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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 Qt`hUyL  
    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! ]-[M&i=+&  
     
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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 #a1zk\R3  
    function z = zernfun(n,m,r,theta,nflag) V~#e%&73FH  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. =$bJ`GpJ  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N (al.7VA;9  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Vb{5-v ;a  
    %   unit circle.  N is a vector of positive integers (including 0), and $cl[Qcw  
    %   M is a vector with the same number of elements as N.  Each element 6P,vGmR  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) j,<3[  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, .CJQ]ECl7p  
    %   and THETA is a vector of angles.  R and THETA must have the same }f rij1/G  
    %   length.  The output Z is a matrix with one column for every (N,M) 5L ]TV\\  
    %   pair, and one row for every (R,THETA) pair. DI9hy/T(  
    % b1+6I_u.  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike t<~WDI|AN  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), EY~b,MIL4  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Dl C\sm  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, D$ X9xtT  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized E} Ir<\  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. RYhaQ &1i  
    % ~kDR9s7  
    %   The Zernike functions are an orthogonal basis on the unit circle. :TU|;(p  
    %   They are used in disciplines such as astronomy, optics, and JA]TO (x  
    %   optometry to describe functions on a circular domain. Q1ox<-  
    % oZM6%-@qi  
    %   The following table lists the first 15 Zernike functions. $qz(9M(m#  
    % yH`4 sd  
    %       n    m    Zernike function           Normalization /"~ D(bw0=  
    %       -------------------------------------------------- {;:QY 1Q T  
    %       0    0    1                                 1 C%c}lv8;^  
    %       1    1    r * cos(theta)                    2 4)]w"z0Pc  
    %       1   -1    r * sin(theta)                    2 l'yX_`*Iq  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) c L+-- $L  
    %       2    0    (2*r^2 - 1)                    sqrt(3) *[ ' n8Z  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) !WTZ =|  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) g>k"R4  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) :ik$@5wp  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) gK&MdF*  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) [G.4S5FX.]  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) xXa* d  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |AosZeO_  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Z`_`^ \"  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) m7~<z>5$  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ]YQ!i@Y  
    %       -------------------------------------------------- #9R[%R7Nz  
    % 4[\$3t.L  
    %   Example 1: 5,Q3#f~!  
    % 7z.(pg=  
    %       % Display the Zernike function Z(n=5,m=1) cImOZx  
    %       x = -1:0.01:1; B/:+(|  
    %       [X,Y] = meshgrid(x,x); ;f]p`!] 3  
    %       [theta,r] = cart2pol(X,Y); FWi c/7  
    %       idx = r<=1; W^o* ^v  
    %       z = nan(size(X)); 4jWzYuI&J  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); \IL;}D{  
    %       figure 6[b?ckvi  
    %       pcolor(x,x,z), shading interp t^8 ii  
    %       axis square, colorbar Mz?xvP?z  
    %       title('Zernike function Z_5^1(r,\theta)') jb~W(8cj  
    % O }ES/<an  
    %   Example 2: Ou26QoT9XI  
    % 0r4,27w  
    %       % Display the first 10 Zernike functions P M x`P B  
    %       x = -1:0.01:1; )+Nm @+B  
    %       [X,Y] = meshgrid(x,x); Z$UPLg3=;_  
    %       [theta,r] = cart2pol(X,Y); -dj9(~?^  
    %       idx = r<=1; v?BVUH>#9  
    %       z = nan(size(X)); Fi7G S;  
    %       n = [0  1  1  2  2  2  3  3  3  3]; `.MY" g9  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; \2UtT@3|C  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; a&sVcsX  
    %       y = zernfun(n,m,r(idx),theta(idx)); #!A'6SgbkM  
    %       figure('Units','normalized') f *Xum[  
    %       for k = 1:10 @yGK $<R  
    %           z(idx) = y(:,k); fbl8:c)I  
    %           subplot(4,7,Nplot(k)) Sckt gp8  
    %           pcolor(x,x,z), shading interp ;)6LX-  
    %           set(gca,'XTick',[],'YTick',[]) # NoY}*  
    %           axis square 3SI~?&HU!/  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) "mbjS(-eg  
    %       end 5l(8{,NDt  
    % )2nx5 "  
    %   See also ZERNPOL, ZERNFUN2. $uPM.mPFE  
    P#8+GN+bF  
    %   Paul Fricker 11/13/2006 2qA"emUM  
    ?{)sdJe  
    ;^[VqFpeS  
    % Check and prepare the inputs: #5Q?Q~E@  
    % ----------------------------- jfLkp>2E'  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +qWrm |O]  
        error('zernfun:NMvectors','N and M must be vectors.') g9T9TQ-O  
    end -a[[1  
    `Kt]i5[ "  
    if length(n)~=length(m) slQxz;t  
        error('zernfun:NMlength','N and M must be the same length.') rXIFCt8J  
    end /k$H"'`j4  
    bu2@~  
    n = n(:); :jKiHeBQu?  
    m = m(:); b0PQ;?R#V  
    if any(mod(n-m,2)) b}f#[* Z  
        error('zernfun:NMmultiplesof2', ... `rwzCwA1  
              'All N and M must differ by multiples of 2 (including 0).') p{V_}:|=Q  
    end ?k 4|;DD  
    @nh* H{  
    if any(m>n) x;F^7c1  
        error('zernfun:MlessthanN', ... j;BMuLTm1  
              'Each M must be less than or equal to its corresponding N.') q2$-U&  
    end V[Z^Z  
    Tc3~~X   
    if any( r>1 | r<0 ) 96VJE,^h  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') D*nNu]|j  
    end Au=9<WB%H  
    kPuI'EPK  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 5z T~/6-(  
        error('zernfun:RTHvector','R and THETA must be vectors.') v M lT  
    end G*`H2-,  
    TJ5g? #Wul  
    r = r(:); ^xNs^wC.  
    theta = theta(:); San=E@3}v!  
    length_r = length(r); Uo~-^w}  
    if length_r~=length(theta) dF`\ewRFn  
        error('zernfun:RTHlength', ... e@`"V,i  
              'The number of R- and THETA-values must be equal.') US.7:S-r"  
    end xn &$qLB  
    en5sqKqh+  
    % Check normalization: ='\Di '*  
    % -------------------- 2w7PwNb*32  
    if nargin==5 && ischar(nflag) `Z' h[-2`  
        isnorm = strcmpi(nflag,'norm'); b3vPGR  
        if ~isnorm 2_i9 q>I  
            error('zernfun:normalization','Unrecognized normalization flag.') 6Hh\ys  
        end 9>OPaL n  
    else O'WB O"  
        isnorm = false; T, z80m}  
    end $;V?xZm[  
    c1wP/?|.>  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1Z$` }a  
    % Compute the Zernike Polynomials \ y^Ho1Fj  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [bK5q;#U4  
    -".q=$f  
    % Determine the required powers of r: MT3TWWtZ:  
    % ----------------------------------- ^'Z?BK  
    m_abs = abs(m); $oo`]R_   
    rpowers = []; Hf#VW^  
    for j = 1:length(n) J}{a&3@Hm  
        rpowers = [rpowers m_abs(j):2:n(j)]; 2C &G' @>  
    end Nr(t5TP^  
    rpowers = unique(rpowers); h,palP6^  
    jMAZ4M  
    % Pre-compute the values of r raised to the required powers, X9S` #N  
    % and compile them in a matrix: ~CRd0T[^  
    % ----------------------------- *Bm7>g6  
    if rpowers(1)==0 \I[f@D-J  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); *URBx"5XZ  
        rpowern = cat(2,rpowern{:}); #J): N  
        rpowern = [ones(length_r,1) rpowern]; gR]NH  
    else JHvawFBN<u  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); d siQ~ [   
        rpowern = cat(2,rpowern{:}); uex m|5|  
    end  A|90Ps  
    }iE!( l  
    % Compute the values of the polynomials: vTk\6o q  
    % -------------------------------------- %RS~>pK1  
    y = zeros(length_r,length(n)); >Hd0l L  
    for j = 1:length(n) H[M(t^GM  
        s = 0:(n(j)-m_abs(j))/2; qrw"z iW  
        pows = n(j):-2:m_abs(j); Z6S?xfhr'{  
        for k = length(s):-1:1 f7y3BWOi]  
            p = (1-2*mod(s(k),2))* ... MJ..' $>TC  
                       prod(2:(n(j)-s(k)))/              ... |}07tUq  
                       prod(2:s(k))/                     ... ~ 7^#.  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... g)M"Cx.  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); u &qFE=5:  
            idx = (pows(k)==rpowers); dW4FMm>|  
            y(:,j) = y(:,j) + p*rpowern(:,idx); /9 ^F_2'_  
        end %vZTD +i  
         Jjr&+Q^3Tu  
        if isnorm (=eJceE!  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 1I#]OY#>  
        end 8rEUZk  
    end -L6YLe%w  
    % END: Compute the Zernike Polynomials cmu|d  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,jc')#]9B  
    U{[ g"_+~  
    % Compute the Zernike functions: qPvWb1H:  
    % ------------------------------ Ix59(g  
    idx_pos = m>0; l =X6m(  
    idx_neg = m<0; 4F=cER6l  
    .VG5 / 6zp  
    z = y; IJQ" *;  
    if any(idx_pos) 7+2DsZ^6MW  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ^lP;JT?  
    end gbvMS*KQz  
    if any(idx_neg) ENhKuX  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); u$%;03hJ  
    end ]K!NLvz  
    ; VH:dg  
    % EOF zernfun
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    function z = zernfun2(p,r,theta,nflag) kes GwMr"e  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. F[5sFk M7  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 'Z}3XVZEN  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive \NF5)]:  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Ej#pM.  
    %   and THETA is a vector of angles.  R and THETA must have the same 'jA>P\@8  
    %   length.  The output Z is a matrix with one column for every P-value,  c_ Dg0  
    %   and one row for every (R,THETA) pair. 3k_bhK zI  
    % AA &>6JB{  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike CY</v,\:#  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ;^f ;<  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) r>,s-T!7  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 kD\7wz,ui  
    %   for all p. Qk[YF  
    % `P *wz<  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 g/(3D  
    %   Zernike functions (order N<=7).  In some disciplines it is ={G0p=~+,p  
    %   traditional to label the first 36 functions using a single mode ,ui=Wi1  
    %   number P instead of separate numbers for the order N and azimuthal MG-#p8  
    %   frequency M. !L3\B_#  
    % r>_40+|&  
    %   Example: m'PU0x  
    % $, vX yZ  
    %       % Display the first 16 Zernike functions ~kp,;!^vr  
    %       x = -1:0.01:1; Rc2JgV  
    %       [X,Y] = meshgrid(x,x); _uq[D`=  
    %       [theta,r] = cart2pol(X,Y); 4d63+iM+}  
    %       idx = r<=1; W!o|0u!D  
    %       p = 0:15; 1%$Z%?  
    %       z = nan(size(X)); PR3&LI;B*  
    %       y = zernfun2(p,r(idx),theta(idx)); 8s9ZY4_  
    %       figure('Units','normalized') t->I# t7  
    %       for k = 1:length(p) 4o3TW#  
    %           z(idx) = y(:,k); ;2}wrX  
    %           subplot(4,4,k) " 7g\X$  
    %           pcolor(x,x,z), shading interp z]HaE|j}S  
    %           set(gca,'XTick',[],'YTick',[]) CNl @8&R  
    %           axis square "L&84^lmf  
    %           title(['Z_{' num2str(p(k)) '}']) MRN=-|fV^  
    %       end 6m mc{kw'  
    % #5yz~&  
    %   See also ZERNPOL, ZERNFUN. (Tv~$\=  
    i5#4@ 4aC  
    %   Paul Fricker 11/13/2006 QX$3"AZ~  
    [9d4 0>e  
    ?)O!(=6%'  
    % Check and prepare the inputs: *X\J[$!  
    % ----------------------------- $!7$0WbC  
    if min(size(p))~=1 &a!MT^anA~  
        error('zernfun2:Pvector','Input P must be vector.') JXQh$hs  
    end 0BP Ubp(  
    ~v6]6+   
    if any(p)>35 p8(Z{TSv  
        error('zernfun2:P36', ... #pnB+h&tE  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ^t Y _ q  
               '(P = 0 to 35).']) c&'5r OY~  
    end &P&VJLAe  
    .~,=?aq^  
    % Get the order and frequency corresonding to the function number: Sb.%B^O  
    % ---------------------------------------------------------------- yrG=2{I  
    p = p(:); 9>r@wK'Pn  
    n = ceil((-3+sqrt(9+8*p))/2); ~cul;bb#  
    m = 2*p - n.*(n+2); $1Qcz,4B|  
    Pos(`ys;  
    % Pass the inputs to the function ZERNFUN:   bKt4  
    % ---------------------------------------- gX]ewbPDQ  
    switch nargin 8EY]<#PN  
        case 3 . "Q}2  
            z = zernfun(n,m,r,theta); c0Yc~&RF  
        case 4 |3G;Rh9w,  
            z = zernfun(n,m,r,theta,nflag); p:zRgwcn  
        otherwise u]Z;Q_=  
            error('zernfun2:nargin','Incorrect number of inputs.') ]Jo}F@\g  
    end k:8NOx|s"  
    <UwYI_OX  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) sh*/wM  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. [1e]_9)p  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of C!U$<_I\2  
    %   order N and frequency M, evaluated at R.  N is a vector of =w/AJ%6  
    %   positive integers (including 0), and M is a vector with the B+z>$6  
    %   same number of elements as N.  Each element k of M must be a }/QtIY#I  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) WAf"|  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Ax*xa6_2  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix +F*h\4ry#  
    %   with one column for every (N,M) pair, and one row for every u.Tknw-X  
    %   element in R. zUq(bD  
    % -vv_6Z L[  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- CA5T3J@vAQ  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is _& r19pY  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to xlH3t&i7  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 T+oOlug  
    %   for all [n,m]. 6!SW]#sD  
    % 4v` G/w  
    %   The radial Zernike polynomials are the radial portion of the _*E j3=u  
    %   Zernike functions, which are an orthogonal basis on the unit -us:!p1T  
    %   circle.  The series representation of the radial Zernike Pn l}<i  
    %   polynomials is |g'ceG-  
    % Z!3R  
    %          (n-m)/2 &Q*  7  
    %            __ qkqtPbQ 7  
    %    m      \       s                                          n-2s Dus!Ki~8(t  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ]Y@_2`  
    %    n      s=0 b,X+*hRt  
    % !E|m'_x*  
    %   The following table shows the first 12 polynomials. gq@8Z AWn  
    % nu Vux5:  
    %       n    m    Zernike polynomial    Normalization #8~ygEa}  
    %       --------------------------------------------- iNc!z A4  
    %       0    0    1                        sqrt(2) 8`6G_:&X  
    %       1    1    r                           2 5H 1N]v+  
    %       2    0    2*r^2 - 1                sqrt(6) *qu5o5Q  
    %       2    2    r^2                      sqrt(6) m&s>Sn+  
    %       3    1    3*r^3 - 2*r              sqrt(8) P-<1vfThH  
    %       3    3    r^3                      sqrt(8) U8-OQ:2.  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) aKE`nA0\B  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) C_JO:$\rE  
    %       4    4    r^4                      sqrt(10) Xpp v  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ":q+"*fy  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) gAVD-]`  
    %       5    5    r^5                      sqrt(12) yL),G*[p\}  
    %       --------------------------------------------- jFip-=T{4  
    % ||ugb6q[6B  
    %   Example: c,O;B_}M]  
    % KuW>^mF(I  
    %       % Display three example Zernike radial polynomials n_:EWm$\  
    %       r = 0:0.01:1; v+}${h9  
    %       n = [3 2 5]; G"tlJ7$myQ  
    %       m = [1 2 1]; 8I Ip,#%v  
    %       z = zernpol(n,m,r); X8ZO } X  
    %       figure G:y+yE4  
    %       plot(r,z) '$eJATtC  
    %       grid on L62%s[  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') aGfp"NtL  
    % <EcxNj1  
    %   See also ZERNFUN, ZERNFUN2. e ;^}@X  
    ,7k-LAA  
    % A note on the algorithm. hg#O_4D  
    % ------------------------ >#'?}@FWQN  
    % The radial Zernike polynomials are computed using the series ~<~ ~C#R  
    % representation shown in the Help section above. For many special hgzNEx%^q  
    % functions, direct evaluation using the series representation can Dv L8}dz  
    % produce poor numerical results (floating point errors), because ?>iUz.];t  
    % the summation often involves computing small differences between 7=5eLc^  
    % large successive terms in the series. (In such cases, the functions Pf<yLT]  
    % are often evaluated using alternative methods such as recurrence qS"#jxc==+  
    % relations: see the Legendre functions, for example). For the Zernike  rV4K@)~  
    % polynomials, however, this problem does not arise, because the 4Gh\T`=  
    % polynomials are evaluated over the finite domain r = (0,1), and %K.rrn M  
    % because the coefficients for a given polynomial are generally all 4T; <`{]  
    % of similar magnitude. WUo\jm[yr  
    % FvYciU!  
    % ZERNPOL has been written using a vectorized implementation: multiple (xoYYO  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] bar=^V)  
    % values can be passed as inputs) for a vector of points R.  To achieve )B)f`(SA"<  
    % this vectorization most efficiently, the algorithm in ZERNPOL aWH  
    % involves pre-determining all the powers p of R that are required to n\V7^N  
    % compute the outputs, and then compiling the {R^p} into a single ipl,{  
    % matrix.  This avoids any redundant computation of the R^p, and Gi#-TP\  
    % minimizes the sizes of certain intermediate variables. V0# Ocq,  
    % k<CbI V  
    %   Paul Fricker 11/13/2006 T:zM]%Xh  
    $IjI{%  
    U*E)y7MY  
    % Check and prepare the inputs: 2  
    % ----------------------------- mw}Bl; - O  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) T:Ee6I 3l  
        error('zernpol:NMvectors','N and M must be vectors.') 6>)nkD32g  
    end V=% ;5/  
    "tl$JbRTY  
    if length(n)~=length(m) X{Hh^H  
        error('zernpol:NMlength','N and M must be the same length.') #.[eZ[  
    end _H@ATut  
    5ya^k{`+ZO  
    n = n(:); w= n(2M56C  
    m = m(:); 2<y9xvp  
    length_n = length(n); 9xhc:@B1J  
    S4[ #[w`=  
    if any(mod(n-m,2)) k4hk* 0Jq  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 3Jt# Mp  
    end (_<,Oj#*S  
    S*|/txE'~Y  
    if any(m<0) =-X-${/  
        error('zernpol:Mpositive','All M must be positive.') M@<9/xPS  
    end /*k_`3L  
    VN`fZ5*d~  
    if any(m>n) X-kOp9/.  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 7pciB}$2  
    end R3l{.{3p2  
    Iz#4!E|<  
    if any( r>1 | r<0 ) `OWHf?t:  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') /e?0Iv" 8>  
    end Z+J4 q9^$  
    D9A%8o  
    if ~any(size(r)==1) i,a"5DR8  
        error('zernpol:Rvector','R must be a vector.') |rwY   
    end %Q0R] Hg  
    D\9-/ p  
    r = r(:); &JqaIJh   
    length_r = length(r); K)~aH  
    gCC7L(1  
    if nargin==4 / + %  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); O0xqA\  
        if ~isnorm t4G$#~  
            error('zernpol:normalization','Unrecognized normalization flag.') y^}u L|=  
        end *wj5(B<y  
    else ktx| c19  
        isnorm = false; <?5|(Q"@:  
    end Q=xXj'W-  
    dCTyfXou[=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Yg3nT:K_Y&  
    % Compute the Zernike Polynomials #0[^jJ3J  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% J:glJ'4E  
    )3:0TFS}}k  
    % Determine the required powers of r: Z%B6J>;uM  
    % ----------------------------------- 3cL iZ%6^  
    rpowers = []; `w\P- q  
    for j = 1:length(n) CCe>*tdf  
        rpowers = [rpowers m(j):2:n(j)]; fM4B.45j  
    end @|c])  
    rpowers = unique(rpowers); )j>U4a  
    jxkjPf?  
    % Pre-compute the values of r raised to the required powers, xi(\=LbhY  
    % and compile them in a matrix: ~r5S{&  
    % ----------------------------- !LwHKCj  
    if rpowers(1)==0 -R:_o1"  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); %csrNf  
        rpowern = cat(2,rpowern{:}); >!xyA;  
        rpowern = [ones(length_r,1) rpowern]; .Lna\Bv  
    else /~{`!30  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ~BmA!BZV`  
        rpowern = cat(2,rpowern{:}); m)Kg6/MV.  
    end 4F6aPo2  
    @ CmKF  
    % Compute the values of the polynomials: nwUz}em?O  
    % -------------------------------------- ZDW=>}~_y  
    z = zeros(length_r,length_n); IUFc_uL@\  
    for j = 1:length_n qGV_oa74  
        s = 0:(n(j)-m(j))/2; <SI|)M,, 3  
        pows = n(j):-2:m(j); ^F+7<$ 2  
        for k = length(s):-1:1 lt&(S)  
            p = (1-2*mod(s(k),2))* ... P$#:$U @  
                       prod(2:(n(j)-s(k)))/          ... 1d<Uwb>  
                       prod(2:s(k))/                 ... 4>>=TJ!M  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... XAU%B-l:  
                       prod(2:((n(j)+m(j))/2-s(k))); PU/<7P*  
            idx = (pows(k)==rpowers); i9DD)Y<  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Xi98:0<=  
        end @I`^\oJ  
         $z jdCg<  
        if isnorm zTw"5N  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); =\WF +r]V  
        end f|~{j(.v  
    end *(F`NJ 3  
    mOsp~|d  
    % 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)  F9K%f&0 a  
    d>QFmsh-  
    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。 3JnpI,By  
    D|- ]<r1"  
    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)