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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 7<xnE]jdq  
    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! 23):OB>S`  
     
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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 !2|Lb'O  
    function z = zernfun(n,m,r,theta,nflag) hObL=^F  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. KOy{?  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N i|^Q{3?o#  
    %   and angular frequency M, evaluated at positions (R,THETA) on the /L^g. ~  
    %   unit circle.  N is a vector of positive integers (including 0), and FHOw ]"#  
    %   M is a vector with the same number of elements as N.  Each element t$!zgUJ  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ]pR?/3  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ,{0Y:/T'  
    %   and THETA is a vector of angles.  R and THETA must have the same Z Ts*Y,  
    %   length.  The output Z is a matrix with one column for every (N,M) R0-0  
    %   pair, and one row for every (R,THETA) pair. DhM=q  
    % 40kAGs>_  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike z0 9Gp}^;  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), v+nXKNL  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral k+h}HCzE  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, :'p)xw4K|  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized M/<ypJ  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 3:WqUb\QK  
    % 8oX1 F(R  
    %   The Zernike functions are an orthogonal basis on the unit circle. _EMI%P& s  
    %   They are used in disciplines such as astronomy, optics, and layxtECP(  
    %   optometry to describe functions on a circular domain. ?Q]&;5o  
    % mo| D  
    %   The following table lists the first 15 Zernike functions. egq,)6>  
    % 6F (z6_<  
    %       n    m    Zernike function           Normalization &nmBsl3Q.  
    %       -------------------------------------------------- Xw4Eti._D  
    %       0    0    1                                 1 D :@W*,  
    %       1    1    r * cos(theta)                    2 agUdI_'~@9  
    %       1   -1    r * sin(theta)                    2 [\ao#f0WR  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) {"wF;*U.V  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 5eTA]  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) tyR?A>F4  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) AIHH@z   
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) -N' (2'  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) KTm^}')C8  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) M}|(:o3Yo  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) #z(:n5$F  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1TZ[i  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) m^ xTV-#l@  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) gNZwD6GMe?  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) nd' D0<%  
    %       -------------------------------------------------- M1Q&)am  
    % ]ae(t`\l^  
    %   Example 1: 1`8s "T  
    % d4b!  r  
    %       % Display the Zernike function Z(n=5,m=1) Km5_P##  
    %       x = -1:0.01:1; [(n5-#1S  
    %       [X,Y] = meshgrid(x,x); 1clzDwW  
    %       [theta,r] = cart2pol(X,Y); ( >}1t!1  
    %       idx = r<=1; `:C1Wo^<  
    %       z = nan(size(X)); j3sz"(  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); \m7\}Nbz0/  
    %       figure H1-DK+Q:  
    %       pcolor(x,x,z), shading interp #*A&jo'E  
    %       axis square, colorbar WM+8<|)n  
    %       title('Zernike function Z_5^1(r,\theta)') ,l&?%H9q  
    % 1| sem(t  
    %   Example 2: )?72 +X  
    % ci;2XLAM  
    %       % Display the first 10 Zernike functions NO*, }aeG  
    %       x = -1:0.01:1; qJ;~ANwt  
    %       [X,Y] = meshgrid(x,x); J`5VE$2M  
    %       [theta,r] = cart2pol(X,Y); *8)?ZZMM  
    %       idx = r<=1; ?i<l7   
    %       z = nan(size(X)); oRbWqN`F.  
    %       n = [0  1  1  2  2  2  3  3  3  3]; nFni1cCD  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; hrniZ^  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Be{@ L  
    %       y = zernfun(n,m,r(idx),theta(idx)); ?^|[Yzk  
    %       figure('Units','normalized')  hE:~~ox  
    %       for k = 1:10 M{L<aYe  
    %           z(idx) = y(:,k); [],[LkS  
    %           subplot(4,7,Nplot(k)) 0Jv6?7]LKa  
    %           pcolor(x,x,z), shading interp dg|+?M^9`  
    %           set(gca,'XTick',[],'YTick',[]) 5j`sJvq  
    %           axis square F>.y>h  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ?h`,@~6u  
    %       end 'wPX.h?  
    % s $(%]~P  
    %   See also ZERNPOL, ZERNFUN2. F.TIdkvp  
    3Y P! B=  
    %   Paul Fricker 11/13/2006 91z=ou  
    ,.Ofv):=  
    xiW}P% bf  
    % Check and prepare the inputs: z"6o|]9I  
    % ----------------------------- lZwjrU| _  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Hl$qmq  
        error('zernfun:NMvectors','N and M must be vectors.') 54z`KX 73  
    end S[y'{;  
    Dml?.-Uv<  
    if length(n)~=length(m) ^fKKsfIf  
        error('zernfun:NMlength','N and M must be the same length.') I e!KIU  
    end moM'RO,M  
    +Vg(2Xt  
    n = n(:); 7NEOaX(J9  
    m = m(:); yMW3mx301j  
    if any(mod(n-m,2)) A#$l;M.3R  
        error('zernfun:NMmultiplesof2', ... QY+{ OCB  
              'All N and M must differ by multiples of 2 (including 0).') h6~xz0,u  
    end oxFd@WV5  
    jYU0zGpj  
    if any(m>n) eZ) |m  
        error('zernfun:MlessthanN', ... LEKE+775  
              'Each M must be less than or equal to its corresponding N.') wPghgjF{  
    end em'3 8L|(  
    Gad&3M0r  
    if any( r>1 | r<0 ) W[QgddR  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') R?:K\  
    end :V8oWMY  
    }!g$k $y  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) LZ#A`&qUd  
        error('zernfun:RTHvector','R and THETA must be vectors.') 2s2KI=6  
    end r(]Gd`]  
    ;Qd'G7+  
    r = r(:); }0R"ZPU1Rw  
    theta = theta(:); ,9|7{j|u  
    length_r = length(r); j; /@A lZl  
    if length_r~=length(theta) QdZHIgh`i  
        error('zernfun:RTHlength', ... 2aivc,m{r  
              'The number of R- and THETA-values must be equal.') [9EL[}  
    end 7OZ0;fK  
    7TX$  
    % Check normalization: #\~m}O,  
    % -------------------- ;|rFP  
    if nargin==5 && ischar(nflag) Uwiy@ T Z  
        isnorm = strcmpi(nflag,'norm'); %Y`)ZKh  
        if ~isnorm ,vi6<C\  
            error('zernfun:normalization','Unrecognized normalization flag.') ;rJ#>7K  
        end Pw|/PfG  
    else '&/Y}]  
        isnorm = false; =w7k@[Bq  
    end {KqW<X6Hp  
    5k_%%><: q  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #I yM`YB0  
    % Compute the Zernike Polynomials 1g!%ej jd  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F8m@mh*8>  
    c1%ki%J#  
    % Determine the required powers of r: "(F:'J} X  
    % ----------------------------------- USf;}F:-C  
    m_abs = abs(m); 7Il /+l(  
    rpowers = []; (>D{"}  
    for j = 1:length(n) aj+I+r"~  
        rpowers = [rpowers m_abs(j):2:n(j)]; My9fbT  
    end ;hDIoSz  
    rpowers = unique(rpowers); D>#Jh>4  
    b#e|#!Je  
    % Pre-compute the values of r raised to the required powers, Y%rC\Ij/i  
    % and compile them in a matrix: ]=^NTm,  
    % ----------------------------- )N ^g0 L  
    if rpowers(1)==0 AQBr{^inH|  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); p t{/|P  
        rpowern = cat(2,rpowern{:}); ``?Z97rH  
        rpowern = [ones(length_r,1) rpowern]; d~d~Cd`V  
    else @n=FSn6 c  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); VN4H+9E  
        rpowern = cat(2,rpowern{:}); ( (mNB]sy  
    end YKj P E  
    oX]c$<w5  
    % Compute the values of the polynomials: [k +fkr]  
    % -------------------------------------- n;dp%SD  
    y = zeros(length_r,length(n)); BI)$aR  
    for j = 1:length(n) gJn_8\,C>Q  
        s = 0:(n(j)-m_abs(j))/2; i*vf(0G  
        pows = n(j):-2:m_abs(j); v/Ei0}e6~  
        for k = length(s):-1:1 tdRnRoB  
            p = (1-2*mod(s(k),2))* ... nIP*yb}5  
                       prod(2:(n(j)-s(k)))/              ... _EZrZB  
                       prod(2:s(k))/                     ... eYjr/`>O  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... _q\w9gN  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); {wf e!f  
            idx = (pows(k)==rpowers); r`'n3#O*  
            y(:,j) = y(:,j) + p*rpowern(:,idx); i%_nH"h  
        end 4THGHS^  
         mm<rdo(`  
        if isnorm ;,]Wtmu)7  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); PT`gAUCw  
        end RIl+QA  
    end hI 1 }^;  
    % END: Compute the Zernike Polynomials of:xj$dQ_  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {#1}YGpiVM  
    j1,ir  
    % Compute the Zernike functions: <yrl_vl{  
    % ------------------------------ PM%Gsy]q  
    idx_pos = m>0; >'lte&  
    idx_neg = m<0; !n/"39KT  
    X}3o  
    z = y; DbkKmv&  
    if any(idx_pos) -d 6B;I<'  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); +lqX;*a=N  
    end _gF )aE  
    if any(idx_neg) 13P8Zmco  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); F\;G'dm  
    end 7fJWb)z!k  
    toCT5E_0=  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) E(oI0*S.5  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 3X&}{M:Qo  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated /-l7GswF  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive #yv_Eb02  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ZLJfSnB  
    %   and THETA is a vector of angles.  R and THETA must have the same PI#xRKt  
    %   length.  The output Z is a matrix with one column for every P-value, y-93 >Y  
    %   and one row for every (R,THETA) pair. bi bjFg   
    % !(soMv  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Q!:J.J  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) gI qYIt  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) nDS mr  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 G.,dP +i  
    %   for all p. z5v)~+"1  
    % io$!z=W  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Ii /#cdgF  
    %   Zernike functions (order N<=7).  In some disciplines it is fKMbOqU_  
    %   traditional to label the first 36 functions using a single mode Lh6G"f(n  
    %   number P instead of separate numbers for the order N and azimuthal spV/+jy{  
    %   frequency M. /,z4tf  
    % 7W6tz\Y  
    %   Example: :Uf\r `a9  
    % Ax4nx!W,   
    %       % Display the first 16 Zernike functions MkG3TODfHB  
    %       x = -1:0.01:1; PG8|w[V1"  
    %       [X,Y] = meshgrid(x,x); lUd/^u`  
    %       [theta,r] = cart2pol(X,Y); ^|?/ y=  
    %       idx = r<=1; 8M;VX3X  
    %       p = 0:15; `Li3=!V[  
    %       z = nan(size(X)); )Ab!R:4  
    %       y = zernfun2(p,r(idx),theta(idx)); $UAmUQg)}_  
    %       figure('Units','normalized') %SL'X`j  
    %       for k = 1:length(p) mVN^X/L(y  
    %           z(idx) = y(:,k); 0Kxc$c  
    %           subplot(4,4,k) .AOf-a  
    %           pcolor(x,x,z), shading interp GQOz\ic  
    %           set(gca,'XTick',[],'YTick',[]) & Me%ZM0  
    %           axis square q/[)Z @&(  
    %           title(['Z_{' num2str(p(k)) '}']) :yo tpa  
    %       end :[oFe/1K!4  
    % '-tiH  
    %   See also ZERNPOL, ZERNFUN. JB~79Lsdz  
    X|)Ox ,(  
    %   Paul Fricker 11/13/2006 _4VF>#b  
    mr]IxTv  
    't:|>;Wx  
    % Check and prepare the inputs: 9pD=E>4?#  
    % ----------------------------- 445}Yw5;9  
    if min(size(p))~=1 FWv-_  
        error('zernfun2:Pvector','Input P must be vector.')  &y/  
    end 4i>sOP3 B  
    2B{~"<  
    if any(p)>35 FOxMt;|M  
        error('zernfun2:P36', ... A\9Q gM  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... !fXwX3B  
               '(P = 0 to 35).']) )54;YK  
    end s)ymm7?  
    =^m,|j|d>4  
    % Get the order and frequency corresonding to the function number: c0.i  
    % ---------------------------------------------------------------- 01VEz 8[\  
    p = p(:); fGDR<t3yiQ  
    n = ceil((-3+sqrt(9+8*p))/2); l(Dkmt>^  
    m = 2*p - n.*(n+2); 6vySOVMj  
    (a0q*iC%  
    % Pass the inputs to the function ZERNFUN: L|Zja*  
    % ---------------------------------------- SnsOuC5Ah  
    switch nargin vs-%J 6}G  
        case 3 ,C%fA>?UF8  
            z = zernfun(n,m,r,theta); <RfPd+</  
        case 4 #;59THdtPk  
            z = zernfun(n,m,r,theta,nflag); pBV_'A}ioh  
        otherwise c|8[$_2  
            error('zernfun2:nargin','Incorrect number of inputs.') AvF:$ kG  
    end M8 oCh  
    $Cw> z^}u  
    % 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) 0J-]  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. cNX,%  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of DK8eFyG^2  
    %   order N and frequency M, evaluated at R.  N is a vector of d4OWnPHv&}  
    %   positive integers (including 0), and M is a vector with the =\;yxl  
    %   same number of elements as N.  Each element k of M must be a w E^6DNh  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) $^|I?5xD  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Id`?yt  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix DU9A3Z  
    %   with one column for every (N,M) pair, and one row for every $2u^z=`b!%  
    %   element in R. /5 rWcX  
    % u~MD?!LV  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- t?J Y@hT*  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is |DAe2RK  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to KUs\7Sb  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 !vNZ- }  
    %   for all [n,m]. 2 MFGKzO  
    % M>H4bU(  
    %   The radial Zernike polynomials are the radial portion of the ?M'_L']N[  
    %   Zernike functions, which are an orthogonal basis on the unit Q"UWh~  
    %   circle.  The series representation of the radial Zernike So &c\Ff  
    %   polynomials is Ul@ Jg    
    % .yp"6S^b  
    %          (n-m)/2 fAMJFHW  
    %            __ WR'm<u  
    %    m      \       s                                          n-2s Z]-C,8MM  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ="fq.Tt  
    %    n      s=0 x!$Dje}  
    % "m^whHj  
    %   The following table shows the first 12 polynomials. J7. }2  
    % MZ_+doN  
    %       n    m    Zernike polynomial    Normalization ~r`~I"ZK7^  
    %       --------------------------------------------- }hT1@I   
    %       0    0    1                        sqrt(2) r Ntc{{3_  
    %       1    1    r                           2 0>D:  
    %       2    0    2*r^2 - 1                sqrt(6) B Z:H$v  
    %       2    2    r^2                      sqrt(6) IT \Pj_  
    %       3    1    3*r^3 - 2*r              sqrt(8) 6`LC(Nv%-n  
    %       3    3    r^3                      sqrt(8) /$OX'L&b  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) W!a~ #R/r-  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) #UN(R  
    %       4    4    r^4                      sqrt(10) F+e J9  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) {V& 2k9*  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ;0}8vs  
    %       5    5    r^5                      sqrt(12) {.7ve<K  
    %       --------------------------------------------- (n{sp  
    % y]OW{5(  
    %   Example: \R& 4Nu2F  
    % Nkfu k  
    %       % Display three example Zernike radial polynomials 9g" 1WZ!  
    %       r = 0:0.01:1; nU"V@_?\  
    %       n = [3 2 5]; -la~p~8  
    %       m = [1 2 1]; =l`xXma  
    %       z = zernpol(n,m,r); 1\d$2N"  
    %       figure [0GM!3YJ7  
    %       plot(r,z) F<q3{}1zR  
    %       grid on P=& Je?  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') (sw1HR  
    % H[%F o  
    %   See also ZERNFUN, ZERNFUN2. 6l#1E#]|  
    @]f"X>  
    % A note on the algorithm. ]?F05!$*  
    % ------------------------ qQR> z  
    % The radial Zernike polynomials are computed using the series )@Yp;=l  
    % representation shown in the Help section above. For many special qR<  
    % functions, direct evaluation using the series representation can amf=uysr  
    % produce poor numerical results (floating point errors), because ,eRl Z3T  
    % the summation often involves computing small differences between =$5[uI2  
    % large successive terms in the series. (In such cases, the functions uPe4Rr  
    % are often evaluated using alternative methods such as recurrence 96F:%|yG  
    % relations: see the Legendre functions, for example). For the Zernike . iq.H  
    % polynomials, however, this problem does not arise, because the U'LO;s04m  
    % polynomials are evaluated over the finite domain r = (0,1), and HSx~Fs^J  
    % because the coefficients for a given polynomial are generally all my")/e  
    % of similar magnitude. : o$ R@l  
    % 1G|Q~%cv  
    % ZERNPOL has been written using a vectorized implementation: multiple @.kv",[{[  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] o]#Q6J  
    % values can be passed as inputs) for a vector of points R.  To achieve bi!4I<E>k  
    % this vectorization most efficiently, the algorithm in ZERNPOL 'O`jV0aa'  
    % involves pre-determining all the powers p of R that are required to ]^gD@].  
    % compute the outputs, and then compiling the {R^p} into a single mU_?}}aK,  
    % matrix.  This avoids any redundant computation of the R^p, and h_]3L/  
    % minimizes the sizes of certain intermediate variables. 'xb|5_D  
    % ;JFE7\-mC  
    %   Paul Fricker 11/13/2006 ,@!8jar@w}  
    nx=#QLi  
    l{#m"S7J^  
    % Check and prepare the inputs: )F? 57eh  
    % ----------------------------- H'I|tPs  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) LH 4-b-  
        error('zernpol:NMvectors','N and M must be vectors.') !U>"H8}dv  
    end Kggf!\MR8  
    .mDqZOpf=4  
    if length(n)~=length(m) &7Ixf?e!K  
        error('zernpol:NMlength','N and M must be the same length.') |k&.1NkZ  
    end $},:z]%D  
    K@n.$g  
    n = n(:); h9+ylHW_cp  
    m = m(:); Dr`\  
    length_n = length(n); R)cns7oW  
    -f9M*7O<gf  
    if any(mod(n-m,2)) O%o#CBf0  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') (%#d._j>fZ  
    end - |[_j$g  
    yN3Tk}{V  
    if any(m<0) (6+6]`c$  
        error('zernpol:Mpositive','All M must be positive.') $%r|V*5  
    end *, *"G?  
    10#!{].#x  
    if any(m>n) ,zXL8T  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') a #@Q.wL  
    end qsvUJU  
    h| UT/:  
    if any( r>1 | r<0 ) k|A!5A2  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') hQL9 Zl~  
    end 5.?O PK6  
    m9G,%]4|  
    if ~any(size(r)==1) A^,(Vyd  
        error('zernpol:Rvector','R must be a vector.') =~=/ dq  
    end 1r~lh#_8  
    1xguG7  
    r = r(:); eaX`S.!jR  
    length_r = length(r); n[4Nu`E9  
    a|nlmH"l  
    if nargin==4 :m&`bq  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); nrt0[E-&~  
        if ~isnorm CCEx>*E6c  
            error('zernpol:normalization','Unrecognized normalization flag.') ik$wS#1+L  
        end # Jdip)  
    else :ZL>JVk  
        isnorm = false; Z;>~<#!4  
    end >^-[Mpa(*  
    H <1?<1^  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t*qA.xc6  
    % Compute the Zernike Polynomials e-e{-pB6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G0Wd"AV+  
    UkqLLzL  
    % Determine the required powers of r: Ra{B8)Q  
    % ----------------------------------- 0H>Fyl2_  
    rpowers = []; mKsj7  
    for j = 1:length(n) _O!D*=I  
        rpowers = [rpowers m(j):2:n(j)]; Q ,;x;QR4  
    end ~HYP:6f  
    rpowers = unique(rpowers); %<M<'jxSca  
    8PEOi  
    % Pre-compute the values of r raised to the required powers, ~zm/n,Epb  
    % and compile them in a matrix: z!3Z^d`  
    % ----------------------------- mefmoZ  
    if rpowers(1)==0 < `r+l5  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); M`>W'<  
        rpowern = cat(2,rpowern{:}); |wLQ)y*  
        rpowern = [ones(length_r,1) rpowern]; :mJM=FeJ  
    else T^nX+;:|  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); l?R_wu,Q  
        rpowern = cat(2,rpowern{:}); aDOH3Ri0K!  
    end J<BdIKCma  
    +.N;h-'  
    % Compute the values of the polynomials: ks r5P~  
    % -------------------------------------- EmUxM_ T/2  
    z = zeros(length_r,length_n); A0]o/IBz  
    for j = 1:length_n Dw #&x/G  
        s = 0:(n(j)-m(j))/2; f.f4<_v'h  
        pows = n(j):-2:m(j); (mD]}{>  
        for k = length(s):-1:1 %om7h$D =`  
            p = (1-2*mod(s(k),2))* ... 41zeN++  
                       prod(2:(n(j)-s(k)))/          ... u!-eP7;7  
                       prod(2:s(k))/                 ... gU~)(|Nu.  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... # _7c>gn  
                       prod(2:((n(j)+m(j))/2-s(k))); X3-1)|g !z  
            idx = (pows(k)==rpowers); 2"MI8EK  
            z(:,j) = z(:,j) + p*rpowern(:,idx); B.G!7>=  
        end eLTNnz  
         &q< 8tTW5  
        if isnorm *Vc=]Z2G^  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); lC4By,1*  
        end C\; 8l}t  
    end h0eo:Ahi  
    ]Bsq?e^  
    % 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)  I0HY#z%  
    gn W~KLqH  
    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。 )~.&bEm\  
    {,f!'i&b@  
    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)