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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 g_vm&~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! {]`p&@  
     
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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 1bSD,;$sQ  
    function z = zernfun(n,m,r,theta,nflag) x=*L-  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ;^3$kF  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 78]gt J  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Im)EDTm$  
    %   unit circle.  N is a vector of positive integers (including 0), and cp%ii'  
    %   M is a vector with the same number of elements as N.  Each element d#>y}H9  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) :=fvZAWD  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, >emcJVYV`[  
    %   and THETA is a vector of angles.  R and THETA must have the same <kbyZXV@K  
    %   length.  The output Z is a matrix with one column for every (N,M) Wi$dZOcSJ  
    %   pair, and one row for every (R,THETA) pair. %Q~CB7ILK  
    % }ZzLs/v%X  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike c-8!#~M(  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), @cv{rr  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral RH[+1z8  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 2"&)W dm  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized f*fE};  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Cq\I''~8  
    %  'Cc(3  
    %   The Zernike functions are an orthogonal basis on the unit circle. &leK}je [  
    %   They are used in disciplines such as astronomy, optics, and $$1qF"GF  
    %   optometry to describe functions on a circular domain. #/ "+  
    % a$JLc a  
    %   The following table lists the first 15 Zernike functions. i9m*g*"2  
    % b{5K2k&,  
    %       n    m    Zernike function           Normalization xs!p|  
    %       -------------------------------------------------- X?o( b/F -  
    %       0    0    1                                 1 ! '0S0a8  
    %       1    1    r * cos(theta)                    2 >/^#Drwb!i  
    %       1   -1    r * sin(theta)                    2 x0Z5zV9  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6)  }roG(  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 1-VT}J(  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) O#_b7i  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) JTW)*q9a  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 6`\ya@  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 2]WE({P  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) P Sx304  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) \Fb| {6+  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (^n*Am;zlH  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) e3m*i}K}  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) uk7'K 0j  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) EKZVF`L  
    %       -------------------------------------------------- ..<3%fL3  
    % :}q\tNY<  
    %   Example 1: i7Z=|&  
    % Ee2c5C!|C  
    %       % Display the Zernike function Z(n=5,m=1) K@:m/Z}|4  
    %       x = -1:0.01:1; z@VP:au  
    %       [X,Y] = meshgrid(x,x); F n|gVR  
    %       [theta,r] = cart2pol(X,Y); <{J5W6  
    %       idx = r<=1; w4:\N U  
    %       z = nan(size(X)); jC, FG'P  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); <4`eQ  
    %       figure |qN'P}L  
    %       pcolor(x,x,z), shading interp y,Q5; $w8  
    %       axis square, colorbar 0ejdKdYN  
    %       title('Zernike function Z_5^1(r,\theta)') ,FQK;BU!lh  
    % & >JDPB?5  
    %   Example 2: #exss=as/  
    % H+C6[W=  
    %       % Display the first 10 Zernike functions 7^:4A'  
    %       x = -1:0.01:1; `a]44es9q  
    %       [X,Y] = meshgrid(x,x); xUWr}j4;  
    %       [theta,r] = cart2pol(X,Y); BavO\{J#|0  
    %       idx = r<=1; r':TMhzHq?  
    %       z = nan(size(X)); bGXR7u&K  
    %       n = [0  1  1  2  2  2  3  3  3  3]; my.`k'  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 8E^@yZo{  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; #-# NqX:  
    %       y = zernfun(n,m,r(idx),theta(idx)); *r,b=8|  
    %       figure('Units','normalized') oFC)  
    %       for k = 1:10 y 'M#z_.z  
    %           z(idx) = y(:,k); ^ 4hO8  
    %           subplot(4,7,Nplot(k)) k|E]YvnfG  
    %           pcolor(x,x,z), shading interp G*}F5.>8(  
    %           set(gca,'XTick',[],'YTick',[]) 1s7^uA$}6  
    %           axis square @y|_d  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ` %?9=h%  
    %       end D!LX?_cD1i  
    % !K0JV|-?t  
    %   See also ZERNPOL, ZERNFUN2. /Z%>ArAx  
    mY&ud>,U:  
    %   Paul Fricker 11/13/2006 F # YPOH  
    TH1B#Y#<J  
    7"v$- Wy  
    % Check and prepare the inputs: u5E]t9~Pq  
    % ----------------------------- S"2qJ!.u  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) dZ(|uC!?  
        error('zernfun:NMvectors','N and M must be vectors.') ^ @=^;nB  
    end ^4$ 'KIq  
    4sF v?W  
    if length(n)~=length(m) 0tz:Wd*<  
        error('zernfun:NMlength','N and M must be the same length.') 2y GOzc  
    end lC?Icn|o  
    sq0 PBEqq  
    n = n(:);  lhLGG  
    m = m(:); WQePSU  
    if any(mod(n-m,2)) P\R27Jd  
        error('zernfun:NMmultiplesof2', ... "4xfrlOc  
              'All N and M must differ by multiples of 2 (including 0).') Zm TDQ`Ix  
    end (!K_Fy@  
    CnF |LTi  
    if any(m>n) MXh "Y*}  
        error('zernfun:MlessthanN', ... K\.5h4k  
              'Each M must be less than or equal to its corresponding N.') vA%^`5  
    end oR#:Nt X@  
    woOy*)@  
    if any( r>1 | r<0 ) }xb=<  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 12`_;[37  
    end udqS'g&  
    Sr.;GS5i  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) x8#ODuH  
        error('zernfun:RTHvector','R and THETA must be vectors.') u=l1s1>  
    end iZ,YxN<R  
    JWO=!^  
    r = r(:); |QZ 58)>  
    theta = theta(:); >v5k{Cbp0  
    length_r = length(r); dj9 ?t  
    if length_r~=length(theta) akuJz  
        error('zernfun:RTHlength', ... jx`QB')kX  
              'The number of R- and THETA-values must be equal.')  -7]Xjb5  
    end = bt]JRU  
    !Jfs?Hy  
    % Check normalization: # '|'r+  
    % -------------------- hsLzj\)6  
    if nargin==5 && ischar(nflag) !b|'Vp^U  
        isnorm = strcmpi(nflag,'norm'); H}0dd"  
        if ~isnorm jFG0`n}I  
            error('zernfun:normalization','Unrecognized normalization flag.') ik,lSTBD  
        end }E^S]hdvz  
    else alFjc.~}  
        isnorm = false; ;&;W T  
    end 7 6fIC  
    ~[PKcEX  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% T5lQIr@a  
    % Compute the Zernike Polynomials )hKS0`$|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tx7~S Ur  
    y6HuN  
    % Determine the required powers of r: VL( <  
    % ----------------------------------- jdqj=Yc  
    m_abs = abs(m); 3ha|0[r9  
    rpowers = []; lT8\}hNI+  
    for j = 1:length(n) t` ^ Vb-  
        rpowers = [rpowers m_abs(j):2:n(j)]; xBnbF[  
    end Lm)\Z P+W  
    rpowers = unique(rpowers); yl]FP@N(  
    M~T.n)x2  
    % Pre-compute the values of r raised to the required powers, cd@.zg'sYn  
    % and compile them in a matrix: q`|CrOzO  
    % ----------------------------- P1zK2sL_  
    if rpowers(1)==0 M@UVpQwgv  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); nY?  
        rpowern = cat(2,rpowern{:}); {OMg d3%14  
        rpowern = [ones(length_r,1) rpowern]; #TJk-1XM*q  
    else rjA@U<o  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); N> Jw  
        rpowern = cat(2,rpowern{:}); 25{ uz  
    end Xo5$X7m  
    5t:8.%<UK  
    % Compute the values of the polynomials: p QE)p  
    % -------------------------------------- E;\M1(\u  
    y = zeros(length_r,length(n)); 7()?C}Ni-  
    for j = 1:length(n) *6ZCDm&N  
        s = 0:(n(j)-m_abs(j))/2; *e.*=$  
        pows = n(j):-2:m_abs(j); +54aO  
        for k = length(s):-1:1 i\}:hU-U  
            p = (1-2*mod(s(k),2))* ... 0`#(Toe{B  
                       prod(2:(n(j)-s(k)))/              ... %"3 )TN4  
                       prod(2:s(k))/                     ... H. ,;-  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... |  FM }  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); m$Y :0_^-  
            idx = (pows(k)==rpowers); yOXO)u1n  
            y(:,j) = y(:,j) + p*rpowern(:,idx); aC=['a>)  
        end |(IO=V4P  
         Q%ad q-B  
        if isnorm oW}!vf3z  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 6o&ZIYJ9k  
        end q%3<Juq~$  
    end N_wp{4 0/  
    % END: Compute the Zernike Polynomials |WQ9a' '  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% g$37;d3Tx  
    ;6;H*Y0,|E  
    % Compute the Zernike functions: s'I)A^i+  
    % ------------------------------ EYzg%\HH  
    idx_pos = m>0; :> -1'HC  
    idx_neg = m<0; Ggm` ~fS  
    >wON\N0V_  
    z = y; |w&~g9   
    if any(idx_pos) xh9qg0d  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); fZryG  
    end 3!9 Z=- tD  
    if any(idx_neg) %HuyK  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); |^n3{m  
    end j+ ::y) $  
    pK_?}~  
    % EOF zernfun
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    function z = zernfun2(p,r,theta,nflag) j'rS&BI G  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 649 !=  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ~UMOT!4}3  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 02b6s&L  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, QTJu7^ O9  
    %   and THETA is a vector of angles.  R and THETA must have the same }0/a\  
    %   length.  The output Z is a matrix with one column for every P-value, JSVeU54T^<  
    %   and one row for every (R,THETA) pair. e<+<lj "  
    % d,#.E@Po  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike  -;c  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) )m#']c:rg  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) hl/itSl$  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 n~r 9!m$<  
    %   for all p. SG]K   
    % D'i6",Z>  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 $Xf(^K  
    %   Zernike functions (order N<=7).  In some disciplines it is 1"E\C/c  
    %   traditional to label the first 36 functions using a single mode KFhG(   
    %   number P instead of separate numbers for the order N and azimuthal F8mC?fbK9  
    %   frequency M. 4br6$  
    % d]B= *7]  
    %   Example: )2j:z#'>  
    % "dCzWFet  
    %       % Display the first 16 Zernike functions &^QPkX@p  
    %       x = -1:0.01:1; pM$ @m]  
    %       [X,Y] = meshgrid(x,x); uH{'gd,q8  
    %       [theta,r] = cart2pol(X,Y); ZwJciT!_~  
    %       idx = r<=1; o}D![/  
    %       p = 0:15; vOT*iax0  
    %       z = nan(size(X)); 3'#%c>_  
    %       y = zernfun2(p,r(idx),theta(idx)); 9D_wG\g  
    %       figure('Units','normalized') zG% |0  
    %       for k = 1:length(p) gA:TL{X0  
    %           z(idx) = y(:,k); L,M+sN  
    %           subplot(4,4,k) &=w|vB)(p  
    %           pcolor(x,x,z), shading interp VTw/_Hf2p  
    %           set(gca,'XTick',[],'YTick',[]) 'D6 bmz  
    %           axis square FmT `Oa>  
    %           title(['Z_{' num2str(p(k)) '}']) SGf9U^ds  
    %       end fPQ|e"?  
    % uLW/f=7 L  
    %   See also ZERNPOL, ZERNFUN. [MmM9J["  
    UxqWnHH.`  
    %   Paul Fricker 11/13/2006 _}=E^/;(  
    -u6#-}S  
    w-rOecwFvu  
    % Check and prepare the inputs: g [L  
    % ----------------------------- .E<Dz  
    if min(size(p))~=1 Uf2:gLrF  
        error('zernfun2:Pvector','Input P must be vector.') G11cNr>*  
    end Q_}n%P:u  
    K2|7%  
    if any(p)>35 \y~)jq:d"  
        error('zernfun2:P36', ... L8P 36]>  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... m5gI~1(9  
               '(P = 0 to 35).']) <Cq"| A  
    end M,..Kw/ }~  
    *.8:'F  
    % Get the order and frequency corresonding to the function number: R}k69-1vL  
    % ---------------------------------------------------------------- M&e8zS  
    p = p(:); P9yw&A  
    n = ceil((-3+sqrt(9+8*p))/2); qJ Gm8^b-  
    m = 2*p - n.*(n+2); .<w)Bmh  
    `o-*Tr  
    % Pass the inputs to the function ZERNFUN: 2PBepgQyPU  
    % ---------------------------------------- H m8y]>$  
    switch nargin 1v]t!}W:6  
        case 3 rw*M&qg!z  
            z = zernfun(n,m,r,theta); jtLn j@,  
        case 4 |[WL2<  
            z = zernfun(n,m,r,theta,nflag); &;U|7l~vl  
        otherwise K+> V|zKuk  
            error('zernfun2:nargin','Incorrect number of inputs.') P#bm uCOS  
    end ?N _)>&b  
    vU 9ek:.l  
    % 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) *vD/(&pQ1:  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. X&,N}9>B  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of tnJ`D4  
    %   order N and frequency M, evaluated at R.  N is a vector of c}'Xoc  
    %   positive integers (including 0), and M is a vector with the .S(^roM;+  
    %   same number of elements as N.  Each element k of M must be a v C-[#]<  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) Crg#6k1~EN  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is  %|bN@@  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix o[imNy~~  
    %   with one column for every (N,M) pair, and one row for every #'KY`&Tw&  
    %   element in R. wRj~Qv~E  
    % l`qP~ k#  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ]%||KC!O  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is Y` q!V=  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to flPZlL  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 wH!}qz /  
    %   for all [n,m]. uzO {{S-  
    % L*IU0Jy>  
    %   The radial Zernike polynomials are the radial portion of the epYj+T  
    %   Zernike functions, which are an orthogonal basis on the unit qb9}&'@:  
    %   circle.  The series representation of the radial Zernike dkETM,  
    %   polynomials is #yX^?+Rc  
    % O/nqNQ?<  
    %          (n-m)/2 a0)vvo=bz  
    %            __ ^Jn|*?+l  
    %    m      \       s                                          n-2s 4fD`M(wv  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Jfe~ ,cI  
    %    n      s=0 Le_CIk 5YL  
    % Y'?Izn b  
    %   The following table shows the first 12 polynomials. D8a)(wm  
    % j#CuR7m  
    %       n    m    Zernike polynomial    Normalization +6uOg,;  
    %       --------------------------------------------- ]y$)%J^T  
    %       0    0    1                        sqrt(2)  ?DJuQFv  
    %       1    1    r                           2 dPRtN@3  
    %       2    0    2*r^2 - 1                sqrt(6) _pJX1_vD  
    %       2    2    r^2                      sqrt(6) W;5N04ko  
    %       3    1    3*r^3 - 2*r              sqrt(8) Q4#\{" N!  
    %       3    3    r^3                      sqrt(8) uAChu]  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 1o(+rR<h9  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) |_!PD$i-  
    %       4    4    r^4                      sqrt(10) Lax9 "xI  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) #3YdjU3w  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) R,uJK)m  
    %       5    5    r^5                      sqrt(12) 69N1 mP  
    %       --------------------------------------------- 0qOM78rE  
    % }`#j;H$i  
    %   Example: u|(aS^H=q  
    % "tpvENz2s  
    %       % Display three example Zernike radial polynomials R%Kl&c  
    %       r = 0:0.01:1; wHq('+{=&  
    %       n = [3 2 5]; qdKh6{  
    %       m = [1 2 1]; 4U_rB9K$  
    %       z = zernpol(n,m,r); \XZU'JIO  
    %       figure :Xb*m85y  
    %       plot(r,z) (M4]#5  
    %       grid on goG] WGVr  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') r7zf+a]  
    % 9t,aT!f  
    %   See also ZERNFUN, ZERNFUN2. Vx0MG{vG1  
    F I80vV7  
    % A note on the algorithm. T]5U_AI@  
    % ------------------------ avF&F  
    % The radial Zernike polynomials are computed using the series 0;. e#(`-  
    % representation shown in the Help section above. For many special b=6ZdN1  
    % functions, direct evaluation using the series representation can >?H_A  
    % produce poor numerical results (floating point errors), because 3 ATN?V@  
    % the summation often involves computing small differences between A{xSbbDk  
    % large successive terms in the series. (In such cases, the functions Rt*-#`I $  
    % are often evaluated using alternative methods such as recurrence **"zDY*?W  
    % relations: see the Legendre functions, for example). For the Zernike fo/(()  
    % polynomials, however, this problem does not arise, because the cuJ / Vc  
    % polynomials are evaluated over the finite domain r = (0,1), and Ut0qr kqF  
    % because the coefficients for a given polynomial are generally all r%O rH-T  
    % of similar magnitude. VKl~oFKXJ  
    % hg)!m\g  
    % ZERNPOL has been written using a vectorized implementation: multiple XyN`BDFi  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] {FrHm  
    % values can be passed as inputs) for a vector of points R.  To achieve mE)x7  
    % this vectorization most efficiently, the algorithm in ZERNPOL %a%+!wX0x  
    % involves pre-determining all the powers p of R that are required to kW*W4{Fth  
    % compute the outputs, and then compiling the {R^p} into a single pZNlcB[Qn-  
    % matrix.  This avoids any redundant computation of the R^p, and C{lB/F/|!  
    % minimizes the sizes of certain intermediate variables. x`&P}4v0  
    % 6'3Ey'drH  
    %   Paul Fricker 11/13/2006 CJ37:w{%*Y  
    B$iMU?B3  
    zwF7DnW<<  
    % Check and prepare the inputs: &k {t0>  
    % ----------------------------- nJnO/~|  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) -M(58/y  
        error('zernpol:NMvectors','N and M must be vectors.') @5uyUSt]  
    end tFb|y+  
    lUm}nsp=X  
    if length(n)~=length(m) [xH2n\7  
        error('zernpol:NMlength','N and M must be the same length.') u!156X?[eU  
    end pw(*X,gj  
    6M2i? c  
    n = n(:); ixUiXP  
    m = m(:); aG8;,H=%,  
    length_n = length(n); 2n\i0?RD  
    O>{t}6o  
    if any(mod(n-m,2)) dRarNW  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') M\x7=*\  
    end l7FZ;%&  
    *HfW(C$  
    if any(m<0) G/^5P5y%@  
        error('zernpol:Mpositive','All M must be positive.') ~=aI2(b  
    end QyBK*uNdV  
    $(!D/bvJ  
    if any(m>n) x<j($iv  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') !3gpiQH{  
    end ui:>eYv  
    c"_H%x<[  
    if any( r>1 | r<0 ) aF_ZV bS  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') KfN`ZZ<  
    end 7kew/8-  
    &dHm!b  
    if ~any(size(r)==1) _*-'yu8#  
        error('zernpol:Rvector','R must be a vector.') +BTNm66Z  
    end 5<>R dLo  
    m0q`A5!)  
    r = r(:);  ;#Bh_f  
    length_r = length(r); 0V>N#P]  
    p{pzOMi6  
    if nargin==4 azATKH+j  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); I'wk/  
        if ~isnorm &t@6qi`d  
            error('zernpol:normalization','Unrecognized normalization flag.') mU0r"\**c3  
        end ` TVcI\W  
    else /A`zy  
        isnorm = false; DqC}f#  
    end ?5+KHG*)  
    `cTsS  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @XSu?+s)  
    % Compute the Zernike Polynomials I]j/ ab7>  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `E1_S  
    cbHn\m)J,  
    % Determine the required powers of r: ?tM].\  
    % ----------------------------------- SwmPP-n  
    rpowers = []; X;[zfEB  
    for j = 1:length(n) 28L'7  
        rpowers = [rpowers m(j):2:n(j)]; 113x9+w[  
    end \(v_",  
    rpowers = unique(rpowers); 2wim P8  
    m"> =QP  
    % Pre-compute the values of r raised to the required powers, OgpH{"  
    % and compile them in a matrix: c#-97"_8  
    % ----------------------------- EG:WE^4  
    if rpowers(1)==0 WoT z'  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); XQoT},C  
        rpowern = cat(2,rpowern{:}); UK9MWC5g9  
        rpowern = [ones(length_r,1) rpowern]; # ;KG6IE  
    else &+|4(d1  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ,0u0 '  
        rpowern = cat(2,rpowern{:}); !/lY q;$R  
    end %<o$ J~l~  
    XjM)/-w  
    % Compute the values of the polynomials: sYW)h$p;D  
    % -------------------------------------- Hbj:CViYq  
    z = zeros(length_r,length_n); 8t 35j   
    for j = 1:length_n ^T1-dw(  
        s = 0:(n(j)-m(j))/2; |@9I5Eg)iE  
        pows = n(j):-2:m(j); ? F), 4Q  
        for k = length(s):-1:1 (6y3"cbe  
            p = (1-2*mod(s(k),2))* ... zNTu j p  
                       prod(2:(n(j)-s(k)))/          ... h+d;`7Z>  
                       prod(2:s(k))/                 ... X!+ a;wr  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... =id $  
                       prod(2:((n(j)+m(j))/2-s(k))); '|R@k_nx  
            idx = (pows(k)==rpowers); D{d$L9.  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ~oR&0et  
        end wg~`Md  
         0\<-R  
        if isnorm J^a"1|  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ~O7(0RsCN  
        end 9<r}s  
    end <R8Z[H:bV  
    ^"#rDP"v  
    % 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)  Uiw7Y\Im|  
    aS G2K0  
    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。 YU(*kC8   
    ^MV%\0o  
    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)