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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 正序阅读 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 |z!q r}i  
    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! 7GBZA=J  
     
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    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)  W@:^aH  
    GIl:3iB49  
    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。 JiKImz  
    pd=7^"[};  
    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)
    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niuhelen
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) "K 8nxnq  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. yxqTm%?y  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ,&0Z]*  
    %   order N and frequency M, evaluated at R.  N is a vector of $H4=QVj6  
    %   positive integers (including 0), and M is a vector with the pH^ z  
    %   same number of elements as N.  Each element k of M must be a {>S4 #^@}  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) VIetcs  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is y*_K=}pk  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix '=$TyiU  
    %   with one column for every (N,M) pair, and one row for every P1$f}K}  
    %   element in R. S 9WawI  
    % bS,etd  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ubD#I{~J  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ?.8<-  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to q5!0\o:  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 Tu==49  
    %   for all [n,m]. D^$]>-^  
    %  X@cSP7b  
    %   The radial Zernike polynomials are the radial portion of the .-J`d=Krp  
    %   Zernike functions, which are an orthogonal basis on the unit Q< dba12  
    %   circle.  The series representation of the radial Zernike FZeP<Ban  
    %   polynomials is 9w zwY[{  
    % q Z#!CPHS  
    %          (n-m)/2 }lH;[+u3  
    %            __ 4AJ9`1d4  
    %    m      \       s                                          n-2s `nKJR'QC  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r $kv@tzO  
    %    n      s=0 Q Qi@>v|d  
    % 0Qt~K#mr/  
    %   The following table shows the first 12 polynomials. y`({ .L  
    % f]c <9Q>*  
    %       n    m    Zernike polynomial    Normalization 9g96 d-  
    %       --------------------------------------------- l4zw]AYk+X  
    %       0    0    1                        sqrt(2) 5|5=Y/   
    %       1    1    r                           2 \` &ej{  
    %       2    0    2*r^2 - 1                sqrt(6) 6`1k ^  
    %       2    2    r^2                      sqrt(6) WBa /IM   
    %       3    1    3*r^3 - 2*r              sqrt(8) _$!`VA%  
    %       3    3    r^3                      sqrt(8) *aI~W^N3  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) J, r Xx:  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) <W?WUF  
    %       4    4    r^4                      sqrt(10) }H5/3be  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) _;#9!"&  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) yk`)Cq%=;  
    %       5    5    r^5                      sqrt(12) ?x'w~;9R/  
    %       --------------------------------------------- sSNCosb  
    % C]M7GHe1q  
    %   Example: *G\=i A  
    % zqY)dk  
    %       % Display three example Zernike radial polynomials (NPxab8e*  
    %       r = 0:0.01:1; !KAsvF,j  
    %       n = [3 2 5]; ,2,W^HJ  
    %       m = [1 2 1]; %iX/y  
    %       z = zernpol(n,m,r); (xbIUz.  
    %       figure i]dz}=j'  
    %       plot(r,z) ;|;iCaD a+  
    %       grid on {-J:4*`  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 1EQvcw #  
    % Q4=|@|U0  
    %   See also ZERNFUN, ZERNFUN2. 9Eu #lV  
    xuF5/(__  
    % A note on the algorithm. Et.j1M|g  
    % ------------------------ !8o\.uyi  
    % The radial Zernike polynomials are computed using the series ZOC#i i`:  
    % representation shown in the Help section above. For many special S{- f $Q*  
    % functions, direct evaluation using the series representation can .8:+MW/  
    % produce poor numerical results (floating point errors), because d[S#Duz<&  
    % the summation often involves computing small differences between r 3|4gG  
    % large successive terms in the series. (In such cases, the functions  9|<Be6  
    % are often evaluated using alternative methods such as recurrence TH YVT%v  
    % relations: see the Legendre functions, for example). For the Zernike %OEq,Tb  
    % polynomials, however, this problem does not arise, because the :SK<2<8h  
    % polynomials are evaluated over the finite domain r = (0,1), and xb]o dYGdW  
    % because the coefficients for a given polynomial are generally all JA< :K0  
    % of similar magnitude. gd_ ^  
    % 4j{oaey  
    % ZERNPOL has been written using a vectorized implementation: multiple `2,a(Sk#  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Ox~ 9_d  
    % values can be passed as inputs) for a vector of points R.  To achieve viJJ e'\2  
    % this vectorization most efficiently, the algorithm in ZERNPOL Oi6Eo~\f  
    % involves pre-determining all the powers p of R that are required to 9{$8\E9*nd  
    % compute the outputs, and then compiling the {R^p} into a single Hg aZbb>'  
    % matrix.  This avoids any redundant computation of the R^p, and /,LfA2^_j{  
    % minimizes the sizes of certain intermediate variables. ;$z7[+M  
    % l0:5q?g  
    %   Paul Fricker 11/13/2006 x^X$M$o,l  
    Hsgy'X%om  
    3(C :X1  
    % Check and prepare the inputs: (![t_r0  
    % ----------------------------- d+Ds9(gV  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +2Z#M  
        error('zernpol:NMvectors','N and M must be vectors.') u0g*O]Y  
    end A=y"x$%-_  
    x~z_,':  
    if length(n)~=length(m) fx]eDA|$e  
        error('zernpol:NMlength','N and M must be the same length.') ZL=N[XW4'  
    end +YuzpuxjJ  
    M!#AfIyB  
    n = n(:); wA631kr  
    m = m(:); NocFvF7\  
    length_n = length(n); ~$Y|ca  
    ewym 1}o  
    if any(mod(n-m,2)) Za0gs @$  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ~#q;bS  
    end M%|f+u&  
    a*s\Em7f  
    if any(m<0) kN.B/itvA  
        error('zernpol:Mpositive','All M must be positive.') 9ad6uTc  
    end UGCox-W"  
    8kS~ENe?o  
    if any(m>n) {@45?L('  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 2f^-~dz  
    end S/fW/W*/}  
    ED/FlL{  
    if any( r>1 | r<0 ) v8~YR'T0`V  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') Fg4@On[,i  
    end ~~q}cywBk  
    as#J qE  
    if ~any(size(r)==1) p-Pz=Cx-  
        error('zernpol:Rvector','R must be a vector.') 7*;^UqGjz  
    end x6%#ws vS  
    !k-` eJ|  
    r = r(:); EHhd;,;O  
    length_r = length(r); 9~~UM<66W  
    h0lu!m#\_  
    if nargin==4 vhpvO >Q  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 8U=A{{0p  
        if ~isnorm 7k~Lttuk  
            error('zernpol:normalization','Unrecognized normalization flag.') Y"*:&E2)r  
        end G0/>8_Q>Nr  
    else :Y^I]`lR"  
        isnorm = false; |xeE3,8  
    end {*$9,  
    fI]bzv;  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mW +tV1XjG  
    % Compute the Zernike Polynomials 0+j}};   
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s!de2z  
    UJn/s;$.e  
    % Determine the required powers of r: ESv:1o`?n  
    % ----------------------------------- ) Fx ?%  
    rpowers = []; SX_4=^  
    for j = 1:length(n) OpQ8\[X+  
        rpowers = [rpowers m(j):2:n(j)]; %t[K36,p  
    end {(Fe7,.S3  
    rpowers = unique(rpowers); ^/a*.cu  
    o|rzN\WJn  
    % Pre-compute the values of r raised to the required powers, k!owl+a   
    % and compile them in a matrix: v ): V  
    % ----------------------------- "lrA%~3%[P  
    if rpowers(1)==0 HTR1)b  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 7=3O^=Q ^Q  
        rpowern = cat(2,rpowern{:}); l[*sHi  
        rpowern = [ones(length_r,1) rpowern]; nh0&'hA  
    else "-0;#&!  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); { i;6vRr  
        rpowern = cat(2,rpowern{:}); *<q4S(l  
    end J3IRP/*z  
    'HB~Dbq`V  
    % Compute the values of the polynomials: ^Plc}W7h  
    % -------------------------------------- EY$?^iS  
    z = zeros(length_r,length_n); 61|B]ei/  
    for j = 1:length_n C0(sAF@  
        s = 0:(n(j)-m(j))/2; >3P9 i ;W  
        pows = n(j):-2:m(j); tT-=hDw  
        for k = length(s):-1:1 enumK\  
            p = (1-2*mod(s(k),2))* ... VYigxhP7  
                       prod(2:(n(j)-s(k)))/          ... x8/us  
                       prod(2:s(k))/                 ... 41}/w3Z4  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... /buWAX 1  
                       prod(2:((n(j)+m(j))/2-s(k))); >UWStzH<  
            idx = (pows(k)==rpowers); N9`97;.X  
            z(:,j) = z(:,j) + p*rpowern(:,idx); iRs V#s  
        end ^1VbH3M  
         OoM_q/oI  
        if isnorm c/'M#h)"  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); X+at%L=  
        end r0Z+ RB^I  
    end xlw 2g<s  
    JY@X2'>v/  
    % EOF zernpol
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) S.hC$0vrj  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. =qX*]  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ymkR!  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive I.9o`Q[8&  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, $}4K`Iu  
    %   and THETA is a vector of angles.  R and THETA must have the same `j:M)2:*y  
    %   length.  The output Z is a matrix with one column for every P-value, ph#efY`a:  
    %   and one row for every (R,THETA) pair. cAibB&`~  
    % Cya5*U0=  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike PY -+Bf  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) gQR1$n0  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) =)*JbwQ   
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 %YCd%lAe,  
    %   for all p. uS-3\$  
    % T<M?PlED  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 1 7i$8  
    %   Zernike functions (order N<=7).  In some disciplines it is z{M8Yf |  
    %   traditional to label the first 36 functions using a single mode oAnigu;  
    %   number P instead of separate numbers for the order N and azimuthal lC2?sD$  
    %   frequency M. e`AUYli"  
    % "uhV|Lk*7  
    %   Example: 0\wiam-  
    % 3KT_AJ4}  
    %       % Display the first 16 Zernike functions {U6"]f%  
    %       x = -1:0.01:1; M8zE3;5  
    %       [X,Y] = meshgrid(x,x); AWL[zixR  
    %       [theta,r] = cart2pol(X,Y); `oVB!eapl  
    %       idx = r<=1; [?I/Uo8  
    %       p = 0:15; (Com,  
    %       z = nan(size(X)); f8#*mQ  
    %       y = zernfun2(p,r(idx),theta(idx)); 7t3X`db  
    %       figure('Units','normalized') z^3Q.4Qc6^  
    %       for k = 1:length(p) o$\tHzB9!A  
    %           z(idx) = y(:,k); V&R$8tpz  
    %           subplot(4,4,k) ctK65h{Eo  
    %           pcolor(x,x,z), shading interp *`1bc'umM;  
    %           set(gca,'XTick',[],'YTick',[]) /6jGt'^U  
    %           axis square zHqhl}  
    %           title(['Z_{' num2str(p(k)) '}']) sbA2W~:  
    %       end gWi{\x8dt  
    % ?~ ?H dv  
    %   See also ZERNPOL, ZERNFUN. pX^=be_  
    K9*IA@xL  
    %   Paul Fricker 11/13/2006 |i u2&p >  
    (Z 8,e  
    6J"(xT  
    % Check and prepare the inputs: eK *W =c#@  
    % ----------------------------- p_9g|B0D  
    if min(size(p))~=1 P>fKX2eQ-  
        error('zernfun2:Pvector','Input P must be vector.') gg(k7e  
    end }\VX^{K j  
    Y-= /,   
    if any(p)>35 (,U7 R^  
        error('zernfun2:P36', ... wsI5F&R,  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... S?2YJ l8B  
               '(P = 0 to 35).']) .8x@IWJD  
    end ]K*GSU  
    =R2l3-HA=  
    % Get the order and frequency corresonding to the function number: F:,#?  
    % ---------------------------------------------------------------- $N dH*  
    p = p(:); V pH|R  
    n = ceil((-3+sqrt(9+8*p))/2); +nzTxpcP@K  
    m = 2*p - n.*(n+2); ZBC@xM&-  
    ([tG y  
    % Pass the inputs to the function ZERNFUN: E$R_rX4x  
    % ---------------------------------------- vU{jda$$#  
    switch nargin ]xYayN!n  
        case 3 x RB7lV*  
            z = zernfun(n,m,r,theta); fRFYJFc n  
        case 4 RJLFj  
            z = zernfun(n,m,r,theta,nflag); W.p66IQwL&  
        otherwise lU& Q^Zj`  
            error('zernfun2:nargin','Incorrect number of inputs.') w_GLC%|7  
    end `^zQ$au'u  
    5)8 .  
    % EOF zernfun2
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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 I>\}}!  
    function z = zernfun(n,m,r,theta,nflag) FU'^n6[<B  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. `9:v*KuM#R  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Z5yt]-WN&  
    %   and angular frequency M, evaluated at positions (R,THETA) on the f x%z| K  
    %   unit circle.  N is a vector of positive integers (including 0), and HuK Aj  
    %   M is a vector with the same number of elements as N.  Each element +A&EKk%$ |  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) {rs6"X^  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, C CLfvex  
    %   and THETA is a vector of angles.  R and THETA must have the same PMD,8]|  
    %   length.  The output Z is a matrix with one column for every (N,M) GCZu<,  
    %   pair, and one row for every (R,THETA) pair. s8{-c^G:R  
    % 1`nc8qC  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike g<0w/n!jmC  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Vvx a.B  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral /E; ;j9  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, MM=W9#  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized B #;s(O  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. VyRW'  
    % (R,NV3m?w  
    %   The Zernike functions are an orthogonal basis on the unit circle. &Jrq5Q C  
    %   They are used in disciplines such as astronomy, optics, and 3zk:59  
    %   optometry to describe functions on a circular domain. "9TxK6  
    % F]hx  
    %   The following table lists the first 15 Zernike functions. ?G2qlna  
    % =ZFcxGo  
    %       n    m    Zernike function           Normalization ;L#L Dk{Za  
    %       -------------------------------------------------- R (t!xf  
    %       0    0    1                                 1 O_qu;Dx!  
    %       1    1    r * cos(theta)                    2 Z3LQl(  
    %       1   -1    r * sin(theta)                    2 .ruqRGe/  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) |^ 2rtI  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ]JkpRaP$  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) mjWp8i  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) l2z`<2mp  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) v+|@}9|Z  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ;a#}fX  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Xi1q]ps  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ';i"?D?NAk  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6RR4L^(m  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) eA3`]XP.`b  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) a*pXrp@  
    %       4    4    r^4 * sin(4*theta)             sqrt(10)  O6M}W_  
    %       -------------------------------------------------- QwKky ^A  
    % =1V>Vd?8.  
    %   Example 1: &':UlzG  
    % :u[ oc.  
    %       % Display the Zernike function Z(n=5,m=1) Lf$Q %eM0  
    %       x = -1:0.01:1; KIXwx98  
    %       [X,Y] = meshgrid(x,x); $8<j5%/ $M  
    %       [theta,r] = cart2pol(X,Y); qk"oFP6  
    %       idx = r<=1; ?,A}E|jZ  
    %       z = nan(size(X)); HV#?6,U}  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); SSSDl$}'t  
    %       figure 6Cop#kW#  
    %       pcolor(x,x,z), shading interp :)^# xE(  
    %       axis square, colorbar 0KWy?6 X  
    %       title('Zernike function Z_5^1(r,\theta)') B}l}Aq8  
    % O2V6UX@&<w  
    %   Example 2: [Gh%nsH  
    % x= vE&9_u  
    %       % Display the first 10 Zernike functions t?3{s\z8+  
    %       x = -1:0.01:1; n1k$)S$iiy  
    %       [X,Y] = meshgrid(x,x); o O{|C&A  
    %       [theta,r] = cart2pol(X,Y); \N'hbT=  
    %       idx = r<=1; PVQ#>_~5  
    %       z = nan(size(X)); XcJ'm{=   
    %       n = [0  1  1  2  2  2  3  3  3  3]; %l9WZ*yZ`2  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; <;TP@-a  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ~/]\iOL  
    %       y = zernfun(n,m,r(idx),theta(idx)); 7(nz<z p  
    %       figure('Units','normalized') )-TeDIfm  
    %       for k = 1:10 b3CspBgC  
    %           z(idx) = y(:,k); '6d D^0dZ  
    %           subplot(4,7,Nplot(k)) `-9*@_ -=M  
    %           pcolor(x,x,z), shading interp Kq@m?h  
    %           set(gca,'XTick',[],'YTick',[]) yNb#Ia  
    %           axis square 9;xL!cy  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) q7)]cY_  
    %       end qxg7cj2  
    % Wq[=}qh~  
    %   See also ZERNPOL, ZERNFUN2. @+T{M:&l  
    Qzs\|KS  
    %   Paul Fricker 11/13/2006 Jnu}{^~  
    /64^5DjTh  
    x]mye  
    % Check and prepare the inputs:  q~:'R  
    % ----------------------------- N('S2yfDR  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) [][:/~q!  
        error('zernfun:NMvectors','N and M must be vectors.') "0!eb3n  
    end hK9t}NE.O  
    t?#vb}_  
    if length(n)~=length(m) qMW%$L\HA  
        error('zernfun:NMlength','N and M must be the same length.') !X v2PdP  
    end 99+/W*C  
    >X\s[d&(  
    n = n(:); j4 &  
    m = m(:); hsQrd%{f  
    if any(mod(n-m,2)) %gne%9nn  
        error('zernfun:NMmultiplesof2', ... _n Iqy&<  
              'All N and M must differ by multiples of 2 (including 0).') U d=gdsL  
    end %RT6~0z  
    2A18hP`^  
    if any(m>n) M#8Ao4 T  
        error('zernfun:MlessthanN', ... :vgh KI  
              'Each M must be less than or equal to its corresponding N.') GqK&'c   
    end P/1UCITq}  
    y uK5r  
    if any( r>1 | r<0 ) c|;|%"Mk  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') \aJ-q?=  
    end &:e}4/G  
    OV@h$fg  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) D=I5[t0c4  
        error('zernfun:RTHvector','R and THETA must be vectors.') 2'UFHiK  
    end z"P,=M6De  
    z7us*8X{  
    r = r(:); lo]B 5_en  
    theta = theta(:); 65e Wu=T  
    length_r = length(r); (k)gZD9~{?  
    if length_r~=length(theta) coP$7Q .  
        error('zernfun:RTHlength', ... /NN[gz  
              'The number of R- and THETA-values must be equal.') $M3A+6["H  
    end w]5f3CIm  
    39a]B`y  
    % Check normalization: Rp%\`'+Xz  
    % -------------------- Qig!NgOM  
    if nargin==5 && ischar(nflag) M]/wei"X  
        isnorm = strcmpi(nflag,'norm'); 52C-D+zCJ  
        if ~isnorm ^D> MDj6  
            error('zernfun:normalization','Unrecognized normalization flag.') YI\Cs=T/  
        end pil*/&pB  
    else !y2h`ZAZ  
        isnorm = false; :7PSZc:xE  
    end 3TvhOC>yG  
    YT%SCaU  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t=pkYq5t8  
    % Compute the Zernike Polynomials  rgvc5p  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q$2taG}  
    ~JmxW;|_x)  
    % Determine the required powers of r: M(]|}%  
    % ----------------------------------- F]&J%i F[  
    m_abs = abs(m); ALt";8Oa  
    rpowers = []; WZ V*J&  
    for j = 1:length(n) #uw*8&%0  
        rpowers = [rpowers m_abs(j):2:n(j)]; HgBEV  
    end wqoN@d  
    rpowers = unique(rpowers); D~`YRbv  
    =z /mI y<  
    % Pre-compute the values of r raised to the required powers, VA r?teY  
    % and compile them in a matrix: zB7dCw  
    % ----------------------------- d?qO`- ~$  
    if rpowers(1)==0 AJ1$$c  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); #?d>S;)+  
        rpowern = cat(2,rpowern{:});   SrU   
        rpowern = [ones(length_r,1) rpowern]; &i}cC4i   
    else *Lk&@(  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); H&Lbdu~E  
        rpowern = cat(2,rpowern{:}); C5z  
    end ,`2xfVa-  
    3eDx@8N }  
    % Compute the values of the polynomials: qmeEUch`  
    % -------------------------------------- 3&d+U)E  
    y = zeros(length_r,length(n)); $gtT5{"PN(  
    for j = 1:length(n) 7Sv5fLu2  
        s = 0:(n(j)-m_abs(j))/2; <YNPhu~5  
        pows = n(j):-2:m_abs(j); 0QSi\: 1f  
        for k = length(s):-1:1 S gsR;)2  
            p = (1-2*mod(s(k),2))* ... (C[S?@S  
                       prod(2:(n(j)-s(k)))/              ... #^ [N4uV  
                       prod(2:s(k))/                     ... aj-uk(r  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ',ybHW%D%i  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); E|}Nj}(*  
            idx = (pows(k)==rpowers); k <Sa<  
            y(:,j) = y(:,j) + p*rpowern(:,idx); x};g!FYfkB  
        end wDTV /"Y  
         Z]+Xh  
        if isnorm L ]'CA^N  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi);  2x J5  
        end zi 14]FWo  
    end e ^& 8x  
    % END: Compute the Zernike Polynomials !7kOw65+0  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% qO'5*d;!d  
    O g~"+IGp  
    % Compute the Zernike functions: @wZ_VE7B  
    % ------------------------------ '(:J|DN  
    idx_pos = m>0; KT?s\w  
    idx_neg = m<0; QlXF:Gx"=  
    R20GjWy=  
    z = y; bL[W.O0  
    if any(idx_pos) IY6S\Gn  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); /[T8/7;_l  
    end cuk}VZ  
    if any(idx_neg) At|tk  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); *zht(~%  
    end srA~gzF  
    #iU/Yg!  
    % EOF zernfun
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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
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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