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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 ?Pnx ~m{%*  
    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! 7E(%9W6P  
     
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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 /W0E(8:C)  
    function z = zernfun(n,m,r,theta,nflag) [,GU5,o  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. /ISLVp%H  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N shNE~TA  
    %   and angular frequency M, evaluated at positions (R,THETA) on the f,JX"  
    %   unit circle.  N is a vector of positive integers (including 0), and T*R{L  
    %   M is a vector with the same number of elements as N.  Each element ?DRR+n _  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) !pl_Ao~(  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, `{<JC{yc?  
    %   and THETA is a vector of angles.  R and THETA must have the same KD=bkZ&  
    %   length.  The output Z is a matrix with one column for every (N,M) $N dH*  
    %   pair, and one row for every (R,THETA) pair. 0:#7M}U  
    % 5v+L';wx[T  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike /vy?L\`)#  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), %b9fW  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral x RB7lV*  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, EzUPah  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ^F&A6{9f/h  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 7~D`b1||  
    % ?jFc@t*\:  
    %   The Zernike functions are an orthogonal basis on the unit circle. 2$3kKY6$e  
    %   They are used in disciplines such as astronomy, optics, and _\!0t  
    %   optometry to describe functions on a circular domain. *.xZfi_|  
    % g&XhQ.aa  
    %   The following table lists the first 15 Zernike functions. mgxz1d  
    % a 1NCVZ  
    %       n    m    Zernike function           Normalization &jFKc0\i@  
    %       -------------------------------------------------- *n,UOHlO  
    %       0    0    1                                 1 Ir^BC!<2>  
    %       1    1    r * cos(theta)                    2 T6;>O`B.r  
    %       1   -1    r * sin(theta)                    2 EL"4E',  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 0T=jR{j!o  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ea>[BB3#  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) BJ"Ay@D*  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) "AV1..mu  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Bg5;Q)  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) S7Qen6lm  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ?F9hDLX  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) UQSX<6"  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |HNQ|r_5S  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) JE/l#Q!  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #DrZ`Aq  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) {7/A  
    %       -------------------------------------------------- zV6AuUIt  
    % !'Gb$l!  
    %   Example 1: an pJAB:1  
    % )H.ubM1  
    %       % Display the Zernike function Z(n=5,m=1)  \\y}DNh  
    %       x = -1:0.01:1; _!| =AIX  
    %       [X,Y] = meshgrid(x,x); @"jmI&hYn  
    %       [theta,r] = cart2pol(X,Y); k\Yu5)  
    %       idx = r<=1; yY-FL`-  
    %       z = nan(size(X)); yp( ?1  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); e?_c[`sg  
    %       figure 8}ii3Py  
    %       pcolor(x,x,z), shading interp D!81(}p  
    %       axis square, colorbar l2z`<2mp  
    %       title('Zernike function Z_5^1(r,\theta)') ,?P8m"  
    % %ZJ),9+  
    %   Example 2: 2<p5_4"-U*  
    % K7)j  
    %       % Display the first 10 Zernike functions -='8_B/75  
    %       x = -1:0.01:1; oHYD_8'f  
    %       [X,Y] = meshgrid(x,x); %4QoF  
    %       [theta,r] = cart2pol(X,Y); !7kAJG g  
    %       idx = r<=1; Dx p>  
    %       z = nan(size(X)); AH"g^ gw~T  
    %       n = [0  1  1  2  2  2  3  3  3  3]; HV#?6,U}  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; l5":[C$  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 8=ukS_?Vy  
    %       y = zernfun(n,m,r(idx),theta(idx)); b/a?\0^  
    %       figure('Units','normalized') hY4)W  
    %       for k = 1:10 n.;5P {V1  
    %           z(idx) = y(:,k); ;] l{D}  
    %           subplot(4,7,Nplot(k)) *il]$i  
    %           pcolor(x,x,z), shading interp \N'hbT=  
    %           set(gca,'XTick',[],'YTick',[]) *SMoodFBS  
    %           axis square te!]9rR  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) IPr*pQ{;c  
    %       end %^Q@*+{:f  
    % ~/]\iOL  
    %   See also ZERNPOL, ZERNFUN2. )-TeDIfm  
    b3CspBgC  
    %   Paul Fricker 11/13/2006 '6d D^0dZ  
    `-9*@_ -=M  
    #J<`p  
    % Check and prepare the inputs: s)`1Rf  
    % ----------------------------- _{Fdw  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) J*^,l`C/  
        error('zernfun:NMvectors','N and M must be vectors.') SSA%1l 2!  
    end ],fwZd[t  
    r(?'Yy  
    if length(n)~=length(m) Fw_bY/WN{  
        error('zernfun:NMlength','N and M must be the same length.') V5(tf'  
    end &t9XK8S  
    l1iF}>F2  
    n = n(:); {Vt^Xc  
    m = m(:); /pSUn"3  
    if any(mod(n-m,2)) dwf #~7h_  
        error('zernfun:NMmultiplesof2', ... 8KGv?^M 6W  
              'All N and M must differ by multiples of 2 (including 0).') 1o5Y9#7  
    end c9cphZ(z  
    ]C!Y~  
    if any(m>n) hq&  
        error('zernfun:MlessthanN', ... -G^t-I  
              'Each M must be less than or equal to its corresponding N.') ;nAg4ll8Q  
    end .9[8H:Fe  
    X T)hPwg.  
    if any( r>1 | r<0 ) X{9JSq  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 'nj&}A'  
    end kVG6\<c]  
    f@xfb ie !  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ^S;RX*  
        error('zernfun:RTHvector','R and THETA must be vectors.') _sf0{/< )  
    end ]%Q]C 8[C  
    kgbr+Yw2X  
    r = r(:); HLyFyv\  
    theta = theta(:); ;5JIY7t  
    length_r = length(r); L]L~TA<D9i  
    if length_r~=length(theta) +(h6{e%)  
        error('zernfun:RTHlength', ... wEHrer  
              'The number of R- and THETA-values must be equal.') O( 5L2G  
    end ]cGz~TN~  
    Z+h7 0,|  
    % Check normalization: 65`'Upu  
    % -------------------- n[cyK$"  
    if nargin==5 && ischar(nflag) PE6u8ZAb"  
        isnorm = strcmpi(nflag,'norm'); V~uA(3\U  
        if ~isnorm p?`|CE@h7  
            error('zernfun:normalization','Unrecognized normalization flag.') >-tH&X^  
        end w or'=byh\  
    else KiRt'  
        isnorm = false; Rcx'a:k  
    end GYb2m"a)  
    >. nt'BQ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Rp%\`'+Xz  
    % Compute the Zernike Polynomials % OfDTs  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% e5/ DCz  
    Mbi+Vv-  
    % Determine the required powers of r: >"$-VY6i  
    % ----------------------------------- /CQQ^/  
    m_abs = abs(m); x8rFMR#S=  
    rpowers = []; 4Z T  
    for j = 1:length(n) (+Nmio  
        rpowers = [rpowers m_abs(j):2:n(j)]; ;x0KaFk  
    end aXid;v,  
    rpowers = unique(rpowers); \$\(9!=  
    '/qe#S  
    % Pre-compute the values of r raised to the required powers, "a`0w9Mm}  
    % and compile them in a matrix: *,*:6^t  
    % ----------------------------- d# ?* 62  
    if rpowers(1)==0 Vx4pP$S  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); bHH}x"d[x  
        rpowern = cat(2,rpowern{:}); PG~m-W+  
        rpowern = [ones(length_r,1) rpowern]; fjZveH0  
    else JU2' ~chh  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); aFc'_FrQ  
        rpowern = cat(2,rpowern{:}); /a/uS3&  
    end S?z j&X Y3  
    *[5#g3  
    % Compute the values of the polynomials: /z-C :k\  
    % -------------------------------------- n,'AFb4AF  
    y = zeros(length_r,length(n)); & I'F-F;  
    for j = 1:length(n) #?d>S;)+  
        s = 0:(n(j)-m_abs(j))/2; P9cI{RI  
        pows = n(j):-2:m_abs(j); ;\&bvGj8V  
        for k = length(s):-1:1 %fSk "%u%<  
            p = (1-2*mod(s(k),2))* ... o!dkS/u-m  
                       prod(2:(n(j)-s(k)))/              ... 1bAp{u&  
                       prod(2:s(k))/                     ... b({b5z.A  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... g$+O<a@n  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ?*5l}y=  
            idx = (pows(k)==rpowers); 4a-F4j'  
            y(:,j) = y(:,j) + p*rpowern(:,idx); }sNZQ89V*v  
        end W)P_t"'@L  
         |;1:$E"  
        if isnorm c+M@{EbuN  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi);  ]mU*Y:<  
        end )Zr0_b"V:e  
    end K<9MK>T  
    % END: Compute the Zernike Polynomials ]CJ>iS!V  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r ($t.iS  
    iQR})=Q  
    % Compute the Zernike functions: 'eXw`kw(  
    % ------------------------------ O9IjU10:  
    idx_pos = m>0; x};g!FYfkB  
    idx_neg = m<0; wDTV /"Y  
    QO^X7A"?X  
    z = y; .Zz7LG{  
    if any(idx_pos) _)H+..=  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Xg#([}b  
    end U"G+su->e  
    if any(idx_neg) g}j>;T  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); *)SgdC/f  
    end  o|im  
    ] :#IZ0#  
    % 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) xeh|u"5  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. s*WfRY*=V  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated DKCy h`  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive k/Ro74f=  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, } ~bOP^'  
    %   and THETA is a vector of angles.  R and THETA must have the same {vlh ,0~  
    %   length.  The output Z is a matrix with one column for every P-value, '.<"jZ  
    %   and one row for every (R,THETA) pair. :Djp\ e6!  
    % $B/cj^3  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 1mM52q.R4  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) }7v2GfEkM  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) j!H?dnE||  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 5X-(@GwN  
    %   for all p. oOz6Er[KO  
    % +rX,Sl`/  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 FZ/&[;E!  
    %   Zernike functions (order N<=7).  In some disciplines it is Vs{sB*:  
    %   traditional to label the first 36 functions using a single mode 0:8'Ov(  
    %   number P instead of separate numbers for the order N and azimuthal Uij$ eBN  
    %   frequency M. gJ7pu N  
    % }y/t~f+  
    %   Example: *?'T8yf^  
    % \H(,'w7H  
    %       % Display the first 16 Zernike functions :gt wvM7/B  
    %       x = -1:0.01:1; B!anY}/U  
    %       [X,Y] = meshgrid(x,x); ?[">%^  
    %       [theta,r] = cart2pol(X,Y); 1vb0G ;a;|  
    %       idx = r<=1; D1k]  
    %       p = 0:15; $!@f{9+  
    %       z = nan(size(X)); &YMj\KmlSg  
    %       y = zernfun2(p,r(idx),theta(idx)); 56dl;Z)  
    %       figure('Units','normalized') ;0E 4S  
    %       for k = 1:length(p) ~3 (>_r  
    %           z(idx) = y(:,k); _,Y79 b6  
    %           subplot(4,4,k) KS_d5NvYl  
    %           pcolor(x,x,z), shading interp G7?EaLsfQ  
    %           set(gca,'XTick',[],'YTick',[]) VGIc|Q=F  
    %           axis square mt'#j"mU  
    %           title(['Z_{' num2str(p(k)) '}']) V5MbWXgR  
    %       end V ZGhF!To  
    % %Et]w  
    %   See also ZERNPOL, ZERNFUN. 6qf`P!7d]M  
    /-lmfpT  
    %   Paul Fricker 11/13/2006 *UC^&5:  
    Mt`.|N;y!  
    UHWun I S  
    % Check and prepare the inputs: +L6$Xm5DAv  
    % ----------------------------- $Izk]o;X~  
    if min(size(p))~=1 E~ kmU{D  
        error('zernfun2:Pvector','Input P must be vector.') #96a7K  
    end -6\9B>qa  
    WYL.J5O  
    if any(p)>35 I%Z &i-33y  
        error('zernfun2:P36', ... Iq*7F5B  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... [<hiOB  
               '(P = 0 to 35).']) JAK*HA  
    end ,D1QJPM  
    H2}i .  
    % Get the order and frequency corresonding to the function number: DS yE   
    % ---------------------------------------------------------------- 3L|k3 `I4  
    p = p(:); QPn c "!  
    n = ceil((-3+sqrt(9+8*p))/2); v:'y&yS  
    m = 2*p - n.*(n+2); L<n_}ucA  
    r%4:,{HF  
    % Pass the inputs to the function ZERNFUN: YeVhWPn@  
    % ---------------------------------------- ORNE>6J H  
    switch nargin r|+Zni]  
        case 3 Bb)J8,LQ  
            z = zernfun(n,m,r,theta); _4+1c5Q!  
        case 4 jBM>Pe^`3  
            z = zernfun(n,m,r,theta,nflag); )I@iW\`7  
        otherwise i2DR}%U  
            error('zernfun2:nargin','Incorrect number of inputs.') "q8wEu,z[  
    end cQjJ9o7  
    ^]HwStn&=  
    % 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) ()iJvf>@  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. `&o|=  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of LxbVRw  
    %   order N and frequency M, evaluated at R.  N is a vector of u[% #/  
    %   positive integers (including 0), and M is a vector with the 2 Ug jH  
    %   same number of elements as N.  Each element k of M must be a Pdv&X*KA  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) E ?-K_p  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is qQb8K+t  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 5HB4B <2  
    %   with one column for every (N,M) pair, and one row for every @U}UCG7+  
    %   element in R. W\Gg!XsLk  
    % FUQT,7CA  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- )S]c'}^  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is uzS57 O%  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to eu~;G H  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 w v1R ]3}  
    %   for all [n,m]. lm+wjhkN  
    % ;J4_8N-  
    %   The radial Zernike polynomials are the radial portion of the 2iUF%>  
    %   Zernike functions, which are an orthogonal basis on the unit ?lTQjw{  
    %   circle.  The series representation of the radial Zernike hX^XtIC=  
    %   polynomials is te`4*t  
    % )_BteLo-  
    %          (n-m)/2 - na]P3 s  
    %            __ )TxhJB5|  
    %    m      \       s                                          n-2s V; ChrmE  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ,HkJ.6KF  
    %    n      s=0 _C##U;e!  
    % @? t)UE  
    %   The following table shows the first 12 polynomials. =[P||  
    % Q5Wb)  
    %       n    m    Zernike polynomial    Normalization G#csN&|,  
    %       --------------------------------------------- ;;2s{{(R  
    %       0    0    1                        sqrt(2) Aoj X)_"z  
    %       1    1    r                           2 p4/D%*G^`  
    %       2    0    2*r^2 - 1                sqrt(6) /rquI y^  
    %       2    2    r^2                      sqrt(6) J[^-k!9M  
    %       3    1    3*r^3 - 2*r              sqrt(8) O)hNHIF  
    %       3    3    r^3                      sqrt(8) 6(eyUgnb  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 1PWDK1GI8  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) {3l] /X3  
    %       4    4    r^4                      sqrt(10) 8garRB{  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) S-im o  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) TG!sck4/-Q  
    %       5    5    r^5                      sqrt(12) ec{pWzAe  
    %       --------------------------------------------- $:>K-4X\}  
    % \KTX{qI"f  
    %   Example: VlK WWQj  
    % M]oaWQu  
    %       % Display three example Zernike radial polynomials ?@tp1?)  
    %       r = 0:0.01:1; -ohqw+D  
    %       n = [3 2 5]; .(! $j-B  
    %       m = [1 2 1]; .}^m8PP  
    %       z = zernpol(n,m,r); . 8k9yk  
    %       figure >1W)J3  
    %       plot(r,z) Obbjl@]  
    %       grid on d}Q;CF3 m:  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') t1D6#JP(a  
    % Nl0*"}`I_  
    %   See also ZERNFUN, ZERNFUN2. 6<gh:vj  
    SI-s:%O  
    % A note on the algorithm. UZqr6A(/H  
    % ------------------------ g$uj<"^  
    % The radial Zernike polynomials are computed using the series V4_ZBeWA  
    % representation shown in the Help section above. For many special tX)^$3A  
    % functions, direct evaluation using the series representation can *!vwW T  
    % produce poor numerical results (floating point errors), because oPl^tzO  
    % the summation often involves computing small differences between o H$4K8j  
    % large successive terms in the series. (In such cases, the functions @2V#bK  
    % are often evaluated using alternative methods such as recurrence {"-uaH>,  
    % relations: see the Legendre functions, for example). For the Zernike u1rT:\G1  
    % polynomials, however, this problem does not arise, because the L)kwMk  
    % polynomials are evaluated over the finite domain r = (0,1), and H|5\c=  
    % because the coefficients for a given polynomial are generally all d7A vx  
    % of similar magnitude. 86oa>#opU  
    % Rkgpa/te"  
    % ZERNPOL has been written using a vectorized implementation: multiple L2+~I<|>  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] sZ_+6+ :  
    % values can be passed as inputs) for a vector of points R.  To achieve [8[g_  
    % this vectorization most efficiently, the algorithm in ZERNPOL I vO#tI  
    % involves pre-determining all the powers p of R that are required to ,-D3tleu`  
    % compute the outputs, and then compiling the {R^p} into a single *{ 6{ZKM  
    % matrix.  This avoids any redundant computation of the R^p, and DYx3 NDX7  
    % minimizes the sizes of certain intermediate variables. 8a)lrIg  
    % O,u$L  
    %   Paul Fricker 11/13/2006 rjz$~(&m6  
    icH\(   
    F@UbUm2o  
    % Check and prepare the inputs: *6<<6f`(  
    % ----------------------------- G?*)0`~W  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ,:;ZzHzR0  
        error('zernpol:NMvectors','N and M must be vectors.') g{RVxGE7  
    end @X5F$=aqZr  
    0.!_k )tu  
    if length(n)~=length(m) z&Cz!HrS  
        error('zernpol:NMlength','N and M must be the same length.') P9c!   
    end ?cF`T/z]"  
    bL-+  
    n = n(:); Dn~c  
    m = m(:); +8h!@  
    length_n = length(n); ;LD!eWSK,  
    4SlEc|'7@  
    if any(mod(n-m,2)) Yv>kToa\^  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') (l}W\iB' d  
    end F!ZE4S_  
    +VT/ c  
    if any(m<0) @L0xU??"|  
        error('zernpol:Mpositive','All M must be positive.') ZW7z[,tk<.  
    end ~>SqJ&-moo  
    RzhAX I=  
    if any(m>n) ~HBQQt  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') ZD~ra7  
    end 07b =Zhh  
    kn %i#Fz  
    if any( r>1 | r<0 ) z[|2od  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 3127 4O  
    end 7 x#QkImQ  
    P@| W \  
    if ~any(size(r)==1) W-D{ cU  
        error('zernpol:Rvector','R must be a vector.') (P'{A>aHl0  
    end As{"B  
    n37P$0  
    r = r(:); O pavno%&  
    length_r = length(r); < #FxI  
    N(s5YX7<hd  
    if nargin==4 Q-<h)WTA  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); !:baG]Y  
        if ~isnorm _59f.FsVR  
            error('zernpol:normalization','Unrecognized normalization flag.') ANWfRtiU#  
        end g|TWoRx:  
    else /Vdu|k=  
        isnorm = false; y7^E`LKK  
    end \:-"?  
    {;5\#VFg  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7w\L<vFm  
    % Compute the Zernike Polynomials t03X/%H  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZSvU1T8  
    fj|X`,TiZ;  
    % Determine the required powers of r: y=!7PB_\|  
    % ----------------------------------- k:[T#/;  
    rpowers = []; t#d{hEr  
    for j = 1:length(n) %-fQ[@5  
        rpowers = [rpowers m(j):2:n(j)]; zt;aB>jz#  
    end ?[?;%Y  
    rpowers = unique(rpowers); 'C7$,H'  
    ?k`UQi]Q  
    % Pre-compute the values of r raised to the required powers, .fAHP 5-  
    % and compile them in a matrix: T].Xx`  
    % ----------------------------- dk/f_m  
    if rpowers(1)==0 >=1Aa,_tc  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); m`BE{%  
        rpowern = cat(2,rpowern{:}); uA4x xY  
        rpowern = [ones(length_r,1) rpowern]; |"g+p)A  
    else # Rhtaq9  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); K2{6{X=  
        rpowern = cat(2,rpowern{:}); ~^bf1W[  
    end fG zx;<0P!  
    !?)aZ |r  
    % Compute the values of the polynomials: i^@hn>s$  
    % -------------------------------------- *b7evU *1  
    z = zeros(length_r,length_n); m<sCRWa-  
    for j = 1:length_n &G!~@\tMg  
        s = 0:(n(j)-m(j))/2; @>*r2=#14  
        pows = n(j):-2:m(j); }Q a  
        for k = length(s):-1:1 /Z~5bb(  
            p = (1-2*mod(s(k),2))* ... O1Ynl` }  
                       prod(2:(n(j)-s(k)))/          ...  s2`}~  
                       prod(2:s(k))/                 ... vM?,#:5  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... mWF\h>]|.  
                       prod(2:((n(j)+m(j))/2-s(k))); O{x-9p  
            idx = (pows(k)==rpowers); CC)Mws+2  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 7jw5'`;)"  
        end NvvD~B b  
         4k!>JQor  
        if isnorm <UY9<o  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); b ,x$wP+  
        end <Uu[nUJ  
    end F9k}zAY\J  
    r{{5@  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  i[wnG)  
    1;!dTh  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 jaIcIc=Pf  
    7mn&w$MS4:  
    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)