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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 #Iz)Mu  
    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! ( uD^_N]3  
     
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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 T"Y#u  
    function z = zernfun(n,m,r,theta,nflag) <7J3tn B  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. S#C-j D  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N :V+rC]0  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ~3:hed7:  
    %   unit circle.  N is a vector of positive integers (including 0), and 6L8nw+mEK  
    %   M is a vector with the same number of elements as N.  Each element u$aK19K/  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) iptA#<Yj  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, / =6_2t#vA  
    %   and THETA is a vector of angles.  R and THETA must have the same _j , Tc*T  
    %   length.  The output Z is a matrix with one column for every (N,M) _r3Y$^!U  
    %   pair, and one row for every (R,THETA) pair. ]w6 F%d  
    % *>=tmW;%  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike /r~2KZE  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), %;QK5L   
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 2Cp4aTGv#  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, mnM]@8^G  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized z]8Mv(eL  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. QZvQ8  
    % [m:cO6DM,  
    %   The Zernike functions are an orthogonal basis on the unit circle. D|ze0A@  
    %   They are used in disciplines such as astronomy, optics, and 5\quh2Q_  
    %   optometry to describe functions on a circular domain. Hu<]*(lK%  
    % -"nkC  
    %   The following table lists the first 15 Zernike functions. nzaDO-2!  
    % *x2!N$b  
    %       n    m    Zernike function           Normalization BGibBF^  
    %       -------------------------------------------------- Qt4mg?X/  
    %       0    0    1                                 1 ]j7`3%4uK  
    %       1    1    r * cos(theta)                    2 F!#)l*OX;  
    %       1   -1    r * sin(theta)                    2 k(H]ILL  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ]" V_`i7Z  
    %       2    0    (2*r^2 - 1)                    sqrt(3) yP$esDP  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) e5bXgmyil  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) n}Z%D-b$  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) G]aey>)  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) W'vekuM  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ^x O](,H  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) o i'iZX  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) }>@SyE'Q  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) fphCQO^#vW  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) M(+Pd_c6  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ^oPFLez56  
    %       -------------------------------------------------- Nxe1^F33  
    % x]wi&  
    %   Example 1: (k!7`<k!Y  
    % Jt]RU+TB  
    %       % Display the Zernike function Z(n=5,m=1) K]$PRg1| 3  
    %       x = -1:0.01:1; k5-4^  
    %       [X,Y] = meshgrid(x,x); Q9OCf"n$  
    %       [theta,r] = cart2pol(X,Y); .S,E=  
    %       idx = r<=1; u $-&Im<  
    %       z = nan(size(X)); }'wZ)N@  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); "|(.W3f1  
    %       figure AAa7)^R  
    %       pcolor(x,x,z), shading interp ((]i}s0S  
    %       axis square, colorbar 3mU~G}ig  
    %       title('Zernike function Z_5^1(r,\theta)') =A,B'n\R  
    % M2cGr  
    %   Example 2: Nxt:U{`T'  
    % *D%w r'!>  
    %       % Display the first 10 Zernike functions )@DDs(q=i  
    %       x = -1:0.01:1; Mu/(Xp62  
    %       [X,Y] = meshgrid(x,x); P,pC Z+H  
    %       [theta,r] = cart2pol(X,Y); 5T.U=_ag  
    %       idx = r<=1; <Mvni z  
    %       z = nan(size(X)); P0>2}/;o  
    %       n = [0  1  1  2  2  2  3  3  3  3]; /Yi4j,8!|  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; mTu>S  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; i;{lY1  
    %       y = zernfun(n,m,r(idx),theta(idx)); 0e0)1;t\  
    %       figure('Units','normalized') :8O T  
    %       for k = 1:10 MkMDI)Y|  
    %           z(idx) = y(:,k); E'4Psx9: =  
    %           subplot(4,7,Nplot(k)) >#:SJ?)`T  
    %           pcolor(x,x,z), shading interp [(Z(8{3i  
    %           set(gca,'XTick',[],'YTick',[]) }y*D(`  
    %           axis square HUjX[w8  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 9Zd\6F,  
    %       end vW eg1  
    % X[ Ufq^fyA  
    %   See also ZERNPOL, ZERNFUN2. [ S  
    RdD>&D$I  
    %   Paul Fricker 11/13/2006 4r4 #u'Om  
    !D['}%  
    s.7=!JQ#]p  
    % Check and prepare the inputs: %C`P7&8m=O  
    % ----------------------------- +0U=UV)U  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) o#6QwbU25  
        error('zernfun:NMvectors','N and M must be vectors.') z<9C-  
    end BNJ0D  
    {E%c%zzQ  
    if length(n)~=length(m)  &ox  
        error('zernfun:NMlength','N and M must be the same length.') |*JMPg?zI  
    end !`N:.+DT  
    'd Be,@  
    n = n(:); kiJ=C2'&  
    m = m(:); S|| W  
    if any(mod(n-m,2)) vrb@::sy0T  
        error('zernfun:NMmultiplesof2', ... rzHBop-8  
              'All N and M must differ by multiples of 2 (including 0).') h(yFr/  
    end V~*>/2+  
    Tk[]l7R~  
    if any(m>n) pW.WJ`Rk  
        error('zernfun:MlessthanN', ... VK*_p EV,}  
              'Each M must be less than or equal to its corresponding N.') })<u ~r  
    end F8<G9#%s\  
    4>oM5Yf8  
    if any( r>1 | r<0 ) d6*84'|!  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ,Tar?&C:  
    end py7Zh%k  
    RiAg:  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) <QvVPE}z   
        error('zernfun:RTHvector','R and THETA must be vectors.') 7+f6?  
    end er24}G8  
    3'x>$5 W  
    r = r(:); 40MKf/9  
    theta = theta(:); s"#N;  
    length_r = length(r); ^_3Ey  
    if length_r~=length(theta) -4+'(3qr  
        error('zernfun:RTHlength', ... QAx9W%  
              'The number of R- and THETA-values must be equal.') :k ?`gm$  
    end B|a<=~  
    D+;4|7s+  
    % Check normalization: \?t8[N\_[(  
    % -------------------- ?bM%#x{e  
    if nargin==5 && ischar(nflag) ,N:^4A  
        isnorm = strcmpi(nflag,'norm'); mD7NQ2:wA  
        if ~isnorm |~%RSS~b*  
            error('zernfun:normalization','Unrecognized normalization flag.') Sak^J.~G[  
        end sE&nEc  
    else > "rM\ Q  
        isnorm = false; 1@{ov!YB]  
    end 7r?,wM  
    $!. [R}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k-3;3Mq  
    % Compute the Zernike Polynomials Xh}q/H<  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2~hdJ/  
    &Qda|  
    % Determine the required powers of r: 5'f_~>1Wt  
    % ----------------------------------- &TRKd)wd  
    m_abs = abs(m); <2@t ~ 9  
    rpowers = []; (BtU\f#d  
    for j = 1:length(n) 1J1Jp|j.  
        rpowers = [rpowers m_abs(j):2:n(j)]; P=EZ6<c3&  
    end TJRp/BP  
    rpowers = unique(rpowers); EsWB|V>  
    {@L{l1|0  
    % Pre-compute the values of r raised to the required powers, p' ^}J$  
    % and compile them in a matrix: !QAndg{;D  
    % ----------------------------- z =H?@z  
    if rpowers(1)==0 **__&X p1  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ?MSZO]Q4+  
        rpowern = cat(2,rpowern{:}); d(t)8k$  
        rpowern = [ones(length_r,1) rpowern]; Bn 8&~  
    else vM5I2C3_>!  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); %P1zb7:8  
        rpowern = cat(2,rpowern{:});  dEXhn  
    end 9Oj b~  
    vh"';L_*37  
    % Compute the values of the polynomials: .T8^>z1/\F  
    % -------------------------------------- )x [=}0C  
    y = zeros(length_r,length(n)); l2W+VBn6  
    for j = 1:length(n) VJK4C8]  
        s = 0:(n(j)-m_abs(j))/2; bny@AP(CY+  
        pows = n(j):-2:m_abs(j); Ke@Bf  
        for k = length(s):-1:1 \I i# R  
            p = (1-2*mod(s(k),2))* ... U[;ECw@  
                       prod(2:(n(j)-s(k)))/              ... !-qk1+<h  
                       prod(2:s(k))/                     ... l]DRJ  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... o/ \o -kC}  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ?xKiN5q"6  
            idx = (pows(k)==rpowers); H h](n<Bs  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 3@eI? (N  
        end |1ry*~  
         xF) .S@  
        if isnorm |af<2(d  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); :W&kl UU"  
        end tZ=|1lM  
    end nq7)0F%e  
    % END: Compute the Zernike Polynomials vQXF$/S  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |#*+#27  
    >@4Ds"Ye"O  
    % Compute the Zernike functions: Brg0:5H   
    % ------------------------------ wAR:GO'n  
    idx_pos = m>0; /-<]v3J  
    idx_neg = m<0; nC/T$ #G  
    <5]_u:  
    z = y; Rbm+V{EF&  
    if any(idx_pos) jj ` 0w@  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ,t1s#*j\!q  
    end _mdJIa0D6k  
    if any(idx_neg) )tV]h#4  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); {QK9pZB  
    end 2 (ux  
    v/KTEM  
    % 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) U_B`SS  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Q"xDRQA  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated nzYFa J+  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive CA"`7<,  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, p-g@c wOu  
    %   and THETA is a vector of angles.  R and THETA must have the same /s:akLBaD  
    %   length.  The output Z is a matrix with one column for every P-value, BYS>"  
    %   and one row for every (R,THETA) pair. p|*b] 36  
    % [iSLn3XXRX  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike gY=+G6;=<  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ER$~kFE2yP  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 1 gRR  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 v#IZSBvuQK  
    %   for all p. T5U(B3j_  
    % ]}0+7Q  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 }Ty_ } 6a5  
    %   Zernike functions (order N<=7).  In some disciplines it is .Qj`_q6=  
    %   traditional to label the first 36 functions using a single mode ]@1ncn7N  
    %   number P instead of separate numbers for the order N and azimuthal |om3*]7  
    %   frequency M. LyXABQ]  
    % bL7mlh  
    %   Example: =WP}RZ{S  
    % }fxH>79g  
    %       % Display the first 16 Zernike functions U(qM( E  
    %       x = -1:0.01:1; 0$49X  
    %       [X,Y] = meshgrid(x,x); X4*/h$48 w  
    %       [theta,r] = cart2pol(X,Y); Y!s/uvRI  
    %       idx = r<=1; BQ&h&57K  
    %       p = 0:15; 1-gX=8]]  
    %       z = nan(size(X)); *m]%eU(  
    %       y = zernfun2(p,r(idx),theta(idx)); {gi"ktgk  
    %       figure('Units','normalized') kQBVx8Uq]  
    %       for k = 1:length(p) ]R]X#jm  
    %           z(idx) = y(:,k); eS Fmx  
    %           subplot(4,4,k) IWpUbD|kC  
    %           pcolor(x,x,z), shading interp WCWBvw4&"{  
    %           set(gca,'XTick',[],'YTick',[]) XJOo.Y  
    %           axis square ]X _&  
    %           title(['Z_{' num2str(p(k)) '}']) p|bpE F=U  
    %       end CGg6nCB  
    % )5V1H WjU  
    %   See also ZERNPOL, ZERNFUN. Cw^)}23R  
    d ly 08 74  
    %   Paul Fricker 11/13/2006 C"mb-n 7s  
    #QDV_ziE5  
    %r,2ZLZ  
    % Check and prepare the inputs: (}qLxZ/U  
    % ----------------------------- 1Q;` <=  
    if min(size(p))~=1 & ='uAw  
        error('zernfun2:Pvector','Input P must be vector.') 6ensNr~ea  
    end <*5D0q#~"  
    )*JTxMQ  
    if any(p)>35 \)"qN^we  
        error('zernfun2:P36', ... 1!NaOfP;@  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... #5I "M WA  
               '(P = 0 to 35).']) ,^,J[F  
    end mLYB6   
    lJ,s}l7  
    % Get the order and frequency corresonding to the function number: |Z/ySAFM  
    % ---------------------------------------------------------------- hg>YOf&RG  
    p = p(:); e)bqE^JP  
    n = ceil((-3+sqrt(9+8*p))/2); Ek. j@79  
    m = 2*p - n.*(n+2); V7v,)a" L  
    Bms?`7}N  
    % Pass the inputs to the function ZERNFUN: =f p(hX"  
    % ---------------------------------------- Y{'G2)e  
    switch nargin DpR%s",Q  
        case 3 [(K^x?\Y0'  
            z = zernfun(n,m,r,theta); \ a<Ye T  
        case 4 LMDa68 s  
            z = zernfun(n,m,r,theta,nflag); Q'Tn+}B&  
        otherwise ZqGq%8\.s  
            error('zernfun2:nargin','Incorrect number of inputs.') `'QPe42  
    end  t|DYz#]  
    0&-sz=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) o_5@R+&  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. b6(yyYdF  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of rG]Xgq"   
    %   order N and frequency M, evaluated at R.  N is a vector of Sq'z<}o  
    %   positive integers (including 0), and M is a vector with the V]2z5u_q  
    %   same number of elements as N.  Each element k of M must be a Sm,$~~iq}  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ]=3O,\  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is A.FI] K@  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix +A3 H#'  
    %   with one column for every (N,M) pair, and one row for every )wRD  
    %   element in R. CAA~VEUL  
    % !|/fVWH  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- [`lAc V<  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is vX\e* v  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to uwka 2aSS  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 2o/`8+eJu  
    %   for all [n,m]. +*q@=P,  
    % on\\;V_/Q  
    %   The radial Zernike polynomials are the radial portion of the 2 *$n?  
    %   Zernike functions, which are an orthogonal basis on the unit 5dk,!Cjg  
    %   circle.  The series representation of the radial Zernike `vSsgG  
    %   polynomials is 4@I]PG  
    % v#FUD-Z  
    %          (n-m)/2 /WfxI>v  
    %            __ (Wqhuw!u  
    %    m      \       s                                          n-2s u"jnEKN0y  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r d8#j@='a*  
    %    n      s=0 S8VR#  
    % ,Mc 2dhq  
    %   The following table shows the first 12 polynomials. rWXW}Yg  
    % 0Pk-FSY|f  
    %       n    m    Zernike polynomial    Normalization 3@L%#]xwi  
    %       --------------------------------------------- h~p}08  
    %       0    0    1                        sqrt(2) ?s]`G'=>V`  
    %       1    1    r                           2 =.a ]?&Yyh  
    %       2    0    2*r^2 - 1                sqrt(6) }TW=eu~  
    %       2    2    r^2                      sqrt(6) pS;dvZ  
    %       3    1    3*r^3 - 2*r              sqrt(8) v2{s2kB=  
    %       3    3    r^3                      sqrt(8) ;>9pJ72r  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) t,,^^ll  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) mtHz6+  
    %       4    4    r^4                      sqrt(10) ~~,<+X:  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) `uHpj`EU  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 3)a29uc:U  
    %       5    5    r^5                      sqrt(12) iurB8~Y  
    %       --------------------------------------------- o=QF>\ \  
    % !%r`'|9y  
    %   Example: s_!Z+D$K  
    % :\_MA^<  
    %       % Display three example Zernike radial polynomials KT71%?P  
    %       r = 0:0.01:1; GcW}<g}  
    %       n = [3 2 5]; hJsC \C,^  
    %       m = [1 2 1]; Y;Gm,  
    %       z = zernpol(n,m,r); u0b-JJ7)BQ  
    %       figure S45>f(!  
    %       plot(r,z) j:3Hm0W3  
    %       grid on YWrY{6M  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') {1|7N GQ  
    % %)Dd{|c  
    %   See also ZERNFUN, ZERNFUN2. )fc"])&8  
     q)+ n2FM  
    % A note on the algorithm. lG>e6[Wc  
    % ------------------------ @!3^/D3  
    % The radial Zernike polynomials are computed using the series Gw^=kzh  
    % representation shown in the Help section above. For many special \|` Pul$  
    % functions, direct evaluation using the series representation can :-" jK w  
    % produce poor numerical results (floating point errors), because g8A{aHb1}  
    % the summation often involves computing small differences between >[4|6k|\x  
    % large successive terms in the series. (In such cases, the functions _C=[bI@  
    % are often evaluated using alternative methods such as recurrence iGVb.=)  
    % relations: see the Legendre functions, for example). For the Zernike bCUh^#]x  
    % polynomials, however, this problem does not arise, because the >Eg. c  
    % polynomials are evaluated over the finite domain r = (0,1), and n@[</E(  
    % because the coefficients for a given polynomial are generally all =3dbw8I  
    % of similar magnitude. J&65B./mD9  
    % eAKQR  
    % ZERNPOL has been written using a vectorized implementation: multiple <L/vNP  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] dt&Lwf/  
    % values can be passed as inputs) for a vector of points R.  To achieve ]Wt6V^M'@  
    % this vectorization most efficiently, the algorithm in ZERNPOL ^Jl!WH=20}  
    % involves pre-determining all the powers p of R that are required to .0W4Dp  
    % compute the outputs, and then compiling the {R^p} into a single X3iRR{< @  
    % matrix.  This avoids any redundant computation of the R^p, and 1'm`SRX#e  
    % minimizes the sizes of certain intermediate variables. ef:$1VIBda  
    % '4lT*KN7\  
    %   Paul Fricker 11/13/2006 [k 7N+W8  
    =M{CZm  
    ` +BaDns  
    % Check and prepare the inputs: yi-"hT`  
    % ----------------------------- rRrW   
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) )wzs~Fn/  
        error('zernpol:NMvectors','N and M must be vectors.') |SukiXJZF  
    end 3nC#$L-   
    AYu'ptDNr  
    if length(n)~=length(m) uNZ>oP>  
        error('zernpol:NMlength','N and M must be the same length.') Y1aF._Z  
    end $o$Ev@mi  
    Q[Sd  
    n = n(:); I+j|'=M  
    m = m(:); ) a\DS yr  
    length_n = length(n); ) wo2GF  
    !^bB/e  
    if any(mod(n-m,2)) q '{<c3&  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ],}afa!A  
    end C\^<v&  
    4[CBW  
    if any(m<0) = *;Xc-_  
        error('zernpol:Mpositive','All M must be positive.') 2{]S_. zV  
    end mx(%tz^t  
    %OTA5  
    if any(m>n) Ed0>R<jR9  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') K!D!b'|bb  
    end pM'IQ3N  
    f'`nx;@X  
    if any( r>1 | r<0 ) [g h[F  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') NuS|X   
    end <AAZ8#^  
    -=t3O#  
    if ~any(size(r)==1) )\D40,p  
        error('zernpol:Rvector','R must be a vector.') [T[9*6Kt  
    end w]Ko/;;^2  
    Y^ZBA\D2,k  
    r = r(:); & kjwIg{  
    length_r = length(r); n:^"[Le  
    Fx[A8G  
    if nargin==4 <X I35\^  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); #C,f/PXfaB  
        if ~isnorm jPYe_y  
            error('zernpol:normalization','Unrecognized normalization flag.') t3#H@0<  
        end ZYA.1VrM  
    else sAD P~xvU  
        isnorm = false; |CZnq-,C  
    end B`?N0t%X  
    Y zBA{FE  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [N95.aD  
    % Compute the Zernike Polynomials C/CfjRzd  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BvZ^^IUb  
    @X|i@{<';  
    % Determine the required powers of r: o ,Tr^e$  
    % ----------------------------------- qzH qj;  
    rpowers = []; v<7Gln  
    for j = 1:length(n) B/sBYVU  
        rpowers = [rpowers m(j):2:n(j)]; 3b?OW7H  
    end Mi/ &$" =  
    rpowers = unique(rpowers); $nfBv f  
    kToVBU$  
    % Pre-compute the values of r raised to the required powers, g<rKV+$6  
    % and compile them in a matrix: `Ge+(1x  
    % ----------------------------- )p!*c,  
    if rpowers(1)==0 [C+Gmu  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ;la#Vf:]  
        rpowern = cat(2,rpowern{:}); e\A(#l@g  
        rpowern = [ones(length_r,1) rpowern]; N7~)qqb  
    else 2i7i\?<.  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); i (%tHa37  
        rpowern = cat(2,rpowern{:}); F[7Kw"~J  
    end Yt/SnF  
    U7]<U-.&  
    % Compute the values of the polynomials: 1(%>`=R8  
    % -------------------------------------- [j=,g-EOA  
    z = zeros(length_r,length_n); $@_<$t  
    for j = 1:length_n U50s!Z t45  
        s = 0:(n(j)-m(j))/2; `s>UU- 9  
        pows = n(j):-2:m(j); UKKSc>D1  
        for k = length(s):-1:1 C?w <$DU  
            p = (1-2*mod(s(k),2))* ... q4 Oxs  
                       prod(2:(n(j)-s(k)))/          ... ] ;pf  
                       prod(2:s(k))/                 ... P>{US1t  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... J+}+ "h~.  
                       prod(2:((n(j)+m(j))/2-s(k))); FI1THzW4J  
            idx = (pows(k)==rpowers); 3qAwBVWa  
            z(:,j) = z(:,j) + p*rpowern(:,idx); tIGVB+g{F  
        end R,x>$n  
         yV J dZI  
        if isnorm jF38kj3O7  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); pK6e/eC  
        end IP30y>\  
    end =mHkXHE~:  
    ySwYV  
    % 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)  /9pbnzn  
    3Y`>6A=  
    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。 0 sZwdO  
    Rb#/qkk/  
    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)