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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 St7ZyN1  
    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! a6wPkf7-H  
     
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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 Hj\>&vMf  
    function z = zernfun(n,m,r,theta,nflag) kEiWE|  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. tk=~b} 8  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ;|7]%Z}%  
    %   and angular frequency M, evaluated at positions (R,THETA) on the a^/j&9  
    %   unit circle.  N is a vector of positive integers (including 0), and FbO\#p s  
    %   M is a vector with the same number of elements as N.  Each element s[6y|{&ze  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) }\Kki  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, o+Cd\D69S  
    %   and THETA is a vector of angles.  R and THETA must have the same Q #!|h:K  
    %   length.  The output Z is a matrix with one column for every (N,M) :+Ti^FF`w  
    %   pair, and one row for every (R,THETA) pair. bit@Kv1<C  
    % DvL/xlN  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike H|@R+  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), >wx1M1  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral )2vkaR  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, $;~  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 4FLL*LCNX  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 'KL!)}B$h  
    % ~Psv[b=]  
    %   The Zernike functions are an orthogonal basis on the unit circle. BhFyEY(  
    %   They are used in disciplines such as astronomy, optics, and o}QtKf)W  
    %   optometry to describe functions on a circular domain. w K)/m`{g  
    % oMdqg4HUF  
    %   The following table lists the first 15 Zernike functions. QxUsdF?p  
    % *F[;D7sZ~  
    %       n    m    Zernike function           Normalization !.\-l2f  
    %       -------------------------------------------------- #>)OLKP  
    %       0    0    1                                 1 |Iq#Q3w  
    %       1    1    r * cos(theta)                    2 ;F3#AO4(  
    %       1   -1    r * sin(theta)                    2 @ootKY`  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) #i6ZY^+ee  
    %       2    0    (2*r^2 - 1)                    sqrt(3) N5m+r.<;  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) [OTZ"XQLI  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ?-.Qv1hs6p  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) [&_c.ti  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ftr?@^  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 7Qoy~=E  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) &v}c3wL]  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) [*i6?5}-  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 'UW]~  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) y*6-?@  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) b Ag>;e(  
    %       -------------------------------------------------- ^j-w^)@T  
    % BZUA/;Hz &  
    %   Example 1: \~ACWF7l  
    % Ic!8$NhRS  
    %       % Display the Zernike function Z(n=5,m=1) ?U^h:n  
    %       x = -1:0.01:1; (bT3 r_  
    %       [X,Y] = meshgrid(x,x); ;_]Z3  
    %       [theta,r] = cart2pol(X,Y); U`25bb1W j  
    %       idx = r<=1; XMJEIG  
    %       z = nan(size(X)); Wu:@+~J.h  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); I[td:9+hK@  
    %       figure uW@o,S0:  
    %       pcolor(x,x,z), shading interp <Oyxzs  
    %       axis square, colorbar 4=b{k,kzgA  
    %       title('Zernike function Z_5^1(r,\theta)') ;8VvpO^G/  
    % ]E8S`[Vn  
    %   Example 2: Gd= l{~  
    % gr&Rkuyfv  
    %       % Display the first 10 Zernike functions +[2X@J  
    %       x = -1:0.01:1; J3;dRW  
    %       [X,Y] = meshgrid(x,x); 0SJ7QRo|K  
    %       [theta,r] = cart2pol(X,Y); cag9f?w@V  
    %       idx = r<=1; O7KR~d  
    %       z = nan(size(X)); gJn_Z7MgJ  
    %       n = [0  1  1  2  2  2  3  3  3  3]; _mi(:s(  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; xQKD1#y  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; n-%8RV  
    %       y = zernfun(n,m,r(idx),theta(idx)); \q |n0>  
    %       figure('Units','normalized') 9S _N*wC.  
    %       for k = 1:10 y%9Q]7&=  
    %           z(idx) = y(:,k); `U~Y{f_!H  
    %           subplot(4,7,Nplot(k)) c[a1 Md&  
    %           pcolor(x,x,z), shading interp C/sDyv$  
    %           set(gca,'XTick',[],'YTick',[]) vW\|% @hW,  
    %           axis square NbDfD3 1GK  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) g;M\4o  
    %       end 5[1#d\QR  
    % jNO8n)a&p  
    %   See also ZERNPOL, ZERNFUN2. ~w>Z !RuhT  
    1|PmZPKq9n  
    %   Paul Fricker 11/13/2006 TLkJZ4}?Q  
    *C0gpEf9S  
    $!msav  
    % Check and prepare the inputs: HJ\CGYmyz  
    % -----------------------------  fK$N|r  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) wG&+*,}  
        error('zernfun:NMvectors','N and M must be vectors.') /G>reG,G  
    end UpA{$@  
    c/c%-=  
    if length(n)~=length(m) w|1Gb[  
        error('zernfun:NMlength','N and M must be the same length.')  W1@Q)i  
    end #=MQE  
    `Al[gG?/!  
    n = n(:); 0H V-e  
    m = m(:); /&+6nOP  
    if any(mod(n-m,2)) !Qg%d&q.Sx  
        error('zernfun:NMmultiplesof2', ... >VAZ^kgi  
              'All N and M must differ by multiples of 2 (including 0).') MKuy?mri~  
    end 7 -(LWH  
    OoFQ@zE7%  
    if any(m>n) <?TJ-   
        error('zernfun:MlessthanN', ... MI!JZI$z5  
              'Each M must be less than or equal to its corresponding N.') L-ZJ[#D  
    end zn4Yo  
    @QAyXwp  
    if any( r>1 | r<0 ) AR}M*sSh  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') h=3156M  
    end x+O}RD*G  
    GMw|@?:{  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ,H3C\.%w\  
        error('zernfun:RTHvector','R and THETA must be vectors.') kUJ\AK  
    end [xXml On!  
    @UO=)PxN3  
    r = r(:); %5_eos&<^)  
    theta = theta(:); zr0_SCh;2  
    length_r = length(r); !d1}IU-h  
    if length_r~=length(theta) RRD\V3C84  
        error('zernfun:RTHlength', ... u+]v. Mt  
              'The number of R- and THETA-values must be equal.') `9Qr kkG+  
    end /HNZwbh]uJ  
    !Xwp;P=  
    % Check normalization: E(T6s^8  
    % -------------------- ;3n0 bKDY  
    if nargin==5 && ischar(nflag) ; y#6Nx,:  
        isnorm = strcmpi(nflag,'norm'); [@}{sH(#Ta  
        if ~isnorm Ii?"`d+JA  
            error('zernfun:normalization','Unrecognized normalization flag.') `>fN? He  
        end ? OBe!NDf  
    else A a2*f[  
        isnorm = false; `d=$9Pi  
    end xDBEs*  
    P,"z  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _"Q +G@@  
    % Compute the Zernike Polynomials E<3hy  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q{UP_6O F  
    `8F%bc54iw  
    % Determine the required powers of r: FhB^E$r%  
    % ----------------------------------- Rg&6J#h  
    m_abs = abs(m); x8T5aS  
    rpowers = []; SaEe7eHd  
    for j = 1:length(n) ]lF'o&v]  
        rpowers = [rpowers m_abs(j):2:n(j)]; gKg2Ntxj  
    end NQ<~$+{  
    rpowers = unique(rpowers); +G&h  
    b?oT|@  
    % Pre-compute the values of r raised to the required powers, }>xgzhdT  
    % and compile them in a matrix: {KL<Hx2M  
    % ----------------------------- Do(7LidC5  
    if rpowers(1)==0 2 G_*Pqc  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); J p .wg  
        rpowern = cat(2,rpowern{:}); 1!\!3xaV  
        rpowern = [ones(length_r,1) rpowern]; gQ h0-Dnw  
    else >TsJ0E?3x  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ',0~\V  
        rpowern = cat(2,rpowern{:}); ?ZSG4La\  
    end  Be2@9  
    ,"PwNv  
    % Compute the values of the polynomials: +byw*Kk  
    % -------------------------------------- @hm %0L  
    y = zeros(length_r,length(n)); .jr1<LE  
    for j = 1:length(n) G=3/PYp  
        s = 0:(n(j)-m_abs(j))/2; ~0fT*lp  
        pows = n(j):-2:m_abs(j); *6Rl[eXS  
        for k = length(s):-1:1 >w9)c|  
            p = (1-2*mod(s(k),2))* ... W.\HfJ74  
                       prod(2:(n(j)-s(k)))/              ... $BE^'5G&4Y  
                       prod(2:s(k))/                     ... g_] u<8&  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 6!bA~"N  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); -p ) l63  
            idx = (pows(k)==rpowers); |.:O$/ Tt[  
            y(:,j) = y(:,j) + p*rpowern(:,idx); C3 0b}2  
        end -baGr;,Cu  
         S#+G?I3w  
        if isnorm Sct-,K%i  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); $t 1]w]}d  
        end 6kT l(+  
    end f\~e&`PV  
    % END: Compute the Zernike Polynomials V{Idj\~Jh  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Q|gun}  
    %8$JL=c  
    % Compute the Zernike functions: ACl:~7;  
    % ------------------------------ Oe$cM=Yf  
    idx_pos = m>0; lIzJO$8cM  
    idx_neg = m<0; 8t}=?:B+{  
    #jr;.;8sQ  
    z = y; 'xStA  
    if any(idx_pos) u{H,i(mx?  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)');  2WE   
    end }jiqUBn%  
    if any(idx_neg) (fh:q2E#  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); >Fx$Rty  
    end cw"x0 RS  
    b daZ{5^{  
    % 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) n#J$=@  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 0^ E!P>  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ` V^#Sb  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive  _&(ij(H  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Go`omh b  
    %   and THETA is a vector of angles.  R and THETA must have the same ziH2<@  
    %   length.  The output Z is a matrix with one column for every P-value, #mkr]K8A4  
    %   and one row for every (R,THETA) pair. Ac7`nvI=  
    % X'?v8\mPK  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike z%Z}vWn  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) d}l^yln  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) >P0AGZ  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 }(o/+H4  
    %   for all p. _L$)~},cT  
    % E0O{5YF^T  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 .k*2T<p$rC  
    %   Zernike functions (order N<=7).  In some disciplines it is 2OT6*+D  
    %   traditional to label the first 36 functions using a single mode _)_XO92~  
    %   number P instead of separate numbers for the order N and azimuthal oC7#6W:@w  
    %   frequency M. b%PVF&C9W  
    % A+F-r_]}db  
    %   Example: ~ml\|  
    %  gA[M  
    %       % Display the first 16 Zernike functions ]#:xl}'LS  
    %       x = -1:0.01:1; _-!6@^+  
    %       [X,Y] = meshgrid(x,x); E,6E-9  
    %       [theta,r] = cart2pol(X,Y); l&|{uk  
    %       idx = r<=1; 2~`dV_  
    %       p = 0:15; <=7)t.  
    %       z = nan(size(X)); @H_LPn  
    %       y = zernfun2(p,r(idx),theta(idx)); ;XtDz  
    %       figure('Units','normalized') rSJ}qRXwU  
    %       for k = 1:length(p) P)\f\yb  
    %           z(idx) = y(:,k); Xj@Kt|&`k  
    %           subplot(4,4,k) F Q k;  
    %           pcolor(x,x,z), shading interp H~~(v52wD  
    %           set(gca,'XTick',[],'YTick',[]) ^Q OvK>W<  
    %           axis square jU#%@d6!#  
    %           title(['Z_{' num2str(p(k)) '}']) ;< ][upn  
    %       end .N'UnKz  
    % 7>~iS@7GV  
    %   See also ZERNPOL, ZERNFUN. ttKfZ0  
    VuBp$H(U  
    %   Paul Fricker 11/13/2006 .XE]vo  
    6h,'#|:d  
    nx +& {hn(  
    % Check and prepare the inputs: \c\=S  
    % ----------------------------- aLq;a  
    if min(size(p))~=1 /<2_K4(-{4  
        error('zernfun2:Pvector','Input P must be vector.') 5 O6MI4:  
    end LtU+w*Gj  
    h/k`+  
    if any(p)>35 UetmO`qju  
        error('zernfun2:P36', ... -)Vj08aP  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... A^Zs?<C-  
               '(P = 0 to 35).']) \mDm *UuG  
    end avz 4 &  
    .=FJ5?:4i%  
    % Get the order and frequency corresonding to the function number: |S8pq4eKJ_  
    % ---------------------------------------------------------------- ;i:7E#@  
    p = p(:); Fi`:G}   
    n = ceil((-3+sqrt(9+8*p))/2); eo80L  
    m = 2*p - n.*(n+2); W9D)QIqbvW  
    hf-S6PEsM  
    % Pass the inputs to the function ZERNFUN: /PCQv_Y&,/  
    % ---------------------------------------- [y:LA ~q  
    switch nargin {h=Ai[|l4Q  
        case 3 p(8\w-6  
            z = zernfun(n,m,r,theta); \[&]kPcDl  
        case 4 KJ~pY<a?  
            z = zernfun(n,m,r,theta,nflag); F)IP~BE-k  
        otherwise 9e5UTJ  
            error('zernfun2:nargin','Incorrect number of inputs.') 3 /e !7  
    end d]^i1  
    k$>T(smh  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) |3L MVN  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 4l*&3Ar  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 7,zARWB!?  
    %   order N and frequency M, evaluated at R.  N is a vector of ZS+2.)A  
    %   positive integers (including 0), and M is a vector with the f/x "yUq  
    %   same number of elements as N.  Each element k of M must be a {V8Pn2mlo  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ?2TH("hV$  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is bq O"k t  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix SD&[K 8-i2  
    %   with one column for every (N,M) pair, and one row for every t/Fe"T[,V  
    %   element in R. "ir*;|  
    % D |o@(V  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- NP8TF*5V  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is PwW@I~@>  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to qAS^5|(b[  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 1N+#(<x@,  
    %   for all [n,m]. X d6y7s  
    % Y"qY@`  
    %   The radial Zernike polynomials are the radial portion of the J.nq[/Q=  
    %   Zernike functions, which are an orthogonal basis on the unit q1y4B`  
    %   circle.  The series representation of the radial Zernike &; \v_5N6  
    %   polynomials is Os{qpR^<I:  
    % d%3BJ+J  
    %          (n-m)/2 5qy}~dQ  
    %            __ R=PzR;8  
    %    m      \       s                                          n-2s tOw 0(-:iq  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r  B]7jg9/  
    %    n      s=0 5WT\0]RUa  
    % 2#3R]zIO  
    %   The following table shows the first 12 polynomials. {rZ"cUm  
    % "tM/`:Qp  
    %       n    m    Zernike polynomial    Normalization }Kt?0  
    %       --------------------------------------------- &pm{7nH  
    %       0    0    1                        sqrt(2) 16|S 0 )  
    %       1    1    r                           2 .TC `\mV  
    %       2    0    2*r^2 - 1                sqrt(6) iC-ABOOu{l  
    %       2    2    r^2                      sqrt(6) L])w-  
    %       3    1    3*r^3 - 2*r              sqrt(8) Q8y|:tb$Y  
    %       3    3    r^3                      sqrt(8) fXe-U='  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) h>:RCpC  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ItADO'M  
    %       4    4    r^4                      sqrt(10) : qRT9n$  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) l{9h8]^  
    %       5    3    5*r^5 - 4*r^3            sqrt(12)  #Uh 5tc  
    %       5    5    r^5                      sqrt(12) pm\X*t}L  
    %       --------------------------------------------- l,wN@Nk  
    % yU,xcq~l  
    %   Example: :N*T2mP  
    % : !3y>bP)  
    %       % Display three example Zernike radial polynomials Bq@wS\W>b}  
    %       r = 0:0.01:1; 070IBAk}_  
    %       n = [3 2 5]; G4' U;  
    %       m = [1 2 1]; 1i:g /H  
    %       z = zernpol(n,m,r); .E#Sm?gK  
    %       figure rz{'X d  
    %       plot(r,z) 9^ p{/Io  
    %       grid on 1 ],, Ar5  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') acQN pT  
    % \_nmfTr!K  
    %   See also ZERNFUN, ZERNFUN2. 8"mW!M  
    e oSM@Isu  
    % A note on the algorithm. @BUqQ9q:  
    % ------------------------ I^(#\vRW  
    % The radial Zernike polynomials are computed using the series }Y`<(V5:  
    % representation shown in the Help section above. For many special 2F @)nh  
    % functions, direct evaluation using the series representation can *Ne&SXg  
    % produce poor numerical results (floating point errors), because u3mT l  
    % the summation often involves computing small differences between siYRRr  
    % large successive terms in the series. (In such cases, the functions h6y4Ii  
    % are often evaluated using alternative methods such as recurrence vUe *  
    % relations: see the Legendre functions, for example). For the Zernike <[:7#Yo g  
    % polynomials, however, this problem does not arise, because the Cfo 8gX*  
    % polynomials are evaluated over the finite domain r = (0,1), and %aBJ+V F  
    % because the coefficients for a given polynomial are generally all ggc?J<Dv  
    % of similar magnitude.  x9"4vp  
    % ;+34g6  
    % ZERNPOL has been written using a vectorized implementation: multiple _/~ ,a  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] (%yc5+f!  
    % values can be passed as inputs) for a vector of points R.  To achieve `(/saq*  
    % this vectorization most efficiently, the algorithm in ZERNPOL qlITQKGG  
    % involves pre-determining all the powers p of R that are required to AAq=,=:R<  
    % compute the outputs, and then compiling the {R^p} into a single  ;c Co+(  
    % matrix.  This avoids any redundant computation of the R^p, and DnsP7k.8T  
    % minimizes the sizes of certain intermediate variables. :dIQV(iW  
    % .#55u+d,  
    %   Paul Fricker 11/13/2006 )nHE$gVM s  
    7[v@*/W@  
    tQ/ #t<4D  
    % Check and prepare the inputs: E\~ KVn  
    % ----------------------------- $KWYe{#  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Qy5Os?9"  
        error('zernpol:NMvectors','N and M must be vectors.') 5%?b5(mnD  
    end IAF;mv}'  
    rp @  
    if length(n)~=length(m) B$TChc3B  
        error('zernpol:NMlength','N and M must be the same length.') S=H_9io  
    end 15KV} ){  
    'nWs0iH.  
    n = n(:); 'K`Rbhy  
    m = m(:); c|}K_~l_  
    length_n = length(n); ct}%Mdg  
    [Z5[~gP3  
    if any(mod(n-m,2)) dG-or  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') AR~$MCR]"k  
    end Ft;u\KT  
     6Z&u  
    if any(m<0) .3&a{IxM]  
        error('zernpol:Mpositive','All M must be positive.') !Wixs]od   
    end YYE8/\+B.  
    A ,-V$[;~D  
    if any(m>n) $HBT%g@UN  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') G_M:0YI@  
    end 2Za ,4'  
    8VuZ,!WH#  
    if any( r>1 | r<0 ) o"#TZB+k  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ZEj!jWP2m  
    end p2x1xv  
    wD6!#t k  
    if ~any(size(r)==1) _2m[(P9d  
        error('zernpol:Rvector','R must be a vector.') 7"F|6JP"$c  
    end Q^lQi\[  
    x*h`VS(?6  
    r = r(:); _}zo /kDA  
    length_r = length(r); s[3![ "^Y  
    J1tzHa6  
    if nargin==4 m0|Ae@g~3  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); n{64g+  
        if ~isnorm au~]  
            error('zernpol:normalization','Unrecognized normalization flag.') 0L:V#y-*  
        end j,=*WG  
    else X a"XB  
        isnorm = false; E7gHi$  
    end tE]5@b,R  
    mQJ4;BJw  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ik2- OM  
    % Compute the Zernike Polynomials QB/7/PW{H\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $N;"}G z  
    V$dJmKg  
    % Determine the required powers of r: 2cCWQ"_,  
    % ----------------------------------- ADYx.8M|9i  
    rpowers = []; @PQrmn6w  
    for j = 1:length(n) W$" Y%^L  
        rpowers = [rpowers m(j):2:n(j)]; [jl2\3*  
    end MSZ!W(7,<  
    rpowers = unique(rpowers); gX@nPZjg  
    u0XP(d H  
    % Pre-compute the values of r raised to the required powers, Ecxj9h,S  
    % and compile them in a matrix: +{xMIl_  
    % ----------------------------- Ap]4QqU  
    if rpowers(1)==0 *o02!EYge  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); vWU%ST  
        rpowern = cat(2,rpowern{:}); >`7OcjLg  
        rpowern = [ones(length_r,1) rpowern]; `'p`PyMt`  
    else e0f":Vct  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ;/@?6T"  
        rpowern = cat(2,rpowern{:}); h/I@_?k+  
    end 9&jQ 35  
    <S]KaDu^  
    % Compute the values of the polynomials: ?}vzLgp  
    % -------------------------------------- lPY@{1W  
    z = zeros(length_r,length_n); S:ls[9G[3  
    for j = 1:length_n nQ5N=l  
        s = 0:(n(j)-m(j))/2; 9nn>O?  
        pows = n(j):-2:m(j); sFLcOPj-%  
        for k = length(s):-1:1 LeY+p]n~  
            p = (1-2*mod(s(k),2))* ... 2;:lK":  
                       prod(2:(n(j)-s(k)))/          ... [%)@|^hw91  
                       prod(2:s(k))/                 ... bxAsV/j  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... hUVk54~l  
                       prod(2:((n(j)+m(j))/2-s(k))); um;:fT+  
            idx = (pows(k)==rpowers); "H>.':c"+3  
            z(:,j) = z(:,j) + p*rpowern(:,idx); {3hqp*xl  
        end qAqoZMpI|;  
         bA}Z0a  
        if isnorm vQUZVq5M  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); %N )e91wC  
        end =H[\%O~?b  
    end RI-A"cc6A  
    BI};"y  
    % 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)  [ic870_  
    r|u MovnV  
    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。 Jl3g{a  
    P!G858V(  
    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)