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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 JYj*.Q0  
    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! \dU.#^ryp  
     
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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 ]r!|@AWrQ\  
    function z = zernfun(n,m,r,theta,nflag) E> pr})^w  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. Oto8?4[n  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N * G*VY#L  
    %   and angular frequency M, evaluated at positions (R,THETA) on the >{(c\oMD  
    %   unit circle.  N is a vector of positive integers (including 0), and du }HTrsC  
    %   M is a vector with the same number of elements as N.  Each element CR.d3!&28  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) yuC$S&Y >!  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 6oQ7u90z*  
    %   and THETA is a vector of angles.  R and THETA must have the same bxPa|s?  
    %   length.  The output Z is a matrix with one column for every (N,M) 7;@YR  
    %   pair, and one row for every (R,THETA) pair. 0sSBwG  
    % vv)w@A:Vn)  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike <t!0{FJ  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), >A]l|#Rz  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral {?^ES*5  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, jTqJ(M}L  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized X} V]3  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. FZU1WBNL%t  
    % ~)$R'=  
    %   The Zernike functions are an orthogonal basis on the unit circle. 4J`-&05O  
    %   They are used in disciplines such as astronomy, optics, and gA_oJW4_  
    %   optometry to describe functions on a circular domain. D1deh=  
    % Fv,c8f  
    %   The following table lists the first 15 Zernike functions. GD}rsBQNkJ  
    %  :Kyr}-  
    %       n    m    Zernike function           Normalization nTsV>lQY,  
    %       -------------------------------------------------- 'HfI~wN  
    %       0    0    1                                 1 :T PG~`k(  
    %       1    1    r * cos(theta)                    2 ":T"Y;  
    %       1   -1    r * sin(theta)                    2 n::i$ZUdK  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) GCQOjqiR  
    %       2    0    (2*r^2 - 1)                    sqrt(3) $l.8  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 1Zk1!> ?  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) `ba<eT':  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) i)cG  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) hx%UZ<a  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) @ >'Wiq!  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) hC{2LLu;n  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Dz.kJ_"Ro  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 8  rE`  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) MwD+'5   
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Q$'\_zV  
    %       -------------------------------------------------- h$~$a;2cR  
    % /^{Q(R(X<  
    %   Example 1: b; ;y|H  
    % N0D5N(kH%  
    %       % Display the Zernike function Z(n=5,m=1) Z$Ps_Ik  
    %       x = -1:0.01:1; ;CL^2{  
    %       [X,Y] = meshgrid(x,x); uVZm9Sp  
    %       [theta,r] = cart2pol(X,Y); <.lN'i;(  
    %       idx = r<=1; @:'E9J06  
    %       z = nan(size(X)); /YwwG;1  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); )i39'0a  
    %       figure ss|n7  
    %       pcolor(x,x,z), shading interp )('{q}JxV  
    %       axis square, colorbar 3!*` hQ;s  
    %       title('Zernike function Z_5^1(r,\theta)') }EfRYE$E  
    % e6gj'GmY  
    %   Example 2: c7?|Tipc  
    % _mQ~[}y+?  
    %       % Display the first 10 Zernike functions A-\n"}4  
    %       x = -1:0.01:1; S=w~bz, /  
    %       [X,Y] = meshgrid(x,x); z}VCiS0  
    %       [theta,r] = cart2pol(X,Y); =5pwNi_S  
    %       idx = r<=1; J{EK}'  
    %       z = nan(size(X)); tUfze9m  
    %       n = [0  1  1  2  2  2  3  3  3  3]; I.6#>=  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ]%Whtj.,x7  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; pek5P4W_  
    %       y = zernfun(n,m,r(idx),theta(idx)); 'HvW&~i(  
    %       figure('Units','normalized') g2r8J0v  
    %       for k = 1:10 ? zic1i  
    %           z(idx) = y(:,k); mp]UUpt  
    %           subplot(4,7,Nplot(k)) :e_yOT}}  
    %           pcolor(x,x,z), shading interp a 6fH*2E  
    %           set(gca,'XTick',[],'YTick',[]) <&M5#:u  
    %           axis square QmPHf*w[  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @yPI$"Ma  
    %       end &19z|Id  
    % a5g1.6hF  
    %   See also ZERNPOL, ZERNFUN2. 7.^1I7O  
    ol4!#4Y&{  
    %   Paul Fricker 11/13/2006 7 Uu  
    C\[g>_J  
    g'eJN  
    % Check and prepare the inputs: )i.\q   
    % ----------------------------- ?=Z0N&}[  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 37,)/8]lG  
        error('zernfun:NMvectors','N and M must be vectors.') `jFvG\aC  
    end 3o__tU)B  
    eY$Q}BcW  
    if length(n)~=length(m) %} Ob~m>P  
        error('zernfun:NMlength','N and M must be the same length.') vr>Rd{dm  
    end %eqL)pC]  
    Q# $dp  
    n = n(:); YC~kq?  
    m = m(:); j~9,Ct  
    if any(mod(n-m,2)) 5adB5)`  
        error('zernfun:NMmultiplesof2', ... A832z`  
              'All N and M must differ by multiples of 2 (including 0).') Uefw  
    end m&#a M8:\  
    uO`YA]  
    if any(m>n) F{aM6I  
        error('zernfun:MlessthanN', ... Ax+q/nvnb  
              'Each M must be less than or equal to its corresponding N.') U5wO;MA  
    end bQM_rqjJGw  
    FmRa]31W  
    if any( r>1 | r<0 ) AU +2'  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') \=/^H  
    end ~cx/>Hu  
    2L_ts=  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) \uV;UH7qe  
        error('zernfun:RTHvector','R and THETA must be vectors.') o93A:fc  
    end G(~"Zt}?  
    K:eP Il{JE  
    r = r(:); MoP 0qNk  
    theta = theta(:); pYs"Y;%  
    length_r = length(r); ojitBo~  
    if length_r~=length(theta) ~m56t5+uw  
        error('zernfun:RTHlength', ... C[O \aW  
              'The number of R- and THETA-values must be equal.') q,a|lH  
    end l0$ +)FKd  
    ;0VE *  
    % Check normalization: S)*eAON9  
    % -------------------- ' RjFWHAp  
    if nargin==5 && ischar(nflag) d98ZC+q  
        isnorm = strcmpi(nflag,'norm'); q|%(47}z  
        if ~isnorm Q04iuhDO:  
            error('zernfun:normalization','Unrecognized normalization flag.') k w!1]N  
        end ATU 2\Y  
    else |EaEdA@T  
        isnorm = false; i.Qy0  
    end { O*maE"  
    c.PPVqx  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,9f$a n  
    % Compute the Zernike Polynomials ZIx,?E+eJ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9c1n  
    5xHl6T+  
    % Determine the required powers of r: h^5'i} @u  
    % ----------------------------------- HBL)_c{/O  
    m_abs = abs(m); ; . c]0  
    rpowers = []; }cE,&n  
    for j = 1:length(n) BS#@ehdig  
        rpowers = [rpowers m_abs(j):2:n(j)]; T%xB|^lf  
    end X] /r'Tz  
    rpowers = unique(rpowers); (6G5UwSt  
    f[!Q R  
    % Pre-compute the values of r raised to the required powers, ;%#@vXH[Oo  
    % and compile them in a matrix: >w?O?&Q$  
    % ----------------------------- SA|f1R2uS  
    if rpowers(1)==0 lfKrd3KS_  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); l 49)Cv/  
        rpowern = cat(2,rpowern{:}); #]|9aVrr  
        rpowern = [ones(length_r,1) rpowern]; C``%<)WC  
    else :(Feg2c  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); XH0R:+s  
        rpowern = cat(2,rpowern{:}); 2Fce| Tn  
    end vpUS(ztvs  
    % j7lLSusX  
    % Compute the values of the polynomials: c|Nv^V*2  
    % -------------------------------------- rj*4ZA?  
    y = zeros(length_r,length(n)); 81/Bn!  
    for j = 1:length(n) +aV>$Y  
        s = 0:(n(j)-m_abs(j))/2; 8KW}XG  
        pows = n(j):-2:m_abs(j); R)#D{/#FW  
        for k = length(s):-1:1 atFj Vk^  
            p = (1-2*mod(s(k),2))* ... ue$\ i=jw  
                       prod(2:(n(j)-s(k)))/              ... Mx-,:a9}  
                       prod(2:s(k))/                     ... pWB)N7x&  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Sg0 _l(  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Ne.W-,X^cL  
            idx = (pows(k)==rpowers);  OXzJ%&h  
            y(:,j) = y(:,j) + p*rpowern(:,idx); >=i47-H  
        end BRV /7ao="  
         9QI\[lT&  
        if isnorm Q4Q*5>  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); `yHV10  
        end Ni{ (=&*=  
    end ' d1E~A  
    % END: Compute the Zernike Polynomials +tOBt("5/  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r 06}@7  
    6lq7zi}'w  
    % Compute the Zernike functions: ^&DHBx"J  
    % ------------------------------ NwuME/C7#  
    idx_pos = m>0; Om{[ <tL  
    idx_neg = m<0; 2[Q*?N  
    6,0pkx&Nv  
    z = y; ZsUxO%jP  
    if any(idx_pos) _pKW($\  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); v)+wr[Qs  
    end 2 ,;+)  
    if any(idx_neg) F)Yn1&a#H  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); }jfU qqFd  
    end 3b{8c8N^  
    mhp5}  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) v (=fV/  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. V_~}7~ I  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ()(@Qcc  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive <=cj)  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, LlRvm/  
    %   and THETA is a vector of angles.  R and THETA must have the same HHCsWe-  
    %   length.  The output Z is a matrix with one column for every P-value, @o44b!i  
    %   and one row for every (R,THETA) pair. q uv`~qn  
    % R/b)hP ~  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ).N}x^  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) JQsS=m7Et  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) } ~=53$+  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ?Hf^& yo  
    %   for all p. y*\ M7}](  
    % &.=d,XKN  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 9 o-T#~i  
    %   Zernike functions (order N<=7).  In some disciplines it is Fmt5"3B  
    %   traditional to label the first 36 functions using a single mode }&rf'E9  
    %   number P instead of separate numbers for the order N and azimuthal ^gH.5L0]gH  
    %   frequency M. ^P:9iu)+]~  
    % OWB^24Z&3  
    %   Example: 0UWLs_k:  
    % W8yr06{]  
    %       % Display the first 16 Zernike functions T {(6*^g<B  
    %       x = -1:0.01:1; ')bx1gc(?  
    %       [X,Y] = meshgrid(x,x); t{!}^{ "5  
    %       [theta,r] = cart2pol(X,Y); %9-).k  
    %       idx = r<=1; -G;4['p  
    %       p = 0:15; {^"c>'R  
    %       z = nan(size(X)); i<bFF03*S  
    %       y = zernfun2(p,r(idx),theta(idx)); C n\'sb{  
    %       figure('Units','normalized') r&1N8o  
    %       for k = 1:length(p) @XDU !<N  
    %           z(idx) = y(:,k); zL3~,z/o  
    %           subplot(4,4,k) x nWapG  
    %           pcolor(x,x,z), shading interp 2y ~]Uo  
    %           set(gca,'XTick',[],'YTick',[]) rA8neO)  
    %           axis square xlgN}M  
    %           title(['Z_{' num2str(p(k)) '}']) )f(#Fn  
    %       end n9t8RcJS:  
    % 3UD_2[aqN(  
    %   See also ZERNPOL, ZERNFUN. I@+dE V`Lf  
    S=krF yFw  
    %   Paul Fricker 11/13/2006 L;fhJ~ r  
    AJ^9[j}  
    F{]dq/{  
    % Check and prepare the inputs: ZaH<\`=%  
    % ----------------------------- "p#mNc  
    if min(size(p))~=1 FFR_1Vf  
        error('zernfun2:Pvector','Input P must be vector.') GLnj& Ve  
    end h+,zfVJu  
    ?%;7k'0"  
    if any(p)>35 .9lx@6]+  
        error('zernfun2:P36', ... 46Nl];g1`  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... WkXa%OZ  
               '(P = 0 to 35).']) -AD3Pd|Y[  
    end Xy_+L_h^  
    }j+ZF'#  
    % Get the order and frequency corresonding to the function number: 2[r#y1ro  
    % ---------------------------------------------------------------- Ls5|4%+&  
    p = p(:); 4 FGcCE3  
    n = ceil((-3+sqrt(9+8*p))/2); MHI0>QsI  
    m = 2*p - n.*(n+2); yGZb  
    y*vs}G'W  
    % Pass the inputs to the function ZERNFUN: 6n  
    % ---------------------------------------- $w)yQ %  
    switch nargin "CT'^d+  
        case 3 zK k;&y|{  
            z = zernfun(n,m,r,theta); db@i*Bf  
        case 4 8nt:peJ$+  
            z = zernfun(n,m,r,theta,nflag); DFVaZN?~  
        otherwise $;@^coz9U  
            error('zernfun2:nargin','Incorrect number of inputs.') Dx4?6  
    end (](:0H  
    yJppPIW^  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) ,gUSW  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 5T:e4U&  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of #i%it  
    %   order N and frequency M, evaluated at R.  N is a vector of Ha[Bf*  
    %   positive integers (including 0), and M is a vector with the Z Mt9'w;  
    %   same number of elements as N.  Each element k of M must be a Urm&4&y  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) vCb3Ra~L`  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is B~D{p t3y  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix E2Q;1Re@  
    %   with one column for every (N,M) pair, and one row for every K#%L6=t$<  
    %   element in R. =$X5O&E3'  
    % p3&/F=T;)  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- V\W?@V9g-  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ~-.}]N+([  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to O6pswMhAc  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 Mi%i_T^i  
    %   for all [n,m]. P%8 Gaa=  
    % fFMGpibkM  
    %   The radial Zernike polynomials are the radial portion of the T&oY:1D,g  
    %   Zernike functions, which are an orthogonal basis on the unit qg7.E+  
    %   circle.  The series representation of the radial Zernike }TzMWdT  
    %   polynomials is V: fz  
    % ?T3zA2  
    %          (n-m)/2 "T=Z/@Vy  
    %            __ MRR5j;4GK  
    %    m      \       s                                          n-2s E2 Q[  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r FIL?nkYEO  
    %    n      s=0 GbUw:I  
    % R9A8)dDz  
    %   The following table shows the first 12 polynomials. IDQ@h`"B  
    % $sTbFY  
    %       n    m    Zernike polynomial    Normalization ;PCnEs  
    %       --------------------------------------------- JR8 b[Oj.S  
    %       0    0    1                        sqrt(2) "1FPe63\*O  
    %       1    1    r                           2 {_&'tXL  
    %       2    0    2*r^2 - 1                sqrt(6) EiQX* v  
    %       2    2    r^2                      sqrt(6) Jz&a9  
    %       3    1    3*r^3 - 2*r              sqrt(8) n+QUT   
    %       3    3    r^3                      sqrt(8) )e(Rf!P{  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) PIR#M('  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) cR{F|0X  
    %       4    4    r^4                      sqrt(10) GlHP`&;UH  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) \.aKxj5  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ECyG$j0  
    %       5    5    r^5                      sqrt(12) x8xz33  
    %       --------------------------------------------- +7Uv|LZ~@  
    % fN1b+ d~*6  
    %   Example: d5>EvK U  
    % ken.#>w  
    %       % Display three example Zernike radial polynomials R XCjYzt  
    %       r = 0:0.01:1; 3ey.r%n  
    %       n = [3 2 5]; q@G}Hjn  
    %       m = [1 2 1]; VbDk44X.W  
    %       z = zernpol(n,m,r); sf0\#Q  
    %       figure ]K3bDU~  
    %       plot(r,z) 04WxV(fo'  
    %       grid on h<Ct[46,S  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') A0>r]<y  
    % dVPY07P  
    %   See also ZERNFUN, ZERNFUN2. 3RX9LJGX  
    !{4'=+  
    % A note on the algorithm. +I\54PBws  
    % ------------------------ {!j)j6(NY  
    % The radial Zernike polynomials are computed using the series Kx?.g#>U;  
    % representation shown in the Help section above. For many special y^e3Gyk  
    % functions, direct evaluation using the series representation can it-]-=mqb  
    % produce poor numerical results (floating point errors), because V.9p4k`  
    % the summation often involves computing small differences between ]WzeJ"r {3  
    % large successive terms in the series. (In such cases, the functions -%0pYB  
    % are often evaluated using alternative methods such as recurrence YV _ 7 .+A  
    % relations: see the Legendre functions, for example). For the Zernike gKY6S?  
    % polynomials, however, this problem does not arise, because the bsm/y+R  
    % polynomials are evaluated over the finite domain r = (0,1), and qqJghV$Oj  
    % because the coefficients for a given polynomial are generally all #sg*GK+|:R  
    % of similar magnitude. rq^%)tR  
    % 8f<y~L_(`  
    % ZERNPOL has been written using a vectorized implementation: multiple t-5K dLB  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ?(U a+*b  
    % values can be passed as inputs) for a vector of points R.  To achieve ie11syhV"  
    % this vectorization most efficiently, the algorithm in ZERNPOL o%f:BJS  
    % involves pre-determining all the powers p of R that are required to Y]=k"]:%  
    % compute the outputs, and then compiling the {R^p} into a single aM xd"cTzx  
    % matrix.  This avoids any redundant computation of the R^p, and H0!$aO  
    % minimizes the sizes of certain intermediate variables. gkX7,J-0  
    % tUuARo7#  
    %   Paul Fricker 11/13/2006 d/T&J=  
    }a/z.&x]V  
    Fg 8lX9L  
    % Check and prepare the inputs: \>>P%EU,  
    % ----------------------------- piH0_7qr  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) pGfGGY>i%  
        error('zernpol:NMvectors','N and M must be vectors.') Q% d1O  
    end sYo&@~T  
    BzzZ.AH~  
    if length(n)~=length(m) ZW9OPwV  
        error('zernpol:NMlength','N and M must be the same length.') ?:M4GY" gV  
    end AAs&P+;  
    |AacV  
    n = n(:); p!~1~q6  
    m = m(:); ' tHa5`  
    length_n = length(n); 0 )cSm"s  
    8MI8~  
    if any(mod(n-m,2)) liG|#ny{  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ;c)( 'k<  
    end Z:<an+v|5  
    Xtfs)"  
    if any(m<0) DRR)mQBb  
        error('zernpol:Mpositive','All M must be positive.') Qclq^|O0  
    end {;E6jw@  
    Cl9rJ oT  
    if any(m>n) dWQB1Y*N  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') y.I&x#(^  
    end *Ti"8^`6  
    |IV7g*J89  
    if any( r>1 | r<0 ) ^iBIp#  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 122s 7A  
    end }#u #m.  
    qrp@   
    if ~any(size(r)==1) ^H7xFd|>  
        error('zernpol:Rvector','R must be a vector.') kxd*B P  
    end tk*-Cx?_  
    g`Cv[Pq?at  
    r = r(:); $i6z)]rjg  
    length_r = length(r); },#7  
    ^e <E/j{~  
    if nargin==4 [FrLxU  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); n}[S  
        if ~isnorm :b(W&iBWhI  
            error('zernpol:normalization','Unrecognized normalization flag.') enZZ+|h  
        end 'fGKRd|)  
    else jwAYlnQ^EM  
        isnorm = false; ypG*41  
    end F[$cE  
    e3W~6P  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1%*\*z  
    % Compute the Zernike Polynomials 9]w?mHslE  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% IQ_s]b;z  
    G"E_4YkJ  
    % Determine the required powers of r: X?[ )e  
    % ----------------------------------- |idw?qCn  
    rpowers = []; ~CkOiWC0  
    for j = 1:length(n) GVJ||0D  
        rpowers = [rpowers m(j):2:n(j)]; E/a2b(,Tg  
    end e2N K7  
    rpowers = unique(rpowers); J ffaT_"\  
    0QW=2rs  
    % Pre-compute the values of r raised to the required powers, j}",+H v  
    % and compile them in a matrix: ZK'46lh  
    % ----------------------------- z)U7  
    if rpowers(1)==0 @`C'tfG/4  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); % g  
        rpowern = cat(2,rpowern{:}); bTrusSAl  
        rpowern = [ones(length_r,1) rpowern]; :&TM0O  
    else Z:7eroZP  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); rvy%8%e?  
        rpowern = cat(2,rpowern{:}); {9}CU~R  
    end W"_<SYVJ  
    ) c\Y!vS  
    % Compute the values of the polynomials: >8k Xa.)84  
    % -------------------------------------- &=jPt%7#M  
    z = zeros(length_r,length_n); 4Ex&AR8  
    for j = 1:length_n e 9RYk:O  
        s = 0:(n(j)-m(j))/2; NT.#U?9c  
        pows = n(j):-2:m(j); h2f8-}fsq  
        for k = length(s):-1:1 $7DW-TA  
            p = (1-2*mod(s(k),2))* ... A2:}bb~H  
                       prod(2:(n(j)-s(k)))/          ...  *0^~@U  
                       prod(2:s(k))/                 ... aMY@**^v  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... :R=6Ku>  
                       prod(2:((n(j)+m(j))/2-s(k))); 0jlM~H  
            idx = (pows(k)==rpowers); A| A#|D  
            z(:,j) = z(:,j) + p*rpowern(:,idx); o>,r<  
        end J'|=J   
         0Q&(j7`^@  
        if isnorm >x;\H(g  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); FUI*nkZY  
        end ^gvTc+|  
    end }8Y! -qX  
    N=<`|I  
    % 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)  zdJPMNHg  
    z!6_u@^-  
    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。 ??nT[bhQ  
    3/vtx9D  
    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)