:RTHvector','R and THETA must be vectors.') '8g/^Y@  
    end .;gK*`G2W)  
    ^Pc>/lY$Q%  
    r = r(:); .f'iod-   
    theta = theta(:); !6:q#B*  
    length_r = length(r); %\=oy=f  
    if length_r~=length(theta) p_hljgOV  
        error('zernfun:RTHlength', ... [oOA@  
              'The number of R- and THETA-values must be equal.') 5u ED  
    end ^/+0L[R  
    >-0b@ +j  
    % Check normalization: 3HsjF5?W  
    % -------------------- phIEz3Fu/  
    if nargin==5 && ischar(nflag) f3h&K}x  
        isnorm = strcmpi(nflag,'norm'); ns.[PJ"8  
        if ~isnorm A:"J&TbBx  
            error('zernfun:normalization','Unrecognized normalization flag.') )r O`K  
        end )N QtjB$  
    else a7G0  
        isnorm = false; dvUBuY^[  
    end l 6.#s3I['  
    ]Y[8|HJ8  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [0GM!3YJ7  
    % Compute the Zernike Polynomials _q([k_4h  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )=\W sQ  
    rN|c0N  
    % Determine the required powers of r: EXz5Rue LV  
    % ----------------------------------- tK&.0)*=  
    m_abs = abs(m); LX<c(i  
    rpowers = []; 0D1yG(ck  
    for j = 1:length(n) Xq&x<td  
        rpowers = [rpowers m_abs(j):2:n(j)]; t;+6>sTu  
    end NEQcEUd?  
    rpowers = unique(rpowers); K[LTw_oE  
    5*1wQlL  
    % Pre-compute the values of r raised to the required powers, .rj FhSr$  
    % and compile them in a matrix: H[%F o  
    % ----------------------------- 6l#1E#]|  
    if rpowers(1)==0 @]f"X>  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ]?F05!$*  
        rpowern = cat(2,rpowern{:}); "r0z( j  
        rpowern = [ones(length_r,1) rpowern]; ~B%EvG7:n  
    else |7Z,z0 ?V  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); maLJ M\C  
        rpowern = cat(2,rpowern{:}); iL1.R+  
    end {+[~;ISL  
    =$5[uI2  
    % Compute the values of the polynomials: uPe4Rr  
    % -------------------------------------- 96F:%|yG  
    y = zeros(length_r,length(n)); o}5:vi]  
    for j = 1:length(n) 4'rWy~` V  
        s = 0:(n(j)-m_abs(j))/2; yy?|q0  
        pows = n(j):-2:m_abs(j); 1Qf21oN{  
        for k = length(s):-1:1 K@VXFV  
            p = (1-2*mod(s(k),2))* ... @M4~,O6-  
                       prod(2:(n(j)-s(k)))/              ... s<qSelj  
                       prod(2:s(k))/                     ... CGg:e:4  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... K G~](4JE(  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); h~elF1dG  
            idx = (pows(k)==rpowers); $X5~9s1Wl  
            y(:,j) = y(:,j) + p*rpowern(:,idx);  L}AR{  
        end 0c1}?$f[?%  
         kETA3(h'  
        if isnorm SPsq][5eR  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); .]ZM2  
        end (?R  
    end n:he`7.6O  
    % END: Compute the Zernike Polynomials UA,&0.7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5S7`gN.  
    iyOd&|.  
    % Compute the Zernike functions: 'KQ]7  
    % ------------------------------ *6*#"#D  
    idx_pos = m>0; Wnl8XHPn  
    idx_neg = m<0; 6u7 (}K  
    !N,Z3p>Q  
    z = y; U^Z[6u  
    if any(idx_pos) N(&FATZUW  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); /db?ltb  
    end D4'? V Iz  
    if any(idx_neg) ao{>.b  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 8)rv.'A((E  
    end t@.gmUUA  
    EyNI]XEj  
    % 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) `r*6P^P  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Y1k/ngH  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated #EHBS~^  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive YGWb!|Z$  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, *~!xeL  
    %   and THETA is a vector of angles.  R and THETA must have the same IU$bP#<  
    %   length.  The output Z is a matrix with one column for every P-value, C2<y(GU[Bh  
    %   and one row for every (R,THETA) pair. f=K1ZD  
    % +crAkb}i  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike IJ4"X#Q/  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) MCh8Q|Yx4  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) a+{g~/z;,Q  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 WP]<\_r2  
    %   for all p. =AD/5E,3  
    % )sV# b  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 T@yH. 4D  
    %   Zernike functions (order N<=7).  In some disciplines it is (la<X <w  
    %   traditional to label the first 36 functions using a single mode \=N tbBL$[  
    %   number P instead of separate numbers for the order N and azimuthal ~Y'e1w$`  
    %   frequency M. 2jhVmK  
    % K,IOD t  
    %   Example: &]LwK5SR  
    % 5?O/Aub  
    %       % Display the first 16 Zernike functions Vj2GK"$v  
    %       x = -1:0.01:1; =rBFMTllM  
    %       [X,Y] = meshgrid(x,x); ,x Tbt4J  
    %       [theta,r] = cart2pol(X,Y); #Ejly2C,  
    %       idx = r<=1; vhL&az  
    %       p = 0:15; 5)nv  
    %       z = nan(size(X)); NWAF4i&$  
    %       y = zernfun2(p,r(idx),theta(idx)); EPa3Yb?BGb  
    %       figure('Units','normalized') Y43#];  
    %       for k = 1:length(p) mCz6&  
    %           z(idx) = y(:,k); a^=4 '.ok  
    %           subplot(4,4,k) chd${ j  
    %           pcolor(x,x,z), shading interp pM46I"  
    %           set(gca,'XTick',[],'YTick',[]) VBH[aIW  
    %           axis square 3Aj_,&X.@(  
    %           title(['Z_{' num2str(p(k)) '}']) . \d0lJSr  
    %       end ?ft_  
    % 6U&Uyd)  
    %   See also ZERNPOL, ZERNFUN. Iyc')\W&  
    4X\*kF%  
    %   Paul Fricker 11/13/2006 3P6'*pZ  
    #>- rKv.A  
    ^m+W  
    % Check and prepare the inputs: f%|S>(   
    % ----------------------------- W^Rb~b^?  
    if min(size(p))~=1 YAPD7hA  
        error('zernfun2:Pvector','Input P must be vector.') _yoG<qI  
    end QE#$bCw  
    'D/AL\1{p(  
    if any(p)>35 <>e<Xd:77{  
        error('zernfun2:P36', ... D/:)rj14b  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... }w-`J5Eq#  
               '(P = 0 to 35).']) AN%.LK  
    end Dw #&x/G  
    TdE_\gEo/R  
    % Get the order and frequency corresonding to the function number: c|.:J]  
    % ---------------------------------------------------------------- YpKai3 B  
    p = p(:); xw-q)u  
    n = ceil((-3+sqrt(9+8*p))/2); >WDpBn:  
    m = 2*p - n.*(n+2); uc<@ Fh(  
    )m'_>-`^:  
    % Pass the inputs to the function ZERNFUN: <+b:  
    % ---------------------------------------- /,>.${,;u  
    switch nargin ]OrFW4tiE  
        case 3 z8!u6odu %  
            z = zernfun(n,m,r,theta); i4pJIb  
        case 4 *Qf }4a0  
            z = zernfun(n,m,r,theta,nflag); YiJu48J  
        otherwise <R(2 9QN  
            error('zernfun2:nargin','Incorrect number of inputs.') P X0#X=$  
    end lC4By,1*  
    EK#m?O:>  
    % 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) B1M/5cr.  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 0/6&2  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of uqUo4z5T  
    %   order N and frequency M, evaluated at R.  N is a vector of v wyDY%B"n  
    %   positive integers (including 0), and M is a vector with the s z\RmX  
    %   same number of elements as N.  Each element k of M must be a =c,gK8C  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) [5VUcXGt*\  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 3DHm9n+/:  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix m?j!0>  
    %   with one column for every (N,M) pair, and one row for every w!}1oy  
    %   element in R. BQU5[8l  
    % *Vl =PNn-  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ;Wa{q.)  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is LasH[:QQQ  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to e%U*~{m+  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 zK 2wLX  
    %   for all [n,m]. *[H+8/n_  
    % F^YIZ,=p!  
    %   The radial Zernike polynomials are the radial portion of the ]~^/w}(K  
    %   Zernike functions, which are an orthogonal basis on the unit nIjQLx  
    %   circle.  The series representation of the radial Zernike 9NP l]iA)  
    %   polynomials is !Ia"pNDf  
    % ;*2e;m~)?  
    %          (n-m)/2 dM UDLr-  
    %            __ VTWE-:r  
    %    m      \       s                                          n-2s #%9]Lq  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 2eU[*x  
    %    n      s=0 lX*;KHT)  
    % m GhJn  
    %   The following table shows the first 12 polynomials. tFaE cP  
    % qN[7zsaj  
    %       n    m    Zernike polynomial    Normalization :\U3bkv+  
    %       --------------------------------------------- H g`{9v  
    %       0    0    1                        sqrt(2) H/k W :k  
    %       1    1    r                           2 .$0Ob<.  
    %       2    0    2*r^2 - 1                sqrt(6) YfwJBz D  
    %       2    2    r^2                      sqrt(6) C._sgO  
    %       3    1    3*r^3 - 2*r              sqrt(8) [M^[61  
    %       3    3    r^3                      sqrt(8) EYxRw  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 4Aew )   
    %       4    2    4*r^4 - 3*r^2            sqrt(10) @;fE%N  
    %       4    4    r^4                      sqrt(10) pRvs;klf  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) LUNs|\&  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ;_/q>DR>,3  
    %       5    5    r^5                      sqrt(12) b 0b9#9x  
    %       --------------------------------------------- Z9&D'n)  
    % B)]{]z0+`  
    %   Example: Qe$>Jv5  
    % K$cIVsfr  
    %       % Display three example Zernike radial polynomials " ^baiN@ac  
    %       r = 0:0.01:1; mRH]'d lD7  
    %       n = [3 2 5]; .w_`d'}  
    %       m = [1 2 1]; 7J;~ &x  
    %       z = zernpol(n,m,r); ^<\} Y  
    %       figure _IV@^v  
    %       plot(r,z) `b")Bx|  
    %       grid on dB+GTq=6f  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') p@Y$eZ:O  
    % c^Jgr(Ow  
    %   See also ZERNFUN, ZERNFUN2. fq'Xy9L  
    AC:s4iacC  
    % A note on the algorithm. ` G=L07  
    % ------------------------ 'ZH<g8:=@  
    % The radial Zernike polynomials are computed using the series Tn0l|GRuZA  
    % representation shown in the Help section above. For many special W)!{U(X  
    % functions, direct evaluation using the series representation can ^:!(jiH  
    % produce poor numerical results (floating point errors), because $U1kP?pR  
    % the summation often involves computing small differences between lF8 dRIav  
    % large successive terms in the series. (In such cases, the functions Rca Os  
    % are often evaluated using alternative methods such as recurrence \678Nx  
    % relations: see the Legendre functions, for example). For the Zernike @k+G Cf  
    % polynomials, however, this problem does not arise, because the %?oU{KzQ@;  
    % polynomials are evaluated over the finite domain r = (0,1), and L-C/Luws  
    % because the coefficients for a given polynomial are generally all 4A/,X>W61  
    % of similar magnitude. 2^ bpH%  
    % NhK(HTsvK  
    % ZERNPOL has been written using a vectorized implementation: multiple As'M3 9*V  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 4@]xn  
    % values can be passed as inputs) for a vector of points R.  To achieve c =N]! ,MO  
    % this vectorization most efficiently, the algorithm in ZERNPOL *_<*bhR<  
    % involves pre-determining all the powers p of R that are required to to!W={S<ol  
    % compute the outputs, and then compiling the {R^p} into a single {9Mdt`WL  
    % matrix.  This avoids any redundant computation of the R^p, and mu6xL QdA  
    % minimizes the sizes of certain intermediate variables. vvsNWA  
    % \+Rwm:lI  
    %   Paul Fricker 11/13/2006 Kt 90mA  
    R_Gq8t$  
    8<-oJs_o+  
    % Check and prepare the inputs: (L0 hS'  
    % ----------------------------- {#*?S>DA  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) CoUd16*"JM  
        error('zernpol:NMvectors','N and M must be vectors.') wEfz2Eq  
    end (: TGev  
    9{%g-u \  
    if length(n)~=length(m) %V%#y $l  
        error('zernpol:NMlength','N and M must be the same length.') r$r&4d Y  
    end bN4&\d*u#  
    &Ev]x2YC  
    n = n(:); < k+fKl  
    m = m(:); 2rj/wakd  
    length_n = length(n); CS;W)F  
    QFNz9c  
    if any(mod(n-m,2)) moVa'1ul  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') }&(E#*>x  
    end Q}Ze-JIL$  
     w{ r(F`  
    if any(m<0) W\>^[c/  
        error('zernpol:Mpositive','All M must be positive.') [)H,zpl  
    end urrO1  
    \ L9?69B~  
    if any(m>n) A)RI:?+  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') T`g?)/  
    end  &6\r  
    ?79SPp)oo  
    if any( r>1 | r<0 ) miG; ]-"^  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 17 i<4f#  
    end v#:+n+y\z  
    ~"JE![XR  
    if ~any(size(r)==1) qP;{3FSkAF  
        error('zernpol:Rvector','R must be a vector.') ^fE\S5P  
    end &N/|(<CB  
    . QQ?w  
    r = r(:); =M1a0i|d  
    length_r = length(r); u+mjguIv  
    BP[CR1Gs  
    if nargin==4 yhSk"e'G  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); V4`:Vci Aw  
        if ~isnorm >}<29Ii  
            error('zernpol:normalization','Unrecognized normalization flag.') I 9{40_  
        end :jk)(=^  
    else 3@PUg(M  
        isnorm = false; 3R<ME c  
    end jU]]:S4xD/  
    u#41osUVW>  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DnNt@e2|  
    % Compute the Zernike Polynomials d,Cz-.'sOf  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% KFTf~!|  
    < v|%K.yd  
    % Determine the required powers of r: zj~nnfoys  
    % ----------------------------------- $f<eq7rRe  
    rpowers = []; "ibK1}-  
    for j = 1:length(n) s6uF5]M;2  
        rpowers = [rpowers m(j):2:n(j)]; g[#k.CuP  
    end zB#_:(1qK  
    rpowers = unique(rpowers); .)i O Du  
    iUeV5cB  
    % Pre-compute the values of r raised to the required powers, WZ^{zFoZ  
    % and compile them in a matrix: }mKGuCoH>  
    % ----------------------------- U8-Q'1IT&  
    if rpowers(1)==0 d98))G~W  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); yxaT7Oqh%  
        rpowern = cat(2,rpowern{:}); ZR!cQ oV=  
        rpowern = [ones(length_r,1) rpowern]; |MTpU@`p5  
    else F^.om2V|9  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Q3'fz 9v  
        rpowern = cat(2,rpowern{:}); +"k.E x0:  
    end aL8p"iSG9  
    TpJg-F  
    % Compute the values of the polynomials: cr76cYq"Q  
    % -------------------------------------- rQ`\JE&`  
    z = zeros(length_r,length_n); RDUT3H6~  
    for j = 1:length_n E|HSwTHe  
        s = 0:(n(j)-m(j))/2; 5~@?>)TBv  
        pows = n(j):-2:m(j); o2;(VSKhS  
        for k = length(s):-1:1 p//T7r s  
            p = (1-2*mod(s(k),2))* ... lo cW_/  
                       prod(2:(n(j)-s(k)))/          ... :d;[DYFLxb  
                       prod(2:s(k))/                 ... <\ y!3;  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... u|(Ux~O  
                       prod(2:((n(j)+m(j))/2-s(k))); J+{Ou rWt  
            idx = (pows(k)==rpowers); ]YCPyc:  
            z(:,j) = z(:,j) + p*rpowern(:,idx); $T"h";M)s  
        end _^_5K(Uq  
         h7bPAW=(  
        if isnorm <_"B}c/2$  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); %9t{Z1$  
        end f"vk# 3  
    end _,DO~L  
    } sf YCz  
    % 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)  g)R1ObpZ  
    RlRs}yF  
    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。 |4mVT&63(  
    ,lm=M 5b  
    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)