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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 V<A$eb>6  
    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! [:BD9V  
     
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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 6Z#$(oC  
    function z = zernfun(n,m,r,theta,nflag) !O 0{ .k  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 9)*218.  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ) #l&BV5  
    %   and angular frequency M, evaluated at positions (R,THETA) on the tjg?zlj  
    %   unit circle.  N is a vector of positive integers (including 0), and M(U<H;Csk  
    %   M is a vector with the same number of elements as N.  Each element @j<Q2z^  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) QAzwNXE+  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, z 4qEC  
    %   and THETA is a vector of angles.  R and THETA must have the same hw({>cH\  
    %   length.  The output Z is a matrix with one column for every (N,M) v\2- %  
    %   pair, and one row for every (R,THETA) pair. QV[#^1  
    % $d*PY_  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike *X /i<  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), <nU8.?\?~  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ?0tm{qP  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, :MihVLF  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized RxE.t[  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?*^HZ~O1  
    % t{-*@8Ke  
    %   The Zernike functions are an orthogonal basis on the unit circle. |OiM(E(  
    %   They are used in disciplines such as astronomy, optics, and x~QZVL=:  
    %   optometry to describe functions on a circular domain. jG`,k*eUrJ  
    % a0&L,7mu<'  
    %   The following table lists the first 15 Zernike functions. $ftxid8  
    % _BoYy JQH  
    %       n    m    Zernike function           Normalization w0X})&,{`m  
    %       -------------------------------------------------- HX{K5+  
    %       0    0    1                                 1 F~sUfqiJ'  
    %       1    1    r * cos(theta)                    2 #T=e p0  
    %       1   -1    r * sin(theta)                    2 q 7-ZPX  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ;}H*|"z;!  
    %       2    0    (2*r^2 - 1)                    sqrt(3) VG_xNM  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 4_-L1WH  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) q"i]&dMr  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) /@64xrvIl=  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ~t1?oJ  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 9-Z ?  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Vn65:" O  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) NJCSo(O  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) v7/k0D .  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ; +1ooeU  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) wf_ $#.;m  
    %       -------------------------------------------------- A=sz8?K+`  
    % NiYT%K%  
    %   Example 1: y&V%xE/  
    % <v!jS=T  
    %       % Display the Zernike function Z(n=5,m=1) pVM1%n:#  
    %       x = -1:0.01:1; :F_>`{  
    %       [X,Y] = meshgrid(x,x); ZnBGNr  
    %       [theta,r] = cart2pol(X,Y); i|rCGa0}  
    %       idx = r<=1; V 4&a+MJ@  
    %       z = nan(size(X)); ibn\&}1  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); \5-Dp9vG  
    %       figure Aho-\9/x%  
    %       pcolor(x,x,z), shading interp w"O{@2B3:H  
    %       axis square, colorbar LLL;SNY  
    %       title('Zernike function Z_5^1(r,\theta)') D&x.io  
    % M8IU[Pz4  
    %   Example 2: a ?\:,5=  
    % 6~l+wu<$  
    %       % Display the first 10 Zernike functions 6tGF  
    %       x = -1:0.01:1; 22*~CIh~x  
    %       [X,Y] = meshgrid(x,x); .Fx3WryF  
    %       [theta,r] = cart2pol(X,Y); >2v<;.  
    %       idx = r<=1; d@tf+_Ih  
    %       z = nan(size(X)); Y$#6%`*#>n  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Tb!FO"o  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; $b[Ha{9(v  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; uPC(|U%  
    %       y = zernfun(n,m,r(idx),theta(idx)); 5j v*C]z  
    %       figure('Units','normalized') Fkg%_v$  
    %       for k = 1:10 9fWR8iV  
    %           z(idx) = y(:,k); RXo6y(^  
    %           subplot(4,7,Nplot(k)) uqD|j:~ =k  
    %           pcolor(x,x,z), shading interp QQ=Kj%R  
    %           set(gca,'XTick',[],'YTick',[]) 1,7 }ah_  
    %           axis square I%b5a`7  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 2.^CIJc  
    %       end x|gYxZ  
    % 2PSkLS&IM  
    %   See also ZERNPOL, ZERNFUN2. O`I}Lg]~q  
    ~pHuh#>  
    %   Paul Fricker 11/13/2006 f\r"7j  
    G.$KP  
    O0s,)8+z5D  
    % Check and prepare the inputs: }=JS d@`_  
    % ----------------------------- o+L [o_er  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) S;u.Ds&  
        error('zernfun:NMvectors','N and M must be vectors.') $$SJLV  
    end J*_^~t  
    \6bvk _  
    if length(n)~=length(m) +_25E.>ml  
        error('zernfun:NMlength','N and M must be the same length.') JDW/Mc1bh  
    end ^/cqE[V~,  
    ?B@3A)a  
    n = n(:); t 1~k+  
    m = m(:); v V;]?  
    if any(mod(n-m,2)) $Ld-lQsL  
        error('zernfun:NMmultiplesof2', ... k2fJ  
              'All N and M must differ by multiples of 2 (including 0).') "a(e2H2&T4  
    end }{kn/m/  
    FS!9 j8  
    if any(m>n) &g>M Z" Z|  
        error('zernfun:MlessthanN', ... ';}:*nZ//_  
              'Each M must be less than or equal to its corresponding N.') vE1:;%Q  
    end B.KK@  
    Spu;   
    if any( r>1 | r<0 ) 0d+b<J,  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') #DARZhU)  
    end \t`VqJLyu  
    4E_u.tJ  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) t~)4f.F:  
        error('zernfun:RTHvector','R and THETA must be vectors.') n*i&o;5  
    end `t44.=%  
    i[MBO`FF  
    r = r(:); ,1cpV|mAr  
    theta = theta(:); -0BxZ AW=  
    length_r = length(r);  !VXy67  
    if length_r~=length(theta) JG&E"j#q  
        error('zernfun:RTHlength', ... kM@e_YtpY  
              'The number of R- and THETA-values must be equal.')  *M$mAy<  
    end N"tX K  
    I2pE}6q  
    % Check normalization: Dx=RLiU9  
    % -------------------- 0M)\([W9&  
    if nargin==5 && ischar(nflag) 2pv by`P4  
        isnorm = strcmpi(nflag,'norm'); ,7Ejb++/M,  
        if ~isnorm Yakrsi/jV}  
            error('zernfun:normalization','Unrecognized normalization flag.') 1 <m.Q*  
        end t:P7ah  
    else }'86hnW  
        isnorm = false; Jr%F#/  
    end h?h)i>  
    }}u`*&,g  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mkPqxzxbrL  
    % Compute the Zernike Polynomials >e(@!\ x  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% O_GHvLO=  
    gwsOw [;k  
    % Determine the required powers of r: L>&{<M_  
    % ----------------------------------- k {vd1,HZ  
    m_abs = abs(m); IP-M)_I  
    rpowers = []; -e?n4YO*\  
    for j = 1:length(n) [6 "5  
        rpowers = [rpowers m_abs(j):2:n(j)]; N})vrB;1  
    end @HnahD  
    rpowers = unique(rpowers); x^i97dZS^"  
    &U CtyCz  
    % Pre-compute the values of r raised to the required powers, ~|"uuA1/#O  
    % and compile them in a matrix: qsN_EMgbdn  
    % ----------------------------- m6H+4@Z-;(  
    if rpowers(1)==0 :8hXkQ  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); R?,v:S&i7;  
        rpowern = cat(2,rpowern{:}); gNZ"Kr o6  
        rpowern = [ones(length_r,1) rpowern]; O'xp"e,  
    else wuxOFlrg  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); %KN2iNq  
        rpowern = cat(2,rpowern{:}); ]/3!t=La  
    end f_;tFP B  
    m*h O@M  
    % Compute the values of the polynomials: ^vv 1cft  
    % -------------------------------------- PI9aKNt  
    y = zeros(length_r,length(n)); cVarvueS  
    for j = 1:length(n) (lq%4h  
        s = 0:(n(j)-m_abs(j))/2; tNOOaj9mw  
        pows = n(j):-2:m_abs(j); 7:=(yBG  
        for k = length(s):-1:1 7L6^IK  
            p = (1-2*mod(s(k),2))* ... MSp) Jc  
                       prod(2:(n(j)-s(k)))/              ... 7|bBC+;(  
                       prod(2:s(k))/                     ... u[4h|*'"|  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... NXz/1ut%  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); "(~fl<;  
            idx = (pows(k)==rpowers); 8/y8tMm]  
            y(:,j) = y(:,j) + p*rpowern(:,idx); :uqEGnEut  
        end G9#3 |B-?  
         M\Wg|gpy  
        if isnorm teLZplC=f  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); E0aFHC[  
        end { i4`- w  
    end : Q2=t!  
    % END: Compute the Zernike Polynomials [Z;H= `  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3RD+;^}q 3  
    Nr"GxezU+A  
    % Compute the Zernike functions: (y\.uPu!  
    % ------------------------------ )(1tDQ`L>  
    idx_pos = m>0; *_Ih@f H  
    idx_neg = m<0; vfVF^ WOd  
    \q^ dhY>)  
    z = y; <h<_''+  
    if any(idx_pos) [iyhrc:@  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); =%u=ma;  
    end B{S^t\T$  
    if any(idx_neg) 31%3&B:Ts  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); onS4ZE3B  
    end }XRfHQk  
    :; La V  
    % 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) 1pN8,[hyR7  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. YW8Odm  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated EIg:@o&Jj  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive n^|7ycB'  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, <BBSC  
    %   and THETA is a vector of angles.  R and THETA must have the same ,W_".aguX  
    %   length.  The output Z is a matrix with one column for every P-value, bQu@.'O!k  
    %   and one row for every (R,THETA) pair. 0f5)]  
    % JxRn)D  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 0nR_I^  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) \@^` G  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) :/fT8KCwo  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 cz$*6P<9J  
    %   for all p. `{}DLaD9  
    % _gCi@uXS3  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 e4.G9(  
    %   Zernike functions (order N<=7).  In some disciplines it is BG]|iHi  
    %   traditional to label the first 36 functions using a single mode COH>B1W@  
    %   number P instead of separate numbers for the order N and azimuthal xR&Le/3+  
    %   frequency M. Dk g-y9  
    % nEik;hAz  
    %   Example: 99b"WH^3$y  
    % iTCY $)J  
    %       % Display the first 16 Zernike functions C}:_&^DQ  
    %       x = -1:0.01:1; ~(^?M  
    %       [X,Y] = meshgrid(x,x); ^Uik{x  
    %       [theta,r] = cart2pol(X,Y); }. V!|R,  
    %       idx = r<=1; 3nUC,T%  
    %       p = 0:15; N_VWA.JHt  
    %       z = nan(size(X)); 8J2U UVA`1  
    %       y = zernfun2(p,r(idx),theta(idx)); zogl2e+  
    %       figure('Units','normalized')  '^,|8A2  
    %       for k = 1:length(p) -TNb=2en(  
    %           z(idx) = y(:,k); =~k#<q1^  
    %           subplot(4,4,k) l<s6Uu"  
    %           pcolor(x,x,z), shading interp KFM)*Icg\8  
    %           set(gca,'XTick',[],'YTick',[]) xK_0@6  
    %           axis square difAQ<`  
    %           title(['Z_{' num2str(p(k)) '}']) !q^2| %  
    %       end j$z!kd+%  
    % ):5H,B+Vr&  
    %   See also ZERNPOL, ZERNFUN. ^4a|gc  
    1?".R]<{2T  
    %   Paul Fricker 11/13/2006 uCf _O~  
    $~1~+s0$  
    vUJQ<D  
    % Check and prepare the inputs: ;n/04z  
    % ----------------------------- ygqWy1C  
    if min(size(p))~=1 Mqmy*m[U  
        error('zernfun2:Pvector','Input P must be vector.') 'L veCi_  
    end /)XN^Jwa;m  
    VyOpPIP  
    if any(p)>35 l3YS_WBSn  
        error('zernfun2:P36', ... ;2,Q:&`   
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 5?Rzyfwk|  
               '(P = 0 to 35).']) 5 r&n  
    end  TsI%M  
    p9*Ak U&]  
    % Get the order and frequency corresonding to the function number: *<ww~^a  
    % ---------------------------------------------------------------- -Dr)+Y  
    p = p(:); .^[{~#Pc*  
    n = ceil((-3+sqrt(9+8*p))/2); p}yp!(l  
    m = 2*p - n.*(n+2); 1&utf0TX6q  
    ""_%u'7t5I  
    % Pass the inputs to the function ZERNFUN: %d#j%=  
    % ---------------------------------------- z_t%n<OvK  
    switch nargin rL6Y4u0e%  
        case 3 w\5;;9_#  
            z = zernfun(n,m,r,theta); _4X3g%nXl  
        case 4 h@D!/PS  
            z = zernfun(n,m,r,theta,nflag); zd {\XW  
        otherwise aHSl_[  
            error('zernfun2:nargin','Incorrect number of inputs.') 4cJka~  
    end |j!U/n.%w  
    t ;bU#THM  
    % 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) w8(z\G_0  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. FYX" q-Z  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of fwz-)?   
    %   order N and frequency M, evaluated at R.  N is a vector of YG#.L}X@C  
    %   positive integers (including 0), and M is a vector with the 9wpV} .(  
    %   same number of elements as N.  Each element k of M must be a W$Op/  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 1 ac;6`  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is B1(T-pr  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix ^(T_rEp  
    %   with one column for every (N,M) pair, and one row for every 'qiDh[ATa  
    %   element in R. oO&R3zA1d  
    % 9{XV=a v  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- )wfqGkr=m!  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is O <"\G!y~  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 9<-7AN}Z  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ]seOc],4  
    %   for all [n,m]. \jHIjFwQ  
    % !A&>Eeai  
    %   The radial Zernike polynomials are the radial portion of the 9?4:},FRmE  
    %   Zernike functions, which are an orthogonal basis on the unit _REAzxe S  
    %   circle.  The series representation of the radial Zernike P,={ C6*  
    %   polynomials is Y3?)*kz%  
    % 7s}E q~  
    %          (n-m)/2 M|}V6F_y  
    %            __ zT ; +akq  
    %    m      \       s                                          n-2s sJ5Ws%q  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r (Kb_/  
    %    n      s=0 p{oc}dWin  
    % wlw`%z-B2  
    %   The following table shows the first 12 polynomials. YzeNr*  
    % + vO; J  
    %       n    m    Zernike polynomial    Normalization ((mR' A|`  
    %       --------------------------------------------- 1Y(NxC0P=g  
    %       0    0    1                        sqrt(2) *8I &|)x  
    %       1    1    r                           2 (KnU-E]L  
    %       2    0    2*r^2 - 1                sqrt(6) (u-eL#@  
    %       2    2    r^2                      sqrt(6) f7oJ6'K  
    %       3    1    3*r^3 - 2*r              sqrt(8) l$g \t]  
    %       3    3    r^3                      sqrt(8)  -wQ@z6R  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 2OsS+6,[x  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) y4j\y ? T8  
    %       4    4    r^4                      sqrt(10) -X_dY>>s  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) <7Ry"z6g;  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ZXC_kmBN/  
    %       5    5    r^5                      sqrt(12) D&!c7_^  
    %       --------------------------------------------- wL~-k  
    % u Xo?  
    %   Example: j kV9$W0  
    %  {B7${AE  
    %       % Display three example Zernike radial polynomials |wGmu&fY  
    %       r = 0:0.01:1; 7&3  
    %       n = [3 2 5]; YWF Hv@  
    %       m = [1 2 1]; \t?rHB3"  
    %       z = zernpol(n,m,r); v?(z4oOD/>  
    %       figure yz^4TqJ  
    %       plot(r,z) tx,q=.(  
    %       grid on XWag+K  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') V2 >+s y  
    % U%rq(`;  
    %   See also ZERNFUN, ZERNFUN2. (Q}ByX  
    BI+x6S>d  
    % A note on the algorithm. `CY c>n"  
    % ------------------------ a7n`(}?Y  
    % The radial Zernike polynomials are computed using the series 2"IDz01ne  
    % representation shown in the Help section above. For many special 0^K2"De  
    % functions, direct evaluation using the series representation can Y@ X>ejk"  
    % produce poor numerical results (floating point errors), because dheobD  
    % the summation often involves computing small differences between B ,U|V  
    % large successive terms in the series. (In such cases, the functions q0L\{  
    % are often evaluated using alternative methods such as recurrence /B)`pF.n  
    % relations: see the Legendre functions, for example). For the Zernike rVZlv3  
    % polynomials, however, this problem does not arise, because the ([dJ'OPx$  
    % polynomials are evaluated over the finite domain r = (0,1), and BKKW3PT  
    % because the coefficients for a given polynomial are generally all @|D#lBm  
    % of similar magnitude. I+W:}}"j  
    % (Rh$0^)A  
    % ZERNPOL has been written using a vectorized implementation: multiple W0XfU`  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] bMB*9<c~  
    % values can be passed as inputs) for a vector of points R.  To achieve HsKq/Oyk  
    % this vectorization most efficiently, the algorithm in ZERNPOL 5Zn:$?7  
    % involves pre-determining all the powers p of R that are required to m\G45%m  
    % compute the outputs, and then compiling the {R^p} into a single F+)g!NQZ  
    % matrix.  This avoids any redundant computation of the R^p, and Egmp8:nZl@  
    % minimizes the sizes of certain intermediate variables. B["jndyr  
    % ZC"a#rQ   
    %   Paul Fricker 11/13/2006 T'!p{Fbg;  
    lP *p7Y '  
    V-dyeb  
    % Check and prepare the inputs: ^Q9K]Vo  
    % ----------------------------- Jw0I$W/  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) )  lofP$  
        error('zernpol:NMvectors','N and M must be vectors.') eh}|Wd7J  
    end IO7cRg'-F  
    ( 'Ha$O72  
    if length(n)~=length(m) 8Y [4JXUK  
        error('zernpol:NMlength','N and M must be the same length.') s*R UYx  
    end VUC_|=?dL  
    QL:Qzr[  
    n = n(:); Ffig0K+ `  
    m = m(:); p ^ ONJL  
    length_n = length(n); + cZC$lo  
    9SXpZ*Sx  
    if any(mod(n-m,2)) X@za4d  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') pZKK7   
    end 49= K]X  
    mFt\xGa  
    if any(m<0) v%7Gh -P  
        error('zernpol:Mpositive','All M must be positive.') e!vWGnY  
    end XZrzG P(  
    w|f@sB>j  
    if any(m>n) JA% y{Wb  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') is=x6G*r  
    end }U?:al/m  
    m[ER~]L/C  
    if any( r>1 | r<0 ) pnUL+UYeM  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 9Zr6 KA{  
    end x"A\ Z-xxz  
    KQ ^E\,@o  
    if ~any(size(r)==1) 4lI&y<F  
        error('zernpol:Rvector','R must be a vector.') LI>Bl  
    end ^UBzX;|p  
    EAHdt=8W{  
    r = r(:); ,(f({l[J}  
    length_r = length(r); ' pIC~  
    .  LeS-  
    if nargin==4 ? M.'YB2  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); h-fm)1S_  
        if ~isnorm qp{~OW3  
            error('zernpol:normalization','Unrecognized normalization flag.') !QC ErE;r  
        end !Wj`U$];  
    else /#j)GlNp:  
        isnorm = false; xl Q]"sm1  
    end L s+zJ1  
    r{f$n  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% # )s +I2  
    % Compute the Zernike Polynomials :lu"14  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >^SQrB   
    TN<"X :x9  
    % Determine the required powers of r: &{q<  
    % ----------------------------------- Ym6v4k!@O  
    rpowers = []; %S^:5#9  
    for j = 1:length(n) c *i,z  
        rpowers = [rpowers m(j):2:n(j)]; ExF6y#Y G<  
    end ^S 45!mSb  
    rpowers = unique(rpowers); $01~G?:]`  
    U G4I @@=  
    % Pre-compute the values of r raised to the required powers, "LH3ZPD  
    % and compile them in a matrix: ru|*xNXKgC  
    % ----------------------------- VxE;tJ>1  
    if rpowers(1)==0 GC_c.|'6[  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Pa"Kk9!o36  
        rpowern = cat(2,rpowern{:}); CZ>Ujw=&k  
        rpowern = [ones(length_r,1) rpowern]; u95D0S  
    else M"-.D;sa1  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ]YOWCFAQot  
        rpowern = cat(2,rpowern{:}); [zSt+K;  
    end /. H(&  
    <U8w#dc  
    % Compute the values of the polynomials: yqR]9 "a  
    % -------------------------------------- yCkX+{ki  
    z = zeros(length_r,length_n); bf.yA:~U  
    for j = 1:length_n OLV3.~T  
        s = 0:(n(j)-m(j))/2; K[x=knFO  
        pows = n(j):-2:m(j); (iIzoEpb8W  
        for k = length(s):-1:1 3Bcv"O,B!{  
            p = (1-2*mod(s(k),2))* ... CWJN{  
                       prod(2:(n(j)-s(k)))/          ... #o,FVYYj  
                       prod(2:s(k))/                 ... oE2VJKs<B  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... gSf >+|  
                       prod(2:((n(j)+m(j))/2-s(k))); 74&{GCL  
            idx = (pows(k)==rpowers); 4~8-^^  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ?y__ Vrw  
        end h iK}&  
         K /%5\h  
        if isnorm (*,R21<%  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); X":2o|R  
        end &wN}<G e6  
    end U#<{RqY  
    Vv+ oq5hf  
    % 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)  c6y>]8_  
    /FC(d5I  
    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。 2^w{Hcf  
    ={;+0Wjb8  
    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)