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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 xo{3r\u?}  
    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! oE:9}]N_  
     
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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 76e%&ZG)Q  
    function z = zernfun(n,m,r,theta,nflag) 9qyA{ |3  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. -$Y@]uf^  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N EPx_xX  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 7WZ).,qxY  
    %   unit circle.  N is a vector of positive integers (including 0), and "4W@p'  
    %   M is a vector with the same number of elements as N.  Each element Oc\Bu6F  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) :e9}k5kdk  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ^`0^|u=  
    %   and THETA is a vector of angles.  R and THETA must have the same FPM@%U  
    %   length.  The output Z is a matrix with one column for every (N,M) #"tHT<8u  
    %   pair, and one row for every (R,THETA) pair. z}I4m  
    % x!6&)T?!n  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike p3?!}VM!y  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), r!/=Iy@  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Rw4"co6  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, =`VA_xVu  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized G$X+g{  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. rn1^6qy)  
    % .xXe *dm%  
    %   The Zernike functions are an orthogonal basis on the unit circle. 4;G:.k!K  
    %   They are used in disciplines such as astronomy, optics, and u\~dsD2)q  
    %   optometry to describe functions on a circular domain. XXbA n-J  
    % EL_rh TWw  
    %   The following table lists the first 15 Zernike functions. |&JCf =  
    % *=]hc@  
    %       n    m    Zernike function           Normalization pJM~'tlHV  
    %       -------------------------------------------------- p-]vf$u  
    %       0    0    1                                 1 ]"'$i4I{R  
    %       1    1    r * cos(theta)                    2 lq2Ah=FuN  
    %       1   -1    r * sin(theta)                    2 u,<#z0R|;$  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) QR'yZ45n4  
    %       2    0    (2*r^2 - 1)                    sqrt(3) z[kz [  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) :W'Yt9v)  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Z i-)PK^  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Cx>iSx  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) xyGk\= S  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) /jJi`'{U  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 4k9O6  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 5GD6%{\O  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) YE<_a;yh1  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) qTM,'7Rwn  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) $[zy|Y(  
    %       -------------------------------------------------- !acm@"Ea  
    % 9NCo0!Fb  
    %   Example 1: a]NQlsE}l  
    % W5a)`%H  
    %       % Display the Zernike function Z(n=5,m=1) J!?hajw7N  
    %       x = -1:0.01:1; IipG?v0z~  
    %       [X,Y] = meshgrid(x,x); YGy.39@31  
    %       [theta,r] = cart2pol(X,Y); :S Tj <  
    %       idx = r<=1; o&2(xI2  
    %       z = nan(size(X)); S{cy|QD  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 6?-vj2,  
    %       figure ?yKW^,q+  
    %       pcolor(x,x,z), shading interp w_-v!s2  
    %       axis square, colorbar 5mNd5IM  
    %       title('Zernike function Z_5^1(r,\theta)') CRy;>UI  
    % ve|:z  
    %   Example 2: H]@M00C  
    % /A3tY"Vn  
    %       % Display the first 10 Zernike functions c}9.Or`?  
    %       x = -1:0.01:1; <"I#lib  
    %       [X,Y] = meshgrid(x,x); 0pP;[7k\  
    %       [theta,r] = cart2pol(X,Y); BElVkb  
    %       idx = r<=1; #DMt<1#:  
    %       z = nan(size(X)); HorFQ?8  
    %       n = [0  1  1  2  2  2  3  3  3  3]; =,B44:`r  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; T;(k  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Wi3:;`>G<p  
    %       y = zernfun(n,m,r(idx),theta(idx)); >;Er[Rywr  
    %       figure('Units','normalized') DyiyH%SSD  
    %       for k = 1:10 v]CH L# |  
    %           z(idx) = y(:,k); Y*-#yG9  
    %           subplot(4,7,Nplot(k)) _97A9wHj  
    %           pcolor(x,x,z), shading interp  6C6<,c   
    %           set(gca,'XTick',[],'YTick',[]) I f9t^T#  
    %           axis square +an.z3?w  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 5c?1JH62o8  
    %       end \W5fcxf  
    % :f?};t+  
    %   See also ZERNPOL, ZERNFUN2. h$`P|#V&  
    s)HLFdis@  
    %   Paul Fricker 11/13/2006 E"p;  
    5 rpX"(  
    z:B4  
    % Check and prepare the inputs: P !:LAb(  
    % -----------------------------  b,] QfC  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) <=;#I_E#E  
        error('zernfun:NMvectors','N and M must be vectors.') '8+<^%c  
    end 92|\`\LP%  
    "M.\Z9BCt  
    if length(n)~=length(m) p8CDFLuV  
        error('zernfun:NMlength','N and M must be the same length.') I^h^QeBis  
    end .t\#>Fe  
    GAK!qLy9  
    n = n(:); sTx23RJ9  
    m = m(:); R"NR-iU  
    if any(mod(n-m,2)) &s.S) 'l4l  
        error('zernfun:NMmultiplesof2', ... IbFS8 *a\  
              'All N and M must differ by multiples of 2 (including 0).') ]"Y? ZS;H  
    end *3;H6   
    ^m ^4LDt  
    if any(m>n) e nsou!l  
        error('zernfun:MlessthanN', ... 7` 113`1  
              'Each M must be less than or equal to its corresponding N.') iTf]Pd'  
    end |KF_h^  
    49vKb(bz{  
    if any( r>1 | r<0 ) [ +w=  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') WCc7 MK  
    end .xnJT2uu'  
    <Co\?h/<  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Gt >*y.]  
        error('zernfun:RTHvector','R and THETA must be vectors.') cB,O"-  
    end HE>6A|rgDr  
    UVND1XV^f  
    r = r(:); =ELl86=CG  
    theta = theta(:); -:mT8'.F-  
    length_r = length(r); WvV!F?uqZ  
    if length_r~=length(theta) - \ {.]KL  
        error('zernfun:RTHlength', ... QrmiQ]d*p  
              'The number of R- and THETA-values must be equal.') v(5zSo  
    end :Fe}.* t  
    NGsG4y^g?z  
    % Check normalization: WX@ a2c.'  
    % -------------------- S6~&g|T,  
    if nargin==5 && ischar(nflag) i7N|p9O.  
        isnorm = strcmpi(nflag,'norm'); 8R G U^&  
        if ~isnorm 6|h~pH  
            error('zernfun:normalization','Unrecognized normalization flag.') O7&6]/`  
        end QU&LC  
    else re\pE2&B  
        isnorm = false; 1|U8DK  
    end F#<$yUf%  
    ,E YB E  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% rd#O ]   
    % Compute the Zernike Polynomials /*v} .fH%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZboY]1L[j  
    Sz z:$!t  
    % Determine the required powers of r: yL ?dC"c  
    % ----------------------------------- X:3W9`s )*  
    m_abs = abs(m); 9d5|rk8VS  
    rpowers = []; 'fIBJ3s[o  
    for j = 1:length(n) g!<=NVhYt  
        rpowers = [rpowers m_abs(j):2:n(j)]; rV *`0hA1  
    end > St]MS  
    rpowers = unique(rpowers); <G+IbUG:  
    ^)\z  
    % Pre-compute the values of r raised to the required powers, -OvzEmI"  
    % and compile them in a matrix: \%=GM J^[p  
    % ----------------------------- h3.6<vM  
    if rpowers(1)==0 bUcq LV  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); x'OYJ>l|  
        rpowern = cat(2,rpowern{:}); VB(S]N)F^  
        rpowern = [ones(length_r,1) rpowern]; y9/x:n&]  
    else AEUXdMo  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 'h]sq {  
        rpowern = cat(2,rpowern{:}); s,~p}A%0  
    end m+3U[KKvG  
    r]6X  
    % Compute the values of the polynomials: !Z0p94L  
    % -------------------------------------- 5Xe1a'n5]  
    y = zeros(length_r,length(n)); qFV }Y0w  
    for j = 1:length(n) xzI?'?duC  
        s = 0:(n(j)-m_abs(j))/2; q!r4"#Y"@Z  
        pows = n(j):-2:m_abs(j); G; onJ>  
        for k = length(s):-1:1 /8$*{ay  
            p = (1-2*mod(s(k),2))* ... :3oLGiL   
                       prod(2:(n(j)-s(k)))/              ... K |Z]  
                       prod(2:s(k))/                     ... 0P?\eoB@8  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... O|ODJOQNol  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 4/_@F>I_  
            idx = (pows(k)==rpowers); !%8|R]d  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 3 $~6+i  
        end *{#l0My  
         :\Pk>a  
        if isnorm &I=27!S  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); v \xuq`  
        end }\-"L/D?+  
    end M@ TXzn!&o  
    % END: Compute the Zernike Polynomials _,G^#$pH  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% e [}m@a  
    &IZthJqV  
    % Compute the Zernike functions: E <O:  
    % ------------------------------ Ho_ 2zx:8b  
    idx_pos = m>0; >sfH[b  
    idx_neg = m<0; 6`V2-zv$  
    :)PAj  
    z = y; 'K8emt$d+  
    if any(idx_pos) 7y/Pch  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); -_4ZT^.Lna  
    end 2u=Nb0  
    if any(idx_neg) O]/BNacS  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); p3f>;|uh_  
    end :t'*fHi~  
    *!W<yNrR  
    % 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) BY&+fK ae  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. w4"4(SR.  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated .dr-I7&!  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive tt%lDr1A)  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ;`(l)X+7  
    %   and THETA is a vector of angles.  R and THETA must have the same :RqTbE4B  
    %   length.  The output Z is a matrix with one column for every P-value, InCJ4D  
    %   and one row for every (R,THETA) pair. <"SOH; w  
    % b5Sgf'B^  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 13lJq:bM  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) "y5LojdCs  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) $ M8ZF(W  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 AD=qB5:  
    %   for all p. P%nN#Qm  
    % yH:gFEJ:x  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 `1Cg)\&[e0  
    %   Zernike functions (order N<=7).  In some disciplines it is = ;!$Qw4  
    %   traditional to label the first 36 functions using a single mode {)c2#h  
    %   number P instead of separate numbers for the order N and azimuthal iFi6,V*PRt  
    %   frequency M. %~$P.Zh  
    % %`F &,!d  
    %   Example: o;#9$j7QP!  
    % B>!OW2q0D  
    %       % Display the first 16 Zernike functions Oosr`e@S  
    %       x = -1:0.01:1; bL)7 /E  
    %       [X,Y] = meshgrid(x,x); W ^MF3  
    %       [theta,r] = cart2pol(X,Y); q!sazVaDp  
    %       idx = r<=1; 6y4&nTq[  
    %       p = 0:15; x KZLXQ'e-  
    %       z = nan(size(X)); IL\2?(&Z  
    %       y = zernfun2(p,r(idx),theta(idx)); '%} k"&t$i  
    %       figure('Units','normalized') h\@\*Xz<v  
    %       for k = 1:length(p) y!,Ly_x$@  
    %           z(idx) = y(:,k); 4J"S?HsW|  
    %           subplot(4,4,k) {okx*]PIc  
    %           pcolor(x,x,z), shading interp SMMsXH  
    %           set(gca,'XTick',[],'YTick',[]) jEkO #xI  
    %           axis square R}S@u@mOE  
    %           title(['Z_{' num2str(p(k)) '}']) Rb8wq.LqD  
    %       end y7?n;3U]CS  
    % @'Y^A  
    %   See also ZERNPOL, ZERNFUN. [J`%i U  
    x  bsk  
    %   Paul Fricker 11/13/2006 5ml#/kE  
    - %5O:n  
    Q=+*OQV29  
    % Check and prepare the inputs: +5>*$L%8T`  
    % ----------------------------- a*oqhOTQ  
    if min(size(p))~=1 t\/i9CBn  
        error('zernfun2:Pvector','Input P must be vector.') ^Qx qv  
    end }ob&d.XZ  
    )rqb<O  
    if any(p)>35 oXQzCjX_   
        error('zernfun2:P36', ... :L E&p[^  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ^>3q@,C]c  
               '(P = 0 to 35).']) hS?pc<~`#  
    end zOEdFU{x  
    )U?O4| \P  
    % Get the order and frequency corresonding to the function number: ry2ZVIFa  
    % ---------------------------------------------------------------- 8H%-/2NW  
    p = p(:); *j9hjq0j  
    n = ceil((-3+sqrt(9+8*p))/2); lHTW e'  
    m = 2*p - n.*(n+2); =FB[<%  
    e)#O-y  
    % Pass the inputs to the function ZERNFUN: 2%|0c\y|z=  
    % ---------------------------------------- 91Fx0(  
    switch nargin ) Ekd  
        case 3 g ss 3e&  
            z = zernfun(n,m,r,theta); (pU@$H  
        case 4 x+=Ko  
            z = zernfun(n,m,r,theta,nflag); t,f)!D$  
        otherwise (iT?uMRz  
            error('zernfun2:nargin','Incorrect number of inputs.') ]WO0v`xh  
    end }u>F}mUa  
    -_p+4tV  
    % 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) X:un4B}O  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 0Bo7EV  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of D.i(Irqw!  
    %   order N and frequency M, evaluated at R.  N is a vector of o `]o(OP  
    %   positive integers (including 0), and M is a vector with the {Xc^-A[~  
    %   same number of elements as N.  Each element k of M must be a kvdiDo  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) `Wes!>Vh!  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is D)]U+Qk  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix u(Y! _  
    %   with one column for every (N,M) pair, and one row for every }%&hxhR^t3  
    %   element in R. Y/3CB  
    % &sbKN[xM  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- zm& D #)  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is tU5Z?QS  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 'T '&OA  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 aZk/\&=6  
    %   for all [n,m]. ae&i]K;  
    % Y`O"+Jr  
    %   The radial Zernike polynomials are the radial portion of the 3!&PI  
    %   Zernike functions, which are an orthogonal basis on the unit wc&`/'<p  
    %   circle.  The series representation of the radial Zernike d>RoH]K4  
    %   polynomials is ="k9 y  
    % (O$PJLI  
    %          (n-m)/2 P ,%IZ.  
    %            __ @y|ZXPC#  
    %    m      \       s                                          n-2s zQ&k$l9  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r P  -O& X  
    %    n      s=0 ?$ft3p}  
    % {.D^2mj |  
    %   The following table shows the first 12 polynomials. vxey $Ir  
    % MHuQGc"e+4  
    %       n    m    Zernike polynomial    Normalization a5)<roWQ  
    %       --------------------------------------------- #|ppW fZQ  
    %       0    0    1                        sqrt(2) 4*)a3jI?  
    %       1    1    r                           2 d4 Hpe>  
    %       2    0    2*r^2 - 1                sqrt(6) K4r"Q*h  
    %       2    2    r^2                      sqrt(6) V!|:rwG2  
    %       3    1    3*r^3 - 2*r              sqrt(8) pG|+\k/B  
    %       3    3    r^3                      sqrt(8) fP:n=A{  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) lBYc(cr  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 'e/= !"T  
    %       4    4    r^4                      sqrt(10) ,y>%m;jL  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) by:"aDGK.  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) L"_l(<g  
    %       5    5    r^5                      sqrt(12) ( <Abw{BTm  
    %       --------------------------------------------- o_(@v2G`  
    % 2 SJ N;A~}  
    %   Example: fcim4dfP  
    % Hv>16W$_  
    %       % Display three example Zernike radial polynomials cC NyW2'  
    %       r = 0:0.01:1; r?:zKj8/u  
    %       n = [3 2 5]; (=T%eJ61  
    %       m = [1 2 1]; =SY`Xkj[  
    %       z = zernpol(n,m,r); Wubvvm8U  
    %       figure }.L\O]~{  
    %       plot(r,z) %u)niY-g  
    %       grid on ; qQ* p  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') VbwB<nQl  
    % hB !>*AsG  
    %   See also ZERNFUN, ZERNFUN2. Xcy Xju#"p  
    6JCq?:#ab  
    % A note on the algorithm. :vsF4  
    % ------------------------ M9t`w-@_w  
    % The radial Zernike polynomials are computed using the series 0m,3''Q5lO  
    % representation shown in the Help section above. For many special -;i vBR  
    % functions, direct evaluation using the series representation can 4P>4d +  
    % produce poor numerical results (floating point errors), because 5Nt40)E}sN  
    % the summation often involves computing small differences between Qw,{"J  
    % large successive terms in the series. (In such cases, the functions ?k}"g$JFn  
    % are often evaluated using alternative methods such as recurrence S5,y!K]C~  
    % relations: see the Legendre functions, for example). For the Zernike %mO.ur>21  
    % polynomials, however, this problem does not arise, because the |([|F|"  
    % polynomials are evaluated over the finite domain r = (0,1), and C{5bG=Sg~  
    % because the coefficients for a given polynomial are generally all kdam]L:9  
    % of similar magnitude. w]% |^:  
    % mF6 U{=  
    % ZERNPOL has been written using a vectorized implementation: multiple <Rno ;  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] q_R^Q>ZIe  
    % values can be passed as inputs) for a vector of points R.  To achieve (L2:|1P)  
    % this vectorization most efficiently, the algorithm in ZERNPOL /`2t$71)  
    % involves pre-determining all the powers p of R that are required to G]$.bq[v  
    % compute the outputs, and then compiling the {R^p} into a single ]bui"-tlK  
    % matrix.  This avoids any redundant computation of the R^p, and (Cc!Iw'0M  
    % minimizes the sizes of certain intermediate variables. (H_YYZ3ZX  
    % gQ0W>\xz  
    %   Paul Fricker 11/13/2006 z0v|%&IK  
    Q&@~<!t  
    : U Yn  
    % Check and prepare the inputs: p bT sn  
    % ----------------------------- egd%,`  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) M?nYplC  
        error('zernpol:NMvectors','N and M must be vectors.') x,+2k6Wn!  
    end dB=aq34l  
    T+q3]&  
    if length(n)~=length(m) .O1Kwu  
        error('zernpol:NMlength','N and M must be the same length.') mZgYR~  
    end |_H{ B+.  
     (H*EZ  
    n = n(:); b14WIgjsl  
    m = m(:); jW!x!8=  
    length_n = length(n); 5TB==Fj ?  
    -!s?d5k")  
    if any(mod(n-m,2)) pAY[XN  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') UD+r{s/%  
    end @5)THYAx4  
    M#5*gWfq9  
    if any(m<0) SBbPO5^](  
        error('zernpol:Mpositive','All M must be positive.') p?#%G`dm  
    end _-mJI+^/  
    N+V_[qr#  
    if any(m>n) sZ7RiH +I  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') $YPQi.  
    end /5s,< 0Kz  
    "+BNas^rF  
    if any( r>1 | r<0 ) D$vP&7pOr4  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') yJMHm8OB7  
    end }vof| (Yh  
    1f/8XxTB  
    if ~any(size(r)==1) 2\'5LL3  
        error('zernpol:Rvector','R must be a vector.') NA<6s]Cs.  
    end flr&+=1?D  
    nWzGb2Y  
    r = r(:); 'y<<ce*   
    length_r = length(r); {-'S#04  
    HEMq4v4  
    if nargin==4 0D=7Mef  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); OZc.Rtgc  
        if ~isnorm I~f8+DE)  
            error('zernpol:normalization','Unrecognized normalization flag.') n@e[5f9?x  
        end E~| XY9U36  
    else 28jm*Cl8  
        isnorm = false; $At,D.mGkb  
    end @?gN &Z)I  
    C\d5t4s  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% rjJ-ZRs\  
    % Compute the Zernike Polynomials +P//p$pE  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% e|~s'{3  
    /EXub U73  
    % Determine the required powers of r: 1$^=M[v  
    % ----------------------------------- M,f|.p{,Y  
    rpowers = []; j1hx{P'  
    for j = 1:length(n) `tjH#W`  
        rpowers = [rpowers m(j):2:n(j)]; Ts~)0  
    end VJ'bS9/T  
    rpowers = unique(rpowers); G1`H H&  
    (8?5REz  
    % Pre-compute the values of r raised to the required powers, /[|ODfY  
    % and compile them in a matrix: h4 X>  
    % ----------------------------- R8K ?! Z  
    if rpowers(1)==0 &8^1:CcE  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); O:>9yZhV  
        rpowern = cat(2,rpowern{:}); % w0Vf$  
        rpowern = [ones(length_r,1) rpowern]; 4\?GA`@  
    else Jc74A=sT  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); N`Q[OFe  
        rpowern = cat(2,rpowern{:}); CWVCYm@!kz  
    end <bwsK,C  
    eB:OvOol*^  
    % Compute the values of the polynomials: D&{ 7Av  
    % -------------------------------------- VOYuog 5o  
    z = zeros(length_r,length_n); -$(,&qyk  
    for j = 1:length_n K)Nbl^6x  
        s = 0:(n(j)-m(j))/2; kF9T 9  
        pows = n(j):-2:m(j); #rZk&q  
        for k = length(s):-1:1 B/i`  
            p = (1-2*mod(s(k),2))* ... wl{Fx+<^3  
                       prod(2:(n(j)-s(k)))/          ...  W t&tu2  
                       prod(2:s(k))/                 ... !@ml^&hP  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... GfAt-huL(  
                       prod(2:((n(j)+m(j))/2-s(k))); y)W.xR  
            idx = (pows(k)==rpowers); gY], (*v  
            z(:,j) = z(:,j) + p*rpowern(:,idx); <}RU37,W  
        end $X %GzrN  
         l-yQ3/:  
        if isnorm !8"516!d|p  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); s D] W/  
        end *f~X wy"  
    end @L%9NqE`O  
    _Cv({m&N  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  IfeG"ua|  
    o;M"C[  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 {HNGohZt  
    Z%6I$KAN8  
    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)