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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 OY$P8y3MY  
    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! iF:`rIC  
     
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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 \gd.Bl  
    function z = zernfun(n,m,r,theta,nflag) <cTusC<  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 3)SO-Bz\  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Y>eypfK"  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 6.fahg?E  
    %   unit circle.  N is a vector of positive integers (including 0), and \A-w,]9^V  
    %   M is a vector with the same number of elements as N.  Each element P<@Yux#  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) \W73W_P&g  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, y7 tK>aD}  
    %   and THETA is a vector of angles.  R and THETA must have the same +f)Nf) \q  
    %   length.  The output Z is a matrix with one column for every (N,M) ;dq AmBG{8  
    %   pair, and one row for every (R,THETA) pair. =1D* JU  
    % u#tLY/KA  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike enu",wC3  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), qGS]2KY  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral "WKE% f  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, -:txmM T  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized zw=as9z1-  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. UH8)r  
    % *.ffyBI*~  
    %   The Zernike functions are an orthogonal basis on the unit circle. V+A1O k )  
    %   They are used in disciplines such as astronomy, optics, and (0%0+vY  
    %   optometry to describe functions on a circular domain. GvQ|+vC  
    % sePOW#|  
    %   The following table lists the first 15 Zernike functions. E|2klA^+*  
    % d7o~$4h|  
    %       n    m    Zernike function           Normalization ~5aq.hF1,A  
    %       -------------------------------------------------- 2fc8w3  
    %       0    0    1                                 1 G7lC'~}  
    %       1    1    r * cos(theta)                    2 _"`wUMee  
    %       1   -1    r * sin(theta)                    2 1a {~B#  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) H9)$ #r6i  
    %       2    0    (2*r^2 - 1)                    sqrt(3) =U3,P%  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 7Kx3G{5ja  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) >M7e'}0 ;  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) `m5cU*@D  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 9\W~5J<7  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) i>bFQ1Rdx  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ;D_6u(IC4:  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ~Ra1Zc$o:  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ;F@dN,Y  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) k07JMS?  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) r]1|I6:&)  
    %       -------------------------------------------------- F]Zg9c{#  
    %  /A|cO   
    %   Example 1: _:om(gL  
    % XQ:HH 8  
    %       % Display the Zernike function Z(n=5,m=1) n }lav  
    %       x = -1:0.01:1; %j=E}J<H5*  
    %       [X,Y] = meshgrid(x,x); ,*.C''  
    %       [theta,r] = cart2pol(X,Y); [_j.pMH/P  
    %       idx = r<=1; T8YqCT"EA<  
    %       z = nan(size(X)); x U1dy*-  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Uc e#v)  
    %       figure R{.wAH(  
    %       pcolor(x,x,z), shading interp avls[Bq  
    %       axis square, colorbar nM8aC&Rd\  
    %       title('Zernike function Z_5^1(r,\theta)') GpF,=:  
    % C78d29  
    %   Example 2: LJZEM;;}  
    % *n?6x!A  
    %       % Display the first 10 Zernike functions =_cWCl^5  
    %       x = -1:0.01:1; "/hs@4{u9  
    %       [X,Y] = meshgrid(x,x); `A80""y:M  
    %       [theta,r] = cart2pol(X,Y); RCNqHYR  
    %       idx = r<=1; y)U8\  
    %       z = nan(size(X)); R4}G@&Q  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ?MeP<5\A  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 2!dIW5I  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; fx.FHhVu  
    %       y = zernfun(n,m,r(idx),theta(idx)); 9>le-}~  
    %       figure('Units','normalized') Mz]LFM  
    %       for k = 1:10 (m3p28Q?  
    %           z(idx) = y(:,k); : M0LAN  
    %           subplot(4,7,Nplot(k)) txr!3-Ne'!  
    %           pcolor(x,x,z), shading interp )'%L#  
    %           set(gca,'XTick',[],'YTick',[]) 6?}8z q[  
    %           axis square IG +nrTY0  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ?Pmj}f  
    %       end  wSV[nK  
    % U$VTk  
    %   See also ZERNPOL, ZERNFUN2. ?h>mrj  
    !0Xes0gK0  
    %   Paul Fricker 11/13/2006 0; V{yh  
    ~`tc|Zu  
    ? dSrY  
    % Check and prepare the inputs: zZ-e2)1v  
    % ----------------------------- hPFIf>%}  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) g;]2'Rj  
        error('zernfun:NMvectors','N and M must be vectors.') .(9IAAwKn  
    end ~>}BDsM  
    >YoK?e6  
    if length(n)~=length(m) 0e vxRcrzz  
        error('zernfun:NMlength','N and M must be the same length.') h"%6tpV-  
    end V^L;Nw5h  
    #C%<g:F8  
    n = n(:); oL }FD !}  
    m = m(:); =K8`[iH  
    if any(mod(n-m,2)) '{( n1es  
        error('zernfun:NMmultiplesof2', ... , {z$M  
              'All N and M must differ by multiples of 2 (including 0).') >47,Hq:2  
    end nk-6W4  
    Y]8l]l 1  
    if any(m>n) Gq-U}r  
        error('zernfun:MlessthanN', ... `q_7rrkO  
              'Each M must be less than or equal to its corresponding N.') ~sSB.g  
    end 5W<BEcV\  
    B0Z*YsbXL  
    if any( r>1 | r<0 ) UQW;!8J#R(  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') i-E&Y*\^9H  
    end ?wwY8e?S  
    ?Cu#(  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) vgE5(fJh  
        error('zernfun:RTHvector','R and THETA must be vectors.') PVEEKKJP]J  
    end O gHWmb  
    yMz@-B  
    r = r(:); e`}|*^-  
    theta = theta(:); 8CEy#%7]}  
    length_r = length(r); cW&OVNj  
    if length_r~=length(theta) yxA0#6so  
        error('zernfun:RTHlength', ... Ti' GSL  
              'The number of R- and THETA-values must be equal.') 8KoPaq   
    end RNvtgZ}k{X  
    ? # G_ &  
    % Check normalization: |Z2_1( ku  
    % -------------------- t]vX9vv+D  
    if nargin==5 && ischar(nflag) [BV{=;iD  
        isnorm = strcmpi(nflag,'norm'); _TX.}167;-  
        if ~isnorm L7Skn-*tnA  
            error('zernfun:normalization','Unrecognized normalization flag.') (\R"v^  
        end AH#e>kU^  
    else ,hOJe=u46  
        isnorm = false; ?on3z  
    end Uc9Uj  
    =ARI*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% oN _% oc  
    % Compute the Zernike Polynomials kc "U)>  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *Li;:b"t  
    T\c dtjk  
    % Determine the required powers of r: VJ1 `&  
    % ----------------------------------- D %5 0  
    m_abs = abs(m); : wn![<`3q  
    rpowers = []; g" M1HxlV  
    for j = 1:length(n) a<\m` Es=  
        rpowers = [rpowers m_abs(j):2:n(j)]; Z)?"pBv'  
    end ,g\.C+.S  
    rpowers = unique(rpowers); Pel3e ~?t  
    kF\ QO [  
    % Pre-compute the values of r raised to the required powers, oEi +S)_  
    % and compile them in a matrix: ]q?<fEG2<  
    % ----------------------------- +F0M?,  
    if rpowers(1)==0 wL%>  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); m*I5 \  
        rpowern = cat(2,rpowern{:}); ^AEg?[q  
        rpowern = [ones(length_r,1) rpowern]; E26ZVFg  
    else =n#xnZ3  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ="I]D I  
        rpowern = cat(2,rpowern{:}); s'K0C8'U  
    end ;#j/F]xG  
    ("9)=x*5  
    % Compute the values of the polynomials: K): )bL(B  
    % -------------------------------------- +$<m;@mZ  
    y = zeros(length_r,length(n)); a{)"KAP  
    for j = 1:length(n) ^Nc\D7( l  
        s = 0:(n(j)-m_abs(j))/2; _|s{G  
        pows = n(j):-2:m_abs(j); 3[Z?`X  
        for k = length(s):-1:1 L:%h]-  
            p = (1-2*mod(s(k),2))* ... ;>Kxl}+R  
                       prod(2:(n(j)-s(k)))/              ... pWQ?pTh  
                       prod(2:s(k))/                     ... 5B@&]-'~  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Y#rao:I  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); kszYbz"  
            idx = (pows(k)==rpowers); NVOY,g=3X  
            y(:,j) = y(:,j) + p*rpowern(:,idx); {cG&l:-r  
        end 46$5f?Z  
         t(s']r  
        if isnorm b2:CFtH5  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Pu}2%P)p  
        end <K2 )v~  
    end #%E~I A%  
    % END: Compute the Zernike Polynomials EW YpYMkm  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Pw+cpM 8<  
    t4 aa5@r  
    % Compute the Zernike functions: Qy9#(596  
    % ------------------------------ 1s1$J2LX  
    idx_pos = m>0; T@f$w/15  
    idx_neg = m<0; >pn?~  
    :]?I|.a  
    z = y; /oh[ Nu1D  
    if any(idx_pos) %)]{*#N4  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); @mw1(J  
    end g.z/%Lp K  
    if any(idx_neg) AC 3 ;i  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 4S+P]U*jW  
    end 1vQ*Br  
    ,.DU)Wi?}  
    % EOF zernfun
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    function z = zernfun2(p,r,theta,nflag) 02?y%  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. v-g2k_ o|  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 2gukK8R$  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive EtKy?]i  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, | [P!9e  
    %   and THETA is a vector of angles.  R and THETA must have the same /_>S0  
    %   length.  The output Z is a matrix with one column for every P-value, a$"3T  
    %   and one row for every (R,THETA) pair. Un@dWf6'  
    % 5_0Eh!sx  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Np+<)q2  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) E%2]c?N5  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) arET2(h  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 :[,-wZiT~6  
    %   for all p. 8FU8E2zo  
    % `Z0FQ( r_  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 (jtrQob  
    %   Zernike functions (order N<=7).  In some disciplines it is b-\ 1D;]  
    %   traditional to label the first 36 functions using a single mode l*":WzRGvF  
    %   number P instead of separate numbers for the order N and azimuthal |+f@w/+  
    %   frequency M. 4Zo.c* BZ  
    % iTwb#Q=  
    %   Example: 6 -N 442  
    % RGf&KV/  
    %       % Display the first 16 Zernike functions k`_sKr]9  
    %       x = -1:0.01:1; \0). ODA(  
    %       [X,Y] = meshgrid(x,x); ACctyGd  
    %       [theta,r] = cart2pol(X,Y); ~5q1zr)E  
    %       idx = r<=1; WB K6Ug  
    %       p = 0:15; <Y:{>=  
    %       z = nan(size(X)); wQEsq<  
    %       y = zernfun2(p,r(idx),theta(idx)); kc-=5l  
    %       figure('Units','normalized') #p*D.We  
    %       for k = 1:length(p) |6v $!wBi  
    %           z(idx) = y(:,k); F2QFQX(j  
    %           subplot(4,4,k) x+EkL3{  
    %           pcolor(x,x,z), shading interp <A^sg?s<'  
    %           set(gca,'XTick',[],'YTick',[]) 3K!(/,`  
    %           axis square O`K2mt\%  
    %           title(['Z_{' num2str(p(k)) '}']) ?I{L^j^#4  
    %       end ]l>LU2 sx  
    % -M5vh~Tp  
    %   See also ZERNPOL, ZERNFUN. d<K2 \:P{}  
    ~@=(#tO.  
    %   Paul Fricker 11/13/2006 Swa0TiT(  
    jVi> 9[rz  
    ,cE yV74  
    % Check and prepare the inputs: FkE)~g  
    % ----------------------------- 0xVw{k}1U  
    if min(size(p))~=1 =gNPS 0H  
        error('zernfun2:Pvector','Input P must be vector.') ,.9k)\/V  
    end J/LsL k  
    d^MRu#]  
    if any(p)>35 ,_iq$I;  
        error('zernfun2:P36', ... <yl%q*gls  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... O,6Wdw3+-3  
               '(P = 0 to 35).']) >Q $ph=  
    end o1`\*]A7J  
    p<1y$=zS  
    % Get the order and frequency corresonding to the function number: A]Bf&+V  
    % ---------------------------------------------------------------- k/P.[5  
    p = p(:); wU6sU]P  
    n = ceil((-3+sqrt(9+8*p))/2); zD)/QFILy  
    m = 2*p - n.*(n+2); }@eIO|  
    8Cs;.>75[  
    % Pass the inputs to the function ZERNFUN: H-vHcqFx3  
    % ---------------------------------------- d~1uK-L]*  
    switch nargin ~8s2p%~  
        case 3 nv0\On7wd  
            z = zernfun(n,m,r,theta); a`L:E'|B9  
        case 4 _%q~K (::  
            z = zernfun(n,m,r,theta,nflag); k&2=-qgVR  
        otherwise 85YUqVi9  
            error('zernfun2:nargin','Incorrect number of inputs.') >H^#!eaqw  
    end |lt]9>|  
    q3AqU?f  
    % 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) Hs*["zFc  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. xNzGp5H  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 7i*eKC`ZqK  
    %   order N and frequency M, evaluated at R.  N is a vector of tLBtE!J$[  
    %   positive integers (including 0), and M is a vector with the <q8@a0e@  
    %   same number of elements as N.  Each element k of M must be a |RFBhB/u  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) MC* Hl`C  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is nq)F$@  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix ,;_+o]  
    %   with one column for every (N,M) pair, and one row for every )V\@N*L`ik  
    %   element in R. 7 !$[XD  
    % CuWJai:nQ;  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- X/yq<_ g  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is _p^ "l2%D/  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to |M7cB$y  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 1Y|a:){G  
    %   for all [n,m]. xSm;~')g  
    % g w" \pD  
    %   The radial Zernike polynomials are the radial portion of the GC{M"q|_  
    %   Zernike functions, which are an orthogonal basis on the unit F!]Sr'UA  
    %   circle.  The series representation of the radial Zernike u.gg N=Z  
    %   polynomials is X H-_tvB  
    % ){*+s RBW  
    %          (n-m)/2 5&ku]l+  
    %            __ $"r9U|6kk  
    %    m      \       s                                          n-2s qI+2,6 sGI  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r uh )S;3|  
    %    n      s=0 Nc;O)K!FH  
    % `#N/]4(j  
    %   The following table shows the first 12 polynomials. -L1785pB85  
    % ~SnUnNDm`  
    %       n    m    Zernike polynomial    Normalization {FNkPX  
    %       --------------------------------------------- w&q[%(G_  
    %       0    0    1                        sqrt(2) 4J2^zx,H  
    %       1    1    r                           2 \84t\jKR  
    %       2    0    2*r^2 - 1                sqrt(6) CkT(\6B-  
    %       2    2    r^2                      sqrt(6) Y@UkP+{f=  
    %       3    1    3*r^3 - 2*r              sqrt(8) Yc:%2KZ"  
    %       3    3    r^3                      sqrt(8) F$C6( C?  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) EY,jy]|#  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) bGPE0}b  
    %       4    4    r^4                      sqrt(10) TSlB.pw%v  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) aD^$v  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) YmziHns`b  
    %       5    5    r^5                      sqrt(12) CKYg!\g(:  
    %       --------------------------------------------- qt@L&v}~j  
    % o~Se[p  
    %   Example: &{}Mds  
    % 9iA rBL"  
    %       % Display three example Zernike radial polynomials :D D<0  
    %       r = 0:0.01:1; 1E+12{~m"i  
    %       n = [3 2 5]; l":W@R  
    %       m = [1 2 1]; Zxa.x?:?n  
    %       z = zernpol(n,m,r); @(3F4Z.i%.  
    %       figure |$RNY``J  
    %       plot(r,z) M/zO|-j&  
    %       grid on Zf'*pp T&q  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') A,}M ^$@  
    % a.Ho>(V/4  
    %   See also ZERNFUN, ZERNFUN2. 3k Ci5C  
    h>-P/  
    % A note on the algorithm. P@{ x@9kI  
    % ------------------------ b;k+N`  
    % The radial Zernike polynomials are computed using the series {]0e=#hw  
    % representation shown in the Help section above. For many special p4`1^}f&Ie  
    % functions, direct evaluation using the series representation can LdPLC':}x|  
    % produce poor numerical results (floating point errors), because oQ$yr^M  
    % the summation often involves computing small differences between [q <'ty  
    % large successive terms in the series. (In such cases, the functions JU 9GJ"  
    % are often evaluated using alternative methods such as recurrence Dw-d`8*  
    % relations: see the Legendre functions, for example). For the Zernike *l2`- gbE  
    % polynomials, however, this problem does not arise, because the y0zMK4b  
    % polynomials are evaluated over the finite domain r = (0,1), and +r:g}iR  
    % because the coefficients for a given polynomial are generally all d9N[f>  
    % of similar magnitude. 34@[ZKJ5  
    % XzUGlrp:Y#  
    % ZERNPOL has been written using a vectorized implementation: multiple ]c.w+<  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Ei;tfB  
    % values can be passed as inputs) for a vector of points R.  To achieve $ [gN#QW%  
    % this vectorization most efficiently, the algorithm in ZERNPOL nbi7r cT  
    % involves pre-determining all the powers p of R that are required to /%wS5IZ^  
    % compute the outputs, and then compiling the {R^p} into a single >=~\b  
    % matrix.  This avoids any redundant computation of the R^p, and La4S/.  
    % minimizes the sizes of certain intermediate variables. +$2{u_m,  
    % N]<(cG&p  
    %   Paul Fricker 11/13/2006 S@qp_!  
    ?#xl3Z ;I  
    O9=/\Kc  
    % Check and prepare the inputs: Shn,JmR  
    % ----------------------------- d2k-MZuT6  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 9T,/R1N8  
        error('zernpol:NMvectors','N and M must be vectors.') Dg&84,bv^  
    end LQ+/|_(.  
     Z>[7#;;  
    if length(n)~=length(m) vOQ% f?%G\  
        error('zernpol:NMlength','N and M must be the same length.') dC11kq qj  
    end =L6#=7hcl  
    s#2t\}/  
    n = n(:); bJ^JK  
    m = m(:); &w@]\7L,:  
    length_n = length(n); $=aO*i  
    ua\t5M5  
    if any(mod(n-m,2)) S-Uod y  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') sZ;|NAx)  
    end X>q`F;W  
    .sMs_ 5D  
    if any(m<0) Z\&f"z?L  
        error('zernpol:Mpositive','All M must be positive.') Nw;qJ58@  
    end h2l;xt  
    C2 N+X(  
    if any(m>n) {#,<)wFV\  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') PEMkx"h +  
    end i"{O~[  
    uuzV,q  
    if any( r>1 | r<0 )   f XD+  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') Q*ITs!~Z  
    end )6|L]'dsZ  
     q+P@2FL  
    if ~any(size(r)==1) 2HbnE&  
        error('zernpol:Rvector','R must be a vector.') rb*|0ST  
    end MKK ^-T  
    @s5=6z]=H  
    r = r(:); tC[ZWL  
    length_r = length(r); blO4)7m  
    /:dLqyQ_V  
    if nargin==4 `~1!nfFD  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); k.J%rRneN  
        if ~isnorm n1[c\1   
            error('zernpol:normalization','Unrecognized normalization flag.') b)w cGBS  
        end m5Bf<E,c  
    else !MbzFs~  
        isnorm = false; :]3X Ez  
    end 3JazQU  
    1wSAwpz  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% bcIae0LZ  
    % Compute the Zernike Polynomials FO{=^I5YA  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C.j+Zb1Z(  
    ]a3$hAcj6"  
    % Determine the required powers of r: i~B?p[  
    % ----------------------------------- *TOdIq&z  
    rpowers = []; 5Xy(za  
    for j = 1:length(n) hp dI5  
        rpowers = [rpowers m(j):2:n(j)]; q'3{M]Tk  
    end lu utyK!  
    rpowers = unique(rpowers); _&KqmQ8$7  
    RTtKf i}  
    % Pre-compute the values of r raised to the required powers, a~o <>H  
    % and compile them in a matrix: yOM/UdWq  
    % ----------------------------- YAi-eL67l  
    if rpowers(1)==0 Mz+I YP`L  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); \Ne`9k  
        rpowern = cat(2,rpowern{:}); # :+Nr  
        rpowern = [ones(length_r,1) rpowern]; d0J /"<  
    else ew;;e|24  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); s 6Wp"V(  
        rpowern = cat(2,rpowern{:}); [9E~=A#  
    end "8za'@D"f  
    R'c*CLaiE  
    % Compute the values of the polynomials: `kKssU<  
    % -------------------------------------- LKN7L kl  
    z = zeros(length_r,length_n); `Fqth^RK?p  
    for j = 1:length_n {X,%GI  
        s = 0:(n(j)-m(j))/2; 8t+eu O  
        pows = n(j):-2:m(j); U1 `5P!ov  
        for k = length(s):-1:1 2- iY:r  
            p = (1-2*mod(s(k),2))* ... e02Hf{eOfw  
                       prod(2:(n(j)-s(k)))/          ... s.1F=u9a  
                       prod(2:s(k))/                 ... Y;w|Fvjj+  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... kUBE+a6#  
                       prod(2:((n(j)+m(j))/2-s(k))); |mT%IR  
            idx = (pows(k)==rpowers); ammi4k/  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Jv~R/qaaD  
        end .jRI $vm  
         i?L=8+9f  
        if isnorm 74e=zW?  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 2H%9l@}u  
        end Ir;JYY!0?  
    end cXXZ'y>FP  
    G1|1Z5r  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  |8f}3R 9  
    hZfj$|<  
    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。 g"748LY>=p  
    7d R?70Sz  
    07年就写过这方面的计算程序了。
    让光学不再神秘,让光学变得容易,快速实现客户关于光学的设想与愿望。
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