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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 9%14k  
    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! F:S>\wG,  
     
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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 )]c3bMVE-  
    function z = zernfun(n,m,r,theta,nflag) EvqAi/(g  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. I#E(r>KW*  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N a<wQzgxG  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 6eYf2sZ;J  
    %   unit circle.  N is a vector of positive integers (including 0), and vF6*c  
    %   M is a vector with the same number of elements as N.  Each element :@%-f:iDj  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) K}E7|gdG  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ;i9<y8Dha  
    %   and THETA is a vector of angles.  R and THETA must have the same ,o@~OTja*  
    %   length.  The output Z is a matrix with one column for every (N,M) A9l})_~i  
    %   pair, and one row for every (R,THETA) pair. wYO"znd  
    % m_!vIUOz  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 4[,B;7  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), koEX4q  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral VMZ]n%XRXW  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ca/o#9:N`:  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized hQ}7Z&O  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. SU%rWH  
    % d9-mWz(V+  
    %   The Zernike functions are an orthogonal basis on the unit circle. s w.AfRQP  
    %   They are used in disciplines such as astronomy, optics, and n^pZXb;Y  
    %   optometry to describe functions on a circular domain. Uy59zB2|=  
    % fQW_YQsb  
    %   The following table lists the first 15 Zernike functions. ke9QT#~p!-  
    % Go\} A:|s  
    %       n    m    Zernike function           Normalization H/Ec^Lc+_  
    %       -------------------------------------------------- (!VMnLlXRK  
    %       0    0    1                                 1 8S1P&+iKs  
    %       1    1    r * cos(theta)                    2 UhSh(E8p>  
    %       1   -1    r * sin(theta)                    2 @bW[J  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) RJRq` T|m  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Uc&6=5~Ys\  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) `o_fUOe8a  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) tSb?]J  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) _iGU|$a  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) C](z#c~c  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) xdL/0 N3  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ,zN3? /7  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) jKj=#O  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 1J-Qh<Q   
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )ew[ Ak|  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) NDRW  
    %       -------------------------------------------------- $K?T=a;z  
    % h eE'S/  
    %   Example 1: &R'w-0k_  
    % ntj`+7mw  
    %       % Display the Zernike function Z(n=5,m=1) 1C0Y0{6,  
    %       x = -1:0.01:1; coF T2Pq  
    %       [X,Y] = meshgrid(x,x); oI_oz0nHk  
    %       [theta,r] = cart2pol(X,Y); *b Ci2mbm@  
    %       idx = r<=1; ,G[r+4|h  
    %       z = nan(size(X)); kUn2RZ6$#  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); *|LbbRu  
    %       figure &0+x2e)7g  
    %       pcolor(x,x,z), shading interp : F7k{~  
    %       axis square, colorbar C#Hcv*D  
    %       title('Zernike function Z_5^1(r,\theta)') |oe!P}u  
    % %XJQ0CE<(  
    %   Example 2: |jahpji6  
    % 7_Ba3+9jpa  
    %       % Display the first 10 Zernike functions 6_R\l@a  
    %       x = -1:0.01:1; `E} p77  
    %       [X,Y] = meshgrid(x,x); (px*R~}  
    %       [theta,r] = cart2pol(X,Y); X~v4"|a  
    %       idx = r<=1; ,4H;P/xsb  
    %       z = nan(size(X)); =5y`(0 I`U  
    %       n = [0  1  1  2  2  2  3  3  3  3]; lo+xo;Nd  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ~@T+mHny  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 8pYyG |\  
    %       y = zernfun(n,m,r(idx),theta(idx)); ^oQekga\l  
    %       figure('Units','normalized') bKk CW  
    %       for k = 1:10 T&1-eq>l  
    %           z(idx) = y(:,k); xClRO,-  
    %           subplot(4,7,Nplot(k)) F2IC$:e M  
    %           pcolor(x,x,z), shading interp AH&9Nye8  
    %           set(gca,'XTick',[],'YTick',[]) 5%<TF .;-J  
    %           axis square Mn]}s:v  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ?. zu2  
    %       end XVQL.A7  
    % O.*jR`l  
    %   See also ZERNPOL, ZERNFUN2. T>#TDMU#Fm  
    <9ma(PFa  
    %   Paul Fricker 11/13/2006 o"|O ]  
    7w "sJ  
    p{NPcT%&  
    % Check and prepare the inputs: C/F@ ]_y  
    % ----------------------------- 6#<Ir @z  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) qE>i,|rP`  
        error('zernfun:NMvectors','N and M must be vectors.') P?^JPbfV  
    end B-!guf rnY  
    ;K3d' U  
    if length(n)~=length(m) <u0*"  
        error('zernfun:NMlength','N and M must be the same length.') K<c2PFo)Q  
    end "?$L'!bM@  
    __ 8&Jv\  
    n = n(:); :I2H&,JT  
    m = m(:); ucw`;<d8  
    if any(mod(n-m,2)) ('=Z }~  
        error('zernfun:NMmultiplesof2', ... #`/bQ~s  
              'All N and M must differ by multiples of 2 (including 0).') avlqDi1l  
    end /x$}D=(CZ  
    $( S*GF$S  
    if any(m>n) 'r~8  
        error('zernfun:MlessthanN', ... w{3ycR  
              'Each M must be less than or equal to its corresponding N.') d>Un J)V}  
    end O{~KR/  
    A*hZv|$0  
    if any( r>1 | r<0 ) vruD U#  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') '}_=kp'X  
    end 5\WUoSgy  
    6VC-KY  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) /w0sj`;"  
        error('zernfun:RTHvector','R and THETA must be vectors.') +vf:z?I8  
    end {t&*>ma6)  
    byafb+x  
    r = r(:); yx2z%E  
    theta = theta(:); DE%fF,Hk3  
    length_r = length(r); sa G8g  
    if length_r~=length(theta) "9w}dQ  
        error('zernfun:RTHlength', ... 6.[)`iF+#  
              'The number of R- and THETA-values must be equal.') /N>} 4Ay  
    end 4h;4!I|  
    \6{LR&  
    % Check normalization: P7Xg{L&@.  
    % -------------------- GLCAiSMz[  
    if nargin==5 && ischar(nflag) / $_M@>  
        isnorm = strcmpi(nflag,'norm'); <KX&zi<L)  
        if ~isnorm syR N4  
            error('zernfun:normalization','Unrecognized normalization flag.') /HZv  
        end tU{\ev$x  
    else e9 *lixh  
        isnorm = false; Ls1B \Aw_  
    end > VP5vkv=  
    6x/s|RWL1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9p4y>3  
    % Compute the Zernike Polynomials Hs$'0:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% KU]ok '  
    4^[ /=J}  
    % Determine the required powers of r: BKay*!'PX  
    % ----------------------------------- eeW`JG-E  
    m_abs = abs(m); h,t:]  
    rpowers = []; <[ZI.+_Wt  
    for j = 1:length(n) ALXTR%f  
        rpowers = [rpowers m_abs(j):2:n(j)]; ^^U%cuKg  
    end b!^@PIX  
    rpowers = unique(rpowers); >g]ON9CGH  
    >La><.z~  
    % Pre-compute the values of r raised to the required powers, 6Hk="$6K  
    % and compile them in a matrix: _w>uI57U  
    % ----------------------------- p?JQ[K7i  
    if rpowers(1)==0 'OD) v  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Wo!;K|~P  
        rpowern = cat(2,rpowern{:}); M?$ZJ-  
        rpowern = [ones(length_r,1) rpowern]; O%&cE*eX  
    else H O*YBL  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); w"~<h;  
        rpowern = cat(2,rpowern{:}); k"0;D-lTZ>  
    end s6n`?,vw  
    pawl|Z'Ez  
    % Compute the values of the polynomials: @PX\{6&  
    % -------------------------------------- nxfoWy  
    y = zeros(length_r,length(n)); [Gtb+'8  
    for j = 1:length(n)  Gq1)1  
        s = 0:(n(j)-m_abs(j))/2; to`mnp9Z  
        pows = n(j):-2:m_abs(j); \f%.n]>  
        for k = length(s):-1:1 \k; n20\u  
            p = (1-2*mod(s(k),2))* ... MA* :<l  
                       prod(2:(n(j)-s(k)))/              ... S)7/0N79A  
                       prod(2:s(k))/                     ... R,,Qt TGB  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 4MLH+/e  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); pRrHuLj^  
            idx = (pows(k)==rpowers); 3{ "O,h  
            y(:,j) = y(:,j) + p*rpowern(:,idx); vy9dAl  
        end :o8MUXH$  
         I2[]A,f ,  
        if isnorm n_23EcSy  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); [E|uY]DR  
        end vFhz!P~  
    end {v,)G)obWw  
    % END: Compute the Zernike Polynomials "@yyXS r  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 24B<[lSK  
    %u!b& 5]e  
    % Compute the Zernike functions: `]<`$71w  
    % ------------------------------ !Z|($21W  
    idx_pos = m>0; HID([Wk  
    idx_neg = m<0; .<YcSG  
    zk}{ dG^M:  
    z = y; kO_5|6  
    if any(idx_pos) ;gB`YNL  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); +}JM&bfK  
    end 76@qHTh }  
    if any(idx_neg) GBQn_(b9I  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)');  rLv;Y  
    end s&Yi 6:J  
    z7T0u.4Ss  
    % 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) l,`!rF_  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. @_yoX(.E&  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated /,tAoa~FA  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 1cC1*c0Z  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, z&}-8JykH  
    %   and THETA is a vector of angles.  R and THETA must have the same ^%<pJMgdF  
    %   length.  The output Z is a matrix with one column for every P-value, {C3Y7<  
    %   and one row for every (R,THETA) pair. g0R[xOS|  
    % 8dO?K*J,H'  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike qoX@@xr1  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 3z8C  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) vjUp *R>h  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 S Xr%kndS  
    %   for all p. ,r^"#C0J}  
    %  $xgBKD  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 TqAPAHg  
    %   Zernike functions (order N<=7).  In some disciplines it is 7Y( 5]A9=  
    %   traditional to label the first 36 functions using a single mode Da1aI]{I  
    %   number P instead of separate numbers for the order N and azimuthal (z7+|JE.  
    %   frequency M. KZ:hKY@q  
    % '7 )"  
    %   Example: !0}\&<8/m  
    % <48<86TP  
    %       % Display the first 16 Zernike functions 0L-!! c3  
    %       x = -1:0.01:1; k$i'v:c|:i  
    %       [X,Y] = meshgrid(x,x); l=m(mf?QBg  
    %       [theta,r] = cart2pol(X,Y); MuI2?:~:*4  
    %       idx = r<=1; RHY4P4B<v>  
    %       p = 0:15; xge7r3i  
    %       z = nan(size(X)); SNpi=K!yn  
    %       y = zernfun2(p,r(idx),theta(idx)); 8Ogv9  
    %       figure('Units','normalized') |U' I/A  
    %       for k = 1:length(p) ; H0{CkH  
    %           z(idx) = y(:,k); >Aq:K^D/3F  
    %           subplot(4,4,k) [iS$JG-  
    %           pcolor(x,x,z), shading interp K|r Lkl9  
    %           set(gca,'XTick',[],'YTick',[]) G4-z3e,crr  
    %           axis square }IaA7f  
    %           title(['Z_{' num2str(p(k)) '}']) )A8v];.]3  
    %       end dJk9@u  
    % 0 p uY"[c  
    %   See also ZERNPOL, ZERNFUN. j?i#L}.I  
    83Ou9E!W  
    %   Paul Fricker 11/13/2006 _e<o7Y@_  
    gFN 9jM  
    k;^ :  
    % Check and prepare the inputs: Y3U9:VB  
    % ----------------------------- V"KS[>>f  
    if min(size(p))~=1 8Cx^0  
        error('zernfun2:Pvector','Input P must be vector.') /n,a?Ft^N)  
    end j;~%lg=)  
    b1?xeG#  
    if any(p)>35 zw@'vncc  
        error('zernfun2:P36', ... EG<s_d?  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... @x&P9M0g  
               '(P = 0 to 35).']) E8[T   
    end L"+$Wc[|  
    I:j3sy  
    % Get the order and frequency corresonding to the function number: (R}ii}&  
    % ---------------------------------------------------------------- R{hf9R,  
    p = p(:); S~OhtHwK  
    n = ceil((-3+sqrt(9+8*p))/2); 3`.P'Fh(k  
    m = 2*p - n.*(n+2); ~l E _L1-c  
    1R%1h9I4'  
    % Pass the inputs to the function ZERNFUN: )7cb6jCU  
    % ---------------------------------------- 7Ke&0eAw  
    switch nargin Z}6^ve  
        case 3 aoW6U{\  
            z = zernfun(n,m,r,theta); TD@v9  
        case 4 L@Nu/(pB=  
            z = zernfun(n,m,r,theta,nflag); afG{lWE)  
        otherwise kAYb!h[`  
            error('zernfun2:nargin','Incorrect number of inputs.') )X+mV  
    end RVw9Y*]b  
    `C E^2  
    % 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) V >-b`e  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. sN=6gCau  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of F"+o@9]  
    %   order N and frequency M, evaluated at R.  N is a vector of L:nXWz  
    %   positive integers (including 0), and M is a vector with the gxNL_(A  
    %   same number of elements as N.  Each element k of M must be a YvL?j  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) z9/G4^qF  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is L71!J0@a#  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix ]jMKC8uz  
    %   with one column for every (N,M) pair, and one row for every C)-^<  
    %   element in R. -NGK@Yk22  
    % k`KGB  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- q<vf,D@{ !  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is fT\:V5-  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to {lG@hN'  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 OTWkUB{  
    %   for all [n,m]. ;i uQ?MR3  
    % t0&@h\K  
    %   The radial Zernike polynomials are the radial portion of the Qq& W3  
    %   Zernike functions, which are an orthogonal basis on the unit ]$-cMX  
    %   circle.  The series representation of the radial Zernike Z4TL6 ]^R  
    %   polynomials is ;zTuKex~  
    % AEirj /  
    %          (n-m)/2 SUCU P<G  
    %            __ eP1nUy=T  
    %    m      \       s                                          n-2s F?+3%>/A @  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r <~U4*  
    %    n      s=0 0rSIfYZa  
    % K]oM8H1  
    %   The following table shows the first 12 polynomials. q}|U4MJm  
    % ,V] ]: eR  
    %       n    m    Zernike polynomial    Normalization Pf_F59"  
    %       --------------------------------------------- `bI)<B  
    %       0    0    1                        sqrt(2) Lz9#A.  
    %       1    1    r                           2 G`h+l<  
    %       2    0    2*r^2 - 1                sqrt(6) Z [Xa%~5>5  
    %       2    2    r^2                      sqrt(6) FVsj;  
    %       3    1    3*r^3 - 2*r              sqrt(8) <~emx'F|  
    %       3    3    r^3                      sqrt(8) UM%o\BiO  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) FwAKP>6*  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) \kIMDg3}  
    %       4    4    r^4                      sqrt(10) t !`Jse>  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) CBT>"sYE1  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ^ZeJ[t&!#  
    %       5    5    r^5                      sqrt(12) km5~Gc}  
    %       --------------------------------------------- I+ l%Sn#\  
    % GOy%^:Xd  
    %   Example: WKM)*@#,  
    % V~MiO.B  
    %       % Display three example Zernike radial polynomials bUy,5gk-  
    %       r = 0:0.01:1; \YJy#2K  
    %       n = [3 2 5]; cR{>IH4^  
    %       m = [1 2 1]; w FtN+  
    %       z = zernpol(n,m,r); Ds8 EMtS  
    %       figure fIC9WbiH-  
    %       plot(r,z) o}Cq.[G4k  
    %       grid on mABe'"8  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') DC+wD Bp;  
    % FN[R(SLbL  
    %   See also ZERNFUN, ZERNFUN2. -<_$m6x"A  
    's x\P[a  
    % A note on the algorithm. 34|a\b}  
    % ------------------------ {i~8 :  
    % The radial Zernike polynomials are computed using the series hjx)D  
    % representation shown in the Help section above. For many special #C*8X+._y  
    % functions, direct evaluation using the series representation can 9.O8/0w7LV  
    % produce poor numerical results (floating point errors), because Bvjl-$m!v  
    % the summation often involves computing small differences between \(UKd v  
    % large successive terms in the series. (In such cases, the functions +#J,BKul  
    % are often evaluated using alternative methods such as recurrence Vn=qV3OE]  
    % relations: see the Legendre functions, for example). For the Zernike j5$BK[p.  
    % polynomials, however, this problem does not arise, because the +V862R4,o  
    % polynomials are evaluated over the finite domain r = (0,1), and ?dZt[vAMn  
    % because the coefficients for a given polynomial are generally all T5Eseesp  
    % of similar magnitude. &:B<Q$g#  
    % 1n*W2:,z  
    % ZERNPOL has been written using a vectorized implementation: multiple pY8q=Kl  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 6 &U+6gb  
    % values can be passed as inputs) for a vector of points R.  To achieve [&S}dQ"  
    % this vectorization most efficiently, the algorithm in ZERNPOL U!w1AY|  
    % involves pre-determining all the powers p of R that are required to C.  MoKa3  
    % compute the outputs, and then compiling the {R^p} into a single ^}yg%+  
    % matrix.  This avoids any redundant computation of the R^p, and 4A`NJ  
    % minimizes the sizes of certain intermediate variables. )x,8D ~p'  
    % n}-3o]ku  
    %   Paul Fricker 11/13/2006 'fwU]Hm  
    u0`o A  
    9?T{}| ?  
    % Check and prepare the inputs: 6~meM@  
    % ----------------------------- Q-TV*FD.  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) L *[K>iW  
        error('zernpol:NMvectors','N and M must be vectors.') + bhym+  
    end [p r"ZQ]  
    2LY=D L7  
    if length(n)~=length(m) Mq%,lJA\  
        error('zernpol:NMlength','N and M must be the same length.') `ejUs]SR  
    end xh@-g|+g  
    !7B\Xl'S  
    n = n(:); ?|;yVew  
    m = m(:); "v*8_El  
    length_n = length(n); _+f+`]iM  
    =;~I_)Pg1  
    if any(mod(n-m,2)) J<n+\F-s  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') :q##fG 'm/  
    end JMBK{JK>  
    pj|pcv^  
    if any(m<0) s0UFym 8  
        error('zernpol:Mpositive','All M must be positive.') rPzQ8<  
    end ~89P[$6  
    6`01EIk  
    if any(m>n) }peBR80tQ  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') /x@RNdKv  
    end Ft{[ae?4  
    T".]m7!  
    if any( r>1 | r<0 ) TTNk r`  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') &(rWwOo6  
    end Nf,Z;5e  
    `rY2up#%  
    if ~any(size(r)==1) jLg@FDb~  
        error('zernpol:Rvector','R must be a vector.') ["<nq`~  
    end OV CR0  
    y9Y1PH7G  
    r = r(:); iyx>q!P  
    length_r = length(r); L7Dh(y=;7  
    "HMP$)d  
    if nargin==4 C}g9'jY  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); Aez2*g3  
        if ~isnorm Tq<2`*Qs  
            error('zernpol:normalization','Unrecognized normalization flag.') Z~G my7h(  
        end `A%^UCd  
    else uw\1b.r'B  
        isnorm = false; Y[ reD  
    end nHFrG =o,  
    RH)EB<PV  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Zzua17  
    % Compute the Zernike Polynomials pI`?(5iK6|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fCAiLkT,C[  
    eZhPu'id\s  
    % Determine the required powers of r: D?jk$^p~m#  
    % ----------------------------------- "pxzntY|  
    rpowers = []; x90*yaw>h  
    for j = 1:length(n) [ Mg8/Oy  
        rpowers = [rpowers m(j):2:n(j)]; l kIn%=Z  
    end b}ODWdJ1  
    rpowers = unique(rpowers); qKS;x@  
    D,l,`jv*  
    % Pre-compute the values of r raised to the required powers, ]6Ug>>x5  
    % and compile them in a matrix: ^yviV Y  
    % ----------------------------- v'2[[u{7*  
    if rpowers(1)==0 `WEZ"5n  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); H14Ic.&  
        rpowern = cat(2,rpowern{:}); G>qZxy`c  
        rpowern = [ones(length_r,1) rpowern]; ;Z[]{SQ  
    else +H/jK@  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); RNVbcd  
        rpowern = cat(2,rpowern{:}); [t\B6XxT  
    end vQVK$n`  
    `i~ Y Fr  
    % Compute the values of the polynomials: ]36sZ *  
    % -------------------------------------- f67NWFX  
    z = zeros(length_r,length_n); 1B>Vt*=  
    for j = 1:length_n <<A`aU^fX  
        s = 0:(n(j)-m(j))/2; 2],_^XBvB  
        pows = n(j):-2:m(j); <3PL@orO  
        for k = length(s):-1:1 EUYCcL'G  
            p = (1-2*mod(s(k),2))* ... %b.UPS@I  
                       prod(2:(n(j)-s(k)))/          ... _#e&t"@GS  
                       prod(2:s(k))/                 ... vh!v MB}}  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 6Z?j AXGSq  
                       prod(2:((n(j)+m(j))/2-s(k))); jdeV|H} u  
            idx = (pows(k)==rpowers); ({0)@+V8  
            z(:,j) = z(:,j) + p*rpowern(:,idx); W) j|rz.  
        end `pZs T ^G[  
         /76 1o\Q  
        if isnorm 6!iJ;1PeE  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); j Ib  
        end ~\nBjM2  
    end v}G]X Z8  
    C) QKPT  
    % 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)  H>Q%"|  
    j.~!dh$mg  
    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。 R_] {2~J+  
    lPH%Do>K  
    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)