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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 y4jiOhF<d  
    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! ^UKY1Q .  
     
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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 /7S g/d%c  
    function z = zernfun(n,m,r,theta,nflag) 5g-1pzP9  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. (G+)v[f  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N RjUrpS[I  
    %   and angular frequency M, evaluated at positions (R,THETA) on the B]yO  
    %   unit circle.  N is a vector of positive integers (including 0), and ,ov$` v  
    %   M is a vector with the same number of elements as N.  Each element bz nMD  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) {f4jE#a>v  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, a U.3  
    %   and THETA is a vector of angles.  R and THETA must have the same [AFR \{  
    %   length.  The output Z is a matrix with one column for every (N,M) k8n9zJ8  
    %   pair, and one row for every (R,THETA) pair. fI;nVRf p  
    % U+B{\38  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike j-/$e,xX  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), m=h/A xW  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ~u0<c:C^  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Y]6d Yq{k  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ?Mo)&,__  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 8v/,< eARJ  
    % mnZfk  
    %   The Zernike functions are an orthogonal basis on the unit circle. b (H J|  
    %   They are used in disciplines such as astronomy, optics, and y]R+/  
    %   optometry to describe functions on a circular domain. e@O]c "  
    % eW<NDI&b  
    %   The following table lists the first 15 Zernike functions. NoF|j57?u'  
    % 3dZj<(.  
    %       n    m    Zernike function           Normalization 3jfAv@I~  
    %       -------------------------------------------------- KIY`3Fl09  
    %       0    0    1                                 1 um/F:rp  
    %       1    1    r * cos(theta)                    2 EFtn !T  
    %       1   -1    r * sin(theta)                    2 mmjWLrhlu  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) *7*cWO=  
    %       2    0    (2*r^2 - 1)                    sqrt(3) zI ^:{]p  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) G 9 &,`  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 4yTgH0(T  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Ed0}$ b  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 8.wtv5eZ  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) mg._c  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) =s.0 f:(  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) vY4}vHH2  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ~S~+'V,d  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) T%"wz3~  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) }3WP:Et  
    %       -------------------------------------------------- Dh}d-m_5  
    % /ioBc}]  
    %   Example 1: W4P\HM>2  
    % +,7vbs3  
    %       % Display the Zernike function Z(n=5,m=1) Fku<|1}&y  
    %       x = -1:0.01:1; 8yOhKEPX  
    %       [X,Y] = meshgrid(x,x); uTO%O}D N  
    %       [theta,r] = cart2pol(X,Y); !%(kMN  
    %       idx = r<=1; XLYGhM  
    %       z = nan(size(X)); /Trbr]lWy  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); @!ja/Y^  
    %       figure G[`2Nd<  
    %       pcolor(x,x,z), shading interp sc-hO9~k  
    %       axis square, colorbar {ktwX\z  
    %       title('Zernike function Z_5^1(r,\theta)') Ur1kb{i  
    % ]d{lS&PRlg  
    %   Example 2: 3 Sf':N`u  
    % .n.N.e  
    %       % Display the first 10 Zernike functions  |#xBC+  
    %       x = -1:0.01:1; C^_m>H3b  
    %       [X,Y] = meshgrid(x,x); iN0'/)ar  
    %       [theta,r] = cart2pol(X,Y); Zf??/+[  
    %       idx = r<=1; 1jBIi  
    %       z = nan(size(X)); lc [)Ev  
    %       n = [0  1  1  2  2  2  3  3  3  3]; PN J&{4wY  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 5TeGdfu @  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; g#1 Y4  
    %       y = zernfun(n,m,r(idx),theta(idx)); ^)`e}}  
    %       figure('Units','normalized') mL#$8wUdt{  
    %       for k = 1:10 211T}a  
    %           z(idx) = y(:,k); [T [] U   
    %           subplot(4,7,Nplot(k)) :1"k`AG  
    %           pcolor(x,x,z), shading interp Bz%wV-  
    %           set(gca,'XTick',[],'YTick',[]) k %sxA  
    %           axis square ;/ |tU o$  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) OYmutq  
    %       end sUiO~<Ozpk  
    % M,oZ_tY%  
    %   See also ZERNPOL, ZERNFUN2. %SCt_9u  
    8NNs_~+x}  
    %   Paul Fricker 11/13/2006 P_p\OK*l]o  
    rAqS;@]0  
    j`Ek:  
    % Check and prepare the inputs: )3 f\H  
    % ----------------------------- qq?o^_^4  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) E*,nKJu'r  
        error('zernfun:NMvectors','N and M must be vectors.') c|k(_#\B  
    end Qk|+Gj  
    8`1]#Vw  
    if length(n)~=length(m) &U([Wd?E2  
        error('zernfun:NMlength','N and M must be the same length.') =E(ed,gH8  
    end /m^G 99N  
    >b:5&s\9  
    n = n(:); 'X4)2iFV  
    m = m(:); *<"{(sAvk  
    if any(mod(n-m,2)) eZhF<<Y  
        error('zernfun:NMmultiplesof2', ... Qs#;sy W@~  
              'All N and M must differ by multiples of 2 (including 0).') i]@k'2N  
    end JnqP`kYbTE  
    :>H{?  
    if any(m>n) COBjJ3  
        error('zernfun:MlessthanN', ... ^LXsU] R  
              'Each M must be less than or equal to its corresponding N.') \PG_i'R  
    end *]Cyc<  
    Be^"sC  
    if any( r>1 | r<0 ) E]a;Ydf~  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') xwHE,ykE  
    end @~5Fcfmm  
    $S2 /*  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) #i*PwgC%_  
        error('zernfun:RTHvector','R and THETA must be vectors.') |942#rM  
    end -Edi"B4K  
    /L|x3RHs  
    r = r(:); -r~9'aEs  
    theta = theta(:); <F-IF7>a  
    length_r = length(r); qwn EVjf  
    if length_r~=length(theta) Dk2Zl  
        error('zernfun:RTHlength', ... jJ'NYG  
              'The number of R- and THETA-values must be equal.') m*i,|{UZ  
    end E7w^A  
    *1:kIi7_  
    % Check normalization: #e@[{s7  
    % -------------------- g 4 $  
    if nargin==5 && ischar(nflag) WYcZD_  
        isnorm = strcmpi(nflag,'norm'); z 9WeOs  
        if ~isnorm Y9st3  
            error('zernfun:normalization','Unrecognized normalization flag.') +;oR_]l  
        end uG YH4  
    else ~xws5n}F  
        isnorm = false; &arJe!K  
    end ,K PrUM}  
    _t4(H))]vG  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;l < amB  
    % Compute the Zernike Polynomials hD,|CQ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% INCD5dihJ  
    ,u- 9e4  
    % Determine the required powers of r: NH=@[t) P,  
    % ----------------------------------- MFWkJbZV  
    m_abs = abs(m); n 1^h;2gz  
    rpowers = []; G"Ey%Q2K  
    for j = 1:length(n) m<*+^JN  
        rpowers = [rpowers m_abs(j):2:n(j)]; 2jkma :$'  
    end A\E ))b9+  
    rpowers = unique(rpowers); 0Xn,q]@Z  
    Z\n^m^Z =  
    % Pre-compute the values of r raised to the required powers, qn}VW0!  
    % and compile them in a matrix: h^14/L=|  
    % ----------------------------- ;.R) uCd{=  
    if rpowers(1)==0 mW,b#'hy  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); IfI:|w}:"r  
        rpowern = cat(2,rpowern{:}); E4_,EeC#  
        rpowern = [ones(length_r,1) rpowern]; ']1a  
    else vuJEPn%  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); z|(<Co8#.  
        rpowern = cat(2,rpowern{:}); !>kg:xV  
    end #2Iw%H2q&  
    pRjrMS  
    % Compute the values of the polynomials: qamq9F$V  
    % -------------------------------------- cBZJ  
    y = zeros(length_r,length(n)); cveQ6 -`K  
    for j = 1:length(n) Cj YI *  
        s = 0:(n(j)-m_abs(j))/2; h2?\A%  
        pows = n(j):-2:m_abs(j); [ThAv Q_$  
        for k = length(s):-1:1 |fgh ryI,  
            p = (1-2*mod(s(k),2))* ... 3RFU  
                       prod(2:(n(j)-s(k)))/              ... WU,b<PU &  
                       prod(2:s(k))/                     ... [.C P,Ly  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... $) qL=kR  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 8;f5;7M n  
            idx = (pows(k)==rpowers); g{2~G6%;0  
            y(:,j) = y(:,j) + p*rpowern(:,idx); E9 @Sc>e  
        end Y&DoA0/y  
         rD !GEU  
        if isnorm GR 1%(,  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); wuSotbc/  
        end h1f 05  
    end {yd(n_PqY  
    % END: Compute the Zernike Polynomials q[+KQ,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% VAqZ`y  
    4#ikdjB;  
    % Compute the Zernike functions: PZ?kv4  
    % ------------------------------ EDF0q i  
    idx_pos = m>0; z"FxKN~Z  
    idx_neg = m<0; 9}a&:QTHR  
    _E/  
    z = y; Q DKY7"H  
    if any(idx_pos) ,<s:* k  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); b+$wx~PLi  
    end .4<lw  
    if any(idx_neg) @`iz0DPG?Y  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ,TYFPulYcp  
    end w`dSc@ :  
    Ip *8R]W  
    % 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) 3[ xHY@c  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ;w._/  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ,&Vir)S  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive P=s3&NDD  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, :dxKcg7  
    %   and THETA is a vector of angles.  R and THETA must have the same -av=5hm  
    %   length.  The output Z is a matrix with one column for every P-value, *V[I&dKq  
    %   and one row for every (R,THETA) pair. O.-A)S@  
    % ,)VAKrSg  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike G+I->n-s4  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) J-5kvQi8  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) IfY?P(P  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 nE bZ8M  
    %   for all p. B^{~,'  
    % `f <w+u  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ]]y,FQ,r  
    %   Zernike functions (order N<=7).  In some disciplines it is #wkSru&LS  
    %   traditional to label the first 36 functions using a single mode tC.etoh  
    %   number P instead of separate numbers for the order N and azimuthal  w U1[/  
    %   frequency M. 6:Y2z!MLO  
    % j'lC]}kH  
    %   Example: {]dvzoE]  
    % \c!e_rZ  
    %       % Display the first 16 Zernike functions en6;I[\  
    %       x = -1:0.01:1; :u)Qs#'29  
    %       [X,Y] = meshgrid(x,x); l j+p}dt  
    %       [theta,r] = cart2pol(X,Y); UXw I?2L  
    %       idx = r<=1; B$S@xD $  
    %       p = 0:15; TKVS%//  
    %       z = nan(size(X)); a :cfr*IsK  
    %       y = zernfun2(p,r(idx),theta(idx)); BH}M]<5  
    %       figure('Units','normalized') ~&"'>C#  
    %       for k = 1:length(p) Z&7Yl(|  
    %           z(idx) = y(:,k); 5@pLGMHT  
    %           subplot(4,4,k) Tl+PRR6D*  
    %           pcolor(x,x,z), shading interp lAGntYv  
    %           set(gca,'XTick',[],'YTick',[]) voJJoy%  
    %           axis square .z*}%,G  
    %           title(['Z_{' num2str(p(k)) '}']) !Y/S2J  
    %       end AOz~@i^  
    % Zcg-i:@  
    %   See also ZERNPOL, ZERNFUN. ID<[=es6  
    ]u:Ij|.'y0  
    %   Paul Fricker 11/13/2006 \!%~( FM  
    8A u W>7_  
    _lC0XDZ  
    % Check and prepare the inputs: _SH~.Mt_!  
    % ----------------------------- &!FI!T -WH  
    if min(size(p))~=1 7|o}m}yVx  
        error('zernfun2:Pvector','Input P must be vector.') 1@F>E;YjL=  
    end  lsgZ  
    &2n 5m&   
    if any(p)>35 !P":z0K4  
        error('zernfun2:P36', ... [<>%I#7ulG  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... c4s,T"H  
               '(P = 0 to 35).']) ZmJ<FF4  
    end M'T[L%AP  
    5,\|XQA5!  
    % Get the order and frequency corresonding to the function number: =Wz)(N  
    % ---------------------------------------------------------------- Ix~rBD9  
    p = p(:); Z|A+\#'  
    n = ceil((-3+sqrt(9+8*p))/2); &v feBth  
    m = 2*p - n.*(n+2); 2tQ?=V(Di  
    Owv}lJ  
    % Pass the inputs to the function ZERNFUN: m @)Ya*=<  
    % ---------------------------------------- E5a7p.  
    switch nargin 8~O0P=  
        case 3 \VypkbE+  
            z = zernfun(n,m,r,theta); }C$D-fH8sW  
        case 4 O:8Ne*L`D  
            z = zernfun(n,m,r,theta,nflag); 0W~1v  
        otherwise G'wyH[ d/  
            error('zernfun2:nargin','Incorrect number of inputs.') 3-)R'  
    end dl6Ju  
    ;{0%Vp{  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) U? ;Q\=>  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Fz| r[  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of O8-Z >;  
    %   order N and frequency M, evaluated at R.  N is a vector of ucJ8l(?Qc  
    %   positive integers (including 0), and M is a vector with the c2]h.G83  
    %   same number of elements as N.  Each element k of M must be a M[e^Z}w.V  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 4~hP25q  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is ?`75ah  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix %H3 iX^}*  
    %   with one column for every (N,M) pair, and one row for every M7YbRl  
    %   element in R. 3~LNz8Z*  
    % Gsz$H_  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- '}q/;}ih  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 'L^M"f^I  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to Q,};O$h  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ![eipOX  
    %   for all [n,m]. w,X J8+B  
    % 7UUu1"|a|  
    %   The radial Zernike polynomials are the radial portion of the 3w/z$bj  
    %   Zernike functions, which are an orthogonal basis on the unit #fXy4iL l  
    %   circle.  The series representation of the radial Zernike q3|SZoN  
    %   polynomials is Ym$`EN  
    % Z}3;Ych  
    %          (n-m)/2 ?iL-2I3*  
    %            __ (Sj<>xgd  
    %    m      \       s                                          n-2s +^.xLTX`$  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r :]LW,Eql  
    %    n      s=0 5H( ]"C  
    % JtF)jRB0,  
    %   The following table shows the first 12 polynomials. Vq^b_^  
    % !|cM<}TF,  
    %       n    m    Zernike polynomial    Normalization ${97G#  
    %       --------------------------------------------- r>rL[`p(2  
    %       0    0    1                        sqrt(2) ]|[mwC4  
    %       1    1    r                           2 =r&i`L{]  
    %       2    0    2*r^2 - 1                sqrt(6) yz)Nco]  
    %       2    2    r^2                      sqrt(6) &0{&4,  
    %       3    1    3*r^3 - 2*r              sqrt(8) ~W3t(\B'  
    %       3    3    r^3                      sqrt(8) y4H/CH$%  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) zY].ZS=7  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) c#@L~<  
    %       4    4    r^4                      sqrt(10) y-Lm^ GW4  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) CYM>4C~>JW  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) v(,YqT>q@U  
    %       5    5    r^5                      sqrt(12) F'ez{ B\AX  
    %       --------------------------------------------- q^L"@Q5;  
    % %mOQIXr1s  
    %   Example: khQ fLA  
    % -@~4:o  
    %       % Display three example Zernike radial polynomials WJ\,Y} J  
    %       r = 0:0.01:1; =~q$k  
    %       n = [3 2 5]; 0 Cyus  
    %       m = [1 2 1]; 59V8cO+qH  
    %       z = zernpol(n,m,r); xSq+>,b  
    %       figure -y/Y%]%0  
    %       plot(r,z) >&T J  
    %       grid on q{!ft9|K\d  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest')  7uzc1}r  
    % rgEN~e'  
    %   See also ZERNFUN, ZERNFUN2. >=3oe.$)  
    |w>d]eA5  
    % A note on the algorithm. C *7x7|z  
    % ------------------------ +;q` A 1  
    % The radial Zernike polynomials are computed using the series G}nj 71=H  
    % representation shown in the Help section above. For many special WqHp23  
    % functions, direct evaluation using the series representation can D$`$4mX@hP  
    % produce poor numerical results (floating point errors), because xk^`4;  
    % the summation often involves computing small differences between S=$ \S9  
    % large successive terms in the series. (In such cases, the functions 2l~qzT-  
    % are often evaluated using alternative methods such as recurrence S4!B;,?AxN  
    % relations: see the Legendre functions, for example). For the Zernike 'Xasd3*Py  
    % polynomials, however, this problem does not arise, because the "rpP  
    % polynomials are evaluated over the finite domain r = (0,1), and )t,efg  
    % because the coefficients for a given polynomial are generally all NQN?CBFQ  
    % of similar magnitude. QjTs$#eMW  
    % ~^o YPd52*  
    % ZERNPOL has been written using a vectorized implementation: multiple k40`,;}9  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Lv#}Gm  
    % values can be passed as inputs) for a vector of points R.  To achieve Q@C  y\l  
    % this vectorization most efficiently, the algorithm in ZERNPOL 1.nYT*  
    % involves pre-determining all the powers p of R that are required to j% '~l#nw  
    % compute the outputs, and then compiling the {R^p} into a single $-39O3  
    % matrix.  This avoids any redundant computation of the R^p, and :6Gf@Z&+  
    % minimizes the sizes of certain intermediate variables. z%$M IC  
    % $Ut1vp1$  
    %   Paul Fricker 11/13/2006 GwmYhG<{  
    %KR2Vlh0  
    Bey9P)_Of  
    % Check and prepare the inputs: [MeFj!(  
    % ----------------------------- ~Vc`AcWP  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 0 R>!jw  
        error('zernpol:NMvectors','N and M must be vectors.') 6Zv-kG  
    end Yb'%J@T}  
    |gfG\fL3V  
    if length(n)~=length(m) +rKV*XX@  
        error('zernpol:NMlength','N and M must be the same length.') YOY2K%o  
    end \CYKj_c  
    q oz[x  
    n = n(:); rW*[sLl3  
    m = m(:); nDz.61$[  
    length_n = length(n); ~vMJ?P@  
    mwF{z.t"  
    if any(mod(n-m,2)) 1\/^X>@W{  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') /WIH#M  
    end N\fj[?f[  
    tl=e!  
    if any(m<0) ?4_ME3$t  
        error('zernpol:Mpositive','All M must be positive.') Hr*xAx  
    end xqv&^,ic  
    %D`j3cEp@  
    if any(m>n) &HqBlRo  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.')  \ns} M3  
    end +O7GgySx  
    BfDC[(n`  
    if any( r>1 | r<0 ) iz27yXHZ~  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') N <M6~  
    end QH kjxj  
    ;/!o0:m^I  
    if ~any(size(r)==1) Dt'bbX'edw  
        error('zernpol:Rvector','R must be a vector.') p R ! m  
    end u/J1Z>0  
    *d-JAE  
    r = r(:); L$@qEsO  
    length_r = length(r); <_SdW 5BF<  
    jN/snU2\0  
    if nargin==4 pwA~?$B1  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 9r 5(  
        if ~isnorm }|],UXk{xB  
            error('zernpol:normalization','Unrecognized normalization flag.') ~eXI}KhBw6  
        end ,2mq}u>WU  
    else z -'e<v;w  
        isnorm = false; s@*i  
    end r%`3*<ALV)  
    R?#.z#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +)_#j/  
    % Compute the Zernike Polynomials l=<},_]{  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =0>[-:Z  
    ]<u%jTQREd  
    % Determine the required powers of r: Tz7|OV_W$  
    % ----------------------------------- P9M%B2DQ6f  
    rpowers = []; EnEaUb?P  
    for j = 1:length(n) )g ?'Nz  
        rpowers = [rpowers m(j):2:n(j)]; }-p-(  
    end k|cP]p4,  
    rpowers = unique(rpowers); L#N ]1#;  
    Qa"4^s  
    % Pre-compute the values of r raised to the required powers, M`?ATmYy  
    % and compile them in a matrix: `~ h8D9G  
    % ----------------------------- pbG v\S F  
    if rpowers(1)==0 fbJa$  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); m>!aI?g  
        rpowern = cat(2,rpowern{:}); rE?B9BF3O  
        rpowern = [ones(length_r,1) rpowern]; HC| ]Au  
    else b6@0?_n  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Q}1qt4xy*  
        rpowern = cat(2,rpowern{:}); YOUX  
    end m(CsO|pz  
    Gyc _B  
    % Compute the values of the polynomials: <WL] (-9I:  
    % -------------------------------------- u|(Iu}sE=  
    z = zeros(length_r,length_n); rfV{+^T;  
    for j = 1:length_n v3cLU7bi?2  
        s = 0:(n(j)-m(j))/2; +; =XiB5R  
        pows = n(j):-2:m(j); fBKN?]BdN  
        for k = length(s):-1:1 /pJr%}sc  
            p = (1-2*mod(s(k),2))* ... }*7Gq  
                       prod(2:(n(j)-s(k)))/          ... WPPD vB  
                       prod(2:s(k))/                 ... jm[f|4\  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... >P-'C^:V=  
                       prod(2:((n(j)+m(j))/2-s(k))); 8]WcW/1r !  
            idx = (pows(k)==rpowers); c&"1Z/tR  
            z(:,j) = z(:,j) + p*rpowern(:,idx); g ~%IA.$c  
        end WmE4TL^8?  
         H^g&e$d0  
        if isnorm uIR   
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ix 5\Y  
        end ^CB@4$!   
    end J,k.*t:  
    6ciA|J'MR  
    % 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)  JI /iq  
    N3w y][bo  
    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。 mv(/M t  
    w7f)v\p  
    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)