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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 ?QMs<  
    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! W.CIyGK  
     
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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 g=%&p?1@E  
    function z = zernfun(n,m,r,theta,nflag) I@B7uFj  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. D'&L wU,o  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Em 7q@  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 'ZDclz9}  
    %   unit circle.  N is a vector of positive integers (including 0), and G1l(  
    %   M is a vector with the same number of elements as N.  Each element Zry>s0  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) kmS8>O  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, QJ /SP  
    %   and THETA is a vector of angles.  R and THETA must have the same c'6H@m#=  
    %   length.  The output Z is a matrix with one column for every (N,M) pA2U+Q@  
    %   pair, and one row for every (R,THETA) pair. y{N9.H2  
    % 2Ar<(v$  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike anvj{1  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), YJy*OS_&  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral u%pief  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, x EBjfn  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized gr;M  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. (pmo[2kg  
    % cNVdGY%&  
    %   The Zernike functions are an orthogonal basis on the unit circle. 1 W0;YcT]  
    %   They are used in disciplines such as astronomy, optics, and H07j&  
    %   optometry to describe functions on a circular domain. ST3qg6Cq2J  
    % Vo%d;>!G\;  
    %   The following table lists the first 15 Zernike functions. VC.?]'OqD  
    % r>Ln*R,9D  
    %       n    m    Zernike function           Normalization Zx_m?C_2_  
    %       -------------------------------------------------- pR"qPSv'  
    %       0    0    1                                 1 q :bKT#\  
    %       1    1    r * cos(theta)                    2 ueR42J%s  
    %       1   -1    r * sin(theta)                    2 @I&"P:E0F;  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) .*j+?  
    %       2    0    (2*r^2 - 1)                    sqrt(3) P5>CSWy%  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) #-;BU{3*  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 9)">()8  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) {UcIt LjY  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) `9ox?|iJ  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) =r~. I  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) HhL%iy1  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0REWbcxd"  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) RVfe}4Stm#  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Bu1z$#AC  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) `y}d)"!  
    %       -------------------------------------------------- gnXjd}  
    % mV,R0olF  
    %   Example 1: o(P:f)B  
    % 9^u?v`!  
    %       % Display the Zernike function Z(n=5,m=1) aJ8pJ{,P  
    %       x = -1:0.01:1; D@^ZpN8r  
    %       [X,Y] = meshgrid(x,x); ' l|_$3  
    %       [theta,r] = cart2pol(X,Y); A-5 +#  
    %       idx = r<=1; Aq!['G  
    %       z = nan(size(X)); WM"^#=+$  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ??Zmj:8E'  
    %       figure lQBM0|n  
    %       pcolor(x,x,z), shading interp Rs`a@ Fn  
    %       axis square, colorbar &r%*_pX  
    %       title('Zernike function Z_5^1(r,\theta)') PoJ$%_a}  
    % F-^HN%  
    %   Example 2: +,Az\aT/%  
    % (GG"'bYk  
    %       % Display the first 10 Zernike functions Ug21d42Z4  
    %       x = -1:0.01:1; h '[vB^  
    %       [X,Y] = meshgrid(x,x); n5.>;N.*  
    %       [theta,r] = cart2pol(X,Y); !dY:S';~  
    %       idx = r<=1; kA__*b}8UK  
    %       z = nan(size(X)); {ah=i8$  
    %       n = [0  1  1  2  2  2  3  3  3  3]; |L;psK  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; (:QQ7xc{}  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Net)l@IB]  
    %       y = zernfun(n,m,r(idx),theta(idx)); [+g@@\X4  
    %       figure('Units','normalized') ;YDF*~9u  
    %       for k = 1:10 t1jlxK  
    %           z(idx) = y(:,k); 6#M0AG  
    %           subplot(4,7,Nplot(k)) %i8>w:@NW  
    %           pcolor(x,x,z), shading interp "<x~{BN?  
    %           set(gca,'XTick',[],'YTick',[]) N?;o_^C  
    %           axis square T-C#xmY(  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) X5Y `(/V  
    %       end <z uE=0P~%  
    % Rt^<xXX$  
    %   See also ZERNPOL, ZERNFUN2. ( 'n8=J  
    #}dVaXY)  
    %   Paul Fricker 11/13/2006 q 9S z7_K  
    A&c@8  
    cTd;p>:>m  
    % Check and prepare the inputs: vt@Us\fI  
    % ----------------------------- EWIc|b:  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) {|Ki^8h/p  
        error('zernfun:NMvectors','N and M must be vectors.') 45sxF?GSwL  
    end DBJA}Cw  
    >}b6J7_  
    if length(n)~=length(m) +RV-VrV  
        error('zernfun:NMlength','N and M must be the same length.') =kh>s$We  
    end t*d >eK`:N  
    ]<T8ZA_Y;  
    n = n(:); fu<2t$Cn>  
    m = m(:); x XM!E 8  
    if any(mod(n-m,2)) EB5_;  
        error('zernfun:NMmultiplesof2', ... ny(GTKoUz  
              'All N and M must differ by multiples of 2 (including 0).') X@qk>/  
    end /;&+ < }  
    ;Q=GJ5`B  
    if any(m>n) b/B`&CIA0"  
        error('zernfun:MlessthanN', ... [OZ=iz.  
              'Each M must be less than or equal to its corresponding N.') u'i%~(:$\)  
    end i*CQor6|z  
    6lmiMU&V  
    if any( r>1 | r<0 ) j;20JA/b  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') {kPe#n>xT  
    end eh:}X}c=J]  
    ~r^5-\[hZ  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ) wY!/&  
        error('zernfun:RTHvector','R and THETA must be vectors.') Sf&?3a+f  
    end hyb +#R  
    4b2mtLn_  
    r = r(:); g[s\~MF@s  
    theta = theta(:); Ji6`-~ k  
    length_r = length(r); E8-fW\!F  
    if length_r~=length(theta) 'DzBp  
        error('zernfun:RTHlength', ... NdsX*o@a  
              'The number of R- and THETA-values must be equal.') zD2.Q%`IM  
    end 0^9:KZ.!  
    J4G> E.8  
    % Check normalization: =1*%>K  
    % -------------------- R6q4 ["  
    if nargin==5 && ischar(nflag) N(:nF5>_  
        isnorm = strcmpi(nflag,'norm'); H 5U x.]y  
        if ~isnorm :YqQlr\  
            error('zernfun:normalization','Unrecognized normalization flag.') Er"R;l]xJ  
        end /z1p/RiX  
    else lC=N:=Mu  
        isnorm = false; \ I^nx+l  
    end [O7w =  
    > X[|c"l.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *O+R|Cdp/  
    % Compute the Zernike Polynomials mN\%f J7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v._Egk0  
    K[uY+!'1  
    % Determine the required powers of r: 4YDT%_h0  
    % ----------------------------------- - J"qrpZ^  
    m_abs = abs(m); "Su b4F`  
    rpowers = []; &_9YLXtMi;  
    for j = 1:length(n) 0{?: FQ#  
        rpowers = [rpowers m_abs(j):2:n(j)]; Cs:+93w  
    end K/vxzHSl  
    rpowers = unique(rpowers); ZT) !8  
    Y^R?Q'  
    % Pre-compute the values of r raised to the required powers, ZD5I5  
    % and compile them in a matrix: d"B@c;dD  
    % ----------------------------- 3s`V)aXP  
    if rpowers(1)==0 }+Rgx@XZ\  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); |*^8~u3J"  
        rpowern = cat(2,rpowern{:}); ?}'N_n ys  
        rpowern = [ones(length_r,1) rpowern]; 7 9Qc`3a  
    else &|Lh38s@$#  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); m$fQ`XzU  
        rpowern = cat(2,rpowern{:}); t_jyyHxoZ:  
    end +"cRhVR  
    UrO=!Gk  
    % Compute the values of the polynomials: _urG_~q  
    % -------------------------------------- *8$>Whr  
    y = zeros(length_r,length(n)); 3ty4D2y  
    for j = 1:length(n) "7=bL7wM&  
        s = 0:(n(j)-m_abs(j))/2; (n=9c%w  
        pows = n(j):-2:m_abs(j); iH-bo@  
        for k = length(s):-1:1 H LjvKE=W  
            p = (1-2*mod(s(k),2))* ... /8xH$n&xoC  
                       prod(2:(n(j)-s(k)))/              ... }m6f^fs}  
                       prod(2:s(k))/                     ... q*\NRq  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... lijB#1<8*  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ,*/Pg 52?  
            idx = (pows(k)==rpowers); 7MY)\aH  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ,{k<JA {  
        end i=oTg  
         \V]t!mZ-}l  
        if isnorm gaQ[3g  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); wJ6_I$>  
        end /"=29sWB  
    end D=$4/D:;  
    % END: Compute the Zernike Polynomials ;0IvF#SJ(.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9%sFJ  
    l+%Fl=Q2em  
    % Compute the Zernike functions: ^6Yd}  
    % ------------------------------ Pp ,Um(  
    idx_pos = m>0; d] U`?A,  
    idx_neg = m<0; ]k[x9,IU\y  
    Hi^35  
    z = y; K[kds`  
    if any(idx_pos) 6DB0ni  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); o&~dGG4J  
    end Y?<)Dg.[  
    if any(idx_neg) z. 'Fv7  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); _=pWG^a  
    end )1WMlG  
    Q|?'(J+  
    % 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) \wYc1M@7V  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 58::h. :  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 1w`2Dt  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive I7~|~<  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ?-f,8Z|h  
    %   and THETA is a vector of angles.  R and THETA must have the same HmMO*k<6@  
    %   length.  The output Z is a matrix with one column for every P-value, Or7 mD  
    %   and one row for every (R,THETA) pair. + ~ "5!  
    % UbO4%YHt  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike |d[5l^6  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) YScvyh?E  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) P;73Hr[E#  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 M 7rIi\4K4  
    %   for all p. :|rPT)yT]  
    % nq1 'F  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 /& r|ec5  
    %   Zernike functions (order N<=7).  In some disciplines it is ,[* ;UR  
    %   traditional to label the first 36 functions using a single mode )qv2)a!H  
    %   number P instead of separate numbers for the order N and azimuthal 6kN:*  
    %   frequency M. )hBE11,PB  
    % wPX*%0]  
    %   Example: dxK9:IX  
    % k2r3dO@q  
    %       % Display the first 16 Zernike functions i)M EK#{  
    %       x = -1:0.01:1; 22&;jpL'?  
    %       [X,Y] = meshgrid(x,x); a7#?h%wf  
    %       [theta,r] = cart2pol(X,Y); X %4Kj[I^  
    %       idx = r<=1; kJT+  
    %       p = 0:15; )"| ||\Iv  
    %       z = nan(size(X)); 5wv fF.v  
    %       y = zernfun2(p,r(idx),theta(idx));  lq>AGw  
    %       figure('Units','normalized') -R b{^/  
    %       for k = 1:length(p) x6W `hpL  
    %           z(idx) = y(:,k); dEp7{jY1O  
    %           subplot(4,4,k) ml0*1Dw  
    %           pcolor(x,x,z), shading interp 'RbQj}@x  
    %           set(gca,'XTick',[],'YTick',[]) [ *>AN7W   
    %           axis square XogVpkA  
    %           title(['Z_{' num2str(p(k)) '}']) s2REt$.q  
    %       end =n+ \\D  
    % XKS8K4"  
    %   See also ZERNPOL, ZERNFUN. pDl3!m  
    v-Qmx-N  
    %   Paul Fricker 11/13/2006 e2cP *J  
    ,[e\cnq[  
    E=$p^s  
    % Check and prepare the inputs: 3I  $>uR  
    % ----------------------------- <%P2qgz5  
    if min(size(p))~=1 -1u9t4+`  
        error('zernfun2:Pvector','Input P must be vector.') ~Lz%.a;o  
    end nB5zNyY4  
    !5Sd2<N  
    if any(p)>35 G8J*Wnwu[K  
        error('zernfun2:P36', ... Pw[g  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Nd@~>&F  
               '(P = 0 to 35).']) ,|h)bg7.  
    end aG%, cQ1  
    -LW[7s$  
    % Get the order and frequency corresonding to the function number: _S`o1^Ad  
    % ---------------------------------------------------------------- S1S;F9F  
    p = p(:); @t*t+Vqw  
    n = ceil((-3+sqrt(9+8*p))/2); ,xfO;yd  
    m = 2*p - n.*(n+2); k{I 01  
    eE@&ze>X  
    % Pass the inputs to the function ZERNFUN: X3%Ic`Lq#  
    % ---------------------------------------- y3G `>  
    switch nargin ~1L:_Sg*  
        case 3 aZ|=(]  
            z = zernfun(n,m,r,theta); sN6N >{  
        case 4 ,|kDsR !  
            z = zernfun(n,m,r,theta,nflag); $I9qgDJ)  
        otherwise >znRyQ~bM  
            error('zernfun2:nargin','Incorrect number of inputs.') n$n 7-7  
    end cyM-)r@YQV  
    $F'>yop2b  
    % 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) y: m_tv0~0  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. xg_D f,  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 4(Iplo*Ys@  
    %   order N and frequency M, evaluated at R.  N is a vector of $-}e; VZb  
    %   positive integers (including 0), and M is a vector with the c(;a=n(E#  
    %   same number of elements as N.  Each element k of M must be a 9}a_:hAy/  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 29CINC  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 91>fqe  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix +l3=3  
    %   with one column for every (N,M) pair, and one row for every } :=Tm]S  
    %   element in R. OCZaQ33  
    % ^sN (  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ABE@n%|`  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is evkH05+;W  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to D%6;^^WyUx  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 )9l^O  
    %   for all [n,m]. B(xN Gs  
    % m<FOu<y  
    %   The radial Zernike polynomials are the radial portion of the 9$`lIy@B  
    %   Zernike functions, which are an orthogonal basis on the unit +)o}c"P!  
    %   circle.  The series representation of the radial Zernike {:@tQdM:i8  
    %   polynomials is iY"l}.7)  
    % H"ZZ.^"5FV  
    %          (n-m)/2 M9zfT !-  
    %            __ sVG(N.y  
    %    m      \       s                                          n-2s [kE."#  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r fd-q3 _f  
    %    n      s=0 {43>m)8+  
    % "HE^v_p  
    %   The following table shows the first 12 polynomials. jck}" N  
    % s(X;Eha  
    %       n    m    Zernike polynomial    Normalization P ;IrBq6|o  
    %       --------------------------------------------- UG=K|OXWJ  
    %       0    0    1                        sqrt(2) a7N!B'y  
    %       1    1    r                           2 o sKKt?^?  
    %       2    0    2*r^2 - 1                sqrt(6) ;2B{9{  
    %       2    2    r^2                      sqrt(6) M1KqY:9E  
    %       3    1    3*r^3 - 2*r              sqrt(8) >jD[X5Y  
    %       3    3    r^3                      sqrt(8) (?nCy HC%g  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) }.s~T#v  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) M Y|w  
    %       4    4    r^4                      sqrt(10) c("_bOAT  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) pAT7)Ch  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) \7CGUB>L  
    %       5    5    r^5                      sqrt(12) KtNY_&xd  
    %       --------------------------------------------- V+O"j^Z_J  
    % lRXK\xIP ,  
    %   Example: itC-4^  
    % rtc9wu  
    %       % Display three example Zernike radial polynomials #8)*1?  
    %       r = 0:0.01:1; @')[FEdW  
    %       n = [3 2 5]; Z?\>JM >;  
    %       m = [1 2 1]; ,G)r=$XU  
    %       z = zernpol(n,m,r); ,c NLkoN  
    %       figure '3uVkp 6tF  
    %       plot(r,z) t.;LnrY  
    %       grid on T?X_c"{8M  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') s_mS^`P7  
    % EAM5{Nc  
    %   See also ZERNFUN, ZERNFUN2. qT+%;(  
    vh$%9ed  
    % A note on the algorithm. b9!FC$^J  
    % ------------------------ L*:jXmUM_~  
    % The radial Zernike polynomials are computed using the series rW=Z>1  
    % representation shown in the Help section above. For many special lv04g} W  
    % functions, direct evaluation using the series representation can j:VbrR  
    % produce poor numerical results (floating point errors), because !jTcsN%  
    % the summation often involves computing small differences between ^jx7@LgS=  
    % large successive terms in the series. (In such cases, the functions jbAx;Xt'=M  
    % are often evaluated using alternative methods such as recurrence .X;3,D[w  
    % relations: see the Legendre functions, for example). For the Zernike >6?__v]9G  
    % polynomials, however, this problem does not arise, because the 2 O%`G+\)  
    % polynomials are evaluated over the finite domain r = (0,1), and mGK|ihYu  
    % because the coefficients for a given polynomial are generally all K57&yVX  
    % of similar magnitude. @<elq'2  
    % ynQ: > tw  
    % ZERNPOL has been written using a vectorized implementation: multiple cH&J{WeZa  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] OOo3G~2r  
    % values can be passed as inputs) for a vector of points R.  To achieve sr{a(4*\  
    % this vectorization most efficiently, the algorithm in ZERNPOL GaK-t*Q  
    % involves pre-determining all the powers p of R that are required to h%uZYsK  
    % compute the outputs, and then compiling the {R^p} into a single Rx}$0c0  
    % matrix.  This avoids any redundant computation of the R^p, and ;'cN<x)% |  
    % minimizes the sizes of certain intermediate variables. .?loO3 m  
    % o{y9r{~A  
    %   Paul Fricker 11/13/2006 =Lf,?"S  
    ^y<<>Y'I  
    VT\F]Oa#  
    % Check and prepare the inputs: H<PtAYFS  
    % ----------------------------- r2,.abo  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) U`2e{>'4t  
        error('zernpol:NMvectors','N and M must be vectors.') xwq+j "  
    end .N ,3 od@  
    _I:/ZF5  
    if length(n)~=length(m) zN^n]N_?  
        error('zernpol:NMlength','N and M must be the same length.') Q$zO83  
    end aWR}R>E  
    Hl{S]]z  
    n = n(:); ;)D];u|_  
    m = m(:); -;^j:L{   
    length_n = length(n); hpO`]  
    JzQ)jdvp  
    if any(mod(n-m,2)) tFp Ygff<  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') pHLB= r  
    end lPw%ErG  
    iO|se:LY<  
    if any(m<0) HTX?,C_  
        error('zernpol:Mpositive','All M must be positive.') ]~'5\58sP  
    end ahJ`$U4n  
    CxwoBuG=?  
    if any(m>n) Mygf T[_  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') Vex{.Vh,"  
    end t gI{`jS%  
    l`#4KCL(  
    if any( r>1 | r<0 ) )48QBz?  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') (|klSz_4LM  
    end ChGYTn`X   
    _`&m\Qe>  
    if ~any(size(r)==1) X=5xh  
        error('zernpol:Rvector','R must be a vector.') 5C o  
    end :~Wrf8 UQ  
    K,+LG7ec  
    r = r(:); &$`P,i 1)  
    length_r = length(r); }dgfqq  
    3@dL /x4A  
    if nargin==4 ,JAx ?Xb  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); "s0)rqf<  
        if ~isnorm (l^3Z3zf&  
            error('zernpol:normalization','Unrecognized normalization flag.') QbkLdM,S*  
        end Br1&8L-|%  
    else Xg|B \ \  
        isnorm = false; WrQDX3  
    end HK,cJah q  
    [>8}J "  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BJ$9v bhZN  
    % Compute the Zernike Polynomials dCi?SIN  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #U! _U+K  
    $rv8K j+  
    % Determine the required powers of r: Q=;U@k@>  
    % ----------------------------------- 2@'oe7E  
    rpowers = []; \<JSkr[h!"  
    for j = 1:length(n) nAW:utTB  
        rpowers = [rpowers m(j):2:n(j)]; ?Y-%'J(  
    end EMwS1~3dD  
    rpowers = unique(rpowers); x$5) ^ud?  
    vQosPS_2L  
    % Pre-compute the values of r raised to the required powers, r e/@D@%  
    % and compile them in a matrix: :ubV};  
    % ----------------------------- Ktb\ bw  
    if rpowers(1)==0 *scVJ  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); q)X$^oE!6  
        rpowern = cat(2,rpowern{:}); zi|+HM  
        rpowern = [ones(length_r,1) rpowern]; [I '0,y  
    else XG{{ 2f  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); i UCXAWP  
        rpowern = cat(2,rpowern{:}); {MtpkUN  
    end G18F&c~  
    wvO|UP H\  
    % Compute the values of the polynomials: ciBP7>'::  
    % -------------------------------------- Ixb=L (V  
    z = zeros(length_r,length_n); bLlKe50  
    for j = 1:length_n K0-ypU*P  
        s = 0:(n(j)-m(j))/2; "+kL )]  
        pows = n(j):-2:m(j); 2D75:@JL}|  
        for k = length(s):-1:1 )j9SGLo  
            p = (1-2*mod(s(k),2))* ... Y2a5bc P  
                       prod(2:(n(j)-s(k)))/          ... cii_U=   
                       prod(2:s(k))/                 ... w~(1%p/  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... !\zWF  
                       prod(2:((n(j)+m(j))/2-s(k))); *w^C"^*  
            idx = (pows(k)==rpowers); =5J7Hw&K  
            z(:,j) = z(:,j) + p*rpowern(:,idx); x2OaPlG,&V  
        end "'c A2~  
         ]NUl9t*N4  
        if isnorm zMj#KA1  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ]RPs|R?  
        end 'n{Nvt.c  
    end `|6'9  
    a|%J=k>>  
    % 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)  fz\9 S  
    [<%H>S1  
    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。 G&i!Hs  
    {[+mpKq  
    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)