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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 Z[*unIk  
    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! 3f^Pr  
     
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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 <!RkkU& 6  
    function z = zernfun(n,m,r,theta,nflag) KX9IC 5pR  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. r craf4%  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N %z@ Z^Jv  
    %   and angular frequency M, evaluated at positions (R,THETA) on the J.h` 0$!  
    %   unit circle.  N is a vector of positive integers (including 0), and FCNYfjB%  
    %   M is a vector with the same number of elements as N.  Each element o%~fJx:]y  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ?SgFD4<~P  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1,  4 Pc-A  
    %   and THETA is a vector of angles.  R and THETA must have the same Q /?`);  
    %   length.  The output Z is a matrix with one column for every (N,M) gNP1UH4m  
    %   pair, and one row for every (R,THETA) pair. Ty&1R?  
    % G(wK(P0j  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 9R8q+2  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ~xxq.rL"  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 5&Yt=)c\  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 2fr%_GNu  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized \u`P(fI!K%  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. k@lJ8(i^qU  
    % D%o(HS\E  
    %   The Zernike functions are an orthogonal basis on the unit circle. G3TS?u8Q  
    %   They are used in disciplines such as astronomy, optics, and u]NsCHKlT  
    %   optometry to describe functions on a circular domain. gq+0t  
    % b>p_w%d[[J  
    %   The following table lists the first 15 Zernike functions. lfM vNv  
    % 1 jB0gNe  
    %       n    m    Zernike function           Normalization u|}\Af  
    %       -------------------------------------------------- 0'*{BAWx  
    %       0    0    1                                 1 m   uO.  
    %       1    1    r * cos(theta)                    2 #1$4<o#M  
    %       1   -1    r * sin(theta)                    2 g^A^@~M  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) / Q@4HV  
    %       2    0    (2*r^2 - 1)                    sqrt(3) w~Q\:<x&~Z  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 6w &<j&V  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) rT4Q^t"  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) j}.gK6Yq*  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 7WmLC  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) cwvJH&%0  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) \wz^Z{U  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) E va&/o?P|  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) gD)M7`4  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) i/_rz.c~3  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) I>.pkf<V  
    %       -------------------------------------------------- Ag0w8F  
    % #\X)|p2  
    %   Example 1: Jm CHwyUK?  
    % i695P}J2  
    %       % Display the Zernike function Z(n=5,m=1) bTeuOpp  
    %       x = -1:0.01:1; geK;r0(f  
    %       [X,Y] = meshgrid(x,x); .?NfV%vv  
    %       [theta,r] = cart2pol(X,Y); b&`~%f-  
    %       idx = r<=1;  )XonFI  
    %       z = nan(size(X)); 'Y2$9qy-L  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); KtAEM;g  
    %       figure _$T !><)y  
    %       pcolor(x,x,z), shading interp _Ml?cT/J.O  
    %       axis square, colorbar cG0)F%?X?  
    %       title('Zernike function Z_5^1(r,\theta)') l,Q`;v5|  
    % X_X7fRC0  
    %   Example 2: <fBJ@>  
    % M/W9"N[ta  
    %       % Display the first 10 Zernike functions ?84f\<"  
    %       x = -1:0.01:1; +?6]Vu&|f  
    %       [X,Y] = meshgrid(x,x); -ABj>y[  
    %       [theta,r] = cart2pol(X,Y); HkRvcX 5  
    %       idx = r<=1; 5u9lKno  
    %       z = nan(size(X)); ph b ;D  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 1 M!4hM Q  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; r:o9:w:  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; W<&/5s  
    %       y = zernfun(n,m,r(idx),theta(idx)); xp:I(  
    %       figure('Units','normalized') Iw[zN[oz  
    %       for k = 1:10 %6fnL~ A  
    %           z(idx) = y(:,k); ]EF"QLNN(  
    %           subplot(4,7,Nplot(k)) $Xo_8SX,  
    %           pcolor(x,x,z), shading interp )M7yj O!  
    %           set(gca,'XTick',[],'YTick',[]) *fi`DiO  
    %           axis square (&*Bl\YoX  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) IW n G@!  
    %       end tpzWi W/  
    % @)Vb?|3  
    %   See also ZERNPOL, ZERNFUN2. hH>a{7V   
    >N! Xey  
    %   Paul Fricker 11/13/2006 qYe`</  
    0K"+u9D^  
    [%LGiCU]  
    % Check and prepare the inputs: F ',1R"/}  
    % ----------------------------- cyd_xB5K  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) P)y2'JKL  
        error('zernfun:NMvectors','N and M must be vectors.') s3ASA.*  
    end >9nVR  
    76[ qFz  
    if length(n)~=length(m) ok,O/|E}?  
        error('zernfun:NMlength','N and M must be the same length.') ByoI+n* U  
    end -|#/KKF  
    \s8h.xjU  
    n = n(:); kQ\l7xd  
    m = m(:); cJm},  
    if any(mod(n-m,2)) B;Z _'.i,d  
        error('zernfun:NMmultiplesof2', ... Q!-"5P X  
              'All N and M must differ by multiples of 2 (including 0).') e"EGqn&!  
    end _{if"  
    -k>k<bDAI  
    if any(m>n) 4Z{R36 {  
        error('zernfun:MlessthanN', ... wk'(g_DP  
              'Each M must be less than or equal to its corresponding N.') Z< C39s  
    end ,lCFe0>k!=  
    HIj:?y  
    if any( r>1 | r<0 ) B[&l<*O-y  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') KvPLA{  
    end Ia9!ucN7DA  
    "7(@I^'t6  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) v8uUv%Hkd  
        error('zernfun:RTHvector','R and THETA must be vectors.') `K$;K8!1  
    end w5-^Py  
    gi:M=  
    r = r(:);  k_^ 4NU  
    theta = theta(:); o}wRgG  
    length_r = length(r); FbdC3G|oA  
    if length_r~=length(theta) 8j]QnH0&  
        error('zernfun:RTHlength', ... 01aw+o  
              'The number of R- and THETA-values must be equal.') D@{m  
    end 1G(wESe  
    \Ym$to  
    % Check normalization: 3uvl'1(%J  
    % -------------------- Pa; *%7  
    if nargin==5 && ischar(nflag) w3fD6$  
        isnorm = strcmpi(nflag,'norm'); y </i1qM  
        if ~isnorm BlqfST#6  
            error('zernfun:normalization','Unrecognized normalization flag.') >9g^-~X;v  
        end 4Im}!q5;:<  
    else )i-`AJK-'v  
        isnorm = false; /3"S_KE1@+  
    end Xn!=/<TIVz  
    +tlbO?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "1P2`Ep;  
    % Compute the Zernike Polynomials q{yzux  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =/xXB  
    2<2a3'pG  
    % Determine the required powers of r: 4g.S!-H@R  
    % ----------------------------------- 5(\[Gke  
    m_abs = abs(m); lvk*Db$  
    rpowers = []; 9 771D  
    for j = 1:length(n) el^<M,7!  
        rpowers = [rpowers m_abs(j):2:n(j)]; #TP Y%  
    end kl&_O8E+K  
    rpowers = unique(rpowers); 7vH4}S\ q  
    jd+HIR  
    % Pre-compute the values of r raised to the required powers, `lO/I+8  
    % and compile them in a matrix: }u5J<*:bZ  
    % ----------------------------- R,zp&L  
    if rpowers(1)==0 $i `@0+:  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); uMC0XE|S  
        rpowern = cat(2,rpowern{:}); $- Z/UHT  
        rpowern = [ones(length_r,1) rpowern]; mL, {ZL ^  
    else M?$tHA~OX  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); SYOU &*  
        rpowern = cat(2,rpowern{:}); 8H SGOs =8  
    end t6+>Zr  
    URTJA<r8D  
    % Compute the values of the polynomials: %ZlnGr  
    % -------------------------------------- G~4|]^`g  
    y = zeros(length_r,length(n)); {\= NZ\  
    for j = 1:length(n) N4 _V  
        s = 0:(n(j)-m_abs(j))/2; J= DD/Gp  
        pows = n(j):-2:m_abs(j); afcyAzIB&  
        for k = length(s):-1:1 9+>%U~U<  
            p = (1-2*mod(s(k),2))* ... -g vS 3`lX  
                       prod(2:(n(j)-s(k)))/              ... Od]wh  
                       prod(2:s(k))/                     ... %A( hmC  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... yD ur9Qd6  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); *-_joAWTG  
            idx = (pows(k)==rpowers); 'V Y\ut  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Fg^zz*e  
        end RKz _GEH)  
         3dI(gm6  
        if isnorm OoAZ t  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); l_=kW!l  
        end SYK?5_804  
    end RQ51xTOL4]  
    % END: Compute the Zernike Polynomials rg+3pX\{  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% YpbJoHiSH  
    Hkj| e6  
    % Compute the Zernike functions: ;W#/;C _h  
    % ------------------------------ o Bp.|8-  
    idx_pos = m>0; $2*&\/;-E!  
    idx_neg = m<0; }(if|skau  
    ok9G9|HA  
    z = y; mZ t:  
    if any(idx_pos) SVyJUd_  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); qS2]|7q?Tc  
    end [$GQ]Y  
    if any(idx_neg) 27jZ~Bp$  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); e%DF9}M  
    end @sb00ad2q  
    ;%aWA  
    % 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) } u7&SU  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ? &;d)TQ  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated h?:Y\DlU'  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 0=J69Yd  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ) N"gW*  
    %   and THETA is a vector of angles.  R and THETA must have the same [?`c>  
    %   length.  The output Z is a matrix with one column for every P-value, <ByDT$E_  
    %   and one row for every (R,THETA) pair. (gv ~Vq  
    % 8M'6Kcr  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike &gR)bNIC_=  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) w]0jq U6  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) \ow0Y >  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 [f'DxZF-  
    %   for all p. J#W>%2 "s  
    % U!x\oLP  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 $yb8..+  
    %   Zernike functions (order N<=7).  In some disciplines it is @V-CG!  
    %   traditional to label the first 36 functions using a single mode i(c'94M  
    %   number P instead of separate numbers for the order N and azimuthal 7u^6`P  
    %   frequency M. w8g36v*+(u  
    % $bKa"T*  
    %   Example: |"Oazll  
    % ZvO:!u0+"  
    %       % Display the first 16 Zernike functions 9?W38EF  
    %       x = -1:0.01:1; .*g;2.-qv&  
    %       [X,Y] = meshgrid(x,x); yMa5?]J  
    %       [theta,r] = cart2pol(X,Y); <cz~q=%v2&  
    %       idx = r<=1; G:rM_q9\u  
    %       p = 0:15; ~dwl7Qc  
    %       z = nan(size(X)); m"vV=6m|\  
    %       y = zernfun2(p,r(idx),theta(idx)); Swua dN  
    %       figure('Units','normalized') .ODtduURe  
    %       for k = 1:length(p) F'#e]/V1  
    %           z(idx) = y(:,k); Ja#idF[V  
    %           subplot(4,4,k) &[KFCn  
    %           pcolor(x,x,z), shading interp c(AjM9s  
    %           set(gca,'XTick',[],'YTick',[]) m%E7V{t  
    %           axis square u;:N 4d=f'  
    %           title(['Z_{' num2str(p(k)) '}']) 6C/D&+4  
    %       end LVLh&9  
    % J:mOg95<  
    %   See also ZERNPOL, ZERNFUN. MJX m7<(  
    tZXtt=M w  
    %   Paul Fricker 11/13/2006 %3VwCuE  
    u-k?ef  
    Rb~Kyy$  
    % Check and prepare the inputs: r/O(EW#=8  
    % ----------------------------- Qg _?..%  
    if min(size(p))~=1 a}c(#ZLs  
        error('zernfun2:Pvector','Input P must be vector.') }RUC#aW1  
    end xD1B50y U  
    8a)EL*LH`  
    if any(p)>35 $ 9DZ5"  
        error('zernfun2:P36', ... z4J-qK~2  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ccm <rZ7  
               '(P = 0 to 35).']) r$\g6m  
    end % e1vq  
    V`WfJ>{;Z  
    % Get the order and frequency corresonding to the function number: vPwDV_zk  
    % ---------------------------------------------------------------- xSOL4  
    p = p(:); Cm ;N5i  
    n = ceil((-3+sqrt(9+8*p))/2); jv|IV  
    m = 2*p - n.*(n+2); >v<}$v6D~  
    "iZ-AG!C  
    % Pass the inputs to the function ZERNFUN: puqH%m+u  
    % ---------------------------------------- ld@f:Zali  
    switch nargin kkOYC?zE?  
        case 3 oG*lU h}  
            z = zernfun(n,m,r,theta); eNNgxQw>m  
        case 4 8w*fg6,=  
            z = zernfun(n,m,r,theta,nflag); 3P\I;xM  
        otherwise :6M0`V;L  
            error('zernfun2:nargin','Incorrect number of inputs.') "?9fL#8f*!  
    end iGU N$  
    ^3yjE/Wi"  
    % 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) 2E1TJ.[BS  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. lb6s3b  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of {oJa8~P  
    %   order N and frequency M, evaluated at R.  N is a vector of :<v$vER,&  
    %   positive integers (including 0), and M is a vector with the \rN_CBM  
    %   same number of elements as N.  Each element k of M must be a ) k2NF="o  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) %8{_;-f  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is yR$_$N+E  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 'OihA^e  
    %   with one column for every (N,M) pair, and one row for every &+^ # `nq  
    %   element in R. y-X'eCUz  
    % i-=ff  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- LK%B6-;~-  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ?7p| F^  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 0l6djN  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 <Ztda !  
    %   for all [n,m]. 6;b 'j\jG  
    % (hV"z;rI  
    %   The radial Zernike polynomials are the radial portion of the ii2Z }qe  
    %   Zernike functions, which are an orthogonal basis on the unit <EUSl|6  
    %   circle.  The series representation of the radial Zernike +UsR  
    %   polynomials is ,Df36-74v5  
    % :58'U|  
    %          (n-m)/2 bOdv]nQ1  
    %            __ (qP$I:Q4]v  
    %    m      \       s                                          n-2s (j I|F-i  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r *u>\&`h=  
    %    n      s=0 prs<ZxbQb  
    % {.OoOqq9  
    %   The following table shows the first 12 polynomials. [-gKkOT8E  
    % 2"'8x?.V  
    %       n    m    Zernike polynomial    Normalization C4uR5U  
    %       --------------------------------------------- }\{1`$*~  
    %       0    0    1                        sqrt(2) M(E_5@?3  
    %       1    1    r                           2 b?NeSiswn  
    %       2    0    2*r^2 - 1                sqrt(6) {4 d$]o0V  
    %       2    2    r^2                      sqrt(6) 4<K`yU]"  
    %       3    1    3*r^3 - 2*r              sqrt(8) I`kp5lGD2  
    %       3    3    r^3                      sqrt(8) 4w6K|v<X  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) D5b _m|7%  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) v`w?QIB]  
    %       4    4    r^4                      sqrt(10) M\6`2q  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) NGs9Jke2  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ko>_@]Jb  
    %       5    5    r^5                      sqrt(12) 7qz-RF#s8  
    %       --------------------------------------------- P0(~~z&%[  
    % xi\RUAW  
    %   Example: }gi' %e  
    % hNo>)$v!s  
    %       % Display three example Zernike radial polynomials EI.Pk>ZIm  
    %       r = 0:0.01:1; ^ks^9*'|j  
    %       n = [3 2 5]; an`(?6d  
    %       m = [1 2 1]; 1n( }Q1fa  
    %       z = zernpol(n,m,r); #jx?uS  
    %       figure DOIWhd5:  
    %       plot(r,z) Lp)8SmN  
    %       grid on y;Ln ao7i  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') XL=R]IC<.  
    % cgeS)C7  
    %   See also ZERNFUN, ZERNFUN2. walRqlo@  
    R<-C>D  
    % A note on the algorithm. :+Ax3  
    % ------------------------ _5w?v~65  
    % The radial Zernike polynomials are computed using the series `>EvT7u  
    % representation shown in the Help section above. For many special *9ub.:EUwV  
    % functions, direct evaluation using the series representation can 7B!Qq/E?g  
    % produce poor numerical results (floating point errors), because c\{}FGC  
    % the summation often involves computing small differences between ydqmuZ%2h#  
    % large successive terms in the series. (In such cases, the functions y]_8. 0zM  
    % are often evaluated using alternative methods such as recurrence g Pj0H&,.  
    % relations: see the Legendre functions, for example). For the Zernike # pB:LPEsK  
    % polynomials, however, this problem does not arise, because the f}Tr$r  
    % polynomials are evaluated over the finite domain r = (0,1), and 1 " 7#|=1/  
    % because the coefficients for a given polynomial are generally all v@uaf=x-  
    % of similar magnitude. 0P40K  
    % cu`J2vm3  
    % ZERNPOL has been written using a vectorized implementation: multiple gNN" H#=2  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] { DQ E7kI  
    % values can be passed as inputs) for a vector of points R.  To achieve L{A-0Ffh  
    % this vectorization most efficiently, the algorithm in ZERNPOL nSQ}yqM)  
    % involves pre-determining all the powers p of R that are required to 7jH`_58  
    % compute the outputs, and then compiling the {R^p} into a single [Um4\QvUx  
    % matrix.  This avoids any redundant computation of the R^p, and ls7A5 <  
    % minimizes the sizes of certain intermediate variables. tZ j,A%<  
    % 51 +M_ ~  
    %   Paul Fricker 11/13/2006 Z;~[@7`  
    q+n1~AT  
    LsQ8sFP_"  
    % Check and prepare the inputs: y2$;t'  
    % ----------------------------- q[OTaSQ~u^  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) V<V\0n!0  
        error('zernpol:NMvectors','N and M must be vectors.') L##8+OJ.L  
    end X/8iJ-KB  
    SKpPR;=q|:  
    if length(n)~=length(m) >-@ U_p  
        error('zernpol:NMlength','N and M must be the same length.') t?nc0;Q9,@  
    end YATdGLTeq  
    1uS-Tx  
    n = n(:); zL},`:(.  
    m = m(:); /3[ 9{r  
    length_n = length(n); RYZE*lWUh  
    51M^yG&M  
    if any(mod(n-m,2)) 1: xnD  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') <VaMUm<2  
    end )ClMw!ZrU  
    "8muMa8Q%  
    if any(m<0) 5nx<,-N*BP  
        error('zernpol:Mpositive','All M must be positive.') aR)en{W  
    end v^c<`i;  
    R,+/A8[j  
    if any(m>n) oCKM5AVWsv  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') .y7)XLC  
    end Sn0gTsZ  
    /Fo/_=FE2  
    if any( r>1 | r<0 ) /"Yx@n  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') Lubs{-5lk  
    end XJ h:U0  
    N8XC~Dh{  
    if ~any(size(r)==1) mheU#&|  
        error('zernpol:Rvector','R must be a vector.') `N,Vs n"  
    end a}Fk x  
    %7]XW2u  
    r = r(:); <m>l-]  
    length_r = length(r); }PFt  
    m@Ip^]9ry  
    if nargin==4 A }-&C  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); O*rmD<L$  
        if ~isnorm (H8C\%g:  
            error('zernpol:normalization','Unrecognized normalization flag.') L8dU (P  
        end IypWVr   
    else m(Y.X=EZr  
        isnorm = false; o'eI(@{F=  
    end !2Y!jz  
    {,Bb"0 \  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7rdw`  
    % Compute the Zernike Polynomials _3.G\/>[K  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7V} ]C>G  
    |)-|2cPRur  
    % Determine the required powers of r: W`N}  
    % ----------------------------------- 2gwZb/'i  
    rpowers = []; 2Io6s '  
    for j = 1:length(n) 7 DY WdDX  
        rpowers = [rpowers m(j):2:n(j)]; 9fNu?dE   
    end 9k\)tWe  
    rpowers = unique(rpowers); 3^\?>C7  
    PEW4J{(W  
    % Pre-compute the values of r raised to the required powers, Vd%v_Ek  
    % and compile them in a matrix: Har~MO?A  
    % ----------------------------- zj>aaY  
    if rpowers(1)==0 "!%wh6`>Md  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ?GO SeV  
        rpowern = cat(2,rpowern{:}); ym|NT0_0  
        rpowern = [ones(length_r,1) rpowern]; FjZc#\^9  
    else |DE%SVZB  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); h(@R]GUX  
        rpowern = cat(2,rpowern{:}); skIiJ'db  
    end V uG?B{  
    )N" Ew0U  
    % Compute the values of the polynomials: yB,{#nM>8  
    % -------------------------------------- gB>imr#e&  
    z = zeros(length_r,length_n); sdJ%S*)5G$  
    for j = 1:length_n UvxSMD:A  
        s = 0:(n(j)-m(j))/2; !K3})& w  
        pows = n(j):-2:m(j); HDmx@E.@  
        for k = length(s):-1:1 y=AsgJ  
            p = (1-2*mod(s(k),2))* ... M{?.hq  
                       prod(2:(n(j)-s(k)))/          ... yiV G ]s  
                       prod(2:s(k))/                 ... j8GY`f#  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... bMN@H\Ek  
                       prod(2:((n(j)+m(j))/2-s(k))); e-X HN  
            idx = (pows(k)==rpowers); SY-ez 91  
            z(:,j) = z(:,j) + p*rpowern(:,idx); p%q.*trUb9  
        end :O $@shV  
         0K<x=-cCB  
        if isnorm v6s\Z\v)Q`  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); d2rL 8jW  
        end TUh&d5a9H  
    end nPcS3!7B#  
    . yZm^&  
    % 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)  q_V0+qH  
    >yB(lKV  
    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。 W/fuKGZi_  
    y7/F _{  
    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)