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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 xKl\:}Ytp  
    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! NH9"89]E  
     
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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 5(5:5q.A/D  
    function z = zernfun(n,m,r,theta,nflag) IJ]rVty  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. O NVhB  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N xO[V>Ud  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ^XX_ qC'1  
    %   unit circle.  N is a vector of positive integers (including 0), and R_W6}  
    %   M is a vector with the same number of elements as N.  Each element /|0xOiib  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) mqtX7rej  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Vx z`  
    %   and THETA is a vector of angles.  R and THETA must have the same P{,A%t  
    %   length.  The output Z is a matrix with one column for every (N,M) ]sTbEw.[  
    %   pair, and one row for every (R,THETA) pair. QUeuN?3X\  
    % ]!q>@b  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike EDT9O  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), iD*21c<kd  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 40%fOu,u`  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, p$=Z0p4%LL  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized NX4G;+6  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 2##;[  
    % GQ(*k)'a  
    %   The Zernike functions are an orthogonal basis on the unit circle. H +' 6*akV  
    %   They are used in disciplines such as astronomy, optics, and Yt[LIn-v:  
    %   optometry to describe functions on a circular domain. cgnMoBIc  
    % nW)?cQ I  
    %   The following table lists the first 15 Zernike functions. ZIN1y;dJ  
    % /!?b&N/d)  
    %       n    m    Zernike function           Normalization EXMW,  
    %       -------------------------------------------------- ,wf:Fr  
    %       0    0    1                                 1 ~R&rQJJeJ  
    %       1    1    r * cos(theta)                    2 7Kf  
    %       1   -1    r * sin(theta)                    2 L{&>,ww  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) S B~opN  
    %       2    0    (2*r^2 - 1)                    sqrt(3) C$p012D1  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ~&?57Sw*m  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) E{0e5.{  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 5dGfO:Dy_  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) NH;e|8  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 0W0GSDx  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) )DmydyQ'  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |8pSMgN  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) "cyRzQ6EH  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =+LIGHIt  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Llkh kq_  
    %       -------------------------------------------------- b@c(Nv  
    % ic5af"/(\  
    %   Example 1: #.rkvoB0N  
    % wz1nV}  
    %       % Display the Zernike function Z(n=5,m=1) No"i6R+  
    %       x = -1:0.01:1; p5jR;nOZ%l  
    %       [X,Y] = meshgrid(x,x); X::@2{-@y  
    %       [theta,r] = cart2pol(X,Y);  )ut$644R  
    %       idx = r<=1; XHxJzYMc  
    %       z = nan(size(X)); vh.-9eD  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); BTD_j&+(  
    %       figure ;vneeW4|  
    %       pcolor(x,x,z), shading interp >fMzUTJ4  
    %       axis square, colorbar & #JYh=#  
    %       title('Zernike function Z_5^1(r,\theta)') L[ZS17 ;*  
    % T$`m!mQ4  
    %   Example 2: `*cqT  
    % qdLzB  
    %       % Display the first 10 Zernike functions }W@refS  
    %       x = -1:0.01:1; (a0(ZOKH  
    %       [X,Y] = meshgrid(x,x); 4qQE9f xdY  
    %       [theta,r] = cart2pol(X,Y); P4HoKoj2`  
    %       idx = r<=1; zJP jsD]  
    %       z = nan(size(X)); "n]x%. *  
    %       n = [0  1  1  2  2  2  3  3  3  3]; GMg! 2CIU  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; k,$/l1D  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; hP8w3gl_  
    %       y = zernfun(n,m,r(idx),theta(idx)); Zr1"'+-  
    %       figure('Units','normalized') #q K.AZi  
    %       for k = 1:10 JN:L%If  
    %           z(idx) = y(:,k); z Ohv>a  
    %           subplot(4,7,Nplot(k)) -8l(eDm"m  
    %           pcolor(x,x,z), shading interp lX%-oRQ/os  
    %           set(gca,'XTick',[],'YTick',[]) wm^1Fn--  
    %           axis square VXiU5n^  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) c]Gs{V]\  
    %       end T*mR9 8i  
    %  pdm(7^  
    %   See also ZERNPOL, ZERNFUN2. gxmo 1  
    unc6 V%  
    %   Paul Fricker 11/13/2006 tvf5b8(Y-  
    b1>]?.  
    *#E_KW1RV  
    % Check and prepare the inputs: qE3Ud:j  
    % ----------------------------- R(pQu! K4  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) n_4.`vs  
        error('zernfun:NMvectors','N and M must be vectors.') +'SL5d*  
    end Kp*3:XK  
    -<k)|]8  
    if length(n)~=length(m) k~so+k&=b  
        error('zernfun:NMlength','N and M must be the same length.') EcX7wrl9x  
    end Go1xyd:k  
    5 =8v\q?)c  
    n = n(:); ]KEE+o  
    m = m(:); C$ K?4$  
    if any(mod(n-m,2)) JBA{i45x  
        error('zernfun:NMmultiplesof2', ... 7D,nxx(`  
              'All N and M must differ by multiples of 2 (including 0).') @I|kY5'c  
    end ?*$uj(  
    p>kny?AJ  
    if any(m>n) ( tq);m&  
        error('zernfun:MlessthanN', ... *Gv:N6  
              'Each M must be less than or equal to its corresponding N.') Q!3-P  
    end n $N M  
    <m^a ?q^  
    if any( r>1 | r<0 ) Ym"^Ds}  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') U+#^>}wc  
    end 43y@9P0  
    6w? GeJ  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) n^$Q^[:Z  
        error('zernfun:RTHvector','R and THETA must be vectors.') </ "Wh4>C  
    end AcEz$wy  
    ^!C  
    r = r(:); ~8 UMwpl-  
    theta = theta(:); aCH;l~+U  
    length_r = length(r); 3QKBuo  
    if length_r~=length(theta) ]@cI_n  
        error('zernfun:RTHlength', ... FeS ,TQ4j  
              'The number of R- and THETA-values must be equal.') =w;-4  
    end N.+A-[7,W  
    Ct?xTFb  
    % Check normalization: j@#RfVx  
    % -------------------- Jw}&[  
    if nargin==5 && ischar(nflag) o\ce|Dzt  
        isnorm = strcmpi(nflag,'norm'); IY6Qd4157  
        if ~isnorm Cq7 uy  
            error('zernfun:normalization','Unrecognized normalization flag.') 3?<A]"X.  
        end q)o;iR  
    else g$mMH  
        isnorm = false; 7)1%Z{Dy  
    end i;/;zG^=_  
    J =8Y D"1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *Q?8OwhJ  
    % Compute the Zernike Polynomials =bP<cC=3b  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A'uaR?  
    mJd8?d  
    % Determine the required powers of r: THX% z `  
    % ----------------------------------- 5M9o(Z\AF  
    m_abs = abs(m); YahW%mv`d  
    rpowers = []; Ake l.&  
    for j = 1:length(n) OAFxf,b  
        rpowers = [rpowers m_abs(j):2:n(j)]; ZwY mR=  
    end Il>o60u1  
    rpowers = unique(rpowers); Y1>OhHuN  
    =Ez@kTvOs  
    % Pre-compute the values of r raised to the required powers, >dgq2ok!u  
    % and compile them in a matrix: ~iiDy;"  
    % ----------------------------- GutiqVP:B  
    if rpowers(1)==0 [-"ZuUG  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); m5] a  
        rpowern = cat(2,rpowern{:}); _,v?rFLE  
        rpowern = [ones(length_r,1) rpowern]; nO'C2)bBSG  
    else q9VBK(,X  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); G#f3 WpD  
        rpowern = cat(2,rpowern{:}); 7rbw_m`12-  
    end K?e16;   
    %dr*dA'  
    % Compute the values of the polynomials: P0_Ymn=&  
    % -------------------------------------- 1#;^ Z3  
    y = zeros(length_r,length(n)); .X(qs1  
    for j = 1:length(n) Khv}q.)F  
        s = 0:(n(j)-m_abs(j))/2; C2zKt/)A  
        pows = n(j):-2:m_abs(j); M&q~e@P  
        for k = length(s):-1:1 `-cw[@uD  
            p = (1-2*mod(s(k),2))* ... E@)'Z6r1  
                       prod(2:(n(j)-s(k)))/              ... *81/q8Az  
                       prod(2:s(k))/                     ... 4bdCbI  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... H/Ql  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); y=+OC1k\8  
            idx = (pows(k)==rpowers); 0t"Iq71/  
            y(:,j) = y(:,j) + p*rpowern(:,idx); B]b/(Q+  
        end 9mn~57`y  
         f-H"|9  
        if isnorm =+?OsH v  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); -vc$I=b;  
        end +>2.O2)%q  
    end ez%:>r4  
    % END: Compute the Zernike Polynomials yA*U^:%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2?:OsA}  
    :yi} CM4  
    % Compute the Zernike functions: "Y5 :{Kj  
    % ------------------------------ P*%P"g  
    idx_pos = m>0; 20haA0s  
    idx_neg = m<0; SS8$.ot  
    P|lDW|}D@  
    z = y; /[/{m]  
    if any(idx_pos) .!lLj1?p  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); XhWo~zh"  
    end 1=9GV+`n  
    if any(idx_neg) CK|AXz+EN  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); cH:&S=>h  
    end -`z%<)!Y  
    O}2/w2n  
    % 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) r~oUln<[  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 1P 'L<z  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated S5Pn6'w  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 7zU~ X,  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, d1t_o2  
    %   and THETA is a vector of angles.  R and THETA must have the same q&NXF (  
    %   length.  The output Z is a matrix with one column for every P-value, E[zq<&P@  
    %   and one row for every (R,THETA) pair. kVt/Hhd9  
    %  rf'A+q  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike w}(pc }^U  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) )$a6l8  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) Fe$o*r,  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 0(Z:QqpU$  
    %   for all p. /P46k4M1U  
    % kJNg>SN*@#  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 3i4m!g5Z?  
    %   Zernike functions (order N<=7).  In some disciplines it is RF -c`C  
    %   traditional to label the first 36 functions using a single mode ]]}iSw'  
    %   number P instead of separate numbers for the order N and azimuthal 7 TM-uA$  
    %   frequency M. 2S[:mnK  
    % t@+e#3P!  
    %   Example: rxJl;!7G  
    % /!6 VP |  
    %       % Display the first 16 Zernike functions #(a;w  
    %       x = -1:0.01:1; ? IlT[yMw  
    %       [X,Y] = meshgrid(x,x); `jhbKgR[  
    %       [theta,r] = cart2pol(X,Y); 10r!p: D  
    %       idx = r<=1; @(N} {om  
    %       p = 0:15; LL+_zBP.   
    %       z = nan(size(X)); \)aFYDq#\  
    %       y = zernfun2(p,r(idx),theta(idx)); *&h]PhY  
    %       figure('Units','normalized') Y-+Kf5_[  
    %       for k = 1:length(p) A5 4u}  
    %           z(idx) = y(:,k); 4W E)2vkS  
    %           subplot(4,4,k) ]+w 27!  
    %           pcolor(x,x,z), shading interp =1)9>=}  
    %           set(gca,'XTick',[],'YTick',[]) H ]](xYy.  
    %           axis square i/!KUbt  
    %           title(['Z_{' num2str(p(k)) '}']) GN5*  
    %       end :8N by$#V  
    % SymlirL  
    %   See also ZERNPOL, ZERNFUN. VtU2&  
    k{|> !(Ax  
    %   Paul Fricker 11/13/2006 W4(  
    R@>^t4#_Q0  
    gd7! +6  
    % Check and prepare the inputs: Dd, &a  
    % ----------------------------- NQiu>Sg  
    if min(size(p))~=1 N693eN!  
        error('zernfun2:Pvector','Input P must be vector.') ]1h9:PF  
    end *Csxf[O  
    {S@, ,  
    if any(p)>35 DM\pi9<m  
        error('zernfun2:P36', ... 8W7ET@`  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... h{jm  
               '(P = 0 to 35).']) ALInJ{X  
    end %Br1b6 V  
    KxFA@3  
    % Get the order and frequency corresonding to the function number: Ia{t/IX\[  
    % ---------------------------------------------------------------- W+s3rS2  
    p = p(:); L$,Kdpj  
    n = ceil((-3+sqrt(9+8*p))/2); 889^P`Q5  
    m = 2*p - n.*(n+2); 5;XU6Rz!  
    c7tO'`q$e  
    % Pass the inputs to the function ZERNFUN: $0~1;@`rQ6  
    % ---------------------------------------- N>sHT =_  
    switch nargin ;uZeYY?   
        case 3 }<'ki ;  
            z = zernfun(n,m,r,theta); lX 50JJwk  
        case 4 IkGM~3e  
            z = zernfun(n,m,r,theta,nflag); oIE3`\xS  
        otherwise 1n.F`%YG  
            error('zernfun2:nargin','Incorrect number of inputs.') ^0I"  
    end ChNT; G<6$  
    !9V; 8g  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) >Wd_?NaI  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. VY=YI}E  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of g<8Oezi 65  
    %   order N and frequency M, evaluated at R.  N is a vector of 52'6wwv6?  
    %   positive integers (including 0), and M is a vector with the PT4iy<  
    %   same number of elements as N.  Each element k of M must be a Jr(Z Ym'  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ? Z2`f6;W4  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is xxC2 h3  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix a`U/|[JM  
    %   with one column for every (N,M) pair, and one row for every = ^%*:iT  
    %   element in R. -V'Y^Df  
    % vnlHUQLO  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- eK\i={va  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is %T}*DC$&S  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to  |vBy=:  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 &IG*;$c!  
    %   for all [n,m]. # 3FsK  
    % |NWHZo  
    %   The radial Zernike polynomials are the radial portion of the ]KUeSg|  
    %   Zernike functions, which are an orthogonal basis on the unit ))7CqN  
    %   circle.  The series representation of the radial Zernike `j 4>  
    %   polynomials is ;2gO(  
    % q5) K  
    %          (n-m)/2 L3*HgkQQ  
    %            __ -x!JTx[K  
    %    m      \       s                                          n-2s {?}^HW9{  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r z)u\(W*\iA  
    %    n      s=0 ey n-bw  
    % Xhpcu1nA  
    %   The following table shows the first 12 polynomials. AU8sU?=  
    % -^< t%{d  
    %       n    m    Zernike polynomial    Normalization JL7;l0#  
    %       --------------------------------------------- }:]CXrdg>  
    %       0    0    1                        sqrt(2) b4(,ls  
    %       1    1    r                           2 +u`4@~D#  
    %       2    0    2*r^2 - 1                sqrt(6) NBw{  
    %       2    2    r^2                      sqrt(6) NjO_Y t  
    %       3    1    3*r^3 - 2*r              sqrt(8) 8RcLs1n/  
    %       3    3    r^3                      sqrt(8) @E"lN  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) j.\0p-,  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) CFu^i|7o  
    %       4    4    r^4                      sqrt(10) Wo5%@C#M  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) \9R=fA18  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) _C,9c7K4  
    %       5    5    r^5                      sqrt(12) 1c*;Lr.K  
    %       --------------------------------------------- 4)p ID`  
    % R}D[ z7  
    %   Example: ]\/"-Y#4Q  
    % /^WOrMR  
    %       % Display three example Zernike radial polynomials oE,TA2  
    %       r = 0:0.01:1; 8zho\'  
    %       n = [3 2 5]; ~1nKL0C6u  
    %       m = [1 2 1]; 64Tb,AL_  
    %       z = zernpol(n,m,r); :OA;vp~$x  
    %       figure -U|Z9sia  
    %       plot(r,z) 5+q dn|9%T  
    %       grid on 'oUTY *  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') FRsp?i K)  
    % !Yz CK*av1  
    %   See also ZERNFUN, ZERNFUN2. n8i: /ypB  
    D/wJF[_  
    % A note on the algorithm. b&RsxW7  
    % ------------------------ 02-% B~oP  
    % The radial Zernike polynomials are computed using the series vTC{  
    % representation shown in the Help section above. For many special k+hl6$:Qj%  
    % functions, direct evaluation using the series representation can }-Jo9dNs  
    % produce poor numerical results (floating point errors), because t~":'le`zr  
    % the summation often involves computing small differences between C)QKodI  
    % large successive terms in the series. (In such cases, the functions ;(Az   
    % are often evaluated using alternative methods such as recurrence 3jHE,5m  
    % relations: see the Legendre functions, for example). For the Zernike 7R,;/3wWjG  
    % polynomials, however, this problem does not arise, because the ^4et; F%  
    % polynomials are evaluated over the finite domain r = (0,1), and |+qsO ;  
    % because the coefficients for a given polynomial are generally all bEmzigN[  
    % of similar magnitude. .0MY$0s  
    % #8y"1I=i&  
    % ZERNPOL has been written using a vectorized implementation: multiple JkKbw&65  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] gLK0L%"5  
    % values can be passed as inputs) for a vector of points R.  To achieve tqjjn5!  
    % this vectorization most efficiently, the algorithm in ZERNPOL }]^/`n  
    % involves pre-determining all the powers p of R that are required to xE!b)@>S  
    % compute the outputs, and then compiling the {R^p} into a single -C* 6>$A  
    % matrix.  This avoids any redundant computation of the R^p, and L^K,YlNBR  
    % minimizes the sizes of certain intermediate variables. D Q c pIV  
    % :NB.ib@*  
    %   Paul Fricker 11/13/2006 Hoi~(Vc.  
    7\gu; [n  
    T # gx2Y  
    % Check and prepare the inputs: ^AERGB\36  
    % ----------------------------- ^oNcZK>  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +Eel|)Z*Q  
        error('zernpol:NMvectors','N and M must be vectors.') Y' 5X4Ks|  
    end RMdU1@  
    &-m}w:j=  
    if length(n)~=length(m) ,bP8"|e  
        error('zernpol:NMlength','N and M must be the same length.') *e:2iM)8~  
    end ?8;WP&  
    ?yu@eo  
    n = n(:); fUPYCw6F  
    m = m(:); Dn#UcMO>W  
    length_n = length(n); qxYCT$1  
    PfGiJ]:V-u  
    if any(mod(n-m,2)) EYi{~  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') y. (m#&T  
    end U /xzl4m6  
    :Y4Sdj  
    if any(m<0) Mky^X,r  
        error('zernpol:Mpositive','All M must be positive.') K cW 5  
    end rje;Bf  
    6rO^ p  
    if any(m>n) ^s$U n6v[  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') *xpPD\{k  
    end 5r d t  
    /+WC6&  
    if any( r>1 | r<0 ) ^ (J%)&_\3  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') q ;_?e_  
    end ^N`KT   
    ce719n$   
    if ~any(size(r)==1) ]I]G3 e  
        error('zernpol:Rvector','R must be a vector.') /UaQ 2h\  
    end }iLi5Qkx  
    /AY q^  
    r = r(:); .k#O[^~]  
    length_r = length(r); dKL9}:oUa  
    6j|~oMYP  
    if nargin==4 1&Ma`M('  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); uzLm TmM+  
        if ~isnorm JV+Uy$P!  
            error('zernpol:normalization','Unrecognized normalization flag.') m~&>+q ^7  
        end <TS ps!(#  
    else :_+U[k(#  
        isnorm = false; MOHw{Vw(  
    end g;:3I\ L  
    TGjxy1A  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $XKUw"%  
    % Compute the Zernike Polynomials S(rnVsW%Ki  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~4c,'k@  
    0BAZWm  
    % Determine the required powers of r: `wSoa#U"@  
    % ----------------------------------- 7 Rc/<,X  
    rpowers = []; F>E_d<m  
    for j = 1:length(n) S'>KGdF  
        rpowers = [rpowers m(j):2:n(j)]; "u{ymJ]t  
    end ?*<1B  
    rpowers = unique(rpowers); u/N_62sk5  
     U8% IpI;  
    % Pre-compute the values of r raised to the required powers, VRHS 4  
    % and compile them in a matrix: &?']EcU5h9  
    % ----------------------------- {yi!vw  
    if rpowers(1)==0 >z,Y%A  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); +LF=oM<  
        rpowern = cat(2,rpowern{:}); 7dlMDHp\Y  
        rpowern = [ones(length_r,1) rpowern]; n"R$b:  
    else YYvX@f  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); |@?='E?h  
        rpowern = cat(2,rpowern{:}); "'>fTk_  
    end g1B P  
    ]]5(:>l  
    % Compute the values of the polynomials: e0#{'_C  
    % -------------------------------------- <YWu/\{KT  
    z = zeros(length_r,length_n); ")fgQ3XZ  
    for j = 1:length_n kjSzu qB  
        s = 0:(n(j)-m(j))/2; SO~pe$c-  
        pows = n(j):-2:m(j); m 7+=w>o  
        for k = length(s):-1:1 `2xt%kC  
            p = (1-2*mod(s(k),2))* ... >as+#rz1p  
                       prod(2:(n(j)-s(k)))/          ... 5Iv"  
                       prod(2:s(k))/                 ... Q0xQx z  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... h^J :k  
                       prod(2:((n(j)+m(j))/2-s(k))); w}29#F\]R  
            idx = (pows(k)==rpowers); kC'm |Y@T  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ~fO#En  
        end Af^9WJ  
         D9n+eZ  
        if isnorm B\`${O(  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); u R!'v  
        end ZV07;`I  
    end Zh?n;n}  
    YT@H^=  
    % 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)  J(x42Q}*S  
    Q 1e hW  
    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。 GAcU8  MD  
    _cXLQ)-  
    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)