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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 #]E(N~  
    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! tnz+bX26  
     
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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 7.Kc:7  
    function z = zernfun(n,m,r,theta,nflag) 23!;}zHp  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. X2|Y  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N nH|,T%  
    %   and angular frequency M, evaluated at positions (R,THETA) on the D*PYr{z'  
    %   unit circle.  N is a vector of positive integers (including 0), and qZv =  
    %   M is a vector with the same number of elements as N.  Each element +rXF{@ l  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) !7bw5H  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, pd[ncL  
    %   and THETA is a vector of angles.  R and THETA must have the same V'Kgdj  
    %   length.  The output Z is a matrix with one column for every (N,M) )D&M2CUw"f  
    %   pair, and one row for every (R,THETA) pair. AK!hK>u`  
    % =sJ _yq0#R  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike x%+{VStA  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), epHJ@W@#  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral j@gMb iu  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, "syh=BC v  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized g7V8D  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?>c=}I#Ui-  
    % F>je4S;  
    %   The Zernike functions are an orthogonal basis on the unit circle. tR0pH8?e"  
    %   They are used in disciplines such as astronomy, optics, and H5CR'Rp  
    %   optometry to describe functions on a circular domain. dy__e^qi  
    % _@mRb^  
    %   The following table lists the first 15 Zernike functions. )tHaB,  
    % J7D}%  
    %       n    m    Zernike function           Normalization cJo\#cr  
    %       -------------------------------------------------- OO dSKf8  
    %       0    0    1                                 1 >_dx_<75&  
    %       1    1    r * cos(theta)                    2 ?3ig)J,e[  
    %       1   -1    r * sin(theta)                    2 E/&Rb*3  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 9E2j!  
    %       2    0    (2*r^2 - 1)                    sqrt(3) >(w2GD?  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 4/kv3rv  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ?bZovRx  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) p(;U@3G  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) {rfF'@[  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 2kAx>R  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) YJg,B\z}  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) GZS1zTwBL  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) h&.wo !  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) &AVpLf:?  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) X"gCR n%tn  
    %       -------------------------------------------------- /+*#pDx/zW  
    % Z/x*Y#0@n  
    %   Example 1: " 96yp4v@  
    % W?yd#j  
    %       % Display the Zernike function Z(n=5,m=1) ^-mRP\5  
    %       x = -1:0.01:1; ah @uUHB  
    %       [X,Y] = meshgrid(x,x); a?|vQ*W  
    %       [theta,r] = cart2pol(X,Y);  ~>O)  
    %       idx = r<=1; UJ-?k &j,  
    %       z = nan(size(X)); WW+l'6.  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); nJ4@I7Sk;  
    %       figure 5D M"0  
    %       pcolor(x,x,z), shading interp T=hho Gn  
    %       axis square, colorbar 7Dnp'*H  
    %       title('Zernike function Z_5^1(r,\theta)') &l$Q^g  
    % |qZko[W}=  
    %   Example 2: }$MN|s  
    % +3s%E{  
    %       % Display the first 10 Zernike functions M8H5K  
    %       x = -1:0.01:1; JN^ &S  
    %       [X,Y] = meshgrid(x,x); j!7`]  
    %       [theta,r] = cart2pol(X,Y); PH"hn]  
    %       idx = r<=1; (feTk72XX  
    %       z = nan(size(X)); &g2 Eptx#  
    %       n = [0  1  1  2  2  2  3  3  3  3]; !fBF|*/  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; p!]6ll^  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; hcVJBK  
    %       y = zernfun(n,m,r(idx),theta(idx)); i)#:qAtP*  
    %       figure('Units','normalized') $^u}a   
    %       for k = 1:10 ,q(&)L$S  
    %           z(idx) = y(:,k); ycFio ,  
    %           subplot(4,7,Nplot(k)) V8eB$in  
    %           pcolor(x,x,z), shading interp ^pM+A6 XY  
    %           set(gca,'XTick',[],'YTick',[]) 98 8]}{w  
    %           axis square Oj<S.fi  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) zlR?,h-[3  
    %       end VG/3xR&y  
    % Ai D[SR  
    %   See also ZERNPOL, ZERNFUN2. BpX6aAx  
    %|G"-%_E  
    %   Paul Fricker 11/13/2006 \{Q?^E  
    Y#!h9F  
    XqM3<~$  
    % Check and prepare the inputs: 2pdvWWh3l  
    % ----------------------------- u?s VcD[  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) olLfko4$*V  
        error('zernfun:NMvectors','N and M must be vectors.') XZw6Xtn  
    end Y>jiXl?&  
    JG{j)O|L  
    if length(n)~=length(m) L 8{\r$  
        error('zernfun:NMlength','N and M must be the same length.') eY{+~|KZ  
    end 7JSNYTH  
    .9O$G2'oh  
    n = n(:); EUsI%p  
    m = m(:); D&HV6#  
    if any(mod(n-m,2)) '+j} >Q  
        error('zernfun:NMmultiplesof2', ... nQ|r"|g  
              'All N and M must differ by multiples of 2 (including 0).') vkLC-Mzm<  
    end gm9mg*aM  
    !n6wWl  
    if any(m>n) 5U_H>oD  
        error('zernfun:MlessthanN', ... h*u`X>!!  
              'Each M must be less than or equal to its corresponding N.') pm{|?R  
    end \M'-O YH_[  
    64:fs?H  
    if any( r>1 | r<0 ) /%lZu^  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') fib}b? vk  
    end qY 4#V k  
    dg4vc][  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) OT'[:|x ;  
        error('zernfun:RTHvector','R and THETA must be vectors.') };'\~g,1  
    end vM_:&j_?``  
    tpuYiL  
    r = r(:); ioPUUUb)  
    theta = theta(:); !bV5Sr^  
    length_r = length(r); h$L"8#  
    if length_r~=length(theta) # p[',$cC  
        error('zernfun:RTHlength', ... y\{%\$  
              'The number of R- and THETA-values must be equal.') NH_<q"gT  
    end {nU=%w"\  
    eW|^tH  
    % Check normalization: %kgkXc~6|x  
    % -------------------- [ @4rjGwB  
    if nargin==5 && ischar(nflag) NWxUn.Gy9  
        isnorm = strcmpi(nflag,'norm'); Le%Z V%,  
        if ~isnorm pKi&[  
            error('zernfun:normalization','Unrecognized normalization flag.') y!]CJigpZ  
        end ,]b~t0|B  
    else }jill+]  
        isnorm = false; WOh|U4vt  
    end &HSq(te  
    )Wb0u0)_  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %u;~kP|S%  
    % Compute the Zernike Polynomials ,]T2$?|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% XV^1tX>f{  
    SM@QUAXO  
    % Determine the required powers of r: tnLAJ+ -M  
    % ----------------------------------- ^wS5>lf7p  
    m_abs = abs(m); "--t e  
    rpowers = []; /> 4"~q)  
    for j = 1:length(n) 0@AAulRl  
        rpowers = [rpowers m_abs(j):2:n(j)]; "W(Q%1!Wi  
    end |g *XK6  
    rpowers = unique(rpowers); H*9~yT' Q  
    qoAj] ")  
    % Pre-compute the values of r raised to the required powers, '}Ri`  
    % and compile them in a matrix: I" KN"v^  
    % ----------------------------- \}]!)}G  
    if rpowers(1)==0 K(q-?n`<  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); <I"S#M7-s  
        rpowern = cat(2,rpowern{:}); FN[{s  
        rpowern = [ones(length_r,1) rpowern]; 1IVuSp`{FU  
    else |<O9Sb_  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 2YDM9`5xs\  
        rpowern = cat(2,rpowern{:}); a5w:u5  
    end )Y)pmjZaG  
    tr7<]Hm:  
    % Compute the values of the polynomials: $HJwb-I  
    % -------------------------------------- gJM`[x`T  
    y = zeros(length_r,length(n)); QD%L0;j  
    for j = 1:length(n) ]7e =fM9V;  
        s = 0:(n(j)-m_abs(j))/2; uIZWO.OdU  
        pows = n(j):-2:m_abs(j); q/n,,!  
        for k = length(s):-1:1 \_B[{e7z  
            p = (1-2*mod(s(k),2))* ... K#"O a h  
                       prod(2:(n(j)-s(k)))/              ... 5<w g 8y  
                       prod(2:s(k))/                     ... )&!&AlLn  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... :^(>YAyHj^  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); p QizJ6  
            idx = (pows(k)==rpowers); >KJ+-QuO&  
            y(:,j) = y(:,j) + p*rpowern(:,idx); yiO. z  
        end ){UcS/GI=  
         RSo& (Uv  
        if isnorm ^yOZArc'r  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); sM9+dh  
        end ]KmO$4  
    end y:6; LZ9[  
    % END: Compute the Zernike Polynomials KGg3 !jY  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z4\=*ic@  
    QqU!Najf  
    % Compute the Zernike functions: r-<F5<H+K@  
    % ------------------------------ LGtIm7  
    idx_pos = m>0; h2D>;k  
    idx_neg = m<0; Ng_!zrx04  
    ye MB0Z*r  
    z = y; 6H7],aMg$A  
    if any(idx_pos) 5;HH4?]p  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); mWvl 38  
    end ynrT a..  
    if any(idx_neg) K1T4cUo  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 6AhM=C  
    end <%" b9T`'  
    5s]. @C8  
    % 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) sL$:"=  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ^t#&@-'(d  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated a5 TioQ  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive l~$+,U&XNe  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, %B.yW`,X  
    %   and THETA is a vector of angles.  R and THETA must have the same J G{3EWXR  
    %   length.  The output Z is a matrix with one column for every P-value, _BP&n  
    %   and one row for every (R,THETA) pair. #n8IZ3+  
    % RtM8yar+sn  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike =aTv! 8</  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) m*kl  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) >: g3k  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 N <Xq]! K-  
    %   for all p. m+T;O/lG0{  
    % 7](KV"%V  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 u@cYw:-C  
    %   Zernike functions (order N<=7).  In some disciplines it is Z '7  
    %   traditional to label the first 36 functions using a single mode <d$x.in  
    %   number P instead of separate numbers for the order N and azimuthal XMu9Uk{|  
    %   frequency M. _y} T/I9  
    % }WH&iES@P  
    %   Example: @y\X R  
    % G\+L~t  
    %       % Display the first 16 Zernike functions M]2]\km  
    %       x = -1:0.01:1; !'9Feoez  
    %       [X,Y] = meshgrid(x,x); VL` z[|e @  
    %       [theta,r] = cart2pol(X,Y); =h5H~G5AT  
    %       idx = r<=1; o9dY9o+Z  
    %       p = 0:15; N@Uy=?)ZJ  
    %       z = nan(size(X)); lSVp%0jR  
    %       y = zernfun2(p,r(idx),theta(idx)); U ^5Kz-5.  
    %       figure('Units','normalized') 7%|~>  
    %       for k = 1:length(p) %/zbgS`  
    %           z(idx) = y(:,k); hFy;ffs.  
    %           subplot(4,4,k) #W.#Hjpp  
    %           pcolor(x,x,z), shading interp e9U9Uu[  
    %           set(gca,'XTick',[],'YTick',[]) [R^i F  
    %           axis square U"^kH|  
    %           title(['Z_{' num2str(p(k)) '}']) 9 %I?).5  
    %       end %|q>pin2  
    % ]\hSI){  
    %   See also ZERNPOL, ZERNFUN. [`"ZjkR_J  
    0vD7v  
    %   Paul Fricker 11/13/2006 1e{IC=  
    ij(B,Y  
    E^/t$M|H  
    % Check and prepare the inputs: < (fRn`)PT  
    % ----------------------------- }o?APvd  
    if min(size(p))~=1 LcTt)rs f  
        error('zernfun2:Pvector','Input P must be vector.') `-J%pEIza  
    end i/`m`qdg  
    qGB{7-ru  
    if any(p)>35 lJ}_G>GJ  
        error('zernfun2:P36', ... ?IqQ-C)6D  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... R\G0'?h >  
               '(P = 0 to 35).']) sHt].gZ  
    end 2q=AEv/  
    zck#tht4 n  
    % Get the order and frequency corresonding to the function number: uXJ;A *  
    % ---------------------------------------------------------------- ;RC{<wBTx  
    p = p(:); R6kD=JY/!  
    n = ceil((-3+sqrt(9+8*p))/2); SwTL|+u  
    m = 2*p - n.*(n+2); QSdHm  
    7e|s wJ>4  
    % Pass the inputs to the function ZERNFUN: Mb|a+,:>3  
    % ---------------------------------------- CUBEW~X}M  
    switch nargin 1Z+8r  
        case 3 yW1)vD7  
            z = zernfun(n,m,r,theta); >>t@}F)  
        case 4 i1NY9br  
            z = zernfun(n,m,r,theta,nflag); z<U-#k7nz  
        otherwise 7vrl'^1  
            error('zernfun2:nargin','Incorrect number of inputs.') w[zjerH3  
    end v1+3}5b'uF  
    IEsEdw]aZE  
    % 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) xW)2<m6C&  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. ]@j*/IP  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of "7'J &^|  
    %   order N and frequency M, evaluated at R.  N is a vector of ?I_s0k I  
    %   positive integers (including 0), and M is a vector with the #%iDT6  
    %   same number of elements as N.  Each element k of M must be a TN!j13,  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) z&#SPH*  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is ;4R$g5-4X  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix vM*-D{  
    %   with one column for every (N,M) pair, and one row for every p Dx1z|@z  
    %   element in R. c }Ft^Il  
    % w28o}$b`  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- z1PBMSG  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is bf ]f=;.+  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 2,$8icM  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 gPNZF\ r  
    %   for all [n,m]. jaTh^L  
    % y I HXg#  
    %   The radial Zernike polynomials are the radial portion of the V+* P2|  
    %   Zernike functions, which are an orthogonal basis on the unit gp};D  
    %   circle.  The series representation of the radial Zernike 0bceI  
    %   polynomials is >BIMi^  
    % $UMFNjL  
    %          (n-m)/2 |%F=po>w  
    %            __ 5,3'=mA6  
    %    m      \       s                                          n-2s >:AARx%  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r |{BIHgMh  
    %    n      s=0 8 ##-EN;ag  
    % 0Is,*Srr  
    %   The following table shows the first 12 polynomials. +X#vVD3"  
    % W]"zctE  
    %       n    m    Zernike polynomial    Normalization oX#Q<2z*  
    %       --------------------------------------------- Q8:Has  
    %       0    0    1                        sqrt(2) &oP +$;Y  
    %       1    1    r                           2 nOt&pq7  
    %       2    0    2*r^2 - 1                sqrt(6) oRm L {UDZ  
    %       2    2    r^2                      sqrt(6) R@58*c:U(  
    %       3    1    3*r^3 - 2*r              sqrt(8) 7L`A{L  
    %       3    3    r^3                      sqrt(8) A;;fACF8e  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) %q/62f7?  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) "Q}#^h]F  
    %       4    4    r^4                      sqrt(10) |'.*K]Yp  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) G"-?&)M#a  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 6LOnU~l,  
    %       5    5    r^5                      sqrt(12) p#01gB  
    %       --------------------------------------------- iqC|G/  
    % oz,np@f)J  
    %   Example: <6EeD5{*  
    % PXK7b2fE.  
    %       % Display three example Zernike radial polynomials +DW~BS3  
    %       r = 0:0.01:1; fV &KM*W*@  
    %       n = [3 2 5]; %}SGl${-  
    %       m = [1 2 1]; 3AHlSX  
    %       z = zernpol(n,m,r); l Q'I  
    %       figure 6 bomh2  
    %       plot(r,z) v0 nj M  
    %       grid on X\`_3=  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') >QjAoDVX?  
    % )<oJnxe]  
    %   See also ZERNFUN, ZERNFUN2. u8gS< \  
    %zA;+s$l  
    % A note on the algorithm. jlkmLcpf  
    % ------------------------ ebm])~ZL  
    % The radial Zernike polynomials are computed using the series @E;=*9ek{u  
    % representation shown in the Help section above. For many special 9E zj"  
    % functions, direct evaluation using the series representation can 4H,`]B8(D  
    % produce poor numerical results (floating point errors), because D N'3QQn  
    % the summation often involves computing small differences between txJr;  
    % large successive terms in the series. (In such cases, the functions ]p _L)  
    % are often evaluated using alternative methods such as recurrence -9%:ilX~  
    % relations: see the Legendre functions, for example). For the Zernike ))R5(R  
    % polynomials, however, this problem does not arise, because the p6 ]7&{>  
    % polynomials are evaluated over the finite domain r = (0,1), and Ov};e  
    % because the coefficients for a given polynomial are generally all D2<fw#  
    % of similar magnitude. C\3y {s  
    % '{a/2 l  
    % ZERNPOL has been written using a vectorized implementation: multiple vX{J' H]u  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] J,V9k[88  
    % values can be passed as inputs) for a vector of points R.  To achieve 7R`M,u~f2^  
    % this vectorization most efficiently, the algorithm in ZERNPOL *?Lv3}E  
    % involves pre-determining all the powers p of R that are required to 1/RsptN"v  
    % compute the outputs, and then compiling the {R^p} into a single XF Patd  
    % matrix.  This avoids any redundant computation of the R^p, and m`v2: S}  
    % minimizes the sizes of certain intermediate variables. PpGL/,]X  
    % EqyeJq .  
    %   Paul Fricker 11/13/2006 V `b2TS  
    Qt iDTr  
    {!.(7wV\  
    % Check and prepare the inputs: SHvq.lYJ  
    % ----------------------------- ZW-yP2  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) D!Q">6_"z  
        error('zernpol:NMvectors','N and M must be vectors.') *v;!-F&8>  
    end ST2.:v;lb  
    C+?Hm1  
    if length(n)~=length(m) ^@ I   
        error('zernpol:NMlength','N and M must be the same length.')  +EFgE1w  
    end ,LTH;<zB)  
    :}yi -/_8!  
    n = n(:); *meZ8DV2DH  
    m = m(:); `k=bL"T>\  
    length_n = length(n); K\>tA)IPSV  
    3Vsc 9B"w  
    if any(mod(n-m,2)) l\BVS)  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') G %N $C  
    end m$`RcwO  
    Jpj}@,  
    if any(m<0) YCdS!&^UN  
        error('zernpol:Mpositive','All M must be positive.') _]04lGx27  
    end /|r^W\DV&x  
    BS /G("oZ[  
    if any(m>n) ;6gDV`Twy  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') z3`-plE  
    end w3#Wh|LQ-  
    g+DzscIT  
    if any( r>1 | r<0 ) F|F]970  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ~1cnE:x;V  
    end l=]cy-H  
    WzAb|&?  
    if ~any(size(r)==1) cnSJ{T  
        error('zernpol:Rvector','R must be a vector.') zw+B9PYqX  
    end H70LhN  
    rE i Ki  
    r = r(:); #?5 (o  
    length_r = length(r); WF2}-NU"  
    <!L>Exh&r  
    if nargin==4 wDcj,:h`  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); s<*XN NE7  
        if ~isnorm $>*3/H  
            error('zernpol:normalization','Unrecognized normalization flag.') wkP#Z"A0~  
        end ;+0t;B!V  
    else L/wD7/ODr  
        isnorm = false; =2Bg9!zW>  
    end F4It/  
    anIAM  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ks=>K(V6  
    % Compute the Zernike Polynomials .9R [ *<  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zEs>b(5u  
    |\QgX%  
    % Determine the required powers of r: #rxVd 7f  
    % ----------------------------------- umD!2 w  
    rpowers = []; %R@X>2l/_  
    for j = 1:length(n) e&7JpT  
        rpowers = [rpowers m(j):2:n(j)]; kdCUORMK  
    end %TX@I$Ba  
    rpowers = unique(rpowers); =jSb'Vu|  
    xm%Um\Pb7  
    % Pre-compute the values of r raised to the required powers, ZPiq-q  
    % and compile them in a matrix: :q=%1~Idla  
    % ----------------------------- +lJG(Qd  
    if rpowers(1)==0 dA@'b5N{"  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Ge,;8N88  
        rpowern = cat(2,rpowern{:}); cj+ FRG~u  
        rpowern = [ones(length_r,1) rpowern]; :80Z6F.k`  
    else %{YN70/  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); n.wF&f'D]  
        rpowern = cat(2,rpowern{:}); $tz;<M7B  
    end WtViW=j'  
    "5;;)\o ~  
    % Compute the values of the polynomials: SfgU`eF%B  
    % -------------------------------------- M(jgd  
    z = zeros(length_r,length_n); f~Fm4 >\(  
    for j = 1:length_n 7s"< 'cx_F  
        s = 0:(n(j)-m(j))/2; kjEEuEv  
        pows = n(j):-2:m(j); ]d,S749(s  
        for k = length(s):-1:1 (:._"jp]  
            p = (1-2*mod(s(k),2))* ... io,M{Ib  
                       prod(2:(n(j)-s(k)))/          ... T6H}/#*tK  
                       prod(2:s(k))/                 ... KC(xb5x Y  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ZX5xF<os8  
                       prod(2:((n(j)+m(j))/2-s(k))); /V8}eZ97  
            idx = (pows(k)==rpowers); ^dP KDrKxh  
            z(:,j) = z(:,j) + p*rpowern(:,idx); GKvN* SU=  
        end 7:9.&W/KE  
         ]04 e1F1J  
        if isnorm  Mu2  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); I *x[:)X8  
        end Vl%^H[]  
    end ~vXaqCX  
    qtiz a~u  
    % 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)  Xe#K{gA  
    9elga"4:'  
    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。 zfc3)7  
    u ~3%bJ]  
    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)