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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 Y5{KtW  
    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! %?`O .W  
     
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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 CcTdLq  
    function z = zernfun(n,m,r,theta,nflag) kQ]4Bo  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. #<~oR5ddlb  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ;Fo7 -kK  
    %   and angular frequency M, evaluated at positions (R,THETA) on the znB+RiV8  
    %   unit circle.  N is a vector of positive integers (including 0), and blLl1Ak  
    %   M is a vector with the same number of elements as N.  Each element <5E)6c_W)  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) xM=ydRu  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Sp6==(:.  
    %   and THETA is a vector of angles.  R and THETA must have the same .]H/u "d  
    %   length.  The output Z is a matrix with one column for every (N,M) 4{Q$^wD+.  
    %   pair, and one row for every (R,THETA) pair. kbL7Xjk  
    % Ee_?aG e&  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike =0L%<@yA  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ScjeAC)  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral yEMM@5W)8  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, F Uz1P  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized >z~_s6#CP  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. u -)ED  
    % S}fQis  
    %   The Zernike functions are an orthogonal basis on the unit circle. S\]9mHJI  
    %   They are used in disciplines such as astronomy, optics, and KWxTN|>  
    %   optometry to describe functions on a circular domain. qzNXz_#+u  
    % /0cm7[a?  
    %   The following table lists the first 15 Zernike functions. _M&n~ r  
    % 15VvZ![$V  
    %       n    m    Zernike function           Normalization mU(v9Jpf7  
    %       -------------------------------------------------- z;?ztpa@  
    %       0    0    1                                 1 2}7_Y6RS*  
    %       1    1    r * cos(theta)                    2 $}IG+ ,L  
    %       1   -1    r * sin(theta)                    2 ck%.D%=  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 'gXD?ARW  
    %       2    0    (2*r^2 - 1)                    sqrt(3) l-cBN^^  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) }9^'etD  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) {\`y)k 7  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) @{U UB=}9  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) e|W;(@$<  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) -[J4nN&N  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) RX%)@e/@  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) auB 931|  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) #6 ni~d&0  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) O8A(OfX  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) &^K(9"  
    %       -------------------------------------------------- #'},/Lm@  
    % HL]J=Gh  
    %   Example 1: P3YM4&6XA  
    % l]~9BPsR  
    %       % Display the Zernike function Z(n=5,m=1) x4PzP  
    %       x = -1:0.01:1; }A]e C  
    %       [X,Y] = meshgrid(x,x); Tt9cX}&&  
    %       [theta,r] = cart2pol(X,Y); K2e68GU  
    %       idx = r<=1; 8(&6*- 7=  
    %       z = nan(size(X)); "+4Jmf9  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); WO{7/h</  
    %       figure :/%Y"0  
    %       pcolor(x,x,z), shading interp Kxa1F,dZ  
    %       axis square, colorbar l.]wBH#RS  
    %       title('Zernike function Z_5^1(r,\theta)') Xn?.Od(  
    % #AP;GoIf"j  
    %   Example 2: 5!S#}=f=  
    % D5oYcGc  
    %       % Display the first 10 Zernike functions 7QnWw0  
    %       x = -1:0.01:1; JWaWOk(t=?  
    %       [X,Y] = meshgrid(x,x); g\q4-  
    %       [theta,r] = cart2pol(X,Y); si)>:e  
    %       idx = r<=1; ?9O#b1f N  
    %       z = nan(size(X)); b{,v?7^4  
    %       n = [0  1  1  2  2  2  3  3  3  3]; A`JE(cIz3  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; pZK 1G  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; N P+ vi@Ud  
    %       y = zernfun(n,m,r(idx),theta(idx)); X`EVjK  
    %       figure('Units','normalized') k H<C9z2=  
    %       for k = 1:10  ^|zag  
    %           z(idx) = y(:,k); xo?'L&%  
    %           subplot(4,7,Nplot(k)) +c!HXX  
    %           pcolor(x,x,z), shading interp MRXw)NAw  
    %           set(gca,'XTick',[],'YTick',[]) K?[Vz[-Fc  
    %           axis square E3Y0@r  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 3uxf n=E  
    %       end  oJ*,a  
    % lw gwdB  
    %   See also ZERNPOL, ZERNFUN2. $Zo|t a^  
    $M4Z_zle)  
    %   Paul Fricker 11/13/2006 +TA~RC d  
    g 8uq6U  
    #]5KWXC'~  
    % Check and prepare the inputs: jIr\.i  
    % ----------------------------- +jZa A/  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) J5F@<vi  
        error('zernfun:NMvectors','N and M must be vectors.') 5@r6'Z  
    end j;b>~_ U%  
    ^X+qut+~  
    if length(n)~=length(m) ) 3"!Q+  
        error('zernfun:NMlength','N and M must be the same length.') LxGD=b  
    end Vt3*~Beb  
    <uS/8MP{  
    n = n(:); 52j3[in  
    m = m(:); 7g]mrI@  
    if any(mod(n-m,2)) Iox)-  
        error('zernfun:NMmultiplesof2', ... 6nE/8m  
              'All N and M must differ by multiples of 2 (including 0).') spm)X-[1  
    end %Vltc4QU  
    <QFayZ$  
    if any(m>n) T?f{.a)  
        error('zernfun:MlessthanN', ... &+@`Si=  
              'Each M must be less than or equal to its corresponding N.') 2)}*'_E9  
    end (0#$%US\  
    w' J`$=  
    if any( r>1 | r<0 ) _0gdt4  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') q78OP}  
    end - EGZ  
    J ;z`bk^  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) #BcUE?K*N  
        error('zernfun:RTHvector','R and THETA must be vectors.') g.di3GGi  
    end *S.FM.r  
    gCPH>8JwS0  
    r = r(:); [pp|*@1T  
    theta = theta(:); n{M Th_C4n  
    length_r = length(r); d7G@Z|R3p  
    if length_r~=length(theta) onRTX|#  
        error('zernfun:RTHlength', ... T:'JA  
              'The number of R- and THETA-values must be equal.') pO7OP"q1  
    end 'Ca;gi !U  
    c%hXj#;  
    % Check normalization: y;fF|t<y  
    % -------------------- $.$nv~f  
    if nargin==5 && ischar(nflag) { V(~  
        isnorm = strcmpi(nflag,'norm'); W!\%v"  
        if ~isnorm a}f /<-L  
            error('zernfun:normalization','Unrecognized normalization flag.') 6@/k|t>OT  
        end Cj4Y, N  
    else ko[d axUB  
        isnorm = false; <yEApWd;  
    end 6Y\TVRR  
    _+aR| AEC  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% hGrX,.zj  
    % Compute the Zernike Polynomials v'?o#_La+  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #"!ga)a%L  
    7bO>[RQB  
    % Determine the required powers of r: b:~#;$g  
    % ----------------------------------- w.J$(o(/  
    m_abs = abs(m); tF-l=ph}`  
    rpowers = []; ;qUB[Kw  
    for j = 1:length(n) j0~c2  
        rpowers = [rpowers m_abs(j):2:n(j)]; z7:* ,X  
    end H<fi,"X^  
    rpowers = unique(rpowers); Yl'8" \HF  
    O%>*=h`P  
    % Pre-compute the values of r raised to the required powers, 0U*f"5F  
    % and compile them in a matrix: 8N"WKBj|_d  
    % ----------------------------- 9)S3{i6w  
    if rpowers(1)==0 $MQ<QP  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); NrXIaN  
        rpowern = cat(2,rpowern{:}); \ILNx^$EL  
        rpowern = [ones(length_r,1) rpowern]; c u";rnj  
    else Da8gOZ  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); wzxV)1jT  
        rpowern = cat(2,rpowern{:}); /({oN1X>i  
    end N;-%:nC  
    J %A=  
    % Compute the values of the polynomials: @73kry v  
    % -------------------------------------- eXnSH$uI  
    y = zeros(length_r,length(n)); wN%lc3[/z2  
    for j = 1:length(n) -R]~kGa6m<  
        s = 0:(n(j)-m_abs(j))/2; H? z~V-8  
        pows = n(j):-2:m_abs(j); FCwE/ 2,  
        for k = length(s):-1:1 ']\SX*z?  
            p = (1-2*mod(s(k),2))* ... L;M@]  
                       prod(2:(n(j)-s(k)))/              ... Z}vDP^rf  
                       prod(2:s(k))/                     ... cU ?F D  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... | Z7 j s"  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); F Xr\  
            idx = (pows(k)==rpowers); U<sGj~"#  
            y(:,j) = y(:,j) + p*rpowern(:,idx); JCBX?rM/  
        end O"o|8 l}M/  
         #*y.C[^5{  
        if isnorm 6m]?*k1HC  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); i4k [#x  
        end McS]aJfrk  
    end /E\04Bs  
    % END: Compute the Zernike Polynomials $n!5JS@40  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^`SEmYb;  
    SYsO>`/ )  
    % Compute the Zernike functions: Hq<4G:#  
    % ------------------------------ pnp8`\cIH  
    idx_pos = m>0; jwLZC  
    idx_neg = m<0; a;IOL  
    FMF  mn|  
    z = y; lo6upir ZX  
    if any(idx_pos) Rsq EAdZw[  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); LQ%QFfC  
    end 9__Q-J  
    if any(idx_neg) TTa3DbFp%  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); [!3cWJCt  
    end +'93%/:  
    $iy!:Did  
    % 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) f9Xa}*  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ZRw^< +  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated F|!=]A<  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Y?K?*`Pkc1  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 8tjWVo  
    %   and THETA is a vector of angles.  R and THETA must have the same FwB xag:u  
    %   length.  The output Z is a matrix with one column for every P-value, ) Kl@dj  
    %   and one row for every (R,THETA) pair. v)|a}5={  
    % | ~>7_:  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike <Pe'&u  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 6?.S-.Mr  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) W]bytsl  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 N:pP@o  
    %   for all p. jg%mWiKwK7  
    % <Tbl |9  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 VE/m|3%t  
    %   Zernike functions (order N<=7).  In some disciplines it is |cuKC \  
    %   traditional to label the first 36 functions using a single mode jJvd!,=)  
    %   number P instead of separate numbers for the order N and azimuthal @sZ' --Y  
    %   frequency M. v;?W|kJ.u  
    % p(4B"[!S  
    %   Example: .) %, R  
    % qSD`S1'2;  
    %       % Display the first 16 Zernike functions "mU2^4q  
    %       x = -1:0.01:1; (Lj*FXmz  
    %       [X,Y] = meshgrid(x,x); [GK## z'5  
    %       [theta,r] = cart2pol(X,Y); "9hD4R  
    %       idx = r<=1; y!S:d  
    %       p = 0:15; m8b-\^eP7  
    %       z = nan(size(X)); k'e1ZAn  
    %       y = zernfun2(p,r(idx),theta(idx)); H0lW gJmi|  
    %       figure('Units','normalized') fo>_*6i74  
    %       for k = 1:length(p) IvQuxs&a  
    %           z(idx) = y(:,k); TL$w~dY  
    %           subplot(4,4,k) Y Fj#{C.  
    %           pcolor(x,x,z), shading interp {H9g&pfv  
    %           set(gca,'XTick',[],'YTick',[]) <pG 4 g  
    %           axis square d%q&[<'jf  
    %           title(['Z_{' num2str(p(k)) '}']) f` -vnh^+  
    %       end tOk=m'aUK  
    % b rDyjh  
    %   See also ZERNPOL, ZERNFUN. U_Mag(^-  
    [?,+DY  
    %   Paul Fricker 11/13/2006 Y37qjV  
    B 'd@ms  
    4pcIH5)z  
    % Check and prepare the inputs: `8\" 3S  
    % ----------------------------- Lew 2Z  
    if min(size(p))~=1 {m,LpI0wG  
        error('zernfun2:Pvector','Input P must be vector.') L KZ<\% X  
    end kxAT  
    # M3d=  
    if any(p)>35 KNP^k$=)3c  
        error('zernfun2:P36', ... 3a U4Z|f~  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 0R]'HA>  
               '(P = 0 to 35).']) G(F=6L~;  
    end Gw6!cp|/  
    [V;u7Z\r-  
    % Get the order and frequency corresonding to the function number: C4-%|+Q i  
    % ---------------------------------------------------------------- -J]N &[  
    p = p(:); .ubZ  
    n = ceil((-3+sqrt(9+8*p))/2); Y~#.otBL&  
    m = 2*p - n.*(n+2); \qG` ts  
    *'{9(Oj  
    % Pass the inputs to the function ZERNFUN: l[WX77bp=  
    % ---------------------------------------- Fy6Lz.baB  
    switch nargin (Nf!E[ }Z  
        case 3 ?AI`,*^  
            z = zernfun(n,m,r,theta); sVnpO$  
        case 4 k%N$eO$  
            z = zernfun(n,m,r,theta,nflag); t""Y -M  
        otherwise -"2%+S{  
            error('zernfun2:nargin','Incorrect number of inputs.') mv/ Nz?  
    end K\lu;   
    2j{T8F\]  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) P}!pmg6V  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. UH5A;SrTqR  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of PL3oV<\4s>  
    %   order N and frequency M, evaluated at R.  N is a vector of pWoeF=+y]W  
    %   positive integers (including 0), and M is a vector with the s|:j~>53  
    %   same number of elements as N.  Each element k of M must be a X#MC|Fzy@  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) wu} Zu  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is iYr*0:M  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix j#H&~f  
    %   with one column for every (N,M) pair, and one row for every M=5hp&=  
    %   element in R. .&KC2#4   
    % 7U&<{U<  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- --7@rxv  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ;s$ P?('  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ^|cax| >  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 9N<TJp,q  
    %   for all [n,m].  I$fm"N  
    % .;:dG  
    %   The radial Zernike polynomials are the radial portion of the `;s#/`c|/  
    %   Zernike functions, which are an orthogonal basis on the unit  S^5Qhv  
    %   circle.  The series representation of the radial Zernike #<-%%  
    %   polynomials is S\2@~*{-8  
    % j>hBNz  
    %          (n-m)/2 AnBD~h h  
    %            __ Nqbm,s  
    %    m      \       s                                          n-2s 9*[!ux7h  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r bI)%g  
    %    n      s=0 <$ ` ^  
    % /jn0Xh  
    %   The following table shows the first 12 polynomials. };>~P%u32  
    % ~8lB#NuN  
    %       n    m    Zernike polynomial    Normalization 7{OD/*|  
    %       --------------------------------------------- hx}X=7w  
    %       0    0    1                        sqrt(2) 0(^ N  
    %       1    1    r                           2 ooN?x31  
    %       2    0    2*r^2 - 1                sqrt(6) ^^[A\'  
    %       2    2    r^2                      sqrt(6) R, U YwI  
    %       3    1    3*r^3 - 2*r              sqrt(8) ZBc8 ^QZ  
    %       3    3    r^3                      sqrt(8) w.-J2%J   
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) TJ0;xn6o  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) Ot~buf'|  
    %       4    4    r^4                      sqrt(10) R& HkWe  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ,mE}#cyY  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) U1=\ `)u;  
    %       5    5    r^5                      sqrt(12) /t _QA  
    %       --------------------------------------------- =n_>7@9l  
    % f"G-',O<  
    %   Example: *7yrm&@nG  
    % p3cb_  
    %       % Display three example Zernike radial polynomials poS=8mN8;  
    %       r = 0:0.01:1; O|&SL03Z8  
    %       n = [3 2 5]; VVi3g  
    %       m = [1 2 1];  4{D^ 4G  
    %       z = zernpol(n,m,r); ua%j}%G(  
    %       figure tAS[T9B  
    %       plot(r,z) rCdTn+O2  
    %       grid on ?#/~ BZR!  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') I=4G+h5p  
    % XWUi_{zn  
    %   See also ZERNFUN, ZERNFUN2. a)rT3gl  
    S 0mt8/ M  
    % A note on the algorithm. pT<I!,~  
    % ------------------------ V`"Cd?R0Z  
    % The radial Zernike polynomials are computed using the series i$XT Qr0K=  
    % representation shown in the Help section above. For many special b'(Hwc\ t  
    % functions, direct evaluation using the series representation can `s_k+ g  
    % produce poor numerical results (floating point errors), because '6f)^DYA'?  
    % the summation often involves computing small differences between `Wp& 'X  
    % large successive terms in the series. (In such cases, the functions 8AmB0W> e  
    % are often evaluated using alternative methods such as recurrence M HKnHPv  
    % relations: see the Legendre functions, for example). For the Zernike )3)fq:[  
    % polynomials, however, this problem does not arise, because the xZ`h8  
    % polynomials are evaluated over the finite domain r = (0,1), and  y7.oy"  
    % because the coefficients for a given polynomial are generally all NV;T*I8O  
    % of similar magnitude. Y]+KsiOL  
    % gq&jNj7V  
    % ZERNPOL has been written using a vectorized implementation: multiple K5(:0Q.5y  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Qa,$_ ,E  
    % values can be passed as inputs) for a vector of points R.  To achieve ;b0;66C8|  
    % this vectorization most efficiently, the algorithm in ZERNPOL !+3nlG4cw  
    % involves pre-determining all the powers p of R that are required to _?;74VWA  
    % compute the outputs, and then compiling the {R^p} into a single ST[E$XL6  
    % matrix.  This avoids any redundant computation of the R^p, and 2%~+c|TH.)  
    % minimizes the sizes of certain intermediate variables. 7y=1\KW(  
    % j.SE'a_  
    %   Paul Fricker 11/13/2006 /3qKsv#  
    XOPiwrg%p  
    G5!!^p~  
    % Check and prepare the inputs: ic?(`6N8  
    % ----------------------------- eZmwF@  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) r.v.y[u  
        error('zernpol:NMvectors','N and M must be vectors.') 3F{R$M}  
    end >$;,1N $bd  
    a]0hB:  
    if length(n)~=length(m) F)=*Ga  
        error('zernpol:NMlength','N and M must be the same length.') 7$Pf  
    end Poxoc-s  
    (kSb74*g  
    n = n(:); (T =u_oe  
    m = m(:); w9CX5Fg  
    length_n = length(n); Gn 1  
    c) _u^Dh  
    if any(mod(n-m,2)) a*!9RQ  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 9K=K,6 b  
    end %H=^U8WB  
    C@9K`N[*  
    if any(m<0) !>6`+$=U  
        error('zernpol:Mpositive','All M must be positive.') (%*~5%l\  
    end O]Q8&(  
    fq !CB]C  
    if any(m>n) *xDV8iu_  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') :qtg`zM/4  
    end FQ]5W |e  
    -5,+gakSk  
    if any( r>1 | r<0 ) .8]=yPm  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') e J:#vX86  
    end DN|+d{^lN  
    Nd**":i$  
    if ~any(size(r)==1) c"NGE  
        error('zernpol:Rvector','R must be a vector.') 'F7UnkKO|  
    end d@{#F"o  
    ,sltB3f  
    r = r(:); {"\pMY'7  
    length_r = length(r); P7;q^jlB  
    s,7 OoLE  
    if nargin==4 h`Xl~=  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); JgcMk]|'  
        if ~isnorm +"PME1  
            error('zernpol:normalization','Unrecognized normalization flag.') *N%)+-   
        end 1c:/c|shQ_  
    else fILD~  
        isnorm = false; L}>ts(!q&  
    end "_ON0._(/  
    ._`?ZJ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &8hW~G>(m  
    % Compute the Zernike Polynomials KZ|p_{0&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }XRRM:B|)(  
    CjLiLB  
    % Determine the required powers of r: W(PNw2  
    % ----------------------------------- @gm!D`YL  
    rpowers = []; *.+N?%sAP)  
    for j = 1:length(n) Qe]aI7Ei  
        rpowers = [rpowers m(j):2:n(j)]; M>@PRb:Oc  
    end /rv=ml pRL  
    rpowers = unique(rpowers); sh}eKwh  
    ccgV-'IG9  
    % Pre-compute the values of r raised to the required powers, lt#3&@<v  
    % and compile them in a matrix: G[Jz(/yNH  
    % ----------------------------- R5fZ }C7  
    if rpowers(1)==0 _sF Ad`  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ?1Uq ud  
        rpowern = cat(2,rpowern{:}); OdtS5:L  
        rpowern = [ones(length_r,1) rpowern]; mWH;-F*%  
    else Ol*|J  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Zu/1:8x  
        rpowern = cat(2,rpowern{:}); )h/Qxf  
    end l,ra24  
    r& nE M6  
    % Compute the values of the polynomials: W! GUA<  
    % -------------------------------------- 1|5TuljTd  
    z = zeros(length_r,length_n); JiRfLB  
    for j = 1:length_n $GIup5  
        s = 0:(n(j)-m(j))/2; #;<dtw  
        pows = n(j):-2:m(j); X/23 /_~L`  
        for k = length(s):-1:1 &dJ\}O[r  
            p = (1-2*mod(s(k),2))* ... xWKUti i  
                       prod(2:(n(j)-s(k)))/          ... > @q4Uez  
                       prod(2:s(k))/                 ... :bz;_DZP  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... }*56 DX  
                       prod(2:((n(j)+m(j))/2-s(k))); "mAMfV0  
            idx = (pows(k)==rpowers); zCSLV>.F  
            z(:,j) = z(:,j) + p*rpowern(:,idx); {e83 A /{  
        end kj'  
          q #X[oVq  
        if isnorm ie<m)  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); q ?m<9`  
        end cDh\$7'b  
    end D~@lpcI  
    )!d_Td\-  
    % 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)  e8<[2J)P&  
    cE0Kvqe`  
    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 L5!H.q  
    *FEY"W+bY  
    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)