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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 #79[Qtkrhm  
    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! [`t ;or  
     
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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 Q_0+N3  
    function z = zernfun(n,m,r,theta,nflag) b;UBvwY_  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ;+E]F8G9r  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N AAlc %d/9  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 7,+eG">0  
    %   unit circle.  N is a vector of positive integers (including 0), and S3ooG14Ls  
    %   M is a vector with the same number of elements as N.  Each element @)6b  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) >] 'oN  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, r6Yd"~ n  
    %   and THETA is a vector of angles.  R and THETA must have the same 1"ZtE\{ "  
    %   length.  The output Z is a matrix with one column for every (N,M) 6+IhI?lI=  
    %   pair, and one row for every (R,THETA) pair. !Ud'(iGa  
    % i *.Y  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike @F<{/|P  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), <J&S[`U!  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral s Z[[ymu8  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ~{/M_ =  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized wS*r<zj  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ~~OFymQ%?q  
    % q5SPyfE[  
    %   The Zernike functions are an orthogonal basis on the unit circle. Kq3c Kp4  
    %   They are used in disciplines such as astronomy, optics, and &L+uu',M0c  
    %   optometry to describe functions on a circular domain. u]IbTJ'  
    % %;k Hnl  
    %   The following table lists the first 15 Zernike functions. 9E2iZt]  
    % 1P!)4W  
    %       n    m    Zernike function           Normalization z3+@[I$  
    %       -------------------------------------------------- >9&31wA_  
    %       0    0    1                                 1 DO*U7V02  
    %       1    1    r * cos(theta)                    2 lA5Dag'  
    %       1   -1    r * sin(theta)                    2 smf"F\W s  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) V%oZT>T3  
    %       2    0    (2*r^2 - 1)                    sqrt(3) \"a{\E,{;  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) P }sr  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) )RJEOl1  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) gm-[x5O"  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) '[{<a Eo  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) N;g@lyo  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) F}nwTras  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) W "'6 M=*  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) @Dh2@2`>  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !{lH*  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) vV}w>Ap[  
    %       -------------------------------------------------- 8F}drK9>F  
    % T$%|=gq  
    %   Example 1: faQmkO  
    % x s{pGQ6Q  
    %       % Display the Zernike function Z(n=5,m=1) jz bq{#  
    %       x = -1:0.01:1; I%3[aBz4  
    %       [X,Y] = meshgrid(x,x); Y$=jAN  
    %       [theta,r] = cart2pol(X,Y); ~lw9sm*2v2  
    %       idx = r<=1; ;o9h|LRs  
    %       z = nan(size(X)); w>%@Ug["  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); _ox+5?>  
    %       figure FJ;I1~??  
    %       pcolor(x,x,z), shading interp h:?^0b!@  
    %       axis square, colorbar oACAC+CP  
    %       title('Zernike function Z_5^1(r,\theta)') w 9dkJo  
    % .Kb3VNgwvm  
    %   Example 2: }UhYwJf89  
    % u:l-qD9=(  
    %       % Display the first 10 Zernike functions ~ bLx2=-"  
    %       x = -1:0.01:1; k;l3^kTy  
    %       [X,Y] = meshgrid(x,x); 3Qy@^"  
    %       [theta,r] = cart2pol(X,Y); <Y]LY_(  
    %       idx = r<=1; }|DspO  
    %       z = nan(size(X)); U) J5K  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 4ijtx)SA  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; J usU5 e|  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; YZol4q|ic  
    %       y = zernfun(n,m,r(idx),theta(idx)); /{^k8 Q  
    %       figure('Units','normalized') ORExI.<`W  
    %       for k = 1:10 n Nt28n@  
    %           z(idx) = y(:,k); 80=0S^gEZ  
    %           subplot(4,7,Nplot(k))  &9y Zfp  
    %           pcolor(x,x,z), shading interp jxog8 E  
    %           set(gca,'XTick',[],'YTick',[])  1MN!  
    %           axis square 3^sbbm.8  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ){;XI2  
    %       end $YSXE :  
    % y\?ey'o  
    %   See also ZERNPOL, ZERNFUN2. g>lZs  
    @-$8)?`q  
    %   Paul Fricker 11/13/2006 U$OZkHA[  
    GKBoSSnV&  
    FdU]!GO- X  
    % Check and prepare the inputs: ZVjB$-do  
    % ----------------------------- `/8@Fj  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ,d#*i  
        error('zernfun:NMvectors','N and M must be vectors.') 5J5?cs-!  
    end 7L!JP:v   
    idI w7hi4  
    if length(n)~=length(m) cQU/z"?+  
        error('zernfun:NMlength','N and M must be the same length.') 5hrI#fpOR  
    end V b0T)C  
     Gl~l  
    n = n(:); )Qbd/zd\U  
    m = m(:); gmGK3am  
    if any(mod(n-m,2)) N^L@MR-  
        error('zernfun:NMmultiplesof2', ... Y}?8  
              'All N and M must differ by multiples of 2 (including 0).') ">H*InF  
    end rAenx Z,tF  
    ~7]V^tG  
    if any(m>n) &2 tfj(ms  
        error('zernfun:MlessthanN', ... a|ufm^ F  
              'Each M must be less than or equal to its corresponding N.') zx.qN  
    end B8@mL-Z-;  
    &LLU@|  
    if any( r>1 | r<0 ) uFkl^2  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') + :MSY p  
    end ":!$Jnj,  
    RZa/la*  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 1Viz`y)^  
        error('zernfun:RTHvector','R and THETA must be vectors.') ~ ld.I4  
    end qmrT d G  
    SDnl^a  
    r = r(:); 3c<aI =$^  
    theta = theta(:); F y+NJSG  
    length_r = length(r); 0Hnj<|HL  
    if length_r~=length(theta) \]X.f&u  
        error('zernfun:RTHlength', ... &jqaW 2  
              'The number of R- and THETA-values must be equal.') 6h:QSVfx  
    end  E]V, @  
    oRV}Nz7hr  
    % Check normalization: u$nzpw0=H  
    % -------------------- y=3 dGOFB  
    if nargin==5 && ischar(nflag) _7c3=f83  
        isnorm = strcmpi(nflag,'norm'); p Cz6[*kC  
        if ~isnorm ^z?b6kTC  
            error('zernfun:normalization','Unrecognized normalization flag.') e(c\U}&  
        end i5e10@Q{  
    else 4 Gu'WbJ  
        isnorm = false; `+H=3`}X  
    end xR+vu>f  
    *$Q>Om]  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% QPlU+5Cx  
    % Compute the Zernike Polynomials &^=Lr:I  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;_}pIO  
    ]S2rqKB  
    % Determine the required powers of r: c{q+h V=  
    % ----------------------------------- E_,/)U8  
    m_abs = abs(m); MO`Y&<g~A  
    rpowers = []; E|O&bUMh  
    for j = 1:length(n) N ,~O+  
        rpowers = [rpowers m_abs(j):2:n(j)]; [,=?e  
    end sI>w#1.m/&  
    rpowers = unique(rpowers); #xI g(nG  
    |#Gxqq'  
    % Pre-compute the values of r raised to the required powers, u~uzKG  
    % and compile them in a matrix: <A3%1 82  
    % ----------------------------- 4I4m4^  
    if rpowers(1)==0 =EJ&=t  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); w-|Rb~XT h  
        rpowern = cat(2,rpowern{:}); 15:9JVH3D  
        rpowern = [ones(length_r,1) rpowern]; {lI}a8DP  
    else ZrN(M p  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); >"W^|2R  
        rpowern = cat(2,rpowern{:}); -E-#@s  
    end H%_^Gy8f  
    j=sfE qN).  
    % Compute the values of the polynomials: EP>u%]#  
    % -------------------------------------- k+QGvgP[4@  
    y = zeros(length_r,length(n)); `z!AjAT-G  
    for j = 1:length(n) FXCBX:LnvU  
        s = 0:(n(j)-m_abs(j))/2; u8f\)m  
        pows = n(j):-2:m_abs(j); *>m[ZJd%=  
        for k = length(s):-1:1 J;4x$BI  
            p = (1-2*mod(s(k),2))* ... XYcZ;Z9:  
                       prod(2:(n(j)-s(k)))/              ... |<W$rzM  
                       prod(2:s(k))/                     ... $QJ3~mG2  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... @-@Coy 4Tt  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); z{XB_j6\=  
            idx = (pows(k)==rpowers); Mc,79Ix"  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ?9 huuJ s7  
        end Ww<Y]H$xZ<  
         ;*%rFt9FK  
        if isnorm [S6u:;7  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); {gD ED  
        end M9"Bx/  
    end ]E9iaq6Z  
    % END: Compute the Zernike Polynomials cU;Bm}U  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I;4quFBlMu  
    C:E f6ZW  
    % Compute the Zernike functions: M;A_'h?Z  
    % ------------------------------ V^7.@BeT  
    idx_pos = m>0; [@i:qB>B  
    idx_neg = m<0; ,TBOEu."4  
    f+e"`80$*C  
    z = y; oW~W(h!  
    if any(idx_pos) A mZXUb  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); f2g tz{r  
    end `KQx#c>'  
    if any(idx_neg) ()lgd7|+  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 7L~ *%j  
    end [6VB&   
     y|LHnNQ  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) a>j}@8[J  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. [8QK @5[  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated hjL;B 'IL  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive VMah3T!  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, N[Z`tk?-  
    %   and THETA is a vector of angles.  R and THETA must have the same s^u  Y   
    %   length.  The output Z is a matrix with one column for every P-value, 66val"^W  
    %   and one row for every (R,THETA) pair. N,Y)'s<  
    % z:Am1B  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike \%7*@&  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) e!VtDJDS  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) [CQR  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ysnW3q!@  
    %   for all p. JBY.er`6C  
    % TFXBN.?9T  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 yTZev|ej@  
    %   Zernike functions (order N<=7).  In some disciplines it is t}+/GSwT  
    %   traditional to label the first 36 functions using a single mode rC8p!e.yL  
    %   number P instead of separate numbers for the order N and azimuthal &nEL}GM)E  
    %   frequency M. C m[}DB  
    % e)sR$]i:v  
    %   Example: ^hLr9k   
    % 2yA)SGri  
    %       % Display the first 16 Zernike functions bq/Aopfr  
    %       x = -1:0.01:1; ~0MpB~ {xd  
    %       [X,Y] = meshgrid(x,x); YTTyMn  
    %       [theta,r] = cart2pol(X,Y); f.w",S^  
    %       idx = r<=1; +byOThuE  
    %       p = 0:15; o/N!l]r  
    %       z = nan(size(X)); ^xe+(83S2?  
    %       y = zernfun2(p,r(idx),theta(idx)); lDL&":t  
    %       figure('Units','normalized') >.meecE?Q  
    %       for k = 1:length(p)  9q"kM  
    %           z(idx) = y(:,k); [=K lDfU=  
    %           subplot(4,4,k) ^ oi']O  
    %           pcolor(x,x,z), shading interp 9q'&tU'a=c  
    %           set(gca,'XTick',[],'YTick',[]) (=j;rfvP  
    %           axis square </[.1&S+\  
    %           title(['Z_{' num2str(p(k)) '}']) {lf{0c$X.  
    %       end ovKM;cRs/  
    % <Y yE1 |  
    %   See also ZERNPOL, ZERNFUN. v0DDim?cc  
    -#ZvjEaey  
    %   Paul Fricker 11/13/2006 >pF*unC;  
    UF+Qx/4h0  
    g~DuK|+  
    % Check and prepare the inputs: 3^/w`(-{@  
    % ----------------------------- M@ U >@x;  
    if min(size(p))~=1 ?c#s}IH  
        error('zernfun2:Pvector','Input P must be vector.') L-|l$Ti"  
    end yCxYFi  
    JI>Y?1i0O  
    if any(p)>35 gGrVpOzBj  
        error('zernfun2:P36', ... 0he3[m}Nr  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... nXT`7  
               '(P = 0 to 35).']) Mq\=pxC@  
    end H$%MIBz>$  
    f"s_dR  
    % Get the order and frequency corresonding to the function number: ^L%_kL_7  
    % ---------------------------------------------------------------- _ /1/{  
    p = p(:); FJ3S  
    n = ceil((-3+sqrt(9+8*p))/2); kyHli~Nr"  
    m = 2*p - n.*(n+2); -PpcFLZ|  
    O=jLZ2os  
    % Pass the inputs to the function ZERNFUN: # 55>?  
    % ---------------------------------------- BA h'H&;V  
    switch nargin YYQvt  
        case 3 +(*HDa|  
            z = zernfun(n,m,r,theta); =+iY<~8  
        case 4 t 'eaR-  
            z = zernfun(n,m,r,theta,nflag);  W9?* ~!  
        otherwise B<d=;V  
            error('zernfun2:nargin','Incorrect number of inputs.') Kn^+kHh:  
    end Tri\5O0lPs  
    0q5J)l:  
    % 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) xUn"XkhP  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. "WKOlfPa  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of p4P=T@:  
    %   order N and frequency M, evaluated at R.  N is a vector of LGhK)]:  
    %   positive integers (including 0), and M is a vector with the c 4AJ`f.5  
    %   same number of elements as N.  Each element k of M must be a PSJj$bt;<+  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) *tv&=  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is OH/9<T?  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 5{HF'1XgZ*  
    %   with one column for every (N,M) pair, and one row for every 2G8w&dtu  
    %   element in R. af|h4.A  
    % )+L|<6JXA  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- E`j' <#V!  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is lc <V_8  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to MnD}i&k[  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 $2uC%er"H  
    %   for all [n,m]. RL` jaS?V  
    % {YiMd oMhg  
    %   The radial Zernike polynomials are the radial portion of the l9 \W=-'  
    %   Zernike functions, which are an orthogonal basis on the unit @@H/q  
    %   circle.  The series representation of the radial Zernike p<r^{y  
    %   polynomials is o<5`uV!f  
    % .'`aX 7{\  
    %          (n-m)/2 i`+w.zJOH8  
    %            __ fnX[R2KZ  
    %    m      \       s                                          n-2s @K <Onh`  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r dy+A$)gY<  
    %    n      s=0 l0'Yq%Nf  
    % 1KUjb@"  
    %   The following table shows the first 12 polynomials. !\#Wq{p>W*  
    % GFT@Pqq  
    %       n    m    Zernike polynomial    Normalization a6gw6jQ  
    %       --------------------------------------------- Zv}F?4T~:  
    %       0    0    1                        sqrt(2) b"X1  
    %       1    1    r                           2 +[ ?!@)  
    %       2    0    2*r^2 - 1                sqrt(6) -. *E<%  
    %       2    2    r^2                      sqrt(6) , >WH)+a  
    %       3    1    3*r^3 - 2*r              sqrt(8) Dspvc  
    %       3    3    r^3                      sqrt(8) F%V|Aa  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) h2'6W)  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 6 5zx<  
    %       4    4    r^4                      sqrt(10) %fF,Fnf2  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 4 Y=0>FlY0  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) (EcP'F*;;y  
    %       5    5    r^5                      sqrt(12) , LwinjHA*  
    %       --------------------------------------------- Osz=OO{  
    % "3VX9{'%@  
    %   Example: fBh"  
    % 4AL,=C3  
    %       % Display three example Zernike radial polynomials ~2>Adp  
    %       r = 0:0.01:1; o1#3A  
    %       n = [3 2 5]; Ar\fA)UQ`  
    %       m = [1 2 1]; BP j?l  
    %       z = zernpol(n,m,r); 6Mk#) ebM  
    %       figure _ uOi:Ti  
    %       plot(r,z) (8 7wWhH  
    %       grid on f&$Bjq  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') KAZ<w~55c  
    % 07SW$INb  
    %   See also ZERNFUN, ZERNFUN2. ;R 6f9tu2  
    U~=?I)Ni  
    % A note on the algorithm. Vl+UC1M}B>  
    % ------------------------ ?8$`GyjS  
    % The radial Zernike polynomials are computed using the series 1M`>;fjYa  
    % representation shown in the Help section above. For many special ;uAh)|;S#  
    % functions, direct evaluation using the series representation can 0m1V@ 3]7>  
    % produce poor numerical results (floating point errors), because dm/3{\ 4  
    % the summation often involves computing small differences between ~Q=;L>Qd  
    % large successive terms in the series. (In such cases, the functions {*bx8*y1  
    % are often evaluated using alternative methods such as recurrence D'[:35z  
    % relations: see the Legendre functions, for example). For the Zernike s2L]H  
    % polynomials, however, this problem does not arise, because the ,]\cf  
    % polynomials are evaluated over the finite domain r = (0,1), and r3x;lICx-  
    % because the coefficients for a given polynomial are generally all "tl{HM5u  
    % of similar magnitude. mKtMI!FR  
    % \\EX'L  
    % ZERNPOL has been written using a vectorized implementation: multiple r]LP=K1  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] nZ=[6?  
    % values can be passed as inputs) for a vector of points R.  To achieve sR$abN+u  
    % this vectorization most efficiently, the algorithm in ZERNPOL qy?$t:*pp  
    % involves pre-determining all the powers p of R that are required to \V'fB5  
    % compute the outputs, and then compiling the {R^p} into a single `\:Ede  
    % matrix.  This avoids any redundant computation of the R^p, and `~\SQ EY$  
    % minimizes the sizes of certain intermediate variables. ?IeBo8  
    % PO5,lcBD<  
    %   Paul Fricker 11/13/2006 8r>\scS  
    b,:^\HKC  
    v8)wu=u  
    % Check and prepare the inputs: KiG19R$  
    % ----------------------------- >#n"r1  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) #3ZAMV  
        error('zernpol:NMvectors','N and M must be vectors.') -`ljKp  
    end "E7<S5 cr  
    ]UIN4E  
    if length(n)~=length(m) kReZch}  
        error('zernpol:NMlength','N and M must be the same length.') P_jav 0j7g  
    end c^"4l 9w  
    prM)t8SE  
    n = n(:); "&XhMw4  
    m = m(:); 7]So=% q  
    length_n = length(n); z z]~IxQ  
    LV9R ]  
    if any(mod(n-m,2)) ({Yfsf,  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') A/9<} m  
    end @rqmDpU  
    Y\<w|LkD8  
    if any(m<0) `[E-V  
        error('zernpol:Mpositive','All M must be positive.') 'N6oXE  
    end z( ^?xv  
    >~7XBb08  
    if any(m>n) kqZRg>1A  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') UazK0{t<f  
    end D$KP>G  
    w9u|E46  
    if any( r>1 | r<0 ) ~9@527m<',  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') $#%R _G]  
    end r]}6iF.  
    s`Z.H5V>\  
    if ~any(size(r)==1) })8D3kzX)  
        error('zernpol:Rvector','R must be a vector.') oFyB-vpYQV  
    end ;L",K?6#  
    r9-ayp#pC  
    r = r(:); 7H6Ge-u  
    length_r = length(r); KN@ [hb7%  
    M-;Mw Lx  
    if nargin==4 0gOca +&  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); \N0wf-qa=  
        if ~isnorm (-1{W^(  
            error('zernpol:normalization','Unrecognized normalization flag.') o=7e8l  
        end Dg~m}La  
    else w"D1mI!L 7  
        isnorm = false; hX,RuI  
    end #v$wjqK5  
    3XUVUd~  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?t}s3P!Q3w  
    % Compute the Zernike Polynomials <  j  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B(FM~TVZ  
    |gk4X%o6  
    % Determine the required powers of r: Y$, ++wx  
    % ----------------------------------- d/+s-g p  
    rpowers = []; `o9:6X?RA  
    for j = 1:length(n) T6?03cSE  
        rpowers = [rpowers m(j):2:n(j)]; E>#@ H  
    end IEM{?  
    rpowers = unique(rpowers); Bk~lM'  
    kwww5p ["  
    % Pre-compute the values of r raised to the required powers, npytb*[|c  
    % and compile them in a matrix: c9@3=6S/  
    % ----------------------------- WAuT`^"u  
    if rpowers(1)==0 2ER_?y  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); rT-.'aQ2t  
        rpowern = cat(2,rpowern{:}); `X ()"Qw  
        rpowern = [ones(length_r,1) rpowern]; j,%i.[8S  
    else 2!9W:I7  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); vG)B}`M  
        rpowern = cat(2,rpowern{:}); Gys-Im6>~@  
    end 7[L%j;)bw  
    }U9dzU14  
    % Compute the values of the polynomials: f]sR4mhO  
    % -------------------------------------- Bn.R,B0PL  
    z = zeros(length_r,length_n); 247vU1  
    for j = 1:length_n gs.+|4dv  
        s = 0:(n(j)-m(j))/2; xHx_! )7  
        pows = n(j):-2:m(j); Az(,Q$"|5  
        for k = length(s):-1:1 Mc8_D,7  
            p = (1-2*mod(s(k),2))* ... ~ e<,GUx(]  
                       prod(2:(n(j)-s(k)))/          ... #PC*l\ )  
                       prod(2:s(k))/                 ... ] fA5D)/m<  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... xIb{*)BUwc  
                       prod(2:((n(j)+m(j))/2-s(k))); [EVyCIcY,h  
            idx = (pows(k)==rpowers); {w ,^Z[<  
            z(:,j) = z(:,j) + p*rpowern(:,idx); .R4,fCN  
        end ^%6f%]_  
         iYdg1  
        if isnorm 'm<L}d  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 4__HH~j?Q  
        end Q?>*h xzoP  
    end 5>H&0> \  
    U5F1m]gFr  
    % 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)  uq]E^#^  
    GS!1K(7  
    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。 \@j3/!=,n%  
    9sB LCZ  
    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)