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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 Y!SE;N&  
    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! -=>sTMWpr  
     
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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 'S*k_vuN  
    function z = zernfun(n,m,r,theta,nflag) cMaOM}mS  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. <YH=3[  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N KFU%DU G  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ^*0'\/N&  
    %   unit circle.  N is a vector of positive integers (including 0), and yrnv!moc%t  
    %   M is a vector with the same number of elements as N.  Each element \9`#]#1bx5  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) c+g@Z"es  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 7b,(\Fm  
    %   and THETA is a vector of angles.  R and THETA must have the same 1yM r~Fo  
    %   length.  The output Z is a matrix with one column for every (N,M) !J3UqS  
    %   pair, and one row for every (R,THETA) pair. L0L2Ns  
    % ;'0=T0\  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike .1#kD M  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ]n;1x1'  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral H>XFz(LWh  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Qs%B'9")  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 2z\e\I  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. BEUK}T K4  
    % H;Ku w  
    %   The Zernike functions are an orthogonal basis on the unit circle. 0J9D"3T)  
    %   They are used in disciplines such as astronomy, optics, and T7[NcZ:I  
    %   optometry to describe functions on a circular domain. bWmw3w  
    % ^nNitF  
    %   The following table lists the first 15 Zernike functions. 6@V~0DG  
    % =^tA_AxVw  
    %       n    m    Zernike function           Normalization V kjuyK  
    %       -------------------------------------------------- P6\6?am  
    %       0    0    1                                 1 Hr^3`@}#1  
    %       1    1    r * cos(theta)                    2 36vgX=}  
    %       1   -1    r * sin(theta)                    2 pr&=n;_ n  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) |gx ~ gG<  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ]{GDS! )  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 69OF_/23  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) x#*QfE/E(@  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) !q' 4D!I  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) S\=1_LDx"  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) AXPMnbUS  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) h&;t.Gdf  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Gh\q^?}  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) =5x&8i  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) b~w=v_[(I  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) WQ6"0*er  
    %       -------------------------------------------------- !h`kX[:  
    % _zMgoc7  
    %   Example 1: aG%, cQ1  
    % -LW[7s$  
    %       % Display the Zernike function Z(n=5,m=1) _S`o1^Ad  
    %       x = -1:0.01:1; -7{ $ Vj  
    %       [X,Y] = meshgrid(x,x); yZ kyC'/  
    %       [theta,r] = cart2pol(X,Y); MTOy8 Im  
    %       idx = r<=1; eOI (6U!  
    %       z = nan(size(X)); i'#Gy,R  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); T~4N+fK  
    %       figure 5d\q-d  
    %       pcolor(x,x,z), shading interp ~Z'w)!h  
    %       axis square, colorbar t2BL( yB  
    %       title('Zernike function Z_5^1(r,\theta)') nNt1C  
    % 4\M.6])_   
    %   Example 2: `bjizS'^  
    % 04U")-\O  
    %       % Display the first 10 Zernike functions }"^'% C8EX  
    %       x = -1:0.01:1; >>{FzR  
    %       [X,Y] = meshgrid(x,x); cV{o?3<:B  
    %       [theta,r] = cart2pol(X,Y); ACq7dLys,B  
    %       idx = r<=1; @]aOyb@  
    %       z = nan(size(X)); 2L?!tBw?1  
    %       n = [0  1  1  2  2  2  3  3  3  3]; {0"YOS`3AX  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; E&$yuW^z  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; umi5Wb<  
    %       y = zernfun(n,m,r(idx),theta(idx)); y|wlq3o  
    %       figure('Units','normalized') }g7]?Ee  
    %       for k = 1:10 ',^+bgs5  
    %           z(idx) = y(:,k); Y!J>U  
    %           subplot(4,7,Nplot(k)) ~{,X3-S_H  
    %           pcolor(x,x,z), shading interp L|@y&di  
    %           set(gca,'XTick',[],'YTick',[]) *3/T;x.  
    %           axis square e [_m< e  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) MY#   
    %       end $-}e; VZb  
    % c(;a=n(E#  
    %   See also ZERNPOL, ZERNFUN2. *jIqAhs0{  
    v[e:qi&fG  
    %   Paul Fricker 11/13/2006 Z_1U9 +,  
    /zDi9W*~1  
    y\dEk:\)  
    % Check and prepare the inputs: L@`ouQ"sa  
    % ----------------------------- Bw%Qbs0Q  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) k@ZLg9  
        error('zernfun:NMvectors','N and M must be vectors.') Suk  
    end yeDsJ/L  
    ,to+oSZE  
    if length(n)~=length(m) D(-yjY8aG  
        error('zernfun:NMlength','N and M must be the same length.') ]0hrRA`  
    end g<{xC_J  
    Wjhvxk  
    n = n(:); ./Q,  
    m = m(:); PxH72hBS  
    if any(mod(n-m,2)) 2MZCw^s>  
        error('zernfun:NMmultiplesof2', ... l2N]a9bq@  
              'All N and M must differ by multiples of 2 (including 0).') $/!{OU.t`  
    end >h0-;  
    `W/sP\3  
    if any(m>n) "BX!  
        error('zernfun:MlessthanN', ... /|6;Z}2  
              'Each M must be less than or equal to its corresponding N.') 3gd&i  
    end J{^RkGF  
    "HE^v_p  
    if any( r>1 | r<0 ) jck}" N  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Y"A/^]  
    end .{y uo{u  
    pPd#N'\*  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 5j~$Mj`  
        error('zernfun:RTHvector','R and THETA must be vectors.') _6 ay-u  
    end a!O0,y  
    @E:,lA  
    r = r(:); V5*OA??k<  
    theta = theta(:); Kq i4hK  
    length_r = length(r); kbM3  
    if length_r~=length(theta) HRB<Y mP@  
        error('zernfun:RTHlength', ... L:@7tc.  
              'The number of R- and THETA-values must be equal.') ,}K<*t[I  
    end /7gOSwY  
    M)SEn/T-  
    % Check normalization: OpHsob~  
    % -------------------- %2v4<icvq  
    if nargin==5 && ischar(nflag) LD!Q8"  
        isnorm = strcmpi(nflag,'norm'); D 9M:^  
        if ~isnorm =UV`.d2[  
            error('zernfun:normalization','Unrecognized normalization flag.') `r?7oxN  
        end 8<Hf" M  
    else cTG|fdgMW  
        isnorm = false; o}ZdTf=  
    end 1dK*y'rx  
    >y,-v:Vy  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% eH{[C*  
    % Compute the Zernike Polynomials 7Hs%Cc"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S\;V4@<Kn  
    %$b:X5$Z  
    % Determine the required powers of r: t<#h$}=:Vt  
    % ----------------------------------- SJHr_bawd  
    m_abs = abs(m); R rda# h^  
    rpowers = []; <)3u6Vky9  
    for j = 1:length(n) o_~eg8  
        rpowers = [rpowers m_abs(j):2:n(j)]; |j7,Mu+  
    end 13>0OKg`#  
    rpowers = unique(rpowers); 5k.oW=  
    jbAx;Xt'=M  
    % Pre-compute the values of r raised to the required powers, .X;3,D[w  
    % and compile them in a matrix: 4T ~}  
    % ----------------------------- 4M2j!Sw  
    if rpowers(1)==0 -PfX0y9n  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);  a24"yT  
        rpowern = cat(2,rpowern{:}); .4E&/w+  
        rpowern = [ones(length_r,1) rpowern]; t;}:waZD  
    else }|pwz   
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); cH&J{WeZa  
        rpowern = cat(2,rpowern{:}); U[ 0=L`0e  
    end z*!%g[3I  
    "/wyZ  
    % Compute the values of the polynomials:  bJX)$G  
    % -------------------------------------- Ys\Wj%6A  
    y = zeros(length_r,length(n)); qHrc9fB  
    for j = 1:length(n) tIuCct-  
        s = 0:(n(j)-m_abs(j))/2; ):[7E(F=  
        pows = n(j):-2:m_abs(j); 32`{7a3!=  
        for k = length(s):-1:1 ]jo1{IcI  
            p = (1-2*mod(s(k),2))* ... IhVO@KJI  
                       prod(2:(n(j)-s(k)))/              ... N u<_}  
                       prod(2:s(k))/                     ... I+tb[*X+  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... )% ~OH  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); : qd`zG3  
            idx = (pows(k)==rpowers); bAx-"Lu  
            y(:,j) = y(:,j) + p*rpowern(:,idx); oY933i@l)P  
        end _I:/ZF5  
         zN^n]N_?  
        if isnorm d^{RQ   
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ]7Tkkw$  
        end ~Vr.J}]J  
    end sTn<#l6  
    % END: Compute the Zernike Polynomials xHD=\,{ig  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )-a'{W/t  
    o!kbK#k  
    % Compute the Zernike functions: m}7iTDJR9  
    % ------------------------------ *%%g{ 3$  
    idx_pos = m>0; ^\4h<M  
    idx_neg = m<0; Z{]0jhUyNh  
    3h$6t7=C  
    z = y; .y!<t}  
    if any(idx_pos) RO 4Z?tz  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ^")Q YE  
    end l1BtI_7p  
    if any(idx_neg) [XEkz#{  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ~?d Nd  
    end ,(EO'T[  
    n*[XR`r}  
    % 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^3Z3zf&  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 1w@(5 ^V  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 7%Gwc?[x  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive RP[{4 Q8  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, e2s]{obf  
    %   and THETA is a vector of angles.  R and THETA must have the same +6HVhoxU#  
    %   length.  The output Z is a matrix with one column for every P-value, ^o3"#r{:+  
    %   and one row for every (R,THETA) pair. a{^m-fSaR"  
    % j*so9M6|c  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike q&s3wDl/  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) $rv8K j+  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) Q=;U@k@>  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 2@'oe7E  
    %   for all p. ]zE;Tw.S  
    % =,spvy'"*C  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 7K,-01-:  
    %   Zernike functions (order N<=7).  In some disciplines it is R!\_rc1/  
    %   traditional to label the first 36 functions using a single mode uki#/GzaO  
    %   number P instead of separate numbers for the order N and azimuthal }vxw*8d?  
    %   frequency M. Qu?R8+"KS  
    % ~.{/0T  
    %   Example: 5na~@-9p  
    % H?<N.Dq  
    %       % Display the first 16 Zernike functions PRu 6xsyA  
    %       x = -1:0.01:1; [Dk=? +  
    %       [X,Y] = meshgrid(x,x); Aw$x;3y  
    %       [theta,r] = cart2pol(X,Y); {> eXR?s/  
    %       idx = r<=1; @$S+Ne[<  
    %       p = 0:15; *6sl   
    %       z = nan(size(X)); i UCXAWP  
    %       y = zernfun2(p,r(idx),theta(idx)); {MtpkUN  
    %       figure('Units','normalized') G18F&c~  
    %       for k = 1:length(p) 1O/+8yw  
    %           z(idx) = y(:,k); _4"mAPt  
    %           subplot(4,4,k) `eE&5.   
    %           pcolor(x,x,z), shading interp @mOH"acGn?  
    %           set(gca,'XTick',[],'YTick',[]) G_;)a]v8)  
    %           axis square HePUWL'  
    %           title(['Z_{' num2str(p(k)) '}']) iHeN9 cl  
    %       end E7t+E)=8  
    % .AR#&mL9  
    %   See also ZERNPOL, ZERNFUN. K&POyOvT  
    .a O,8M  
    %   Paul Fricker 11/13/2006 Rp.Sj{<2  
    7mI:| G  
    /Y9>8XSc  
    % Check and prepare the inputs: !}YAdZJ  
    % ----------------------------- KK&rb~  
    if min(size(p))~=1 aZ2!i  
        error('zernfun2:Pvector','Input P must be vector.') %eX{WgH  
    end h].<t&  
    ;YA(|h<  
    if any(p)>35 o< |cA5f\  
        error('zernfun2:P36', ... ![`Ay4AZ@a  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... L^E[J`  
               '(P = 0 to 35).']) w`4=_J=GO  
    end Huy5-[)15  
    S=5<^o^h3  
    % Get the order and frequency corresonding to the function number: (U&tt]|  
    % ---------------------------------------------------------------- *@Lp`thq  
    p = p(:); .Zn^Nw3  
    n = ceil((-3+sqrt(9+8*p))/2); "fG8?)d;  
    m = 2*p - n.*(n+2); 9!6f-K  
    kE:nsXI )  
    % Pass the inputs to the function ZERNFUN: DK$X2B"cV  
    % ---------------------------------------- (\\eo  
    switch nargin kDEPs$^  
        case 3 I;e=0!9U  
            z = zernfun(n,m,r,theta); PH1p2Je  
        case 4 d ^^bke$~  
            z = zernfun(n,m,r,theta,nflag); 6g 5#TpCh  
        otherwise S)cLW~=z  
            error('zernfun2:nargin','Incorrect number of inputs.') Id_2PkIN$~  
    end E)TN,@%  
    NG--6\  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) Sm)Ha:[4  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Fpm|_f7  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of '?!zG{x  
    %   order N and frequency M, evaluated at R.  N is a vector of YUx.BZf7  
    %   positive integers (including 0), and M is a vector with the gYNjzew'  
    %   same number of elements as N.  Each element k of M must be a |uX,5Q#6  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) W ?qmp|YD  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 5 xppKt  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix mR&H9 NG  
    %   with one column for every (N,M) pair, and one row for every v>$'iT~l  
    %   element in R. j"}*T  
    % ,VCyG:dw  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- Rtb7|  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is le1}0 L  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 'm4W}F  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 !qv ea,vw  
    %   for all [n,m]. ^Y@\1fX 4e  
    % xC{qV,   
    %   The radial Zernike polynomials are the radial portion of the :ctu5{"UJ  
    %   Zernike functions, which are an orthogonal basis on the unit U@HK+C"M|  
    %   circle.  The series representation of the radial Zernike )we}6sE"  
    %   polynomials is fuWO*  
    % <QA6/Ef7  
    %          (n-m)/2 H=g`hF]`  
    %            __ M!/Cknm  
    %    m      \       s                                          n-2s <}E!w_yi  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r d/ARm-D  
    %    n      s=0 P,xKZ{(  
    % qHuZcht  
    %   The following table shows the first 12 polynomials. JTr vnA  
    % *QwY]j%^  
    %       n    m    Zernike polynomial    Normalization J&M o%"[)  
    %       --------------------------------------------- $ {O#  
    %       0    0    1                        sqrt(2) ~Lm$i6E <  
    %       1    1    r                           2 vd [}Gd  
    %       2    0    2*r^2 - 1                sqrt(6) ,quoRan  
    %       2    2    r^2                      sqrt(6) P0W*C6&71|  
    %       3    1    3*r^3 - 2*r              sqrt(8) G_0( |%  
    %       3    3    r^3                      sqrt(8) >+JqA7K  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) [U5\bX@$  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) VKq=7^W  
    %       4    4    r^4                      sqrt(10) } ud0&Oe{  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) M-1ngI0H;  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) EK;YiJ  
    %       5    5    r^5                      sqrt(12) YE|SKx@  
    %       --------------------------------------------- mVVD!  
    % V!lZ\)  
    %   Example: ]^lw*724'>  
    % }|g\ 8jq  
    %       % Display three example Zernike radial polynomials $6mX  
    %       r = 0:0.01:1; ?AJKBW^  
    %       n = [3 2 5]; 2 lj'"nm  
    %       m = [1 2 1]; .!f$ \1l  
    %       z = zernpol(n,m,r); Y8m1M-#w  
    %       figure j6Yy6X]  
    %       plot(r,z) @6wFst\t  
    %       grid on do*EKo  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ;$smH=I  
    % 3QG7C{  
    %   See also ZERNFUN, ZERNFUN2. U.DDaT1  
    l038%U~U!  
    % A note on the algorithm. ujlY! -GM  
    % ------------------------ I aGq]z  
    % The radial Zernike polynomials are computed using the series jN[`L%Qm   
    % representation shown in the Help section above. For many special \.-}adKg  
    % functions, direct evaluation using the series representation can x4E7X_  
    % produce poor numerical results (floating point errors), because 7]blrN]  
    % the summation often involves computing small differences between D|e uX7b  
    % large successive terms in the series. (In such cases, the functions \QYFAa  
    % are often evaluated using alternative methods such as recurrence ~]nSSD)\  
    % relations: see the Legendre functions, for example). For the Zernike CIb2J)qev  
    % polynomials, however, this problem does not arise, because the Dp)=0<$y  
    % polynomials are evaluated over the finite domain r = (0,1), and bgK'{_o-  
    % because the coefficients for a given polynomial are generally all f@Zszt  
    % of similar magnitude. aX5 z&r:{  
    % n/+.s(7c  
    % ZERNPOL has been written using a vectorized implementation: multiple D;;!ODX$?  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ] Hztb  
    % values can be passed as inputs) for a vector of points R.  To achieve Uk^B"y_  
    % this vectorization most efficiently, the algorithm in ZERNPOL @GdbTd  
    % involves pre-determining all the powers p of R that are required to n=y[CKS  
    % compute the outputs, and then compiling the {R^p} into a single [_1G@S6Ex  
    % matrix.  This avoids any redundant computation of the R^p, and dwDcR,z?a  
    % minimizes the sizes of certain intermediate variables. b:tob0TB  
    % G#d{,3Gq1  
    %   Paul Fricker 11/13/2006 X!9 B2w  
    ~N<4L>y<  
    W g02 A\  
    % Check and prepare the inputs: Jl#%uU/sx  
    % ----------------------------- whi`Z:~  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) s'%R  
        error('zernpol:NMvectors','N and M must be vectors.') *X+79vG:  
    end 5N/%v&1  
    :lf+W  
    if length(n)~=length(m) #~C]ZrK  
        error('zernpol:NMlength','N and M must be the same length.') Qo;zHZ'  
    end Exc9` 7%.  
    v(ZYS']d2  
    n = n(:); L"o>wYx  
    m = m(:); +yk24 ` >  
    length_n = length(n); j4|N- :  
    ykV 5  
    if any(mod(n-m,2)) Y]/% t{Y  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') - jb0o/:  
    end ?0v-qj+  
    #xX5,r0  
    if any(m<0)  %oZ6l*  
        error('zernpol:Mpositive','All M must be positive.') 0K`#>}W#X  
    end n1ly y0%u  
    7UVzp v  
    if any(m>n) OY;*zk  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') Xq_5Qv  
    end f !I[>&n  
    DU5c=rxW  
    if any( r>1 | r<0 ) lku[dQdk  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') IC1NKn<k  
    end lDYyqG4  
    aMWmLpv4'  
    if ~any(size(r)==1) vms|x wb  
        error('zernpol:Rvector','R must be a vector.') =&ks)MH-  
    end Y2n!>[[.  
    fI{&#~f4C  
    r = r(:); mS(fgq6  
    length_r = length(r); m1B+31'>^  
    d1AioQ9  
    if nargin==4 nbm&wa[  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); j|U#)v/  
        if ~isnorm ++6`sMJ  
            error('zernpol:normalization','Unrecognized normalization flag.') G,o6292hj  
        end Cd,jDPrw  
    else X*/ho  
        isnorm = false; gtk7)Uh  
    end @k,z:~[C=  
    ?OcJ )5C4  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CE#gfP  
    % Compute the Zernike Polynomials xe{ !wX  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /178A;J y  
    D8Ykg >B;&  
    % Determine the required powers of r: #gz M|  
    % ----------------------------------- n>ULRgiT:o  
    rpowers = []; o]yl ;I  
    for j = 1:length(n) F Sw\_[^CQ  
        rpowers = [rpowers m(j):2:n(j)]; piPR=B+  
    end . $BUw  
    rpowers = unique(rpowers); :~2vJzp@?  
    gp>3I!bo[K  
    % Pre-compute the values of r raised to the required powers, `UD/}j@  
    % and compile them in a matrix: sH{4Y-J  
    % ----------------------------- -w9pwB  
    if rpowers(1)==0 &dM. d!  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); YC++& Nk  
        rpowern = cat(2,rpowern{:}); (s.0P O`  
        rpowern = [ones(length_r,1) rpowern]; OGK}EI  
    else |bTPtrT8  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); sDPs G5q<  
        rpowern = cat(2,rpowern{:}); w,#>G07D  
    end zHA!%>%'  
    ED=V8';D  
    % Compute the values of the polynomials: /]K^ rw[  
    % -------------------------------------- K2TcOFQ  
    z = zeros(length_r,length_n); B2>H_dmQ  
    for j = 1:length_n 'u*D A|HC  
        s = 0:(n(j)-m(j))/2; z@!`:'ak  
        pows = n(j):-2:m(j); %j.0G`x9 +  
        for k = length(s):-1:1 B3We|oe!  
            p = (1-2*mod(s(k),2))* ... */sS`/Lx  
                       prod(2:(n(j)-s(k)))/          ... b$N 2z  
                       prod(2:s(k))/                 ... X{5vXT\/y  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... eD,.~Y#?=  
                       prod(2:((n(j)+m(j))/2-s(k))); GeyvId03H  
            idx = (pows(k)==rpowers); ]{3)^axW;  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Of*Pw[vD  
        end C 3^JAP  
         fObg3S92  
        if isnorm s'!Cp=xQF"  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); =bfJ^]R  
        end /rK}?U  
    end HVi'eNgo  
    ??^5;P{yx  
    % 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)  Um }  
    S$f9m  
    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。 gcii9vz `  
    m-%E-nr  
    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)