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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 d3W0-INL  
    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! +V{7")px6  
     
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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 /s-A?lw^2  
    function z = zernfun(n,m,r,theta,nflag) #U*_1P0h  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. PG8^.)]M  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N =1P6Vk  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 6Z`R#d #I  
    %   unit circle.  N is a vector of positive integers (including 0), and y$3;$ R^  
    %   M is a vector with the same number of elements as N.  Each element .`7cBsXH  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ,ZQZ}`x(  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 0QvT   
    %   and THETA is a vector of angles.  R and THETA must have the same 0W3i()  
    %   length.  The output Z is a matrix with one column for every (N,M) i 9g>9  
    %   pair, and one row for every (R,THETA) pair. 4Q IE8f Y  
    % >Bs#Xb_B]  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike R/^u/~<  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), pGSai &  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral  49d@!  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, |A%<Z(  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized }gkM^*$:%  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. \`, [)`  
    % vsL[*OeI  
    %   The Zernike functions are an orthogonal basis on the unit circle. wBQF~WY  
    %   They are used in disciplines such as astronomy, optics, and &QG6!`fK}3  
    %   optometry to describe functions on a circular domain. q %0Cg=  
    % G60R9y47c  
    %   The following table lists the first 15 Zernike functions. Iyd?|f"  
    % '+ xu#R  
    %       n    m    Zernike function           Normalization t8+_/BXv  
    %       -------------------------------------------------- ,-+"^>  
    %       0    0    1                                 1 OEPa|rb  
    %       1    1    r * cos(theta)                    2 `xiCm':  
    %       1   -1    r * sin(theta)                    2 GabYfUkO  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) PyA&ZkX>  
    %       2    0    (2*r^2 - 1)                    sqrt(3) (~$/$%b  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) q~L^au8  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) aF|d^  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) <xJ/y|{  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) v+e|o:o#  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) dq IlD!  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) eUl/o1~mXa  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) n6(i`{i  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) x f4{r+  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) kAM1TWbaVQ  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) DMF -Y-h  
    %       -------------------------------------------------- 9s}Kl($  
    % |0{u->+ )  
    %   Example 1: {k5X*W  
    % XhdSFxW}  
    %       % Display the Zernike function Z(n=5,m=1) :K?0e `  
    %       x = -1:0.01:1; +,50q N:%[  
    %       [X,Y] = meshgrid(x,x); `.#@@5e  
    %       [theta,r] = cart2pol(X,Y); ds[QwcV9-  
    %       idx = r<=1; .Hc(y7HV  
    %       z = nan(size(X)); hh~n#7w~IR  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); O+=vEp(  
    %       figure H0a/(4/xg  
    %       pcolor(x,x,z), shading interp i)Lp7m z  
    %       axis square, colorbar O[9-:,B{w  
    %       title('Zernike function Z_5^1(r,\theta)') :Vg}V"QR  
    % x90jw$\%7  
    %   Example 2: uhV0J97  
    % )'Wb&A'  
    %       % Display the first 10 Zernike functions  ==/n(LBD  
    %       x = -1:0.01:1; < Fs-3(V+\  
    %       [X,Y] = meshgrid(x,x); 9kKnAf4Z  
    %       [theta,r] = cart2pol(X,Y);  n i  
    %       idx = r<=1; IMQ]1uq0$  
    %       z = nan(size(X)); [oc~iDx%W  
    %       n = [0  1  1  2  2  2  3  3  3  3]; `\<37E\N}  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Je4Z(kj 0  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; gM>=%/.  
    %       y = zernfun(n,m,r(idx),theta(idx)); v&g0ta@  
    %       figure('Units','normalized') 'mdMq=VI  
    %       for k = 1:10 'f/Lv@]a  
    %           z(idx) = y(:,k); ql5x2n  
    %           subplot(4,7,Nplot(k)) W[NEe,.>  
    %           pcolor(x,x,z), shading interp ?IX!+>.H  
    %           set(gca,'XTick',[],'YTick',[]) ZX b}91rzt  
    %           axis square R*1kR|*_)  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) j1Yq5`ia  
    %       end ,]Zp+>{  
    % Aox3s?  
    %   See also ZERNPOL, ZERNFUN2. y?30_#[dN  
    ,/&Zw01dGN  
    %   Paul Fricker 11/13/2006 :^C'<SY2Gs  
    ,6<"  
    h5|.Et  
    % Check and prepare the inputs: -%IcYzyA  
    % ----------------------------- kvsA]tK.  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) FM^9}*  
        error('zernfun:NMvectors','N and M must be vectors.') Gie@JX  
    end Y9r3XhVI  
    $2z _{@Z  
    if length(n)~=length(m) ~3WL)%  
        error('zernfun:NMlength','N and M must be the same length.') ED![^=  
    end NWmtwS+@  
    *QE<zt  
    n = n(:); yno('1B@  
    m = m(:); <o:@dS  
    if any(mod(n-m,2)) 9w;?-  
        error('zernfun:NMmultiplesof2', ... T bE:||r?^  
              'All N and M must differ by multiples of 2 (including 0).') dc0@Y  
    end H!IDV }dn  
    d<o.o?Vc  
    if any(m>n) f1{z~i9@$  
        error('zernfun:MlessthanN', ... n l/UdgI  
              'Each M must be less than or equal to its corresponding N.') Yq'4e[i  
    end s<T?pH  
    h.tY 'F  
    if any( r>1 | r<0 )  5K56!*Y  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') #] KgUc5B  
    end p5 ]_}I`+2  
    eE:&qy^  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) e(\I_  
        error('zernfun:RTHvector','R and THETA must be vectors.') ;q#]-^  
    end V+B71\x<  
    b^V'BC3  
    r = r(:); "-i#BjZl/  
    theta = theta(:); %l9$a`&  
    length_r = length(r); A[/I#Im7  
    if length_r~=length(theta) p6 xPheD  
        error('zernfun:RTHlength', ... EZr6oO@Nc  
              'The number of R- and THETA-values must be equal.') Z>A{i?#m  
    end 2:v<qX  
    |KG&HN fP-  
    % Check normalization: y8s=\`~PR  
    % -------------------- LPE)  
    if nargin==5 && ischar(nflag) i Q`]ms+  
        isnorm = strcmpi(nflag,'norm'); k 'zat3#f  
        if ~isnorm a5wDm  
            error('zernfun:normalization','Unrecognized normalization flag.') !s IwFv )  
        end ;El <%{(  
    else +g\;bLT  
        isnorm = false; OeTu?d&N  
    end h W.2p+  
    Gbb \h  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% VWvoQf^+  
    % Compute the Zernike Polynomials LdWc X`K  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F1u)i  
    5:gj&jt;)7  
    % Determine the required powers of r: PW[6/7  
    % ----------------------------------- YF[$Q=7.  
    m_abs = abs(m); !$kR ;Q"/  
    rpowers = []; R^{xwI  
    for j = 1:length(n) dtW0\^ .L  
        rpowers = [rpowers m_abs(j):2:n(j)]; ToU.mM?f^  
    end ~iTxv_\=6u  
    rpowers = unique(rpowers); F' BdQk3o  
    :EB,{|m  
    % Pre-compute the values of r raised to the required powers, )/%S=c  
    % and compile them in a matrix: ~mA7pOHj  
    % ----------------------------- do' ORcZ  
    if rpowers(1)==0 s-6:N9-  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); k^*$^;z  
        rpowern = cat(2,rpowern{:}); YBylyVZ  
        rpowern = [ones(length_r,1) rpowern]; {%7<"  
    else _t.FL@3e  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); A'g,:8Ou  
        rpowern = cat(2,rpowern{:}); SfDQ;1?  
    end OOLe[P3J3  
    5b fb!7-[i  
    % Compute the values of the polynomials: nEVbfNo0  
    % -------------------------------------- 84Zgo=P}  
    y = zeros(length_r,length(n)); uC[d%v`  
    for j = 1:length(n) /co%:}ln  
        s = 0:(n(j)-m_abs(j))/2; )>$^wT  
        pows = n(j):-2:m_abs(j); dpn3 (  
        for k = length(s):-1:1 `vEqj v  
            p = (1-2*mod(s(k),2))* ... Hja^edLj  
                       prod(2:(n(j)-s(k)))/              ... !aeNq82  
                       prod(2:s(k))/                     ... j |td,82.  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... B/_6Ieb+  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); C1ZyB"{  
            idx = (pows(k)==rpowers); b7v dk  
            y(:,j) = y(:,j) + p*rpowern(:,idx); %BICt @E  
        end H5p5S\g-)  
         DPeVKyjU  
        if isnorm '>]&rb09|  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); R-C5*$  
        end bX&e_Pd  
    end X 7&U3v  
    % END: Compute the Zernike Polynomials u*k*yWdr  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LaT8l?q q  
    -pX|U~a[  
    % Compute the Zernike functions: x\]z j!  
    % ------------------------------ T^NJ4L4#  
    idx_pos = m>0; 9<Ag1l  
    idx_neg = m<0; TK %< a/  
    id4]|jb  
    z = y; -fQX4'3R  
    if any(idx_pos) 3.~h6r5-  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); g z`*|h  
    end )-)pYRlO  
    if any(idx_neg) #{~7G%GPY5  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 9b%|^ .B  
    end qxSs ~Qc  
    wO!% q[  
    % 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) <B=[hk!  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. IuDT=A  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated )*Vj3Jx  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive p&QmIX]BZ  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, $)l2G;&  
    %   and THETA is a vector of angles.  R and THETA must have the same f!EOYowW  
    %   length.  The output Z is a matrix with one column for every P-value, i<bxc  
    %   and one row for every (R,THETA) pair. YV4 : 8At1  
    % ~@?"' !U  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Z$1.^H.Db  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) x g{VP7  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ,5=kDw2  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 *^ey]),f54  
    %   for all p.  c,.0d  
    % i<J^:7  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 !2#\| NJk  
    %   Zernike functions (order N<=7).  In some disciplines it is 'Z59<Ya&x  
    %   traditional to label the first 36 functions using a single mode 98h :X%  
    %   number P instead of separate numbers for the order N and azimuthal &+,:u*%  
    %   frequency M. T^d#hl.U  
    % G I&qwA  
    %   Example: CH55K[{<  
    % wG;}TxrLS  
    %       % Display the first 16 Zernike functions WI0QLR'  
    %       x = -1:0.01:1; I7A7X*  
    %       [X,Y] = meshgrid(x,x); %-\FVKX  
    %       [theta,r] = cart2pol(X,Y); }+*w.X}L  
    %       idx = r<=1; 4v0dd p  
    %       p = 0:15; +jv }\Jt  
    %       z = nan(size(X)); L,E-z_<p  
    %       y = zernfun2(p,r(idx),theta(idx)); `S5>0r5[  
    %       figure('Units','normalized') 9) jo7,VM  
    %       for k = 1:length(p) W;q#ZD(;  
    %           z(idx) = y(:,k); 8I<_w4fC  
    %           subplot(4,4,k) =jsx (3V   
    %           pcolor(x,x,z), shading interp Av@& hD\  
    %           set(gca,'XTick',[],'YTick',[]) tN[St  
    %           axis square yB 1I53E  
    %           title(['Z_{' num2str(p(k)) '}'])  ^8b~ZX  
    %       end sWp{Y.  
    % >-UD]?>  
    %   See also ZERNPOL, ZERNFUN. =VCi8jDkP  
    1d=0q?nH  
    %   Paul Fricker 11/13/2006 PG\\V$}A(  
    dko[  
    UIO6|*ka  
    % Check and prepare the inputs: 7qW.h>%WE  
    % ----------------------------- Gs^(YGtU  
    if min(size(p))~=1 _e;N'DZ  
        error('zernfun2:Pvector','Input P must be vector.') H<v c\r  
    end rat=)n)"t  
    32M6EEmPG  
    if any(p)>35 JJ^iy*v  
        error('zernfun2:P36', ... Ngn\nkf  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... eL>K2Jxq  
               '(P = 0 to 35).']) j7QBU  
    end z\X60T  
    w g$D@E7  
    % Get the order and frequency corresonding to the function number: H4P\hOK7r  
    % ---------------------------------------------------------------- Q,OkO?uY  
    p = p(:); ['/;'NhdlY  
    n = ceil((-3+sqrt(9+8*p))/2); ;+S2h-4  
    m = 2*p - n.*(n+2); ON:LPf>"-  
    tpN]evp|  
    % Pass the inputs to the function ZERNFUN: =4YbVA+(  
    % ---------------------------------------- #N<s^KYG-  
    switch nargin _bN))9 3  
        case 3 ~5-~q0Ge  
            z = zernfun(n,m,r,theta); P&qy.0  
        case 4 `=_7I?  
            z = zernfun(n,m,r,theta,nflag); }`+O$0A  
        otherwise *uk \O]  
            error('zernfun2:nargin','Incorrect number of inputs.') ]hCWe0F  
    end !X \Sp}  
    4v rm&k  
    % 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) d +*T@k]>M  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. $i1>?pb3  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Xm_Ub>N5  
    %   order N and frequency M, evaluated at R.  N is a vector of <8F->k1"3  
    %   positive integers (including 0), and M is a vector with the _*OaiEL+:  
    %   same number of elements as N.  Each element k of M must be a Zq2H9^![y~  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) hr/xpQW  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is $6 Q2)^LJ  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix DYy@t^sC  
    %   with one column for every (N,M) pair, and one row for every #R"9)vHp  
    %   element in R. jk WBw.(  
    % ~|$) 1  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- Rx_,J%0Fq  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is bi4^ zaCEE  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to J8`1V `$  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 m\R@.jkZ  
    %   for all [n,m]. ;&7qw69k  
    % <uWJ>sg^ 6  
    %   The radial Zernike polynomials are the radial portion of the (AyRs7Dkn  
    %   Zernike functions, which are an orthogonal basis on the unit YQfZiz}Fv  
    %   circle.  The series representation of the radial Zernike r`mzsO-'  
    %   polynomials is o$qFa9|Ec?  
    % .q'FSEkMJ  
    %          (n-m)/2 &L[8Mju6  
    %            __ v]d?6g  
    %    m      \       s                                          n-2s t&p:vXF2  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r f6/\JVi)-  
    %    n      s=0 >E 2WZHzd2  
    % T^f&58{ 7  
    %   The following table shows the first 12 polynomials. T8M[eSbZ  
    % Ay[9k=q]  
    %       n    m    Zernike polynomial    Normalization lN]X2 4t  
    %       --------------------------------------------- E`sapk  
    %       0    0    1                        sqrt(2) 0lJBtk9wn  
    %       1    1    r                           2 \sXm Mc  
    %       2    0    2*r^2 - 1                sqrt(6) <iY 9cV|}3  
    %       2    2    r^2                      sqrt(6) S quqaX+<  
    %       3    1    3*r^3 - 2*r              sqrt(8) }gi`?58J6  
    %       3    3    r^3                      sqrt(8) = -a?oH-  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) B=nx8s  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ./5MsHfbxt  
    %       4    4    r^4                      sqrt(10) aV92.Z_Ku  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) @% 5F^Vbd  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 7u&H*e7  
    %       5    5    r^5                      sqrt(12) 3`C3+  
    %       --------------------------------------------- iaq:5||,  
    % R.n`R|NOd  
    %   Example: aX%g+6t2  
    % rqG6Ll`=+  
    %       % Display three example Zernike radial polynomials )I0g&e^Tzy  
    %       r = 0:0.01:1; T J"{nB  
    %       n = [3 2 5]; B1AF4}~5  
    %       m = [1 2 1]; 0$dY;,Q.  
    %       z = zernpol(n,m,r); X#\P.$  
    %       figure g]hn@{[  
    %       plot(r,z) W1M/Z[h6)5  
    %       grid on BRQ5  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 9"l%tq_  
    % s t3]Yy  
    %   See also ZERNFUN, ZERNFUN2. C8%nBa /  
    L:];[xa%  
    % A note on the algorithm. #IciNCIrG  
    % ------------------------ {Ac3/UM/  
    % The radial Zernike polynomials are computed using the series s9)8{z  
    % representation shown in the Help section above. For many special tZ6v@W  
    % functions, direct evaluation using the series representation can BtDgv.;GH  
    % produce poor numerical results (floating point errors), because l=.InSuLT  
    % the summation often involves computing small differences between 9R<J$e  
    % large successive terms in the series. (In such cases, the functions [E&"9%K  
    % are often evaluated using alternative methods such as recurrence )}k`X<~k  
    % relations: see the Legendre functions, for example). For the Zernike vN[m5)aT  
    % polynomials, however, this problem does not arise, because the uc `rt"  
    % polynomials are evaluated over the finite domain r = (0,1), and cVt$#A)  
    % because the coefficients for a given polynomial are generally all [+5SEr}  
    % of similar magnitude.  6-E4)0\  
    % 8CHf.SXh  
    % ZERNPOL has been written using a vectorized implementation: multiple JZ*?1S>  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] )I1V 2k$n  
    % values can be passed as inputs) for a vector of points R.  To achieve vskM;  
    % this vectorization most efficiently, the algorithm in ZERNPOL B?>#cpW j  
    % involves pre-determining all the powers p of R that are required to 7 5cr!+  
    % compute the outputs, and then compiling the {R^p} into a single ` G/QJH{I  
    % matrix.  This avoids any redundant computation of the R^p, and t3kh]2t  
    % minimizes the sizes of certain intermediate variables. L&F\"q9q71  
    % kKTED1MW&W  
    %   Paul Fricker 11/13/2006 Sr-^faL  
    SF}L3/C&h  
    SX<>6vH&  
    % Check and prepare the inputs: @ F"ShT0  
    % ----------------------------- (:hmp"S  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) a}8>(jtSt  
        error('zernpol:NMvectors','N and M must be vectors.') uY#58?>'j  
    end R yIaT  
    v nT  
    if length(n)~=length(m) `rFGSq$9  
        error('zernpol:NMlength','N and M must be the same length.') +/ d8d  
    end l<:)rg^,  
    TEy.zzt  
    n = n(:); S| ?--vai_  
    m = m(:); GAe_Z( T  
    length_n = length(n); +R jD\6bJb  
    [75e\=wK  
    if any(mod(n-m,2)) k{$"-3ed  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') p~ VW3u]  
    end Q? |MBTo  
    5s2}nIe  
    if any(m<0) Y  .X-8  
        error('zernpol:Mpositive','All M must be positive.') {.[,ee-)9  
    end ]$%4;o4O  
    n!,TBCNX  
    if any(m>n) @G&xq "Fg7  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 2G*#Czr"  
    end x6mq['_  
    Qpu2RfP  
    if any( r>1 | r<0 ) ;-wPXXR  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') >TVd*S  
    end ;&:Et  
    H_^u_ %:e  
    if ~any(size(r)==1) mLdyt-1  
        error('zernpol:Rvector','R must be a vector.') `'t;BXedz/  
    end JH,fg K+[  
    gG?*Fi  
    r = r(:); G(,~{N||  
    length_r = length(r); ~ZIRCTQ"  
    im%3*bv-  
    if nargin==4 4E94W,1%,Y  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); n]g,)m  
        if ~isnorm W- B[_  
            error('zernpol:normalization','Unrecognized normalization flag.') |I<-x)joIK  
        end \?I wR]@y  
    else gDBQ\vM8  
        isnorm = false; tUnVdh6L.B  
    end 69w"$V k  
    q.Mck9R7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +VFwYdW,  
    % Compute the Zernike Polynomials ^vz@d+\Kd  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _ o3}Ly}  
    QAw,XZ.K^  
    % Determine the required powers of r: *g}&&$b0  
    % ----------------------------------- CzbNG^+  
    rpowers = []; :cz]8~i\  
    for j = 1:length(n) j ZafwBi  
        rpowers = [rpowers m(j):2:n(j)]; `h]f(  
    end 1Ac1CsK*  
    rpowers = unique(rpowers); Z*B(L@H  
    I8:&Btf  
    % Pre-compute the values of r raised to the required powers, VAzJclB  
    % and compile them in a matrix: |ZzBCL8q  
    % ----------------------------- d+)L\ `4  
    if rpowers(1)==0 _ h-X-s Y  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); W.<<azi  
        rpowern = cat(2,rpowern{:}); ^!tI+F{n{  
        rpowern = [ones(length_r,1) rpowern]; o\gQYi   
    else QQcj"s  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); m^+ ~pC5  
        rpowern = cat(2,rpowern{:}); pdu  
    end w\4m -Z{  
    MPc=cLv  
    % Compute the values of the polynomials: =hlu, By  
    % -------------------------------------- &d;$k  
    z = zeros(length_r,length_n); a06q-3zw  
    for j = 1:length_n xUoY|$fI  
        s = 0:(n(j)-m(j))/2; c^k. <EA  
        pows = n(j):-2:m(j); <rF  
        for k = length(s):-1:1 3I(M<sB}  
            p = (1-2*mod(s(k),2))* ... oPBjsQ  
                       prod(2:(n(j)-s(k)))/          ... *RM'0[1F4  
                       prod(2:s(k))/                 ... ]vErF=[U,  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... &o.SmkJI  
                       prod(2:((n(j)+m(j))/2-s(k))); {xH@8T$DX  
            idx = (pows(k)==rpowers); :Aw VeX@  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ARo5 Ss{  
        end Zv93cv  
         -GjJrYOU  
        if isnorm Awh)@iTL  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); P!EX;+7+x  
        end sxtGl^,mU:  
    end `Mg "!n`  
    U4JN,`p{  
    % 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)  F?UL0Q|uv  
    CTwP{[%Pk  
    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。 :7R\"@V4  
    ?}W:DGudZ  
    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)