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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 iCz,|;w%  
    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! o.y4&bC14;  
     
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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)  n |.- :Zy  
    M> 1V3 sM  
    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。 $c  f?`k  
    dI'C[.zp[  
    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) [-Q"A 6!Zd  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. I0)iC[s8;  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of t@)~{W {  
    %   order N and frequency M, evaluated at R.  N is a vector of sM~CP zMa  
    %   positive integers (including 0), and M is a vector with the X3 a:*1N  
    %   same number of elements as N.  Each element k of M must be a \k;raQR4t*  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) kv`x  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is _k6N(c2Nd  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix /Rt/Efu  
    %   with one column for every (N,M) pair, and one row for every -pkeEuwv{  
    %   element in R. t}*teo[  
    % -pX/Tt6  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- nC>#@*+jK  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is Z < uwqA  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to i-niRu<  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 {L<t6A  
    %   for all [n,m]. S :(1=@  
    % #gsAwna3  
    %   The radial Zernike polynomials are the radial portion of the _);1dcnR  
    %   Zernike functions, which are an orthogonal basis on the unit .fQDj{  
    %   circle.  The series representation of the radial Zernike d@#=cvW  
    %   polynomials is _>3GNvS  
    % yd>kJk^~/  
    %          (n-m)/2 _^&oNm1  
    %            __ X*FK6,Y|(  
    %    m      \       s                                          n-2s a#G7pZX/I}  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r +{Q\B}3cj1  
    %    n      s=0 'OF)`5sj  
    % _$Z46wHmB  
    %   The following table shows the first 12 polynomials. [nG/>Z]W  
    % 2.; OHQTE  
    %       n    m    Zernike polynomial    Normalization ncS^NH(&  
    %       --------------------------------------------- ixfkMM ,W  
    %       0    0    1                        sqrt(2) R`s /^0  
    %       1    1    r                           2 @6t3Us~/  
    %       2    0    2*r^2 - 1                sqrt(6) NK,)"WE  
    %       2    2    r^2                      sqrt(6) 6 t A?<S  
    %       3    1    3*r^3 - 2*r              sqrt(8) *sL'6"#Cre  
    %       3    3    r^3                      sqrt(8) gs0,-)  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) >@EQarD  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) wBeOMA  
    %       4    4    r^4                      sqrt(10) %M'"%Yn@(y  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Kz^aW  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) w8@MUz}/#  
    %       5    5    r^5                      sqrt(12) b[BSUdCB  
    %       --------------------------------------------- #uey1I@"9  
    % ",~3&wx  
    %   Example: pCpj#+|_)  
    % xFyMg&  
    %       % Display three example Zernike radial polynomials U?>zq!C&R  
    %       r = 0:0.01:1; @ ?e;Jp9  
    %       n = [3 2 5]; IXz ad  
    %       m = [1 2 1]; q)@.f.  
    %       z = zernpol(n,m,r); T,H]svN5p  
    %       figure c~$ipX   
    %       plot(r,z) tgrQ$Yjk  
    %       grid on -R&h?ec  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') XWB>' UDQ#  
    % /~AwX8X  
    %   See also ZERNFUN, ZERNFUN2. \&e+f#!u  
    YjdH7.js  
    % A note on the algorithm. `5q`ibyPI  
    % ------------------------ &4{%3w_/  
    % The radial Zernike polynomials are computed using the series G&3j/5V  
    % representation shown in the Help section above. For many special =U,;/f  
    % functions, direct evaluation using the series representation can !;R{-  
    % produce poor numerical results (floating point errors), because *DG*&Me  
    % the summation often involves computing small differences between ?BWWb   
    % large successive terms in the series. (In such cases, the functions -lAA,}&+!  
    % are often evaluated using alternative methods such as recurrence kWoy%?|RRa  
    % relations: see the Legendre functions, for example). For the Zernike tX)]ZuEi$  
    % polynomials, however, this problem does not arise, because the Z?v9ub~%  
    % polynomials are evaluated over the finite domain r = (0,1), and ^[ id8  
    % because the coefficients for a given polynomial are generally all x,p|n  
    % of similar magnitude. kxf'_Nzy  
    % H;$w^Tr  
    % ZERNPOL has been written using a vectorized implementation: multiple XP(q=Mw  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] XeZv%` ?  
    % values can be passed as inputs) for a vector of points R.  To achieve KDn`XCnk,  
    % this vectorization most efficiently, the algorithm in ZERNPOL (tVY /(~#  
    % involves pre-determining all the powers p of R that are required to @j^qT-0M  
    % compute the outputs, and then compiling the {R^p} into a single `qfVgT=2  
    % matrix.  This avoids any redundant computation of the R^p, and 'fg`td  
    % minimizes the sizes of certain intermediate variables. BJ&>'rc  
    % 67n1s  
    %   Paul Fricker 11/13/2006 if `/LJsa  
    Hq%`DWus\  
    .Qi`5C:U  
    % Check and prepare the inputs: s"sX# l[J  
    % ----------------------------- u\Xi]pZ@X]  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }. ,xhF[  
        error('zernpol:NMvectors','N and M must be vectors.') *XNvb ^<  
    end I/Vlw-  
    wef QmRK  
    if length(n)~=length(m) K IqF"5  
        error('zernpol:NMlength','N and M must be the same length.') bBDgyFSI <  
    end yV`!Fq 1k  
    !\!fd(BN  
    n = n(:); !_c<j4O  
    m = m(:); p*dez!  
    length_n = length(n); }Y-f+qX*  
    Y RA[qc  
    if any(mod(n-m,2)) =-vk}O0C  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ^ 0TJys%  
    end 1x\%VtO>\b  
    !Ug J^v  
    if any(m<0) rW1 > t+  
        error('zernpol:Mpositive','All M must be positive.') ls/:/x(5d  
    end ;JAe=wt^'I  
    2 3>lE}^G  
    if any(m>n) kmP0gT{Sj  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') i#Wl?(-i  
    end v#nFPB=z  
    no;Yu  
    if any( r>1 | r<0 ) &[kwM3 95  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') .JH3,L"S^  
    end a?D\H5TF-  
    Z9!goI  
    if ~any(size(r)==1) us5`?XeX]  
        error('zernpol:Rvector','R must be a vector.') S"}FsS;k<?  
    end ,ciNoP*-~%  
    t#<q O6&B  
    r = r(:); F1/f:<}  
    length_r = length(r); O?{pln  
    os#j;C]l  
    if nargin==4 ZPMX19  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); m_St"`6 .  
        if ~isnorm j)J4[j  
            error('zernpol:normalization','Unrecognized normalization flag.') qOk4qbl[  
        end rT"8e*LT  
    else G q0~&6  
        isnorm = false; P= S)V   
    end g3Ff<P P  
    i{ %~&!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !8P#t{2_|  
    % Compute the Zernike Polynomials e[{LNM{/#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .Sb|+[{  
    xat)9Yb}0  
    % Determine the required powers of r: 9K)OQDv%6D  
    % ----------------------------------- W_kJb  
    rpowers = []; &jg,8  
    for j = 1:length(n) y0rT=kU  
        rpowers = [rpowers m(j):2:n(j)]; bC)<AG@Z\  
    end ]YwIuz6]  
    rpowers = unique(rpowers); 8U=M.FFp  
    2{{M{#}S.  
    % Pre-compute the values of r raised to the required powers, mu:Q2t^  
    % and compile them in a matrix: ( XE`,#  
    % ----------------------------- SHh g&~B  
    if rpowers(1)==0 }*? e w  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 5*4P_q(AxD  
        rpowern = cat(2,rpowern{:}); m ;[z)-&"  
        rpowern = [ones(length_r,1) rpowern]; ~L4"t_-  
    else r^Gl~sX  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); E9 q8tE}  
        rpowern = cat(2,rpowern{:}); Te5_T&1Z  
    end PhW#=S  
    6 rmK_Y  
    % Compute the values of the polynomials: )_+#yaC  
    % -------------------------------------- LfF<wDvXf  
    z = zeros(length_r,length_n); a eP4%h  
    for j = 1:length_n #7'ww*+  
        s = 0:(n(j)-m(j))/2; @ZT25CD  
        pows = n(j):-2:m(j); J }JT%S W  
        for k = length(s):-1:1 M0_K%Z(zaR  
            p = (1-2*mod(s(k),2))* ... Y B)1dzU  
                       prod(2:(n(j)-s(k)))/          ... I ][8[UZ  
                       prod(2:s(k))/                 ... [0_Kz"|  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... f~"3#MaV  
                       prod(2:((n(j)+m(j))/2-s(k))); A$|> Jt  
            idx = (pows(k)==rpowers); `[Lap=.' .  
            z(:,j) = z(:,j) + p*rpowern(:,idx);  rro,AS}  
        end 6G1Z"9<2*  
         ~r|.GY  
        if isnorm +F 5Dc  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); V ;>{-p  
        end {J|P2a[  
    end 1 w\Y ._jK  
    kv)LH{  
    % 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) yv'mV=BMJ!  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. /:%^Vh3XF  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated '^P Ud`  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive So!1l7b  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, =OjzBiHR  
    %   and THETA is a vector of angles.  R and THETA must have the same iUl{_vb  
    %   length.  The output Z is a matrix with one column for every P-value, # &M  
    %   and one row for every (R,THETA) pair. 8V4Qyi|@F  
    % ;tKL/eI  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike c#G(7.0MU  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) l~f +h?cF  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) O<%U*:B  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 bfa5X<8  
    %   for all p. e HOm^.gd  
    % JWxPH5L  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 4.VEE~sH$  
    %   Zernike functions (order N<=7).  In some disciplines it is <[pU rJfTr  
    %   traditional to label the first 36 functions using a single mode 29Gej Lg |  
    %   number P instead of separate numbers for the order N and azimuthal =z@'vu$Fh  
    %   frequency M. Ufo- AeQo  
    % 7 s[ ATu  
    %   Example: DR=>la}!  
    % v4Nb/Y  
    %       % Display the first 16 Zernike functions PUlb(3p `  
    %       x = -1:0.01:1; J(l6(+8  
    %       [X,Y] = meshgrid(x,x); ;=e A2  
    %       [theta,r] = cart2pol(X,Y); r2xlcSn%  
    %       idx = r<=1; Y ,}p  
    %       p = 0:15; fc!%W#-  
    %       z = nan(size(X)); hSg: Rqnk  
    %       y = zernfun2(p,r(idx),theta(idx)); (@&|  
    %       figure('Units','normalized') eueXklpg+  
    %       for k = 1:length(p) 3K#e]zoI  
    %           z(idx) = y(:,k); 1,pg:=N9  
    %           subplot(4,4,k) OB"QWdh  
    %           pcolor(x,x,z), shading interp }f({03$  
    %           set(gca,'XTick',[],'YTick',[]) -(1e!5_-@  
    %           axis square \64(`6>  
    %           title(['Z_{' num2str(p(k)) '}']) 134wK]d^  
    %       end [hFyu|I !  
    % #b8/gRfS  
    %   See also ZERNPOL, ZERNFUN. j5ui  
    )}6:Ke)  
    %   Paul Fricker 11/13/2006 w=f8UtY9@A  
    x3WY26e  
    *Pq`~W_M7  
    % Check and prepare the inputs: I#A`fJ  
    % ----------------------------- @-MrmF)<U  
    if min(size(p))~=1 5 wc&0h  
        error('zernfun2:Pvector','Input P must be vector.') 16aaIK  
    end *}2o \h6Q  
    /\\C&Px  
    if any(p)>35 Xt~/8)&  
        error('zernfun2:P36', ... IjrTM{f  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... "#JoB X@yE  
               '(P = 0 to 35).']) &V2G <gm0  
    end q?t>!1c  
    %M^bZ?  
    % Get the order and frequency corresonding to the function number: ?9PNCd3$d  
    % ---------------------------------------------------------------- t8^*s<O  
    p = p(:); RP(FV<ot  
    n = ceil((-3+sqrt(9+8*p))/2); |Z "h q  
    m = 2*p - n.*(n+2); [S9nF  
    s&tr84u|  
    % Pass the inputs to the function ZERNFUN: \LS%bO,Y|  
    % ---------------------------------------- @B[=`9KF[  
    switch nargin /Pf7=P  
        case 3 XM_S"  
            z = zernfun(n,m,r,theta); 6!gGWn5>}  
        case 4 f|apk,o_  
            z = zernfun(n,m,r,theta,nflag); )lW<: ?k  
        otherwise 2OZdj  
            error('zernfun2:nargin','Incorrect number of inputs.') JUXK}0d%eN  
    end t71 0sWh{  
    F 'h[g.\}  
    % EOF zernfun2
    离线niuhelen
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    只看该作者 3楼 发表于: 2011-03-12
    回 2楼(phility) 的帖子
    非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 EM([N*8o  
    function z = zernfun(n,m,r,theta,nflag) 2xjS;lpw  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. z#-&MJ  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 9w AP%xh  
    %   and angular frequency M, evaluated at positions (R,THETA) on the S5uV\Y/A  
    %   unit circle.  N is a vector of positive integers (including 0), and c[;I\g  
    %   M is a vector with the same number of elements as N.  Each element XCt}>/"s\h  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) !WIL|\jbh  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, }ShZ4 xMz  
    %   and THETA is a vector of angles.  R and THETA must have the same U 26Iz  
    %   length.  The output Z is a matrix with one column for every (N,M) Y${ $7+@  
    %   pair, and one row for every (R,THETA) pair. JY_' d,O  
    % QX8N p{g-  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 00 $W>Gr  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 8T2$0  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 2R1W[,Ga!  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, jy1*E3vQ  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized !X,=RR `zT  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ME7JU|@Z  
    % =6%0pu]0  
    %   The Zernike functions are an orthogonal basis on the unit circle. 4f/8APA  
    %   They are used in disciplines such as astronomy, optics, and LOOv8'%O8  
    %   optometry to describe functions on a circular domain. yX)2 hj:s  
    % ?vk&k(FT  
    %   The following table lists the first 15 Zernike functions. uH7u4f1Q  
    % KQ2]VN"?_  
    %       n    m    Zernike function           Normalization fa6L+wt4O  
    %       -------------------------------------------------- sNNt0q(  
    %       0    0    1                                 1 6x.#K9@q4  
    %       1    1    r * cos(theta)                    2 3_D$6/i  
    %       1   -1    r * sin(theta)                    2 &-&6ARb7o  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) :0vNg:u+  
    %       2    0    (2*r^2 - 1)                    sqrt(3) S3n$  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) u''(;U[  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 3c ^_IuW-  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) iaR'):TD  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) kdv>QZ  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) } $OQw'L[  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) \75%[;.  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ANWa%%\T  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) gE%-Pf~  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ok,hm.|  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 0Uybh.dC  
    %       -------------------------------------------------- Iw48+krm>  
    % ,qC_[PUT  
    %   Example 1: BG=h1ybz  
    % Dn9Ta}miTO  
    %       % Display the Zernike function Z(n=5,m=1) 3s$m0  
    %       x = -1:0.01:1; oS]XE!^M  
    %       [X,Y] = meshgrid(x,x); gB&'MA!  
    %       [theta,r] = cart2pol(X,Y); iJ#sg+  
    %       idx = r<=1; +nZx{d,wt  
    %       z = nan(size(X)); 2"2b\b}my  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 5Rc 5/m  
    %       figure xro  
    %       pcolor(x,x,z), shading interp TMq\}k-I5  
    %       axis square, colorbar f v}h;?C  
    %       title('Zernike function Z_5^1(r,\theta)') j'v2m6/  
    % *)"`v]  
    %   Example 2: )<!y_;$A  
    % |>d5 6  
    %       % Display the first 10 Zernike functions :xv"m {8+  
    %       x = -1:0.01:1; #N7@p }P  
    %       [X,Y] = meshgrid(x,x); $n>.;CV  
    %       [theta,r] = cart2pol(X,Y); 9.>v ;:vL  
    %       idx = r<=1; (L q^C=  
    %       z = nan(size(X)); 3d \bB !  
    %       n = [0  1  1  2  2  2  3  3  3  3]; <w 8*Ly:L  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; r"k\G\,%  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; &i6WVNGy  
    %       y = zernfun(n,m,r(idx),theta(idx)); z$S)|6Q  
    %       figure('Units','normalized') 8 \%*4L'  
    %       for k = 1:10 U Tw\_s  
    %           z(idx) = y(:,k); ix6j=5{  
    %           subplot(4,7,Nplot(k)) # bP1rQ0  
    %           pcolor(x,x,z), shading interp qm8[ ^jO&  
    %           set(gca,'XTick',[],'YTick',[]) =C u !  
    %           axis square O?rVa:\  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Y}ITA=L7  
    %       end @G^ l`%  
    % 7H9&\ur9+  
    %   See also ZERNPOL, ZERNFUN2. "Q-TLN5(  
    #2/k^N4r  
    %   Paul Fricker 11/13/2006 _6xC4@~h*  
    ':6`M  
    <`n T+c  
    % Check and prepare the inputs: ^vfp;  
    % ----------------------------- QGn3xM66  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) %^kBcId  
        error('zernfun:NMvectors','N and M must be vectors.') W(Xb]t=19  
    end SfEgmp-m  
    48W$ ,  
    if length(n)~=length(m) X\V1c$13CK  
        error('zernfun:NMlength','N and M must be the same length.') ~#pQWa5  
    end hvwKhQ}wX  
    Y{6y.F*Q#  
    n = n(:); `ZC_F! E  
    m = m(:); +?DP r  
    if any(mod(n-m,2)) j/ow8Jmc*  
        error('zernfun:NMmultiplesof2', ... y)CnH4{  
              'All N and M must differ by multiples of 2 (including 0).') nj]l'~Y0  
    end .T#h5[S2x  
    ko2?q  
    if any(m>n) sZxf.  
        error('zernfun:MlessthanN', ... h3[^uY e  
              'Each M must be less than or equal to its corresponding N.') :Z3Tyj}4  
    end Xy5#wDRC  
    g\ilK:r}  
    if any( r>1 | r<0 ) P uYAoKG  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.')  dtTQY  
    end F-D9nI4{X  
    j0_)DG  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) S ( e]@  
        error('zernfun:RTHvector','R and THETA must be vectors.') *6IytW OX5  
    end iGlg@  
    =ss(~[  
    r = r(:); {(Jbgsxm  
    theta = theta(:); 1Tm,#o  
    length_r = length(r); 9kZ[Z ,=>  
    if length_r~=length(theta) NGIt~"e7R4  
        error('zernfun:RTHlength', ... ;&RBg+Pr  
              'The number of R- and THETA-values must be equal.') Ymt.>8L  
    end }M7{~ov#s  
    @komb IK  
    % Check normalization: |zd+ \o  
    % -------------------- = hL;Q@inb  
    if nargin==5 && ischar(nflag) %k3A`ClW  
        isnorm = strcmpi(nflag,'norm'); N hG?@N  
        if ~isnorm r}T(?KGx  
            error('zernfun:normalization','Unrecognized normalization flag.') x *:v]6y  
        end z{$2bV  
    else V7DMn@Ckw  
        isnorm = false; eO%w i.Q  
    end @:s (L]  
    = j)5kY`  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZP-^10  
    % Compute the Zernike Polynomials u]0{#wu;g  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wB'GV1|jL  
    Y2$wL9">  
    % Determine the required powers of r: H. o=4[  
    % ----------------------------------- `O,^oD4  
    m_abs = abs(m); Q%>6u@'  
    rpowers = []; 7C / ^ Gw  
    for j = 1:length(n) b,h@.s  
        rpowers = [rpowers m_abs(j):2:n(j)]; t9l]ie{"o.  
    end <Fo~|Nh|  
    rpowers = unique(rpowers); 6K Cv  
    8SGqDaRt  
    % Pre-compute the values of r raised to the required powers,  /dI8o  
    % and compile them in a matrix: 7! sR%h5p  
    % ----------------------------- ly9tI-E  
    if rpowers(1)==0 `@3{}  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @V}!elV  
        rpowern = cat(2,rpowern{:}); 6K7DZ96L  
        rpowern = [ones(length_r,1) rpowern]; _|jEuif  
    else Nb3uDA5R  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); V |cPAT%  
        rpowern = cat(2,rpowern{:}); n?(sn  
    end N++ ;}j  
    yOTC>?p%  
    % Compute the values of the polynomials: L$t.$[~L  
    % -------------------------------------- )Szn,  
    y = zeros(length_r,length(n)); >q&X#E<w  
    for j = 1:length(n) -y|*x-iZ  
        s = 0:(n(j)-m_abs(j))/2; l~ Hu#+O  
        pows = n(j):-2:m_abs(j); A<}nXHs-  
        for k = length(s):-1:1 ^#gJf*'UE  
            p = (1-2*mod(s(k),2))* ... gT_tR_g  
                       prod(2:(n(j)-s(k)))/              ... -JfqY?Ue_2  
                       prod(2:s(k))/                     ... N(J'h$E  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... #J'V,_ wH  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Xl,707  
            idx = (pows(k)==rpowers); PiIP%$72O  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Og-v][  
        end 2WUl8?f2Y  
         oM^VtH=>  
        if isnorm .^xQtnq  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); f = 'AI  
        end " M3S  
    end a9_KoOa.H  
    % END: Compute the Zernike Polynomials (M# m BS  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 50e vWD  
    De ([fC  
    % Compute the Zernike functions: <:>[24LJ{  
    % ------------------------------ qTz5P  
    idx_pos = m>0; yZ-Ql1 1  
    idx_neg = m<0; eGW h]%  
    /$d #9Uv  
    z = y; 9 K>~9Za  
    if any(idx_pos) Nd He::  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); cTja<*W^xv  
    end 0nPg`@e.  
    if any(idx_neg) weMufT  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 4axuE]  
    end  c?*x2Vk  
    SveP:uJA[  
    % 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