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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 # $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! tCF,KP?  
     
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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 |bM?Q$>~  
    function z = zernfun(n,m,r,theta,nflag) y88lkV4a  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. +gh*n,:|  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N -]-?>gkN5  
    %   and angular frequency M, evaluated at positions (R,THETA) on the R)Y*<Na  
    %   unit circle.  N is a vector of positive integers (including 0), and ? 3t]9z  
    %   M is a vector with the same number of elements as N.  Each element kKHGcm^r  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) )*m#RqLQ8  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, G?e\w+}Pj@  
    %   and THETA is a vector of angles.  R and THETA must have the same qN@-H6D1=  
    %   length.  The output Z is a matrix with one column for every (N,M) *S?vw'n  
    %   pair, and one row for every (R,THETA) pair.  F<Y>  
    % %gbvX^E?  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike M!#[(:  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), TPi=!*$&  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral >$/PfyY7@#  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, vUD>+*D  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized [CAV"u)0  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. xU(yc}vw,  
    % ){M)0,:  
    %   The Zernike functions are an orthogonal basis on the unit circle. ,^m;[Dl7  
    %   They are used in disciplines such as astronomy, optics, and h;RKF\U:"  
    %   optometry to describe functions on a circular domain. J12hjzk6@  
    % H vezi>M  
    %   The following table lists the first 15 Zernike functions. |\# 6?y[o  
    % qCUn. mI  
    %       n    m    Zernike function           Normalization vq_v;$9}  
    %       -------------------------------------------------- O @)D%*;v  
    %       0    0    1                                 1 cpJ(77e  
    %       1    1    r * cos(theta)                    2  #-^y9B  
    %       1   -1    r * sin(theta)                    2 .G/2CVMj  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) /)LI1\ o  
    %       2    0    (2*r^2 - 1)                    sqrt(3)  +}-Ecr  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 9i`sSi8   
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) vN8Xq+  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Ip&Q'"HYj  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) jC3)^E@:"  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) kM o7mkV  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) r_EuLFMA  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) TQiDbgFo  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) |h{#r7H0  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !3J YG  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) TxDzGC  
    %       -------------------------------------------------- zZ})$Ny(  
    % ^Ss4<  
    %   Example 1: +u[?8D7Y  
    % oH vVZ  
    %       % Display the Zernike function Z(n=5,m=1) dxwH C\"5  
    %       x = -1:0.01:1; ??g`c=R!V  
    %       [X,Y] = meshgrid(x,x); 18{" @<wIs  
    %       [theta,r] = cart2pol(X,Y); /'WIgP  
    %       idx = r<=1; A3cW8 OClz  
    %       z = nan(size(X)); O9Fg_qfuT_  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); lMW4SRk1C  
    %       figure ~V?3A/]  
    %       pcolor(x,x,z), shading interp <-%OXEG  
    %       axis square, colorbar #nS[]UbwZ  
    %       title('Zernike function Z_5^1(r,\theta)') 0{'%j~"  
    % #5a'Z+  
    %   Example 2: { kF"<W  
    % A\S1{JrR  
    %       % Display the first 10 Zernike functions dX vp-oi  
    %       x = -1:0.01:1; ZA! yw7~  
    %       [X,Y] = meshgrid(x,x); Or9`E(  
    %       [theta,r] = cart2pol(X,Y); x O gUX6n  
    %       idx = r<=1; wNt-mgir-Q  
    %       z = nan(size(X)); yccF#zU  
    %       n = [0  1  1  2  2  2  3  3  3  3]; DTi\ 4&41  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Dw-i!dq  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; #Emz9qTsce  
    %       y = zernfun(n,m,r(idx),theta(idx)); RLtIn!2OU  
    %       figure('Units','normalized') rh%-va9  
    %       for k = 1:10 b( qO fek  
    %           z(idx) = y(:,k); `E4OgO  
    %           subplot(4,7,Nplot(k)) jh3X G  
    %           pcolor(x,x,z), shading interp UC{Tmf  
    %           set(gca,'XTick',[],'YTick',[]) sM0o,l(5  
    %           axis square i rRe}  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) MV.$Ay  
    %       end sKU?"|G81G  
    % v?S~ =$.  
    %   See also ZERNPOL, ZERNFUN2. LG6k KG  
    ;p U=>  
    %   Paul Fricker 11/13/2006 'CkN  
    j^&{5s  
    |Vq&IfP  
    % Check and prepare the inputs: h~zG*B5F  
    % ----------------------------- |'bRVqJ  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) f}_d`?K  
        error('zernfun:NMvectors','N and M must be vectors.') ; D a[jFP  
    end rt5eN:'qY  
    <GthJr>1D  
    if length(n)~=length(m) N)rf /E0  
        error('zernfun:NMlength','N and M must be the same length.') 3jG #<4;J  
    end ^%<t^sE  
    AT6:&5_`  
    n = n(:); G>q16nS~KP  
    m = m(:); m=7Z8@sX},  
    if any(mod(n-m,2)) O{F)|<L(G  
        error('zernfun:NMmultiplesof2', ... NcVsQV  
              'All N and M must differ by multiples of 2 (including 0).') iH#b"h{w  
    end 3-T}8VsiP  
    ag \d4y6  
    if any(m>n) 3>I   
        error('zernfun:MlessthanN', ... :1O1I2L0  
              'Each M must be less than or equal to its corresponding N.') )f6:{ma  
    end BL&D|e  
    <P"4Mk7`s  
    if any( r>1 | r<0 ) P4~=_Hh  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') p>c`GDU  
    end 5cza0CriJ  
    aYyUe>  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) '\iWp?`$  
        error('zernfun:RTHvector','R and THETA must be vectors.') X%>Sio  
    end m@_m"1_;  
    mm5y'=#  
    r = r(:); @^)aUOe  
    theta = theta(:); i47xF7y\  
    length_r = length(r); G\U'_G>  
    if length_r~=length(theta) {ta0dS;1  
        error('zernfun:RTHlength', ... UOg4 E  
              'The number of R- and THETA-values must be equal.') 22<T.c  
    end v FL\O  
    |4F 3Gu  
    % Check normalization: {D(_"  
    % -------------------- dK45&JHoW^  
    if nargin==5 && ischar(nflag) %!>~2=Q2*  
        isnorm = strcmpi(nflag,'norm'); $YyN-C  
        if ~isnorm 2+Tu"oG;rB  
            error('zernfun:normalization','Unrecognized normalization flag.') nnZ|oEF  
        end DjX*2O  
    else 7fOk]Yl[  
        isnorm = false; 0uf'6<fR  
    end $:bU<  
    g`skmHS89  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V0Z\e _I  
    % Compute the Zernike Polynomials bLfbzkNV\1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c QjzI#  
    KvM}g2"  
    % Determine the required powers of r: $:YJ<HvG<  
    % ----------------------------------- \(C_t1  
    m_abs = abs(m); $1CAfSgKw  
    rpowers = []; t1)~J  
    for j = 1:length(n) |^ao,3h#  
        rpowers = [rpowers m_abs(j):2:n(j)]; oM@X)6P_  
    end |Q'l&Gt6  
    rpowers = unique(rpowers); zLs[vg.(  
    T|h/n\fx)a  
    % Pre-compute the values of r raised to the required powers, S'I{'jP5  
    % and compile them in a matrix: {ER%r'(4Z  
    % ----------------------------- 8qEK6-  
    if rpowers(1)==0 t^ =6czk  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); bIP'(B#1K  
        rpowern = cat(2,rpowern{:}); ;plzJ6>  
        rpowern = [ones(length_r,1) rpowern]; [S}o[v\  
    else B@,L83  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ?\QEK  
        rpowern = cat(2,rpowern{:}); }<EA)se"  
    end T4\F=iw4  
    S)@95pb  
    % Compute the values of the polynomials: O1.a=O  
    % -------------------------------------- $?l?  
    y = zeros(length_r,length(n)); FZM9aA  
    for j = 1:length(n) dnby&-+T  
        s = 0:(n(j)-m_abs(j))/2; FuZ7xM,  
        pows = n(j):-2:m_abs(j); M~/%V NX  
        for k = length(s):-1:1 }Om+,!_d  
            p = (1-2*mod(s(k),2))* ... Z7eD+4gD  
                       prod(2:(n(j)-s(k)))/              ... !cs +tm3  
                       prod(2:s(k))/                     ... iB Ld*B|#K  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... D3LW 49  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); b@OL !?JP  
            idx = (pows(k)==rpowers); }ST9&w i~  
            y(:,j) = y(:,j) + p*rpowern(:,idx); (9N75uCa  
        end  H4HWr6  
         "RG.27  
        if isnorm acWm+  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); GdqT4a\S  
        end [TPr  
    end U!"+~d)  
    % END: Compute the Zernike Polynomials 2WjQ-mM#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% N/A.1W  
    Z6}B}5@y  
    % Compute the Zernike functions: `}s$cgEG  
    % ------------------------------ Ks.pb !r  
    idx_pos = m>0; T4`.rnzyRb  
    idx_neg = m<0; E%M~:JuKd?  
    yfS`g-j{~  
    z = y; a G^kL  
    if any(idx_pos) &v+8RY^F=  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); eKlh }v  
    end bJD2c\qoc  
    if any(idx_neg) 1"r6qYN!>  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ~n#rATbxf  
    end FAVw80?5k  
    t)74(  
    % 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) 6R1){,8  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. A-vK0l+  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated Wyf+xr'Ky  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive i~}[/^  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, &[}T41  
    %   and THETA is a vector of angles.  R and THETA must have the same Z |uII#lq  
    %   length.  The output Z is a matrix with one column for every P-value, '{j.5~4y  
    %   and one row for every (R,THETA) pair. w{3 B  
    % IiV:bHUE}0  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike *p{wC r  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) D}l^ow  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) |s :b9sfA  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 5QU7!jb I  
    %   for all p. Wa%Zt*7  
    % }3cOZd_,t  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 /GNLZm^  
    %   Zernike functions (order N<=7).  In some disciplines it is D^To:N 7U  
    %   traditional to label the first 36 functions using a single mode KwOn<0P  
    %   number P instead of separate numbers for the order N and azimuthal iCCY222:  
    %   frequency M. +rbj%v}Fh  
    % A,rgN;5fb  
    %   Example: b(dIl)Y4 :  
    % 8 ~.|^no  
    %       % Display the first 16 Zernike functions S+A'\{f  
    %       x = -1:0.01:1; #L.fGTb  
    %       [X,Y] = meshgrid(x,x); MK 7S*N1  
    %       [theta,r] = cart2pol(X,Y); #*;(%\q}  
    %       idx = r<=1; g5/%}8[- 2  
    %       p = 0:15; Rm 1obP  
    %       z = nan(size(X)); -J4?Km  
    %       y = zernfun2(p,r(idx),theta(idx)); " B Z6G`  
    %       figure('Units','normalized') DoAK]zyJA  
    %       for k = 1:length(p) 1SeDrzLA  
    %           z(idx) = y(:,k); =YkJS%)M)  
    %           subplot(4,4,k) ?as)vYP  
    %           pcolor(x,x,z), shading interp N7k<q=r-  
    %           set(gca,'XTick',[],'YTick',[]) $}r*WZ  
    %           axis square oX@nWQBc_  
    %           title(['Z_{' num2str(p(k)) '}']) 7. 9n  
    %       end H[ocIw  
    % FW(y#Fmqs  
    %   See also ZERNPOL, ZERNFUN. T1c2J,+}R  
    C,+6g/{  
    %   Paul Fricker 11/13/2006 RRig  
    48Y5ppcS  
    lG,/tMy  
    % Check and prepare the inputs: 7$x~}*u  
    % ----------------------------- %m1k^  
    if min(size(p))~=1 1[o] u:m9U  
        error('zernfun2:Pvector','Input P must be vector.') .j:[R.  
    end !fr /WxJ  
    Lc(D2=%  
    if any(p)>35 ~,KAJ7O_  
        error('zernfun2:P36', ... j7=I!<w V  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... LivPk`[  
               '(P = 0 to 35).']) 6*%3O=*  
    end 4Waot  
    Gjq:-kX\  
    % Get the order and frequency corresonding to the function number: ~fB}v  
    % ---------------------------------------------------------------- Y ::\;s  
    p = p(:); |9ro&KA  
    n = ceil((-3+sqrt(9+8*p))/2); j|!.K|9B  
    m = 2*p - n.*(n+2); 6z@OGExmd#  
    p% ESp&  
    % Pass the inputs to the function ZERNFUN: B8-v!4b0`  
    % ---------------------------------------- b{qeu$G R  
    switch nargin -=,%9r  
        case 3 q/ -8sO}q  
            z = zernfun(n,m,r,theta); :" Q!Q@>  
        case 4 l^4[;%*f#l  
            z = zernfun(n,m,r,theta,nflag); wOQ-sp0q0  
        otherwise 9k=-8@G9  
            error('zernfun2:nargin','Incorrect number of inputs.') D7cOEL<  
    end nB2AmS  
    }LQ*vD-Jj  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) ,GEMc a,`  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. AH`D&V  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of e'aKI]>a  
    %   order N and frequency M, evaluated at R.  N is a vector of Ib.`2@ o&  
    %   positive integers (including 0), and M is a vector with the [^t"Hf  
    %   same number of elements as N.  Each element k of M must be a UN_f2  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) )k0e}  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is \V`O-wcJ]S  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix lM Gz"cym  
    %   with one column for every (N,M) pair, and one row for every b1{~j]"$L  
    %   element in R. 06*R)siC  
    % #EtS9D'd+  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ERQ a,h/  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is (WiA  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to "T5?<c  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 t+Mr1e  
    %   for all [n,m]. 0K7]<\)  
    % JbzYr] k  
    %   The radial Zernike polynomials are the radial portion of the D.)$\Caq  
    %   Zernike functions, which are an orthogonal basis on the unit nS[0g^}  
    %   circle.  The series representation of the radial Zernike l;U9dO}/[  
    %   polynomials is $1oU^V Y  
    % o1"U'y-9V  
    %          (n-m)/2 w(`X P  
    %            __ tp$NT.z  
    %    m      \       s                                          n-2s ^ef:cS$;  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r KrVF>bq+  
    %    n      s=0 X obiF  
    % #0`2wuo {  
    %   The following table shows the first 12 polynomials. aW %ulZ  
    % <Sm -Z,|  
    %       n    m    Zernike polynomial    Normalization N-jFA8n  
    %       --------------------------------------------- Svn|vH  
    %       0    0    1                        sqrt(2) .{HU1/!  
    %       1    1    r                           2 :Y y+%  
    %       2    0    2*r^2 - 1                sqrt(6) h3a HCr E  
    %       2    2    r^2                      sqrt(6) K>Fo+f  
    %       3    1    3*r^3 - 2*r              sqrt(8) *kgbcUf8  
    %       3    3    r^3                      sqrt(8) MR%M[SK1  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) =Xm [  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 3nd02:GF  
    %       4    4    r^4                      sqrt(10) sq*R)cZ  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) w|>:mQnU  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) wED~^[]f  
    %       5    5    r^5                      sqrt(12) 4a>z]&s  
    %       --------------------------------------------- =NAL*4c+  
    % eL^,-3JA(]  
    %   Example: !fwLC"QC  
    % NeCTEe|V  
    %       % Display three example Zernike radial polynomials CcgCKT  
    %       r = 0:0.01:1; T'\ lntN  
    %       n = [3 2 5]; Vj0`*nC)/  
    %       m = [1 2 1]; E~2}rK+#)  
    %       z = zernpol(n,m,r); |=l;UqB  
    %       figure p(Y'fd}  
    %       plot(r,z) /4w&! $M-  
    %       grid on 93)&  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') M<"&$qZ$R  
    % n0@e%=H)I  
    %   See also ZERNFUN, ZERNFUN2. 0Vh|UJ'&7  
    zjM+F{P8  
    % A note on the algorithm. }OI;M^5L  
    % ------------------------ rxIfatp^  
    % The radial Zernike polynomials are computed using the series +_06{7@h  
    % representation shown in the Help section above. For many special ?t'O\n)M  
    % functions, direct evaluation using the series representation can zvdtP'&uj  
    % produce poor numerical results (floating point errors), because A"I:cw"KY  
    % the summation often involves computing small differences between b0tr)>d  
    % large successive terms in the series. (In such cases, the functions ZR6KE_  
    % are often evaluated using alternative methods such as recurrence IBQmm(+v  
    % relations: see the Legendre functions, for example). For the Zernike _yJd@  
    % polynomials, however, this problem does not arise, because the kE UfQLbn  
    % polynomials are evaluated over the finite domain r = (0,1), and !R[o6V5T  
    % because the coefficients for a given polynomial are generally all :I+%v  
    % of similar magnitude. F,P,dc  
    % $wdIOfaH  
    % ZERNPOL has been written using a vectorized implementation: multiple U"aFi  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] U:#9!J?41  
    % values can be passed as inputs) for a vector of points R.  To achieve y2TJDb1  
    % this vectorization most efficiently, the algorithm in ZERNPOL \H*"UgS  
    % involves pre-determining all the powers p of R that are required to nr! kx)j  
    % compute the outputs, and then compiling the {R^p} into a single `V$i*{c:#  
    % matrix.  This avoids any redundant computation of the R^p, and {O]Cj~}  
    % minimizes the sizes of certain intermediate variables. RnIL>Akp  
    % |bO}|X  
    %   Paul Fricker 11/13/2006 = = cAL"Z  
    rKg5?.  
    R8W{[@  
    % Check and prepare the inputs: H`9E_[  
    % ----------------------------- }RO Cj,|  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) aHXd1\6m  
        error('zernpol:NMvectors','N and M must be vectors.') =BMON{K  
    end "/S-+Ufn  
    bvUjH5.7  
    if length(n)~=length(m) nkG1&wiX  
        error('zernpol:NMlength','N and M must be the same length.') lOCMKaCD  
    end \@N8[  
    qdx(wGG  
    n = n(:); ,?3r-bM  
    m = m(:); D2#3fM6  
    length_n = length(n); X"z^4?Aj+  
    z|Q)^  
    if any(mod(n-m,2)) 7YV}F9h4  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') Nz'fMdaX,  
    end Kyq/'9`  
    @>}!g9c  
    if any(m<0) h#dp_#  
        error('zernpol:Mpositive','All M must be positive.') bX Q*d_]WT  
    end \uQ yp*P1s  
    1Mn=m w  
    if any(m>n) _D1Uc|  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') -|bnvPmE  
    end s/"&k  
    ]0p] u d&  
    if any( r>1 | r<0 ) 2Nrb}LH  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 3k(A&]~v  
    end ` $*I%oT;  
    (HbA?Aja  
    if ~any(size(r)==1) 8s16yuM  
        error('zernpol:Rvector','R must be a vector.') C jISU$O  
    end [1nfSW  
    1Bp?HyCR  
    r = r(:); 6|:K1bI)  
    length_r = length(r); 2n+tc  
     vX1 8 ]  
    if nargin==4 2T 3tKX  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); (SGX|,5X7  
        if ~isnorm Y\75cfD  
            error('zernpol:normalization','Unrecognized normalization flag.') |`eHUtjH  
        end q>|&u  
    else c||EXFS}O  
        isnorm = false;  B0 E`C  
    end D8X~qt/  
    Fi)(~ji:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f|f)Kys%5  
    % Compute the Zernike Polynomials gZq _BY_U  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (;VlK#rnC  
    $,u>,  
    % Determine the required powers of r: B']-4X{SGa  
    % ----------------------------------- DwmK?5p  
    rpowers = []; %\"<lyD  
    for j = 1:length(n) 6mV-+CnYC  
        rpowers = [rpowers m(j):2:n(j)]; iq^F?$gFk  
    end ^?J3nf{  
    rpowers = unique(rpowers); ZSWZz8  
    Zzn N"Si,  
    % Pre-compute the values of r raised to the required powers, ~Y/:]&wF  
    % and compile them in a matrix: |PtfG2Ty?  
    % ----------------------------- rjK]zD9  
    if rpowers(1)==0 36e  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); j+9;Rvt2  
        rpowern = cat(2,rpowern{:}); W{m_yEOf  
        rpowern = [ones(length_r,1) rpowern]; =U7P\s w2  
    else +Tt.5>N  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); n g%~mt  
        rpowern = cat(2,rpowern{:}); \&_pI2X  
    end v]Fw~Y7l!  
    t!-\:8n  
    % Compute the values of the polynomials: dBw7l}  
    % -------------------------------------- c=,HLHpFO(  
    z = zeros(length_r,length_n); `>0%Ha   
    for j = 1:length_n O^J=19Ri  
        s = 0:(n(j)-m(j))/2; %oB0@&!mS  
        pows = n(j):-2:m(j); ,eGguNA9  
        for k = length(s):-1:1 en>n\;U  
            p = (1-2*mod(s(k),2))* ... X1HEeJ|  
                       prod(2:(n(j)-s(k)))/          ... E.%V 0}  
                       prod(2:s(k))/                 ... R_D&"&   
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 4a0Ud !Qcs  
                       prod(2:((n(j)+m(j))/2-s(k))); Mw3$QRM  
            idx = (pows(k)==rpowers); !2Gua1z!CJ  
            z(:,j) = z(:,j) + p*rpowern(:,idx); qV9}N-sS  
        end DIABR%0  
         15@2h  
        if isnorm cLEd -{x  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); LC4VlfU  
        end wPEK5=\4Ob  
    end jzJQ/ZFS  
    uwQgu!|x  
    % 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)  A.7lo  
    wpC .!T  
    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。 }#M>CNi'PU  
    @c"s6h&  
    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)