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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 "k)}qI{  
    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! vtvF)jlX  
     
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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 vx:MLmZ.  
    function z = zernfun(n,m,r,theta,nflag) $$U Mc-Pq  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 7MRu=Z.-b  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 'S_kD! BO  
    %   and angular frequency M, evaluated at positions (R,THETA) on the XCQS_'D  
    %   unit circle.  N is a vector of positive integers (including 0), and ~]+-<O^U~  
    %   M is a vector with the same number of elements as N.  Each element u/`jb2eEU:  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) @x9DV{j)V  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, N|Cx";,|FZ  
    %   and THETA is a vector of angles.  R and THETA must have the same K k 5 vC{  
    %   length.  The output Z is a matrix with one column for every (N,M) W<J".2D  
    %   pair, and one row for every (R,THETA) pair. W/z\j/Rgc  
    % *?;<buJb?  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Ix+===6  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), RYU(z;+0p  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ~Wh} W((L  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, eY3l^Su1  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized kOv2E]  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 5hNjJqu  
    % K\Oz ~,z  
    %   The Zernike functions are an orthogonal basis on the unit circle. 4vri=P 2%  
    %   They are used in disciplines such as astronomy, optics, and h'{}eYb+   
    %   optometry to describe functions on a circular domain. 5F@7A2ZR  
    % 9fk@C/$  
    %   The following table lists the first 15 Zernike functions. VieX 5  
    % |K},f,  
    %       n    m    Zernike function           Normalization czMu<@c [  
    %       -------------------------------------------------- #+mt}w/  
    %       0    0    1                                 1 6pkZ8Vp:  
    %       1    1    r * cos(theta)                    2 %s.hqr,I  
    %       1   -1    r * sin(theta)                    2 fz%I'+!  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) "AN2K  
    %       2    0    (2*r^2 - 1)                    sqrt(3) =[wVRQ?  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ;]ojfR=?%  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) %O 5 k+~9  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) dXAKk[uf  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) "CYh"4]@rD  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) v 4@=>L  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) :D-xa!7  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) nC^|83  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 2o0.ttBAqZ  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) f/spJ<B).4  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) (kpn"]^'  
    %       -------------------------------------------------- ML6V,V/e  
    % 7X3<8:%  
    %   Example 1: }-3| v<d  
    % ;#np~gL  
    %       % Display the Zernike function Z(n=5,m=1)  &!I^m  
    %       x = -1:0.01:1; Evd>s  
    %       [X,Y] = meshgrid(x,x); Da#|}m0>  
    %       [theta,r] = cart2pol(X,Y); 1}#(4tw)  
    %       idx = r<=1; *9"L?S(X#  
    %       z = nan(size(X)); 19)fN-0Z  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); [al,UO  
    %       figure d*%-r2K  
    %       pcolor(x,x,z), shading interp Am~ NBQ7  
    %       axis square, colorbar fH_G;#q  
    %       title('Zernike function Z_5^1(r,\theta)') M8Y\1#~  
    % \cq gCab/2  
    %   Example 2: B_FfXFQm<  
    % @Q:5{?  
    %       % Display the first 10 Zernike functions ,E]u[7A  
    %       x = -1:0.01:1; %|(~k*s4  
    %       [X,Y] = meshgrid(x,x); PV?XpT  
    %       [theta,r] = cart2pol(X,Y); 0sjw`<ic  
    %       idx = r<=1; pg3B^  
    %       z = nan(size(X)); 9*!C|gC9Ia  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 8l|v#^v  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; )A]E:]2  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; "hRw_<  
    %       y = zernfun(n,m,r(idx),theta(idx)); zx7*Bnu0  
    %       figure('Units','normalized') {7^7)^@  
    %       for k = 1:10 . e2qa  
    %           z(idx) = y(:,k); ?#@JH  
    %           subplot(4,7,Nplot(k)) H-%)r&"vn  
    %           pcolor(x,x,z), shading interp *&X.  
    %           set(gca,'XTick',[],'YTick',[]) &gc8"B@V  
    %           axis square a jy.K'B*  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) uMm/$#E  
    %       end '>:mEXK}w  
    % }{*((@GY}  
    %   See also ZERNPOL, ZERNFUN2. /p~Wk4'  
    Qh%(yL!  
    %   Paul Fricker 11/13/2006 ]JQk,<l5E  
    [3`T/Wm  
    1nh2()QI[  
    % Check and prepare the inputs: tN|sHgs  
    % ----------------------------- G!~[+B  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) L2>UA<@mZ  
        error('zernfun:NMvectors','N and M must be vectors.') q|J3]F !n  
    end jREj]V>  
    \M>+6m@w  
    if length(n)~=length(m) pyK|zvr-r  
        error('zernfun:NMlength','N and M must be the same length.') sMAc+9G9k  
    end >j1\]uo  
    '>(R'g42n  
    n = n(:); 84[T!cDk  
    m = m(:);  eWO^n>Y  
    if any(mod(n-m,2)) mLM$dk3  
        error('zernfun:NMmultiplesof2', ... L{$ZL&  
              'All N and M must differ by multiples of 2 (including 0).') ^.Y"<oZSS  
    end o"@y=n/  
    2BOe,giy  
    if any(m>n) 't=\YFQ*v  
        error('zernfun:MlessthanN', ... ADRjCk}I  
              'Each M must be less than or equal to its corresponding N.') =p>"PqJ/7n  
    end ~o`I[-g)  
    q#B^yk|Y  
    if any( r>1 | r<0 ) nf!RB-orF  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 4cK6B)X  
    end qPdNI1 |  
    b7>^w<ki  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ' >[KVvm  
        error('zernfun:RTHvector','R and THETA must be vectors.') {Q8DPkW  
    end iZ+\vO?|  
    bL 5z%bV  
    r = r(:); 2A@9jl s  
    theta = theta(:); XtfO;`   
    length_r = length(r); }*l V  
    if length_r~=length(theta) TEOV>Tt  
        error('zernfun:RTHlength', ... W#|]m=2W  
              'The number of R- and THETA-values must be equal.') N1WP  
    end ?iG}Qj@5  
    ?}%Gr,tj2  
    % Check normalization: FQ?,&s$Bmd  
    % -------------------- z<rdxn,9  
    if nargin==5 && ischar(nflag) V#!ihL/>  
        isnorm = strcmpi(nflag,'norm'); HGmgQ>q@M$  
        if ~isnorm 9z 5K  -s  
            error('zernfun:normalization','Unrecognized normalization flag.') ws5x53K  
        end J=6 7As  
    else /_E:sI9(  
        isnorm = false; 0B)l"$W[)/  
    end f&t]O$  
    VtF^; f  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xI'<4lo7Z  
    % Compute the Zernike Polynomials >%+ "-bY  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dz.]5R  
    ]@1YgV  
    % Determine the required powers of r: DR/qe0D  
    % ----------------------------------- ?_[xpK()  
    m_abs = abs(m); o#E 3{zM  
    rpowers = []; Ea1{9> S  
    for j = 1:length(n) =nOV!!  
        rpowers = [rpowers m_abs(j):2:n(j)]; HyXw^ +tsj  
    end EDvK9J  
    rpowers = unique(rpowers); tA$,4B?  
    ~6@zXHAS  
    % Pre-compute the values of r raised to the required powers, 8 f%@:}H  
    % and compile them in a matrix: { yU1db^  
    % ----------------------------- I})la!9   
    if rpowers(1)==0 _:0<]<x?  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); *=dFTd"#  
        rpowern = cat(2,rpowern{:}); 4NbX! "0  
        rpowern = [ones(length_r,1) rpowern]; )eGGA6G  
    else bv0B  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); n1o/-UY  
        rpowern = cat(2,rpowern{:}); CmRn  
    end AL! ^1hCF  
    y4) M,+O5  
    % Compute the values of the polynomials: g^8dDY[%  
    % -------------------------------------- ,Ihuo5>/z  
    y = zeros(length_r,length(n)); Pca~V>Hd  
    for j = 1:length(n) pOD|  
        s = 0:(n(j)-m_abs(j))/2; 8-cG[/|0  
        pows = n(j):-2:m_abs(j); " e g`3v  
        for k = length(s):-1:1 !`\W8JT+  
            p = (1-2*mod(s(k),2))* ... ^G= wRtS  
                       prod(2:(n(j)-s(k)))/              ... 'T7JXV5  
                       prod(2:s(k))/                     ... Gk,{{:M:5  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... jpyV52  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); WM: ~P$%cx  
            idx = (pows(k)==rpowers); _`/0/69  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 5. :To2  
        end JWy$` "{  
         tu77Sb  
        if isnorm Nv*x^y]  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); :q~qRRmjBe  
        end SDiZOypS  
    end _baqN!N  
    % END: Compute the Zernike Polynomials |`s}PcV  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B+);y  
    = Ii@-C  
    % Compute the Zernike functions: swG^L$r`  
    % ------------------------------ cGkl=-oQ'  
    idx_pos = m>0; riZFcVsB  
    idx_neg = m<0; 0ang~_  
    ' F`*(\#  
    z = y;  g}Hk4+  
    if any(idx_pos) jp8=>mk  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); S n.I ]:l  
    end #"ayq,GC<  
    if any(idx_neg) vKAHf;1  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); oHI~-{m3)  
    end 2P$lXGjh  
    r{)d?Ho=  
    % 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) tyP-J4J  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. .~v~~VL1NS  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated +Jt"JJ>%k  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Cb=r8C  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, lU Uq|Qr  
    %   and THETA is a vector of angles.  R and THETA must have the same W{ eu_  
    %   length.  The output Z is a matrix with one column for every P-value, 8o-?Y.2  
    %   and one row for every (R,THETA) pair. JsnavI6  
    % Da-F(^E  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike hp-< 8Mf  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) G]P4[#5  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ?jqZeO#W7  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 G8u8&|  
    %   for all p. e"r}I!.  
    % H7Y}qP5X  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 4bAgbx-^  
    %   Zernike functions (order N<=7).  In some disciplines it is Z6D4VZVF  
    %   traditional to label the first 36 functions using a single mode T:)>Tcv}:  
    %   number P instead of separate numbers for the order N and azimuthal u:HKmP;  
    %   frequency M. 7IK<9i4O  
    % {)b`fq  
    %   Example: Jk{>*jYk`  
    % ~%#?;hJ  
    %       % Display the first 16 Zernike functions !-N!8 0  
    %       x = -1:0.01:1; |o!<@/iH=  
    %       [X,Y] = meshgrid(x,x); "b1_vA]03  
    %       [theta,r] = cart2pol(X,Y); 44%H? ,d  
    %       idx = r<=1;  u`bWn  
    %       p = 0:15; GK&yP%Z3  
    %       z = nan(size(X)); xR_]^Get  
    %       y = zernfun2(p,r(idx),theta(idx)); Ku&(+e  
    %       figure('Units','normalized') {_q2kk  
    %       for k = 1:length(p) uXhp+q\  
    %           z(idx) = y(:,k); ` 4k;`a  
    %           subplot(4,4,k) 2~ 'Q#(  
    %           pcolor(x,x,z), shading interp s|,]Nb=z/  
    %           set(gca,'XTick',[],'YTick',[]) G!rcY5!J  
    %           axis square  fx;5j;  
    %           title(['Z_{' num2str(p(k)) '}']) 3M@>kIT8  
    %       end OW-+23)sj  
    % z 9D2,N.  
    %   See also ZERNPOL, ZERNFUN. 5Q%#Z L/'  
    E_H1X'|qS4  
    %   Paul Fricker 11/13/2006 qS2%U?S7  
    ?0?'  
    c<H4rB  
    % Check and prepare the inputs: I* bjE '  
    % ----------------------------- N$y4>g  
    if min(size(p))~=1 )j9FB  
        error('zernfun2:Pvector','Input P must be vector.') ze 4/XR  
    end Fe=4^.  
    RU{}qPs?  
    if any(p)>35 t[Q^Xp  
        error('zernfun2:P36', ... TM"-X\e~{  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... JfVay I=  
               '(P = 0 to 35).']) WEsH@ [  
    end .Z^g 7 *s  
    J BwTmOvQ  
    % Get the order and frequency corresonding to the function number: #ERn 8k  
    % ---------------------------------------------------------------- H!Od.$ZIX  
    p = p(:); NNfCJ|  
    n = ceil((-3+sqrt(9+8*p))/2); v4v+;[a%  
    m = 2*p - n.*(n+2); S5d{dTPq  
    uZYeru"w  
    % Pass the inputs to the function ZERNFUN: S1B/ClKWq  
    % ---------------------------------------- %bimcRX#W  
    switch nargin 0)/214^&  
        case 3 )F~_KD)7jJ  
            z = zernfun(n,m,r,theta); Y{O&- 5H^|  
        case 4 1z`,*eD7  
            z = zernfun(n,m,r,theta,nflag); zJsoenU  
        otherwise 6 %=BYDF  
            error('zernfun2:nargin','Incorrect number of inputs.') AzV5Re8M  
    end s{IoL_PJP  
    7d<v\=J}  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) BmM,vllO  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. R#`itIYh  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of j*zK"n  
    %   order N and frequency M, evaluated at R.  N is a vector of N:<O  
    %   positive integers (including 0), and M is a vector with the 5_`}$"<~  
    %   same number of elements as N.  Each element k of M must be a Ocb2XEF  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ,,J3 h  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is f8 ja Mn9o  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix j{^(TE  
    %   with one column for every (N,M) pair, and one row for every c`+ITNV  
    %   element in R. y(dS1.5F  
    % 3/AUV%+  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- K$.zO4  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is md`ToU  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to /OP*ARoC21  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 e ?YbG.(E9  
    %   for all [n,m]. V4-=Ni]k  
    % F[u%t34'  
    %   The radial Zernike polynomials are the radial portion of the -1:Z^&e/  
    %   Zernike functions, which are an orthogonal basis on the unit 6ZR0_v;TD  
    %   circle.  The series representation of the radial Zernike _E;Y ~I,i  
    %   polynomials is ETOc4hMO  
    % NM@An2  
    %          (n-m)/2 FNuu',:  
    %            __ w b[(_@eZ  
    %    m      \       s                                          n-2s @>]3xHE6#=  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r kut|A  
    %    n      s=0 !_) ^bRd  
    % @QG1\W'  
    %   The following table shows the first 12 polynomials. M@kZ(Rkv  
    % d]3sC  
    %       n    m    Zernike polynomial    Normalization @"8QG^q8de  
    %       --------------------------------------------- m'tk#C  
    %       0    0    1                        sqrt(2) 3\+p1f4  
    %       1    1    r                           2 hBhkb ~Oky  
    %       2    0    2*r^2 - 1                sqrt(6) sQZ8<DpB  
    %       2    2    r^2                      sqrt(6) }L!`K"^O&  
    %       3    1    3*r^3 - 2*r              sqrt(8) CiI: uU  
    %       3    3    r^3                      sqrt(8) \2pFFVT  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) |esjhf}H>v  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) G 7]wg>*  
    %       4    4    r^4                      sqrt(10) )^H9C"7T  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) W~3tQ!  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) k_>{"Rc  
    %       5    5    r^5                      sqrt(12) m;f?}z_\$  
    %       --------------------------------------------- H4NEB1 TO>  
    % %KF:- w  
    %   Example: )|R9mW=k9P  
    % Ra5'x)m36)  
    %       % Display three example Zernike radial polynomials >8fH5  
    %       r = 0:0.01:1; UwkX[u  
    %       n = [3 2 5]; <UJJ],)^1A  
    %       m = [1 2 1]; v4_OUA>z,  
    %       z = zernpol(n,m,r); yrAzD=  
    %       figure "5:f{GfO#v  
    %       plot(r,z) ATM:As:<@  
    %       grid on hxVM]e[  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') oh~ vo!  
    % g@i>R>  
    %   See also ZERNFUN, ZERNFUN2. U!:!]DX(  
    O/9%"m:i  
    % A note on the algorithm. V2{#<d-T!  
    % ------------------------ %D(prA_w  
    % The radial Zernike polynomials are computed using the series  |7zP 8  
    % representation shown in the Help section above. For many special Treh{s  
    % functions, direct evaluation using the series representation can 'S7@+kJ  
    % produce poor numerical results (floating point errors), because /}`/i(k  
    % the summation often involves computing small differences between d[e:}1  
    % large successive terms in the series. (In such cases, the functions z-G7Y#  
    % are often evaluated using alternative methods such as recurrence $H-D9+8 7  
    % relations: see the Legendre functions, for example). For the Zernike eD{ @0&   
    % polynomials, however, this problem does not arise, because the &17,]#3  
    % polynomials are evaluated over the finite domain r = (0,1), and H r^15  
    % because the coefficients for a given polynomial are generally all E`uaE=Mdq  
    % of similar magnitude. `)=A !x y  
    % R g0 XW6  
    % ZERNPOL has been written using a vectorized implementation: multiple s>@#9psm  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] U++~3e@l  
    % values can be passed as inputs) for a vector of points R.  To achieve I0w@S7  
    % this vectorization most efficiently, the algorithm in ZERNPOL rw8J:?0x  
    % involves pre-determining all the powers p of R that are required to  pO/SV6N  
    % compute the outputs, and then compiling the {R^p} into a single cxD}t'T  
    % matrix.  This avoids any redundant computation of the R^p, and p~q_0Pg%  
    % minimizes the sizes of certain intermediate variables. 7t-*L}~WA  
    % B2G5h baA  
    %   Paul Fricker 11/13/2006 Krr?`n  
    0?F@iB~1F  
    oBj>9I;  
    % Check and prepare the inputs: I,<>%Z|'  
    % ----------------------------- RZd4(7H=q  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) /%uZKG P  
        error('zernpol:NMvectors','N and M must be vectors.') I#S~  
    end qXB03}] G  
    hr<7l C  
    if length(n)~=length(m) yF/< :  
        error('zernpol:NMlength','N and M must be the same length.') (Gi+7GMV'  
    end LZQFj/,Jg  
    V+=*2?1  
    n = n(:); v$)@AE  
    m = m(:); JMirz~%ib  
    length_n = length(n); yL ;o{ G  
    YMj7  
    if any(mod(n-m,2)) s3Krob`C5  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ;WvYzd9  
    end fXu~69_  
    9B+ zJ Vte  
    if any(m<0) KkdG.c'  
        error('zernpol:Mpositive','All M must be positive.') ''(fH$pY  
    end vn0cKz@  
    us\%BxxI9  
    if any(m>n) ZSF=  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') vJTfo#C|  
    end ol?z<53X]  
    l&6U|q`  
    if any( r>1 | r<0 ) (:.Q\!aZ1  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') r,u<y_YW  
    end 5vs`uUzr  
    >9o,S3  
    if ~any(size(r)==1) oh7#cFZZ0  
        error('zernpol:Rvector','R must be a vector.') io t.E%G  
    end O1x0[sy  
    Y!Uu173  
    r = r(:); O)R7t3t  
    length_r = length(r); 4Vu'r?  
    'a;ini  
    if nargin==4 "pSH!0Ap\  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); f MY;  
        if ~isnorm 8!u/   
            error('zernpol:normalization','Unrecognized normalization flag.') E8T"{ R80  
        end ,+ns {ppn  
    else gdoJ4b  
        isnorm = false; Y!++C MzU  
    end !lQ#sL`  
    ?ID* /u|X  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% IF&g.R  
    % Compute the Zernike Polynomials Sni&?tcY  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >]Mq)V9  
    *.kj]BoO  
    % Determine the required powers of r: hl AR[]  
    % ----------------------------------- KWFyw>*)  
    rpowers = []; Sk8%(JD7  
    for j = 1:length(n) \We"?1^  
        rpowers = [rpowers m(j):2:n(j)]; gB(9vhj $  
    end &mh Ln4^  
    rpowers = unique(rpowers); 0zeUP {MQ  
    uk`T+@K  
    % Pre-compute the values of r raised to the required powers, [ 3$.*   
    % and compile them in a matrix: t{_!Z(Rt5)  
    % ----------------------------- -'80>[}q/  
    if rpowers(1)==0 f!5F]qP>-  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Q,zC_  
        rpowern = cat(2,rpowern{:}); ' 2>l  
        rpowern = [ones(length_r,1) rpowern]; -1Djo:y  
    else |'ZN!2u  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); B }6Kd  
        rpowern = cat(2,rpowern{:}); pG0!ALT  
    end j.k@6[ R>?  
    /VRUz++K  
    % Compute the values of the polynomials: e Wc_N  
    % -------------------------------------- E;9Z\?P  
    z = zeros(length_r,length_n); VVs{l\$=ZV  
    for j = 1:length_n vGXWwQ.1Tp  
        s = 0:(n(j)-m(j))/2; @Ppo &>  
        pows = n(j):-2:m(j); ?sV[MsOsC  
        for k = length(s):-1:1 S*4f%!  
            p = (1-2*mod(s(k),2))* ... q#;BhPc  
                       prod(2:(n(j)-s(k)))/          ... a*V9_Px$&  
                       prod(2:s(k))/                 ... $v FrUv  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... F vj{@B!  
                       prod(2:((n(j)+m(j))/2-s(k))); .FuA;:@%\  
            idx = (pows(k)==rpowers); 5!<o-{J[(=  
            z(:,j) = z(:,j) + p*rpowern(:,idx); K6E}";;  
        end F#6cF=};@  
         uii7b 7[w  
        if isnorm =KV@&Y^x4  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); <lFdexH"T  
        end 8fnR1mWG  
    end ]K7`-p~T  
    (/'h4KS@  
    % 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)  p)3nyN=|_  
    ia4k:\  
    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。 6peyh_  
    P^[/Qi}j  
    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)