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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 5;" $X 1{  
    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!  @9_mk@  
     
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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 ri8=u$!  
    function z = zernfun(n,m,r,theta,nflag) hDB(y4/  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. $%DoLpE>  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 2?q>yL!Gz  
    %   and angular frequency M, evaluated at positions (R,THETA) on the TaYl[I  
    %   unit circle.  N is a vector of positive integers (including 0), and 2yn"K|  
    %   M is a vector with the same number of elements as N.  Each element {v]L|e%{  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) B <r0y  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ef '?O  
    %   and THETA is a vector of angles.  R and THETA must have the same NO[A00m|OL  
    %   length.  The output Z is a matrix with one column for every (N,M) Ro9:kEG$  
    %   pair, and one row for every (R,THETA) pair. Ot-P J i  
    % duEXp]f!  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike |=YK2};  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), T~/>U&k}J  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ohKoX$|p~  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, o5&b'WUJ=  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ZYWGP:Y  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. VNT?  
    % t{iRCj  
    %   The Zernike functions are an orthogonal basis on the unit circle. 2@Yu: |d4U  
    %   They are used in disciplines such as astronomy, optics, and $%bd`d*S  
    %   optometry to describe functions on a circular domain. &t8,326;  
    % Yl&[_ l  
    %   The following table lists the first 15 Zernike functions. 5\h 6"/6Df  
    % G) KI{D  
    %       n    m    Zernike function           Normalization }FS_"0  
    %       -------------------------------------------------- 59 g//;35@  
    %       0    0    1                                 1 bi+M28m  
    %       1    1    r * cos(theta)                    2 ]vf0f,F  
    %       1   -1    r * sin(theta)                    2 t27UlFX  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ,i}EGW,9q  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 2#5Q~  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) QObVJg,GD  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) c]x-mj =  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Z ;rM@x  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) \K\eq>@6  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) :n13v @q  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) kZ@UQ{>`  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) D6@ c|O{Q  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Ey: ?!  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `=hCS0F  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) iYT?6Y|+  
    %       -------------------------------------------------- 0'F/z%SMj  
    % PQl A(v+S  
    %   Example 1: s) s9Z,HY  
    % YFu,<8"swe  
    %       % Display the Zernike function Z(n=5,m=1) In?+  
    %       x = -1:0.01:1; [>dDRsZ  
    %       [X,Y] = meshgrid(x,x); 7P3/Ky@6  
    %       [theta,r] = cart2pol(X,Y); g`'!Vgd?M[  
    %       idx = r<=1; ,}W|cm>  
    %       z = nan(size(X)); <& PU%^Ha  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); x f{`uHa8  
    %       figure B<BS^waU  
    %       pcolor(x,x,z), shading interp &@@PJ!&  
    %       axis square, colorbar 6BA$v-VVU  
    %       title('Zernike function Z_5^1(r,\theta)') g#74c'+  
    % VOr: G85*s  
    %   Example 2: 30WOH 'n  
    % (=u!E+N  
    %       % Display the first 10 Zernike functions &8i$`6wY  
    %       x = -1:0.01:1; t=}]4&Yp  
    %       [X,Y] = meshgrid(x,x); *ilVkV"U  
    %       [theta,r] = cart2pol(X,Y); _/Ve~( "  
    %       idx = r<=1; 3HuocwWbz  
    %       z = nan(size(X)); L7 <30"7  
    %       n = [0  1  1  2  2  2  3  3  3  3]; o9| OL  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; u =L Dfn  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; {_(R?V]w,  
    %       y = zernfun(n,m,r(idx),theta(idx)); TDk[,4  
    %       figure('Units','normalized') P-T@'}lW  
    %       for k = 1:10 ;&|I/MVm  
    %           z(idx) = y(:,k); cz/ E  
    %           subplot(4,7,Nplot(k)) z0\ $# r^I  
    %           pcolor(x,x,z), shading interp 2jhJXM=~  
    %           set(gca,'XTick',[],'YTick',[]) dr"$@  
    %           axis square ?;UR9f|!  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) "[]oWPOj  
    %       end ]Zh$9YK  
    % aO}hE 2]  
    %   See also ZERNPOL, ZERNFUN2. '")'h  
    `'iO+/;GY  
    %   Paul Fricker 11/13/2006 J?#vL\8  
    I__b$  
    0OG 3#pE  
    % Check and prepare the inputs: j|[$P4w}U  
    % ----------------------------- R73@!5N%  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Yg5o!A  
        error('zernfun:NMvectors','N and M must be vectors.') 99:.j=  
    end V!. Y M)B  
    E71H=C 4  
    if length(n)~=length(m) m#[c]v{  
        error('zernfun:NMlength','N and M must be the same length.') 6:}n}q,V  
    end _0u=}tc  
    T}?b,hNl$  
    n = n(:); <f}:YDY'  
    m = m(:); }@ U}c6/  
    if any(mod(n-m,2)) ,ko#z}Z4r,  
        error('zernfun:NMmultiplesof2', ... $;=^|I4E  
              'All N and M must differ by multiples of 2 (including 0).') 1Z_w2D*  
    end C%<Dq0j  
    {I0!q"sF  
    if any(m>n) _-{=Z=?6}  
        error('zernfun:MlessthanN', ... ]QY-L O(  
              'Each M must be less than or equal to its corresponding N.') _?felxG[  
    end WRbdv{ 1E  
    80%"2kG  
    if any( r>1 | r<0 ) 7~1Fy{tc  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 9-{.WZ  
    end 4@F8-V3q4  
    $Sy}im\H  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) N@Ap|`Ei  
        error('zernfun:RTHvector','R and THETA must be vectors.') $aT '~|?  
    end "UY34a^I  
    8f~*T  
    r = r(:); # ^,8JRA  
    theta = theta(:); =s:kC`O  
    length_r = length(r); r&v!2A]:  
    if length_r~=length(theta) P^Og(F8;  
        error('zernfun:RTHlength', ... s H'FqV,)  
              'The number of R- and THETA-values must be equal.') Zd-QZ<c";t  
    end H9BqE+  
    PQF 40g1}  
    % Check normalization: @$p6w  
    % -------------------- h0 %M+g  
    if nargin==5 && ischar(nflag) &l`_D?{<#  
        isnorm = strcmpi(nflag,'norm'); V$$9Rh  
        if ~isnorm Xe`$SNM  
            error('zernfun:normalization','Unrecognized normalization flag.') _a$5"  
        end VJ&-Z |  
    else g=v'[JPd  
        isnorm = false; uJ1oo| sn  
    end *<{hLf  
    K",Xe>  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ESIeZhXVH  
    % Compute the Zernike Polynomials =b)!l9TX  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d{WOO)j  
    Y nTx)uW  
    % Determine the required powers of r: -c0*  
    % ----------------------------------- *fyaAv  
    m_abs = abs(m); 6PWw^Cd  
    rpowers = []; .hf%L1N%F  
    for j = 1:length(n) ]-heG'y]{  
        rpowers = [rpowers m_abs(j):2:n(j)]; 8c%N+E]  
    end K-.%1d@$y  
    rpowers = unique(rpowers); 8 f~M6  
    h6`VU`pPI  
    % Pre-compute the values of r raised to the required powers, |a\,([aU  
    % and compile them in a matrix: 1!+0]_8K  
    % ----------------------------- #w^Ot*{!N  
    if rpowers(1)==0 RWDPsZC  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 3-0jxx(  
        rpowern = cat(2,rpowern{:}); Z]Z&PbP  
        rpowern = [ones(length_r,1) rpowern]; YWANBM(v+  
    else X2np.9hie  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); }LWrtmc  
        rpowern = cat(2,rpowern{:}); Vd) %qw  
    end "x:-#2+h  
    @@!]Raj=  
    % Compute the values of the polynomials: h^{ aG])  
    % -------------------------------------- o/RGzPR  
    y = zeros(length_r,length(n)); PI *Z>VE?  
    for j = 1:length(n) OMjx,@9  
        s = 0:(n(j)-m_abs(j))/2; g'-hSV/@}@  
        pows = n(j):-2:m_abs(j); !.q#X^@>L  
        for k = length(s):-1:1 xTZJ5iZ17  
            p = (1-2*mod(s(k),2))* ... `Y '-2Fv  
                       prod(2:(n(j)-s(k)))/              ... ']X0g{%  
                       prod(2:s(k))/                     ... PIsXX#`7;  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... !0X"^VB  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); !83 N#Y_Mz  
            idx = (pows(k)==rpowers); Us>n`Lj@  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Sn;q:e3i{A  
        end 2:[G4  
         `;Fs  
        if isnorm z5f3T D6,  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi);  )Z:maz  
        end `V[ hE r|  
    end [Fd[(  
    % END: Compute the Zernike Polynomials U!lWP#m  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q6 4bP4K  
    ?/Aql_?3  
    % Compute the Zernike functions: .MxMBrM  
    % ------------------------------ @]],H0  
    idx_pos = m>0; fAT M?  
    idx_neg = m<0; E3_ 5~>  
    DeN$YE#*  
    z = y; 1!ijRr  
    if any(idx_pos) vb\R~%@T,  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); +ldgT"  
    end Xu{S4#1  
    if any(idx_neg) "[ >ql1t{b  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); OZl0I#@A  
    end W%#LHluP  
    0n)UvJ  
    % 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) uo^tND4a;j  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. kc"SUiy/  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated Ktf lbI!  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive )G$0:-J-  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, `S/;S<';  
    %   and THETA is a vector of angles.  R and THETA must have the same gG46hO-M%x  
    %   length.  The output Z is a matrix with one column for every P-value, myWa>Mvb  
    %   and one row for every (R,THETA) pair. >Co5_sCe  
    % '$be+Z32  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 3C;nC?]K  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 0$q)uip  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) nOUF<DNQ  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 E#+|.0*!s  
    %   for all p. ~@ hiLW  
    % RD'i(szi?  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 oyo V1jO  
    %   Zernike functions (order N<=7).  In some disciplines it is :+}Eo9  
    %   traditional to label the first 36 functions using a single mode Ha20g/ UN.  
    %   number P instead of separate numbers for the order N and azimuthal ^y&sKO  
    %   frequency M. #vvQ 1ub  
    % [e`6gGO  
    %   Example: BjCg!6`XF  
    % Z"'tJ3Y.~  
    %       % Display the first 16 Zernike functions ,qO2D_  
    %       x = -1:0.01:1; ;_= +h,n  
    %       [X,Y] = meshgrid(x,x); 8Ir = @  
    %       [theta,r] = cart2pol(X,Y); +`~6Weay  
    %       idx = r<=1; yixAG^<  
    %       p = 0:15; 5KDN8pJN  
    %       z = nan(size(X)); SpX6PwM  
    %       y = zernfun2(p,r(idx),theta(idx)); f^kH[C  
    %       figure('Units','normalized') $n@B:kv5p  
    %       for k = 1:length(p) Lkl ^ `  
    %           z(idx) = y(:,k); :B]yreg  
    %           subplot(4,4,k) K-drN)o  
    %           pcolor(x,x,z), shading interp R3%&\<a)9  
    %           set(gca,'XTick',[],'YTick',[]) |4|j5<5  
    %           axis square d;O4)8 >  
    %           title(['Z_{' num2str(p(k)) '}']) O4fl$egQU  
    %       end ua>YI  
    % mR6hnKa_53  
    %   See also ZERNPOL, ZERNFUN. z1 P=P%F  
    lcYjwA  
    %   Paul Fricker 11/13/2006 dw]jF=u  
    6E@qZvQ  
    *3]_Huw<  
    % Check and prepare the inputs: N.@@ebuE  
    % ----------------------------- <m X EX`?  
    if min(size(p))~=1 rGb<7b%  
        error('zernfun2:Pvector','Input P must be vector.') B(h%>mT[  
    end 2Bg0 M  
    A 2Rp  
    if any(p)>35 C4^o= 6{  
        error('zernfun2:P36', ... !omf>CW;ud  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... #:LI,t  
               '(P = 0 to 35).']) Yb;$z'  
    end c}r"O8M  
    #cy;((zuB  
    % Get the order and frequency corresonding to the function number: 5isqBu  
    % ---------------------------------------------------------------- T.?}iz=ZEq  
    p = p(:); Ty;P`Uv]r  
    n = ceil((-3+sqrt(9+8*p))/2); q aZQ1<e  
    m = 2*p - n.*(n+2); YecV+ K'p:  
    A{Dy3tm=  
    % Pass the inputs to the function ZERNFUN: Ny2. C?2  
    % ---------------------------------------- @m+2e C77  
    switch nargin @[. 0,  
        case 3 0l+[[ZTV  
            z = zernfun(n,m,r,theta); @K=C`N_22  
        case 4 Cu&y',ee~  
            z = zernfun(n,m,r,theta,nflag); vA&MJD{  
        otherwise e-Ma8+X\  
            error('zernfun2:nargin','Incorrect number of inputs.') BH\!yxK  
    end h1REL^!c  
    "cDMFu  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) x|`BF%e/v  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. e82xBLxR%  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of )0?u_Z]w9  
    %   order N and frequency M, evaluated at R.  N is a vector of _?v&\j  
    %   positive integers (including 0), and M is a vector with the W:8pmI  
    %   same number of elements as N.  Each element k of M must be a ex6 QHUQ  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ]8f$&gw&A  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is {R8)DK  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Z;~7L*|  
    %   with one column for every (N,M) pair, and one row for every \=uD)9 V  
    %   element in R. pS+hE4D  
    % QWwdtk  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- TpcJ1*t  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is N$N 7aE$  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 9";qR,  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 N"8'=wB  
    %   for all [n,m]. _E2W%N  
    % # 1 1<=3Yj  
    %   The radial Zernike polynomials are the radial portion of the M$s9   
    %   Zernike functions, which are an orthogonal basis on the unit s"5wnp6pW  
    %   circle.  The series representation of the radial Zernike V 5D8z  
    %   polynomials is IoZ _zz0  
    % ~JHEr48  
    %          (n-m)/2 bT15jNa  
    %            __ >|aVGY  
    %    m      \       s                                          n-2s 0+T:};]  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 089v; d 6  
    %    n      s=0 UM2yv6:/  
    % ~9Qd83`UH  
    %   The following table shows the first 12 polynomials. N\anjG  
    % 2Mu@P8O&  
    %       n    m    Zernike polynomial    Normalization 'x6rU"e$J  
    %       --------------------------------------------- ipyc(u6Z5  
    %       0    0    1                        sqrt(2) SP"t2LTP  
    %       1    1    r                           2 @,m 7%,  
    %       2    0    2*r^2 - 1                sqrt(6) XhUVDmeUMb  
    %       2    2    r^2                      sqrt(6) 9[R+m3V/`  
    %       3    1    3*r^3 - 2*r              sqrt(8) dU-nE5  
    %       3    3    r^3                      sqrt(8) Vsr"W@k_  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) p<+Y;,+  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) OwPXQ 3S  
    %       4    4    r^4                      sqrt(10) Nq1YFI>W  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) yz"hU  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) k}C4:?AT  
    %       5    5    r^5                      sqrt(12) mt~E&Z(A  
    %       --------------------------------------------- <99/7>#  
    % a4n5i.;  
    %   Example: p8FXlTk  
    % (TU/EU5  
    %       % Display three example Zernike radial polynomials oqo7Ge2  
    %       r = 0:0.01:1; ~G1B}c]  
    %       n = [3 2 5]; <G'M/IR a  
    %       m = [1 2 1]; Z*Rg ik  
    %       z = zernpol(n,m,r); Xl:.`{5L  
    %       figure qh+&Zx~  
    %       plot(r,z) ]FgKL0  
    %       grid on ;iW>i8  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 1Tr%lO5?6  
    % Ym.{ {^=  
    %   See also ZERNFUN, ZERNFUN2. "T*1C=  
    gVrfZ&XF84  
    % A note on the algorithm. h_]*|[g  
    % ------------------------ Y<V$3h  
    % The radial Zernike polynomials are computed using the series kj6H+@ {  
    % representation shown in the Help section above. For many special G[6i\Et   
    % functions, direct evaluation using the series representation can T;]Ob3(BpW  
    % produce poor numerical results (floating point errors), because p[ &b@U#  
    % the summation often involves computing small differences between a?xZsR  
    % large successive terms in the series. (In such cases, the functions n5z|@I`S_  
    % are often evaluated using alternative methods such as recurrence F]fXS-@ c  
    % relations: see the Legendre functions, for example). For the Zernike 34Cnbtq^  
    % polynomials, however, this problem does not arise, because the j#xGB]  
    % polynomials are evaluated over the finite domain r = (0,1), and vCXmu_S4^>  
    % because the coefficients for a given polynomial are generally all WZTAXOw  
    % of similar magnitude. 'rTJ*1i  
    % "l hj1zZ  
    % ZERNPOL has been written using a vectorized implementation: multiple fjy7gC2  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] x4(8 =&Z  
    % values can be passed as inputs) for a vector of points R.  To achieve *(qj!U43  
    % this vectorization most efficiently, the algorithm in ZERNPOL B3pjli  
    % involves pre-determining all the powers p of R that are required to @AM11v\:  
    % compute the outputs, and then compiling the {R^p} into a single ahQY-%>  
    % matrix.  This avoids any redundant computation of the R^p, and O8cZl1C3  
    % minimizes the sizes of certain intermediate variables. Ud7Z7?Ym  
    % kBxEp/y  
    %   Paul Fricker 11/13/2006 q!W=U8`  
    8]oolA:^4s  
    IMBjI#\  
    % Check and prepare the inputs: wa~zb!y<  
    % ----------------------------- R:3=!zav  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) {> <1K6t  
        error('zernpol:NMvectors','N and M must be vectors.') z~==7:Os  
    end 5S, Kq35$(  
    y/:%S2za>  
    if length(n)~=length(m) C"$~w3A k  
        error('zernpol:NMlength','N and M must be the same length.') vCNq2l^CW  
    end I~^Xw7  
    xcn~KF8  
    n = n(:); >rJ**y  
    m = m(:); EeT 69o  
    length_n = length(n); H%etYpD  
    TZ `Ypi7r  
    if any(mod(n-m,2)) `6lOqH  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') seRf q&  
    end ' Ttsscv  
    n#}~/\P6  
    if any(m<0) ~( 0bqt3c  
        error('zernpol:Mpositive','All M must be positive.') 1X7GM65#  
    end Srz8sm;  
    mRm}7p  
    if any(m>n) .9WOT ti  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') NbTaI{r  
    end -FI)o`AE  
    y:^o ._  
    if any( r>1 | r<0 ) r>7 +&s*yk  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') %l14K_  
    end *^Ges;5 $"  
    /-i m g^^  
    if ~any(size(r)==1) !icI Rqcf=  
        error('zernpol:Rvector','R must be a vector.') 2K{'F1"RM  
    end _ E-\aS{  
    \HkBp& bqK  
    r = r(:);  @;$cX2  
    length_r = length(r); rsLkH&aM  
    9P)!v.,T/  
    if nargin==4 +RJKJ:W  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); EI7n|X a1q  
        if ~isnorm x;$ESPPg  
            error('zernpol:normalization','Unrecognized normalization flag.') _P!b0x~\  
        end :o8|P  
    else RgUQ:  
        isnorm = false; a/J Mg   
    end `/`iLso& -  
    }Hq3]LVE  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6W{Nw<  
    % Compute the Zernike Polynomials (,jsZ!sl  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% m;\nMdn  
    !=PH5jTY  
    % Determine the required powers of r: o $W@@aM  
    % ----------------------------------- 4w=v /WDo  
    rpowers = []; F6111Q </  
    for j = 1:length(n) :aomDK*  
        rpowers = [rpowers m(j):2:n(j)]; ZO cpF1y  
    end yYYP;N?g4k  
    rpowers = unique(rpowers); WeaT42*Q{  
    9#:fQ!3`  
    % Pre-compute the values of r raised to the required powers, nW"O+s3  
    % and compile them in a matrix: O ylUuYy~j  
    % ----------------------------- ]Idwy|eG  
    if rpowers(1)==0 HcJ!(  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 2uN3:_w  
        rpowern = cat(2,rpowern{:}); A[^#8evaK  
        rpowern = [ones(length_r,1) rpowern]; wK7w[Xt  
    else XHj%U  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ,Yx<"2 W  
        rpowern = cat(2,rpowern{:}); 0C> _aj  
    end U5wh( vi  
    }2LWDQ;po  
    % Compute the values of the polynomials: gaz",kK<  
    % -------------------------------------- %J9u?-~  
    z = zeros(length_r,length_n); UkpTK8>&  
    for j = 1:length_n .\T!oSb4[  
        s = 0:(n(j)-m(j))/2; q[W6I9  
        pows = n(j):-2:m(j); e)B1)c8s  
        for k = length(s):-1:1 8 yi#] 5`Q  
            p = (1-2*mod(s(k),2))* ... 2ua!<^,  
                       prod(2:(n(j)-s(k)))/          ... *EvW: <  
                       prod(2:s(k))/                 ... .Z'NH wCy  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ,$ /Ld76U  
                       prod(2:((n(j)+m(j))/2-s(k))); jT$J~M pHh  
            idx = (pows(k)==rpowers); p7-\a1P3  
            z(:,j) = z(:,j) + p*rpowern(:,idx); JxwKTFU'3O  
        end ^.iRU'{  
         %a WRXW@c  
        if isnorm <=GZm}/]N  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 1uN;JN `_  
        end AVw oOv J  
    end .O'~s/h  
    }Vob)r{R@  
    % 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)  qhTVsZ:{C  
    /DHV-L  
    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。 h,&{m*q&  
    m*B4a9 f  
    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)