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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 P@B]  
    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! X_h}J=33Q  
     
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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 4Nsp<Kn>  
    function z = zernfun(n,m,r,theta,nflag) 1qA;/-Zr<o  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. H:| uw  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N })%{AfDRF  
    %   and angular frequency M, evaluated at positions (R,THETA) on the `c$V$/IT  
    %   unit circle.  N is a vector of positive integers (including 0), and 2^7`mES  
    %   M is a vector with the same number of elements as N.  Each element @yYkti;4-  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) !a\^Sk /  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ? J0y|  
    %   and THETA is a vector of angles.  R and THETA must have the same !nnC3y{G  
    %   length.  The output Z is a matrix with one column for every (N,M) [/r(__.  
    %   pair, and one row for every (R,THETA) pair. L4W5EO$  
    % Pm7}"D'/  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike E1 2uZ$X  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), L/K(dkx  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 8s@3hXD&  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, PKz':_|  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized At;LO9T3z  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ;uGv:$([g  
    % R;LP:,)  
    %   The Zernike functions are an orthogonal basis on the unit circle. %cn<ych G  
    %   They are used in disciplines such as astronomy, optics, and {qVZNXDn  
    %   optometry to describe functions on a circular domain. -~w'Xo#  
    % KI.hy2?e  
    %   The following table lists the first 15 Zernike functions. omx=  
    % .%-8 t{dt  
    %       n    m    Zernike function           Normalization y~V(aih}D  
    %       -------------------------------------------------- xE}>,O|'q  
    %       0    0    1                                 1 53 h0UL  
    %       1    1    r * cos(theta)                    2 dE3) | %  
    %       1   -1    r * sin(theta)                    2 ;tf=gdX;  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) HzJz+ x:  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 6A ah9   
    %       2    2    r^2 * sin(2*theta)             sqrt(6) |w=zOC;v  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Z\sDUJ  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) l]SX@zTb  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) XjBD{m(  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) |s_GlJV.  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 9gIrt 6  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /bmN\I  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) :4|4=mkr  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 46;uW{EY  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) LP=)~K<  
    %       -------------------------------------------------- @4#vm@Yf_  
    % lTsjxw o  
    %   Example 1: zuCSj~  
    % %iB,IEw  
    %       % Display the Zernike function Z(n=5,m=1) O6Y0XL  
    %       x = -1:0.01:1; V]^$S"Tv  
    %       [X,Y] = meshgrid(x,x); `vV7c`K?  
    %       [theta,r] = cart2pol(X,Y); h+,@G,|D  
    %       idx = r<=1; /L 3:  
    %       z = nan(size(X)); rN>R|].  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); \2z>?i)  
    %       figure [F7hu7zY8  
    %       pcolor(x,x,z), shading interp Ys7]B9/1O  
    %       axis square, colorbar p ll)Y  
    %       title('Zernike function Z_5^1(r,\theta)') $cg cX  
    % "N#Y gSr  
    %   Example 2: H?w6C):]  
    % dr"1s-D4IQ  
    %       % Display the first 10 Zernike functions |j|rS5  
    %       x = -1:0.01:1; i/.6>4tE:  
    %       [X,Y] = meshgrid(x,x); ~#/  
    %       [theta,r] = cart2pol(X,Y); 1~gCtBRM  
    %       idx = r<=1; HOi`$vX }N  
    %       z = nan(size(X)); wuBPfb  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Y-9I3?ar  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; $~kA B8z  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; TqQ[_RKg2  
    %       y = zernfun(n,m,r(idx),theta(idx)); +`15le`R  
    %       figure('Units','normalized') OrW  
    %       for k = 1:10 $;PMkUE  
    %           z(idx) = y(:,k); @VI@fN  
    %           subplot(4,7,Nplot(k)) EX"yxZ~  
    %           pcolor(x,x,z), shading interp `0svy}  
    %           set(gca,'XTick',[],'YTick',[]) N>E_%]Ch  
    %           axis square gDzK{6Z}  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) p4QU9DF  
    %       end {{1G`;|v 9  
    % G/W>S,(  
    %   See also ZERNPOL, ZERNFUN2. O0:q;<>z  
    _v:SP LU  
    %   Paul Fricker 11/13/2006 QWU-m{@~&  
    7$#u  
    (?];VG  
    % Check and prepare the inputs: y>LBl]  
    % ----------------------------- =|9!vzG4  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) &3&HY:yF  
        error('zernfun:NMvectors','N and M must be vectors.') F[MFx^sT{  
    end eH,or,r  
    z!\*Y =e  
    if length(n)~=length(m) Xc.`-J~Il  
        error('zernfun:NMlength','N and M must be the same length.') ?4uL-z](V  
    end "jCu6Rjd  
    !~Z"9(v'C  
    n = n(:); m+9#5a-  
    m = m(:); SWLo|)@[/  
    if any(mod(n-m,2)) q\)-BXw:  
        error('zernfun:NMmultiplesof2', ... Zd&S@Z  
              'All N and M must differ by multiples of 2 (including 0).')  lRQYpc\  
    end 2zpr~cB=  
    HT@=evV  
    if any(m>n) $ Q0n  
        error('zernfun:MlessthanN', ... *ui</+  
              'Each M must be less than or equal to its corresponding N.') !9x}  
    end xD$\,{  
    5-M-X#(  
    if any( r>1 | r<0 ) (sj,[  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') V8(-  
    end \NC3'G:Ii  
    u:EiwRW  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ^Dx&|UwiZa  
        error('zernfun:RTHvector','R and THETA must be vectors.') ;}t(Wnu.  
    end Ho%CDz z  
    .(vwIb8\_  
    r = r(:); @ P|y{e6  
    theta = theta(:); Dh*n!7lD`  
    length_r = length(r); v0y(58Rz.  
    if length_r~=length(theta) Xr{v~bf  
        error('zernfun:RTHlength', ... n`KY9[0U=  
              'The number of R- and THETA-values must be equal.') #;<Y[hR{P  
    end ~ K=b\xc^  
    }\LQ3y"[  
    % Check normalization: 1eKT^bgM  
    % -------------------- svSVG:48  
    if nargin==5 && ischar(nflag) t&p|Ynz?i  
        isnorm = strcmpi(nflag,'norm'); = / 8cp  
        if ~isnorm > P)w?:k  
            error('zernfun:normalization','Unrecognized normalization flag.') cZ06Kx..  
        end cNH7C"@GVu  
    else g=rbPbu  
        isnorm = false; s@C}P  
    end `{Ul!  
    [ 3HfQ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7 d vnupLh  
    % Compute the Zernike Polynomials yHGADH0B  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  @8 6f  
    ;=N# `l  
    % Determine the required powers of r: ;PH~<T  
    % ----------------------------------- n*$ g]G$  
    m_abs = abs(m); He)%S]RLk  
    rpowers = []; BuwY3F\-O  
    for j = 1:length(n) DrQ`]]jj7  
        rpowers = [rpowers m_abs(j):2:n(j)]; W4N{S.#!  
    end u&NV,6Fj2[  
    rpowers = unique(rpowers); B1STGL`nK  
    he4(hX^  
    % Pre-compute the values of r raised to the required powers, *8Z32c+C  
    % and compile them in a matrix: M_8{]uo  
    % ----------------------------- g5yJfRLxp  
    if rpowers(1)==0 fIF8%J ^3  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); kP"9&R`E  
        rpowern = cat(2,rpowern{:}); "}!G!k:  
        rpowern = [ones(length_r,1) rpowern]; HV.t6@\};  
    else =Uh$&m  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ;aBG,dr}i  
        rpowern = cat(2,rpowern{:}); ]tD]Wx%  
    end RZ7@cQY  
    ys~x $  
    % Compute the values of the polynomials: Dj+f]~  
    % -------------------------------------- TNth   
    y = zeros(length_r,length(n)); &vJH$R  
    for j = 1:length(n) c:0L+OF}xY  
        s = 0:(n(j)-m_abs(j))/2; PdCEUh\>y  
        pows = n(j):-2:m_abs(j); TN.rrop`#g  
        for k = length(s):-1:1 ! z**y}<T  
            p = (1-2*mod(s(k),2))* ... Z7#+pPt!  
                       prod(2:(n(j)-s(k)))/              ... /ouPg=+Nl  
                       prod(2:s(k))/                     ... ,'+kBZOv  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... IU[ [ H#  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); i$@:@&(~Y  
            idx = (pows(k)==rpowers); G#CXs:1pd+  
            y(:,j) = y(:,j) + p*rpowern(:,idx); k\IbIv7?i  
        end s>en  
         p[-O( 3Y  
        if isnorm :svq E+2  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); +:f"Y0  
        end KP"+e:a%  
    end +%&yJ4-  
    % END: Compute the Zernike Polynomials yr6V3],Tp  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <[phnU^ 8  
    @oNXZRg6  
    % Compute the Zernike functions: ?(PKeq6  
    % ------------------------------ IcEdG(  
    idx_pos = m>0; \lY_~*J  
    idx_neg = m<0; VQs5"K"  
    I*&8^ r:A  
    z = y; ),)lzN%!  
    if any(idx_pos) ;j7#7MN2_E  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); C+]I@Go'Tk  
    end /{[o ~:'p  
    if any(idx_neg) 5\v3;;A[  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); s.#`&Sd>  
    end 92c HwWZ!  
    omFz@  
    % 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) H+#FSdy#  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. {_}I!`opr$  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated r^ XVB`v  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive gr{ DWCK  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, |:o4w  
    %   and THETA is a vector of angles.  R and THETA must have the same _GPe<H  
    %   length.  The output Z is a matrix with one column for every P-value, 3R/bz0 V>  
    %   and one row for every (R,THETA) pair. >_TZ'FT  
    % ,+VGSd  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 0_/[k*Re  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) >~f]_puT  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) TvM~y\s  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 "tZe>>I  
    %   for all p. m'U0'}Ld};  
    % +t.b` U`-  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 IBGrt^$M  
    %   Zernike functions (order N<=7).  In some disciplines it is h1RSVp+?n  
    %   traditional to label the first 36 functions using a single mode 54,er$$V  
    %   number P instead of separate numbers for the order N and azimuthal / 1RpM]d  
    %   frequency M. jdN` mosJ  
    % =wJX 0A|  
    %   Example: F@t3!bj9  
    % ,6/V" kqIP  
    %       % Display the first 16 Zernike functions #4:?gfIj  
    %       x = -1:0.01:1; Sdo-nt  
    %       [X,Y] = meshgrid(x,x); s"|Pdc4  
    %       [theta,r] = cart2pol(X,Y);  z} <^jgJ  
    %       idx = r<=1; x;S @bY  
    %       p = 0:15; :s,Z<^5a)g  
    %       z = nan(size(X)); W_=f'yb:E  
    %       y = zernfun2(p,r(idx),theta(idx)); OI*H,Z "  
    %       figure('Units','normalized') hp2t"t  
    %       for k = 1:length(p) 3$tdwe$S  
    %           z(idx) = y(:,k); v19-./H^ j  
    %           subplot(4,4,k) 3Vwh|1?  
    %           pcolor(x,x,z), shading interp (Z*!#}z`  
    %           set(gca,'XTick',[],'YTick',[]) #E?4E1bnB  
    %           axis square 1oS/`)  
    %           title(['Z_{' num2str(p(k)) '}']) PCvWS.{  
    %       end ?[AD=rUC  
    % 0sqFF[i  
    %   See also ZERNPOL, ZERNFUN. }C:r 9? T  
    W!X@  
    %   Paul Fricker 11/13/2006 9x8fhAy}4  
    7_L;E~\  
    LLo;\WGZ  
    % Check and prepare the inputs: Y73C5.dNcE  
    % ----------------------------- a[C@  
    if min(size(p))~=1 ok[i<zl; '  
        error('zernfun2:Pvector','Input P must be vector.') 1x)J[fyId  
    end +0&/g&a\R  
    ` A>@]d  
    if any(p)>35 6<]lW  
        error('zernfun2:P36', ... x Ar\gu  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... -~0^P,yQ  
               '(P = 0 to 35).']) S!UaH>Rh  
    end H)?z #x  
    Wri<h:1  
    % Get the order and frequency corresonding to the function number: )UR7i8]!0  
    % ---------------------------------------------------------------- %;_MGae  
    p = p(:); ZH8,K Y"  
    n = ceil((-3+sqrt(9+8*p))/2);  &HW9Jn  
    m = 2*p - n.*(n+2); Ie_wHcM<  
    t!XwW$@  
    % Pass the inputs to the function ZERNFUN: ;'|Ey  
    % ---------------------------------------- Tx D#9]Q`  
    switch nargin +2{Lh7Ks  
        case 3 m@c)Xci  
            z = zernfun(n,m,r,theta); }j%5t ~Qa  
        case 4 Y|n"dMrL  
            z = zernfun(n,m,r,theta,nflag); UVP vOtZj  
        otherwise N['  .BN  
            error('zernfun2:nargin','Incorrect number of inputs.') =  [E  
    end +whDU2 "  
    siI;"?  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) (zk"~Ud  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. q m}@!z^  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of A"]YM'.  
    %   order N and frequency M, evaluated at R.  N is a vector of &Jj<h: *  
    %   positive integers (including 0), and M is a vector with the >6T8^Nt  
    %   same number of elements as N.  Each element k of M must be a >7|VR:U?B  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) eFgA 8kY)  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 7dWS  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix K0~rN.C!0  
    %   with one column for every (N,M) pair, and one row for every It(_v  
    %   element in R. 4 KiY6)  
    % dN q$}  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- K1KreYlF  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is By |4 m  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to Xvu(vA  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 3`g^  
    %   for all [n,m]. *@5@,=d  
    % <)9y{J}s:  
    %   The radial Zernike polynomials are the radial portion of the 6Mf0`K  
    %   Zernike functions, which are an orthogonal basis on the unit 1zv'.uu.,  
    %   circle.  The series representation of the radial Zernike ?}oFg#m-<L  
    %   polynomials is th_oJcS  
    % **%37  
    %          (n-m)/2 }vuO$j  
    %            __ 0J9x9j`&j  
    %    m      \       s                                          n-2s V gWRW7Se  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r @"A4$`Xi3  
    %    n      s=0 iS^QTuk3%  
    % C dn J&N{  
    %   The following table shows the first 12 polynomials. 6x`t{g]f,  
    % L *wYx|  
    %       n    m    Zernike polynomial    Normalization 3og.y+.=U.  
    %       --------------------------------------------- [txE .7p  
    %       0    0    1                        sqrt(2) t.<i:#rj>l  
    %       1    1    r                           2 Z.,MVcd  
    %       2    0    2*r^2 - 1                sqrt(6) i1UsIT  
    %       2    2    r^2                      sqrt(6) XFl 6M~ c  
    %       3    1    3*r^3 - 2*r              sqrt(8) WWY6ha  
    %       3    3    r^3                      sqrt(8) 3] Ct6  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) Txu/{ M,  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) j2k"cmsKh  
    %       4    4    r^4                      sqrt(10) !$ JT e  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) kiEa<-]  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) HMXE$d=[  
    %       5    5    r^5                      sqrt(12) -7ep{p-  
    %       --------------------------------------------- 5pX6t  
    % {}9a6.V;}  
    %   Example: YK_ 7ip.a[  
    % =_CzH(=f#  
    %       % Display three example Zernike radial polynomials %9"H  
    %       r = 0:0.01:1; /ZX }Nc g  
    %       n = [3 2 5]; hN_]6,<\  
    %       m = [1 2 1]; OUnA;_  
    %       z = zernpol(n,m,r); 4W75T2q#  
    %       figure F9^S"qv$  
    %       plot(r,z) E .h*g8bXe  
    %       grid on F,kZU$  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') a?1Wq  
    % KNl$3nX  
    %   See also ZERNFUN, ZERNFUN2. >*bvw~y,  
    l1I#QB@5n  
    % A note on the algorithm. @7 }W=HB  
    % ------------------------ X$ D6Ey  
    % The radial Zernike polynomials are computed using the series [),ige  
    % representation shown in the Help section above. For many special h[ ZN+M  
    % functions, direct evaluation using the series representation can &{:-]g\  
    % produce poor numerical results (floating point errors), because P}iE+Z 3  
    % the summation often involves computing small differences between R2NZ{"h  
    % large successive terms in the series. (In such cases, the functions sO Y:e/_F  
    % are often evaluated using alternative methods such as recurrence Iu{V,U  
    % relations: see the Legendre functions, for example). For the Zernike 9r9NxKuAO  
    % polynomials, however, this problem does not arise, because the (7Qo  
    % polynomials are evaluated over the finite domain r = (0,1), and DU^loB+  
    % because the coefficients for a given polynomial are generally all ceA9) {  
    % of similar magnitude. SbZ6t$"  
    % u*R_\*j@  
    % ZERNPOL has been written using a vectorized implementation: multiple \8tsDG(1 '  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M]  ]~-r} `]  
    % values can be passed as inputs) for a vector of points R.  To achieve }H2 R3icE  
    % this vectorization most efficiently, the algorithm in ZERNPOL "@kaHIf[  
    % involves pre-determining all the powers p of R that are required to { w_e9Wbi  
    % compute the outputs, and then compiling the {R^p} into a single 4i bc  
    % matrix.  This avoids any redundant computation of the R^p, and K3C<{#r  
    % minimizes the sizes of certain intermediate variables. ]9-\~Mwh  
    % bt *k.=p  
    %   Paul Fricker 11/13/2006 N`i/mP  
    nN;u,}e  
    =N@t'fOr  
    % Check and prepare the inputs: :k"]5>(^  
    % ----------------------------- *Ex|9FCt$  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) L];b< *d  
        error('zernpol:NMvectors','N and M must be vectors.') ESs\O?nO  
    end 5;?yCWc  
    YIE<pX4Q7)  
    if length(n)~=length(m) ^Cmyx3O^  
        error('zernpol:NMlength','N and M must be the same length.') 0:+E-^X  
    end zDp2g)  
    49P 4b<1  
    n = n(:); QJNFA}*>  
    m = m(:); B!yr!DWv  
    length_n = length(n); /?!u{(h}  
    (hsl~Jf  
    if any(mod(n-m,2)) ^aQ"E9  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ijcm2FJcG  
    end c,22*.V/  
    +p^u^a  
    if any(m<0) <#.g=ay  
        error('zernpol:Mpositive','All M must be positive.') =sFTxd_"iQ  
    end !wNO8;(  
    <VcQ{F  
    if any(m>n) d _ e WcI  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') wzaV;ac4K  
    end Mtv?:q  
    $(9U@N9E  
    if any( r>1 | r<0 ) U.TA^S]`g  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') Jwp7gYZ  
    end pp2~Meg  
    \9d$@V  
    if ~any(size(r)==1) /xQPTT  
        error('zernpol:Rvector','R must be a vector.') JRFtsio*  
    end 5;S.H#YOpO  
    K^$=dLp  
    r = r(:); "3hMq1NQ`g  
    length_r = length(r); ;=@0'xPEa-  
    }Lv;!  
    if nargin==4 vy/-wP|1  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 5r_|yu  
        if ~isnorm _U0f=m  
            error('zernpol:normalization','Unrecognized normalization flag.') /bEAK-  
        end fh{`Mz,o  
    else Ie^l~ Gb  
        isnorm = false; :LTN!jj  
    end KG@8RtHsQ  
    V1?]|HTQcT  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zJXplvaL;  
    % Compute the Zernike Polynomials j9,P/K$:w  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !c-*O<Y  
    *kVV+H<X|b  
    % Determine the required powers of r: {T8Kk)L  
    % ----------------------------------- Y~Ifj,\  
    rpowers = []; ':}\4j&{E  
    for j = 1:length(n) jtc~DL  
        rpowers = [rpowers m(j):2:n(j)]; b2]Kx&!  
    end Mlq.?-QgIL  
    rpowers = unique(rpowers); e%6QTg5#  
    BD-AI  
    % Pre-compute the values of r raised to the required powers, W`&hp6Jq  
    % and compile them in a matrix: P&q7|ST%N  
    % -----------------------------  9a kH  
    if rpowers(1)==0 m3ff;,  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); .G^YqJ 4  
        rpowern = cat(2,rpowern{:}); + )?J#g  
        rpowern = [ones(length_r,1) rpowern]; '!$%> ||S  
    else qa6,z.mQ  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); d1kJRJ   
        rpowern = cat(2,rpowern{:}); ap~^Ty<>  
    end v@Ox:wl>  
    SB7c.H,  
    % Compute the values of the polynomials: Il.K"ll  
    % -------------------------------------- +*^H#|!  
    z = zeros(length_r,length_n); tjnIN?YT  
    for j = 1:length_n I0a<%;JJW  
        s = 0:(n(j)-m(j))/2; v LZoa-w:  
        pows = n(j):-2:m(j); Vg23!E  
        for k = length(s):-1:1 o14cwb  
            p = (1-2*mod(s(k),2))* ... akT6^cP^  
                       prod(2:(n(j)-s(k)))/          ... Z6pUZ[j,  
                       prod(2:s(k))/                 ... |)81Lz  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... j.kG};f  
                       prod(2:((n(j)+m(j))/2-s(k))); z ]Ue|%K  
            idx = (pows(k)==rpowers); JLi|Td "1%  
            z(:,j) = z(:,j) + p*rpowern(:,idx); `XB 9Mi=  
        end Z/K{A`  
         n(|^SH4$b  
        if isnorm 0^ibNiSP  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); H.2QKws^F  
        end 0RK!/:'  
    end Z)\@i=m  
    T^v}mWCZ  
    % 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)  y[_Q-   
    ,tJ" 5O3-  
    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。 nSAdCJ;4  
    Y<ql49-X  
    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)