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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 7kwG_0QO  
    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! Q)}\4&4  
     
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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 IjRmpVcwN  
    function z = zernfun(n,m,r,theta,nflag) c+{4C3z  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. q{ 1U  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ;$E[u)l  
    %   and angular frequency M, evaluated at positions (R,THETA) on the #dt2'V- ,  
    %   unit circle.  N is a vector of positive integers (including 0), and o5@ jMU;  
    %   M is a vector with the same number of elements as N.  Each element Ft rw3OxN  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 8'[wa  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, M!l5,ycF  
    %   and THETA is a vector of angles.  R and THETA must have the same r97[!y1gt  
    %   length.  The output Z is a matrix with one column for every (N,M) `4q}D-'TF8  
    %   pair, and one row for every (R,THETA) pair. v`w?QIB]  
    % NXNon*"  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ;wxt<   
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), S: uEK  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral a0.3$  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, +"cyOC  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized {wXN kq  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. K@~#Gdnl  
    % EM/+1 _u  
    %   The Zernike functions are an orthogonal basis on the unit circle. q$rA-`jw  
    %   They are used in disciplines such as astronomy, optics, and rM=A"  
    %   optometry to describe functions on a circular domain. K-C,+eI  
    % PI \,`^)y  
    %   The following table lists the first 15 Zernike functions. hF7mJ\  
    % <'_GQM`G  
    %       n    m    Zernike function           Normalization jGhg~-m  
    %       -------------------------------------------------- 8#u_+;,p  
    %       0    0    1                                 1 Ux T[  
    %       1    1    r * cos(theta)                    2 4Z9 3 g {  
    %       1   -1    r * sin(theta)                    2 ZC*d^n]x.  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) I=y j  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Hk*cO;c  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) <&%1pZ/6.  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) $#FlnM<=  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) $ ].k6,%{p  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) yN<fmi};c  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) %=8(B.I!  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) XWQ0V  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) xt,L* B  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) /B\-DP3K  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?mR[A`J58  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ]"g >>N  
    %       -------------------------------------------------- vW-`=30  
    % sg"D;b:X  
    %   Example 1: `$SEkYdt  
    % uEGPgYY(  
    %       % Display the Zernike function Z(n=5,m=1) lO:{tV  
    %       x = -1:0.01:1; *F*jA$aY  
    %       [X,Y] = meshgrid(x,x); WriN]/yD  
    %       [theta,r] = cart2pol(X,Y); ls7A5 <  
    %       idx = r<=1; q;zf|'&*7C  
    %       z = nan(size(X)); xqC<p`?4  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); )c >B23D  
    %       figure ~)a ;59<$  
    %       pcolor(x,x,z), shading interp n=o'ocdS)  
    %       axis square, colorbar =&VXn{e  
    %       title('Zernike function Z_5^1(r,\theta)') n_n|^4 w  
    % mhLRi\[c )  
    %   Example 2: d74g|`/  
    % 3!9 yuf  
    %       % Display the first 10 Zernike functions }t%>_  
    %       x = -1:0.01:1; T|s0qQi  
    %       [X,Y] = meshgrid(x,x); CCh8?sM  
    %       [theta,r] = cart2pol(X,Y); Ji:iKkI  
    %       idx = r<=1; 8{<[fZyC  
    %       z = nan(size(X)); .`& /QiD  
    %       n = [0  1  1  2  2  2  3  3  3  3]; /Ej]X`F  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; *Z]WaDw  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; (5q%0|RzRs  
    %       y = zernfun(n,m,r(idx),theta(idx)); sK%Hx`  
    %       figure('Units','normalized') ^_KD&%M6  
    %       for k = 1:10 l \^nC2  
    %           z(idx) = y(:,k); )ozcr^  
    %           subplot(4,7,Nplot(k))  _7#tgZyv  
    %           pcolor(x,x,z), shading interp Ryq"\Q>+  
    %           set(gca,'XTick',[],'YTick',[]) LJ(n?/z%  
    %           axis square Lcs{OW,  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) y /:T(tk$  
    %       end xOL)Pjo /m  
    % CC>fm 1#i\  
    %   See also ZERNPOL, ZERNFUN2. uB <F.!3  
    WTD49_px  
    %   Paul Fricker 11/13/2006 KHlIK`r  
    .K@x4 /1  
    hygnC`|  
    % Check and prepare the inputs: xe6_RO%  
    % ----------------------------- 9S1Ti6A  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 5':Gu}Vq  
        error('zernfun:NMvectors','N and M must be vectors.') .FKJ yzL  
    end G?ugMl}  
    U$A7EFK'  
    if length(n)~=length(m) wVp4c?s  
        error('zernfun:NMlength','N and M must be the same length.') $,;S\JmWP  
    end P YF.#@":&  
    Aa`MK$29F  
    n = n(:); wt}%2x} x  
    m = m(:); qqe2,X?  
    if any(mod(n-m,2)) N2tkCkl^x9  
        error('zernfun:NMmultiplesof2', ... [X }@Ct6  
              'All N and M must differ by multiples of 2 (including 0).') Jh\: X<q  
    end G*(K UG>  
    =a9etF%B  
    if any(m>n) g%\$ !b  
        error('zernfun:MlessthanN', ... *"5N>F[L  
              'Each M must be less than or equal to its corresponding N.') t$K@%yU2  
    end AbF(MK=i  
    ~ThVap[*  
    if any( r>1 | r<0 ) ;v1NL@w*  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') o9ctJf=qn  
    end oQ%\[s$  
    +mc [S  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 5pM&h~M  
        error('zernfun:RTHvector','R and THETA must be vectors.') \L ]   
    end ^XBzZ!h|  
    PUP"ky^q"  
    r = r(:); KZF0rW  
    theta = theta(:); [0&'cu>  
    length_r = length(r); %AG1oWWc>.  
    if length_r~=length(theta) i*S|qX7``  
        error('zernfun:RTHlength', ... dI^IK  
              'The number of R- and THETA-values must be equal.') E.J 0fwyT  
    end !/j,hO4Z4  
    }!%JYG^!D  
    % Check normalization: S9 G+#[.|  
    % -------------------- Tm)GC_  
    if nargin==5 && ischar(nflag) GIm " )}W  
        isnorm = strcmpi(nflag,'norm'); (#6AKr9K  
        if ~isnorm MzQ\rg_B7  
            error('zernfun:normalization','Unrecognized normalization flag.') 22`oFXb'  
        end bVoU|`c  
    else N0Efw$u  
        isnorm = false; r{\BbUnf)  
    end TN35CaSmq  
    IKi{Xh]\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M{?.hq  
    % Compute the Zernike Polynomials ~x|aoozL  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *u>lx!g  
    90/vJN  
    % Determine the required powers of r: "z^(dF|  
    % ----------------------------------- KD% TxK  
    m_abs = abs(m); i;o}o *=  
    rpowers = []; _eJXi,  
    for j = 1:length(n) J I<3\=:+  
        rpowers = [rpowers m_abs(j):2:n(j)]; ,~4H{{<j  
    end n /QfdAg  
    rpowers = unique(rpowers); Y1{B c<tC  
    ^GMJ~[]  
    % Pre-compute the values of r raised to the required powers, |3}5:k  
    % and compile them in a matrix: <B6[i*&  
    % ----------------------------- 01udlW.  
    if rpowers(1)==0 "8N"Udu  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ]3*P:$Rq  
        rpowern = cat(2,rpowern{:}); iF!mV5#  
        rpowern = [ones(length_r,1) rpowern]; i S%  
    else } h[>U  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); M`GP^Ta  
        rpowern = cat(2,rpowern{:}); *'D=1{WZ!  
    end ''IoC j  
    3VmI0gsm.>  
    % Compute the values of the polynomials: LVnHt}  
    % -------------------------------------- s]U4B<q  
    y = zeros(length_r,length(n)); h#i\iK&A  
    for j = 1:length(n) !(Krf  
        s = 0:(n(j)-m_abs(j))/2; IU@_)I+6  
        pows = n(j):-2:m_abs(j); 9UwLF`XM  
        for k = length(s):-1:1 W9Nmx3ve  
            p = (1-2*mod(s(k),2))* ... z3a-+NjDm  
                       prod(2:(n(j)-s(k)))/              ... Bv $UFTz  
                       prod(2:s(k))/                     ... 6vto++  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... bAf,aV/C&|  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); <I; 5wv  
            idx = (pows(k)==rpowers); #~^btL'dHF  
            y(:,j) = y(:,j) + p*rpowern(:,idx); UVz/n68\k7  
        end +$47v$p  
         "PMQyzl  
        if isnorm fN-Gk(Ic  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 38~PWKt  
        end n@!wp/J,  
    end Z%}4bJ  
    % END: Compute the Zernike Polynomials &D]p,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -4w%Iy  
    brh=NAzt  
    % Compute the Zernike functions: uWjN2#&,  
    % ------------------------------ `>6T&  
    idx_pos = m>0;  ~,"N[Q  
    idx_neg = m<0; 4KXc~eF[M"  
    ,_jC$  
    z = y; c%z'xM  
    if any(idx_pos) vJ"i.:Gf4  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); )%mg(O8uL  
    end hQRL,?  
    if any(idx_neg) =E(#YCx  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); RRADg^}l|"  
    end NjX[;e-u  
    BcQEG *N  
    % 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) \OQkZ.cU;  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. _V2xA88  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated s$w;q\1z  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive =Mj 0:rW  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ;),"M{"v  
    %   and THETA is a vector of angles.  R and THETA must have the same ^{F_ a  
    %   length.  The output Z is a matrix with one column for every P-value, "l7NWqfB  
    %   and one row for every (R,THETA) pair. h|D0z_f  
    % f>2MI4nMG  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike u5B:^.:p  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) WBr59@V  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 5'KA'>@  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 Vi *e@IP/  
    %   for all p. -TO\'^][X  
    % [;A[.&6  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 :nA.j"@  
    %   Zernike functions (order N<=7).  In some disciplines it is / 3!fA=+  
    %   traditional to label the first 36 functions using a single mode NX[4PKJ0C  
    %   number P instead of separate numbers for the order N and azimuthal dOFD5}_   
    %   frequency M. 6gH{ R$7L=  
    % 0hY{<^"Y  
    %   Example: ^7^N}x@  
    % - +> 1r  
    %       % Display the first 16 Zernike functions :|+Qe e  
    %       x = -1:0.01:1; S >yLqPp  
    %       [X,Y] = meshgrid(x,x); $q$7^ r@  
    %       [theta,r] = cart2pol(X,Y); JH8}Ru%Z  
    %       idx = r<=1; `=UWqb(K_  
    %       p = 0:15; a5Y IUVCv  
    %       z = nan(size(X)); *.#oxcll  
    %       y = zernfun2(p,r(idx),theta(idx)); rV6/Tdy  
    %       figure('Units','normalized') bN$r k|  
    %       for k = 1:length(p) =WIJ>#Go<  
    %           z(idx) = y(:,k); rG}\Zjn{  
    %           subplot(4,4,k) Lfog {Vzs  
    %           pcolor(x,x,z), shading interp A A^{B  
    %           set(gca,'XTick',[],'YTick',[]) (wRgus  
    %           axis square y5r4+2B  
    %           title(['Z_{' num2str(p(k)) '}']) ?e[lr>-  
    %       end <-'$~G j  
    % 5 Xn.CBd]  
    %   See also ZERNPOL, ZERNFUN. ]\ !5}L  
    c;?fMX  
    %   Paul Fricker 11/13/2006 */:uV B,b2  
    9h&R]yz;  
    ",6M)3{|c  
    % Check and prepare the inputs: eI45PMP  
    % ----------------------------- [NMVoBvG  
    if min(size(p))~=1 :cB=SYcC%  
        error('zernfun2:Pvector','Input P must be vector.') 0k)rc$eDF+  
    end kF(n!2"W  
    [AMAa]^  
    if any(p)>35 >NYW{(j  
        error('zernfun2:P36', ... [e[<p\]  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... BTgG4F/)  
               '(P = 0 to 35).']) 4}+/F}TbJ5  
    end y!tC20Q   
    ktIi$v  
    % Get the order and frequency corresonding to the function number: *(C(tPhC  
    % ---------------------------------------------------------------- ) e5 @  
    p = p(:); n3A aZp[  
    n = ceil((-3+sqrt(9+8*p))/2); ;AO#xv+#  
    m = 2*p - n.*(n+2); svxw^ 0~a  
    YIw1  
    % Pass the inputs to the function ZERNFUN: `]Q:-h  
    % ---------------------------------------- hxQqa 0B  
    switch nargin Po%LE]v,  
        case 3  (x/k.&  
            z = zernfun(n,m,r,theta); VD_$$Gn*q  
        case 4 2hzsKkrA {  
            z = zernfun(n,m,r,theta,nflag); _ODbY;M  
        otherwise _S>JKz  
            error('zernfun2:nargin','Incorrect number of inputs.') QQWadVQo  
    end }zhGS!fO  
    ULMu19>  
    % 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) I:mr}mv=i  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. dq]0X?[6  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of {{QELfH2  
    %   order N and frequency M, evaluated at R.  N is a vector of w@{=nD4p  
    %   positive integers (including 0), and M is a vector with the @nux9MX<9  
    %   same number of elements as N.  Each element k of M must be a +0OLc2 )w  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) )ubiB^g'm  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is HSc~*Q  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 9jw\s P@  
    %   with one column for every (N,M) pair, and one row for every thI F&  
    %   element in R. 3.*8)NW  
    % 0N $v"uX@  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- dw*_(ys  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is !O<)\ )|g  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to (L/_^!ZX  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 h2XfC. f  
    %   for all [n,m]. "thdPZ  
    % ]"?)Z  
    %   The radial Zernike polynomials are the radial portion of the [k/@E+;  
    %   Zernike functions, which are an orthogonal basis on the unit sG F aL  
    %   circle.  The series representation of the radial Zernike hpD!2 K3>  
    %   polynomials is V=o t-1,j7  
    % 0@%v1Oja  
    %          (n-m)/2 |>dI/_'  
    %            __ . QBF`Rz  
    %    m      \       s                                          n-2s 5fd]v<  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r _sTROd)Vh  
    %    n      s=0 $7^o#2 B  
    % ppVHLrUh  
    %   The following table shows the first 12 polynomials. =Jyu4j *}  
    % (&F ,AY3A  
    %       n    m    Zernike polynomial    Normalization R8a3 1&  
    %       --------------------------------------------- KV0]m^@x  
    %       0    0    1                        sqrt(2) woK&q7Vn  
    %       1    1    r                           2 G$F<$  
    %       2    0    2*r^2 - 1                sqrt(6) q1d}{DU  
    %       2    2    r^2                      sqrt(6) ;z$(nhJ  
    %       3    1    3*r^3 - 2*r              sqrt(8) ! t?iXZ  
    %       3    3    r^3                      sqrt(8) Z/Dx,zIR  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ?iWi  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ,)Znb=  
    %       4    4    r^4                      sqrt(10) 7`DBS^O]dG  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) |}Nn!Sj>#;  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 5>D>% iaHv  
    %       5    5    r^5                      sqrt(12) $Avjnm  
    %       --------------------------------------------- Dv5D~on{  
    % {#?N  
    %   Example: %N>%!m  
    % Lh!J >  
    %       % Display three example Zernike radial polynomials CYY=R'1:G{  
    %       r = 0:0.01:1; q_MG?re  
    %       n = [3 2 5]; svcK?^ HTe  
    %       m = [1 2 1]; L.*M&Ry  
    %       z = zernpol(n,m,r); 'h|DO/X~L  
    %       figure 128EPK  
    %       plot(r,z) KBx6NU?;PO  
    %       grid on ~j}cyHg  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') g| I6'K!<  
    % :V RNs  
    %   See also ZERNFUN, ZERNFUN2. !Prg_6 `  
    \ `~Ly-  
    % A note on the algorithm. >UiYL}'br6  
    % ------------------------ '7hu 2i5  
    % The radial Zernike polynomials are computed using the series f!Y?S  
    % representation shown in the Help section above. For many special 9PXFRxGA  
    % functions, direct evaluation using the series representation can =Y|VgV  
    % produce poor numerical results (floating point errors), because cUKE   
    % the summation often involves computing small differences between Ug>~Rq]  
    % large successive terms in the series. (In such cases, the functions ?PeJlpYzV  
    % are often evaluated using alternative methods such as recurrence /Kql>$I  
    % relations: see the Legendre functions, for example). For the Zernike Lo{ E:5q  
    % polynomials, however, this problem does not arise, because the +hg3I8q:  
    % polynomials are evaluated over the finite domain r = (0,1), and /R]U}o^/(%  
    % because the coefficients for a given polynomial are generally all Z'`<5A%;  
    % of similar magnitude. ^c"jH'#.L  
    % [8 ]z|bM  
    % ZERNPOL has been written using a vectorized implementation: multiple xpV|\2C  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ype$ c  
    % values can be passed as inputs) for a vector of points R.  To achieve U)fc*s  
    % this vectorization most efficiently, the algorithm in ZERNPOL 89pEfl j2  
    % involves pre-determining all the powers p of R that are required to <J< {l  
    % compute the outputs, and then compiling the {R^p} into a single ]!?;@$wx  
    % matrix.  This avoids any redundant computation of the R^p, and J=9FRC  
    % minimizes the sizes of certain intermediate variables. e$<0 7Oc  
    % ^a0um/+M}  
    %   Paul Fricker 11/13/2006 N.,X<G.H  
    h \fjBDU^  
    +~m46eI  
    % Check and prepare the inputs: I&#| w"/"U  
    % ----------------------------- n{n52][J]  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) )WNzWUfn=z  
        error('zernpol:NMvectors','N and M must be vectors.') _mqL8ho  
    end lA| 5E?  
    V,lOt4b  
    if length(n)~=length(m) Z]>O+  
        error('zernpol:NMlength','N and M must be the same length.') KKj a/p  
    end MU@UfB|;u  
    3 oF45`3FV  
    n = n(:); _<n~n]%  
    m = m(:); >?2M }TV3  
    length_n = length(n); Ta ZmRL  
    0gW"i&7c  
    if any(mod(n-m,2)) j/323Za+  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') s Ep"D+f  
    end (9''MlGd%  
    @K  &GJ  
    if any(m<0) g0xuxK;9c  
        error('zernpol:Mpositive','All M must be positive.') ~ (|5/ p7t  
    end 7OcW C-<  
    1F5XvQl  
    if any(m>n) | k&Ck  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') >,V9H$n  
    end [\^ n=  
    pA'4|ffwe  
    if any( r>1 | r<0 ) a#cCpE  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') u@Ih GME  
    end dP9qSwTa  
    >:74%D0UF  
    if ~any(size(r)==1) u 1J0$  
        error('zernpol:Rvector','R must be a vector.') ^n*)7K[  
    end |^&b8  
    pNG:0  
    r = r(:); 2%DSUv:H%  
    length_r = length(r); [<lHCQXJ/  
    5Yg'BkEr  
    if nargin==4 ((YMVe  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); H4WP~(__  
        if ~isnorm >6ni")Q9  
            error('zernpol:normalization','Unrecognized normalization flag.') +SP{hHa^  
        end xT3BHnQ(  
    else ? ^0:3$La  
        isnorm = false; \\iQEy<i  
    end FvaUsOy "  
    ^h(ew1:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]AINK UI0  
    % Compute the Zernike Polynomials iOd&B B6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ak7bJ~)X=  
    j@n)kPo,1  
    % Determine the required powers of r: rMdt:`  
    % ----------------------------------- >nJ\BPx  
    rpowers = []; hr#M-K  
    for j = 1:length(n) \T4v|Pw\  
        rpowers = [rpowers m(j):2:n(j)]; OV1_|##LC  
    end -R|,9o^  
    rpowers = unique(rpowers); 5dx$HE&b)  
    &fDIQISC  
    % Pre-compute the values of r raised to the required powers, .<->C?#  
    % and compile them in a matrix: iowTLq!?  
    % ----------------------------- 0pZ4BZdT|  
    if rpowers(1)==0 'j`=if  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 8v:T.o;<  
        rpowern = cat(2,rpowern{:}); J4k=A7^N  
        rpowern = [ones(length_r,1) rpowern]; W,K;6TZhh  
    else J9\Cm!H  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); GB23\Yv  
        rpowern = cat(2,rpowern{:}); M92dZ1+6  
    end eQ$Y0qH1E  
    KI.q@zO6|  
    % Compute the values of the polynomials: j b!x:  
    % -------------------------------------- T`9lV2x*P  
    z = zeros(length_r,length_n); Xe3U`P7(  
    for j = 1:length_n }fp-pe69z  
        s = 0:(n(j)-m(j))/2; G4Q[Th  
        pows = n(j):-2:m(j); .eS<Dbku<  
        for k = length(s):-1:1 i[O& )N,c  
            p = (1-2*mod(s(k),2))* ... ncZ+gzK|"  
                       prod(2:(n(j)-s(k)))/          ... NN%*b yK  
                       prod(2:s(k))/                 ... |-fg j'  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... +>8'mf  
                       prod(2:((n(j)+m(j))/2-s(k))); =J18eH!]  
            idx = (pows(k)==rpowers); E@z<:pG{  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Df}A^G >X  
        end S-a]j;U  
         DF&(8NoX~  
        if isnorm ny]?I  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); evya7^,F  
        end a[>/h3  
    end WltQ63u  
    qFicBpB  
    % 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)  '-QwssE  
    Cmu@4j&  
    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。 [<7@{;r  
    #akpXdXs  
    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)