切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11408阅读
    • 9回复

    [求助]ansys分析后面型数据如何进行zernike多项式拟合? [复制链接]

    上一主题 下一主题
    离线niuhelen
     
    发帖
    19
    光币
    28
    光券
    0
    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 ! FVD_8  
    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! Py`7)S  
     
    分享到
    离线phility
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    19
    光币
    28
    光券
    0
    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 og1Cj{0  
    function z = zernfun(n,m,r,theta,nflag) dP<i/@21Wm  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. tiy#b8  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N J|@O4 g   
    %   and angular frequency M, evaluated at positions (R,THETA) on the E<p<"UjcCJ  
    %   unit circle.  N is a vector of positive integers (including 0), and L<G6)'5W  
    %   M is a vector with the same number of elements as N.  Each element &gP1=P,!  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) v5I5tzt*%H  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, UhXVeGO  
    %   and THETA is a vector of angles.  R and THETA must have the same y#P _ }Kfo  
    %   length.  The output Z is a matrix with one column for every (N,M) "AlR%:]24~  
    %   pair, and one row for every (R,THETA) pair. [U$`nnp  
    % F ~e}=Nb  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike pf#R]  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), f*EDSJu\  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral H? %I((+  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, p`S~UBcL.  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Gx|/ Jq  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 29W`L2L  
    % -j^G4J  
    %   The Zernike functions are an orthogonal basis on the unit circle. @7sHFwtar?  
    %   They are used in disciplines such as astronomy, optics, and iA4VT,  
    %   optometry to describe functions on a circular domain. R0yp9icS  
    % <899r \  
    %   The following table lists the first 15 Zernike functions. ]>0$l _V  
    % Qqd+=mgc  
    %       n    m    Zernike function           Normalization }5d|y*  
    %       -------------------------------------------------- {;38&Izwz  
    %       0    0    1                                 1 Q@s G6 iz  
    %       1    1    r * cos(theta)                    2 m[w~h\FS  
    %       1   -1    r * sin(theta)                    2 'h> l_A  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) [C3wjYi  
    %       2    0    (2*r^2 - 1)                    sqrt(3) }]pOR&o  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) cr!sq.)s  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) $wcV~'fM  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) G[ q<P  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 9x14I2  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) OSK:Cb.-?F  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) $cGV)[KWp@  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ZO\bCrk  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) c;'7o=rr  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) s="cg0PD  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) G)=+Nt\ *  
    %       -------------------------------------------------- WWA!_  
    % Tt{ft?H71  
    %   Example 1: 5?TjuGc  
    % LfsOGC  
    %       % Display the Zernike function Z(n=5,m=1) CasFj9,  
    %       x = -1:0.01:1; 8yGo\\=T  
    %       [X,Y] = meshgrid(x,x); |H8UT S X+  
    %       [theta,r] = cart2pol(X,Y); } ejc  
    %       idx = r<=1; >kV=h?]Y  
    %       z = nan(size(X)); V/8yW3]Xy  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); FFc?Av?_  
    %       figure z6OJT6<'  
    %       pcolor(x,x,z), shading interp .a|ROjd!  
    %       axis square, colorbar a{iG0T.{Yh  
    %       title('Zernike function Z_5^1(r,\theta)') e)4L}a  
    % P' k`H  
    %   Example 2: p{JE@TM  
    % &wB?ks  
    %       % Display the first 10 Zernike functions WoWBZ;+U  
    %       x = -1:0.01:1; GV SVNT}I  
    %       [X,Y] = meshgrid(x,x); WtbOm  
    %       [theta,r] = cart2pol(X,Y); ="[6Z$R  
    %       idx = r<=1; E"%G@,|3*  
    %       z = nan(size(X)); I\VC2U  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ,,(BW7(  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; "\kr;X'  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; f>+:UGmP  
    %       y = zernfun(n,m,r(idx),theta(idx)); YwF\  
    %       figure('Units','normalized') _lG\_6oJ,  
    %       for k = 1:10 jF%l\$)/  
    %           z(idx) = y(:,k); +|Qe/8Q  
    %           subplot(4,7,Nplot(k)) ;gW?Fnry;  
    %           pcolor(x,x,z), shading interp Y.8mgy>   
    %           set(gca,'XTick',[],'YTick',[]) j=w`%nh4"f  
    %           axis square j*1O(p+  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) iLkP@OYgQ  
    %       end ON$-g_s>)  
    % 4";[Xr{pW  
    %   See also ZERNPOL, ZERNFUN2. a.g:yWL\  
    ,m.IhnCV\  
    %   Paul Fricker 11/13/2006 Z+x`q#ZQr  
    1)h+xY  
    ,xIWyI.  
    % Check and prepare the inputs: (~n0,$  
    % ----------------------------- @c{b\is2  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) @&]%%o+  
        error('zernfun:NMvectors','N and M must be vectors.') KfLp cV  
    end Uzd\#edxJ  
    =Qw`F0t  
    if length(n)~=length(m) +wg|~Lef h  
        error('zernfun:NMlength','N and M must be the same length.') [ f`V_1d3  
    end j*N:Kdzvl  
    m-!Uy$yM  
    n = n(:); u:D,\`;)  
    m = m(:); (SYSw%v$A  
    if any(mod(n-m,2)) 'x!5fAy  
        error('zernfun:NMmultiplesof2', ... k M' :.QT  
              'All N and M must differ by multiples of 2 (including 0).') D.R 7#^.  
    end V$fvf#T  
    8_F5c@7  
    if any(m>n) 6'qC *r   
        error('zernfun:MlessthanN', ... [!#<nY/C  
              'Each M must be less than or equal to its corresponding N.') ;-X5#  
    end X Sw0t8  
    -.X-02  
    if any( r>1 | r<0 ) }e*OprF  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') l>O~^41[  
    end [Dq!t1  
    r .b!3CoQ  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 8z h{?0  
        error('zernfun:RTHvector','R and THETA must be vectors.') T#e ;$\  
    end qA6;Q$  
    ?ydqmj2[F  
    r = r(:); [q{[Avqf  
    theta = theta(:); Q)s[ls  
    length_r = length(r); $.2#G"|  
    if length_r~=length(theta) %Cz&7qf"  
        error('zernfun:RTHlength', ... 7U\GX  
              'The number of R- and THETA-values must be equal.') $kef_*BQg  
    end N8^ AH8l  
    [~<X|_L G  
    % Check normalization: &{c.JDO  
    % -------------------- kq kj.#u  
    if nargin==5 && ischar(nflag) usR: -1{  
        isnorm = strcmpi(nflag,'norm'); VgO:`bDF  
        if ~isnorm '=2/0-;Jf  
            error('zernfun:normalization','Unrecognized normalization flag.') 3,<$z1Jm  
        end z.q^`01/H  
    else RrGFGn{  
        isnorm = false; vP{22P  
    end T!*lTzNHm  
    RHc-kggk!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fGtUr _D  
    % Compute the Zernike Polynomials VNcxST15a  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% YxUC.2V|7$  
    )E.!jL:g  
    % Determine the required powers of r: S_VZ^1X]  
    % ----------------------------------- [Grd?mc#  
    m_abs = abs(m); aI l}|n"  
    rpowers = []; 5QR=$?K  
    for j = 1:length(n) Xv%1W? >@/  
        rpowers = [rpowers m_abs(j):2:n(j)]; {m )$b  
    end N%k6*FBp~  
    rpowers = unique(rpowers); #ONad0T;  
    <n)J~B^  
    % Pre-compute the values of r raised to the required powers, [%alnY  
    % and compile them in a matrix: ,X05&'@Z  
    % ----------------------------- U$fh ~w<[  
    if rpowers(1)==0 ([r4N#lx  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ]c.1&OB7o  
        rpowern = cat(2,rpowern{:}); 1'[RrJ$Q  
        rpowern = [ones(length_r,1) rpowern]; ke sg]K  
    else PYYK R  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); gua +-##)  
        rpowern = cat(2,rpowern{:}); O"$uw  
    end LK}Ih@ f  
    IezOal  
    % Compute the values of the polynomials: PtUea  
    % -------------------------------------- WPmH4L>T  
    y = zeros(length_r,length(n)); iz&$q]P8  
    for j = 1:length(n) 'edd6yTd  
        s = 0:(n(j)-m_abs(j))/2; 0@K?'6  
        pows = n(j):-2:m_abs(j); M?i U$qI  
        for k = length(s):-1:1 3 ?1qI'5  
            p = (1-2*mod(s(k),2))* ... H6Mqy}4W  
                       prod(2:(n(j)-s(k)))/              ... h~]G6>D9)>  
                       prod(2:s(k))/                     ... *v}8n95*2  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... mIK-a{?G  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 6QwVgEnSf  
            idx = (pows(k)==rpowers); H\Y5Fd9)  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 7hs1S|  
        end lTe7n'y^^  
         }9k/Y/.  
        if isnorm )"W(0M] >  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ^usZ&9"@P  
        end o=t@83Fh5  
    end Fgf5OHX  
    % END: Compute the Zernike Polynomials tai=2,'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% h%9>js^~  
    _6b?3[Xz  
    % Compute the Zernike functions: i'w8Li  
    % ------------------------------ tl 0_Sd  
    idx_pos = m>0; S_E-H.d"  
    idx_neg = m<0; e;+6U"Jx*  
    L\cd=&b`  
    z = y; [1-1^JY  
    if any(idx_pos) SXYH#p  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); CFm( yFk  
    end 6zo'w Wc3  
    if any(idx_neg) 9{D u)k  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 8T7ex(w  
    end 64)Fz}  
    {XHAQ9'  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) /5Od:n  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. |fL|tkGEa  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated :U6"HP+?g-  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive ?Uq;>  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, iyA=d{S;V  
    %   and THETA is a vector of angles.  R and THETA must have the same fH-fEMyW  
    %   length.  The output Z is a matrix with one column for every P-value, prHM}n{0  
    %   and one row for every (R,THETA) pair. s6q6)RD"  
    % Vu0d\l^$  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike q7}rD$  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 'YKzs;y$  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) lOp7rW]$  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 X#ZQpo'h  
    %   for all p. .wU0F  
    % 4YV 0v,z  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 \;!}z3Ww  
    %   Zernike functions (order N<=7).  In some disciplines it is &$$o=Yg,  
    %   traditional to label the first 36 functions using a single mode D*%?0  
    %   number P instead of separate numbers for the order N and azimuthal _#UiY ffa*  
    %   frequency M. fY4I(~Q  
    % %Z8' h\|  
    %   Example: $ Jz(Lb{  
    % ~+A(zlYr~  
    %       % Display the first 16 Zernike functions x|b52<dLL&  
    %       x = -1:0.01:1; v)b_bU]Hx  
    %       [X,Y] = meshgrid(x,x); .+07 Ui]I!  
    %       [theta,r] = cart2pol(X,Y); _Gu;=H,~&  
    %       idx = r<=1; |rgp(;iO  
    %       p = 0:15; lZ'WFFWLE  
    %       z = nan(size(X)); bu?4$O  
    %       y = zernfun2(p,r(idx),theta(idx)); !K8Kw W|X  
    %       figure('Units','normalized')  )>=!</@  
    %       for k = 1:length(p) x>cl$41!W  
    %           z(idx) = y(:,k); Vktc  
    %           subplot(4,4,k) 9\zasa  
    %           pcolor(x,x,z), shading interp pjN4)y>0  
    %           set(gca,'XTick',[],'YTick',[]) 5K:'VX  
    %           axis square 2 rFjYx8D!  
    %           title(['Z_{' num2str(p(k)) '}']) E/3i _R  
    %       end Sx0/Dm  
    % t]CA!i`  
    %   See also ZERNPOL, ZERNFUN. E0*KKo%  
    Cqs+ o^q  
    %   Paul Fricker 11/13/2006 ~Ydm"G  
    @!Z1*a.  
    $} @gR] Z  
    % Check and prepare the inputs: o"V+W  
    % ----------------------------- Ob&m&2s,  
    if min(size(p))~=1 O#do\:(b  
        error('zernfun2:Pvector','Input P must be vector.') 8\X-]Gh\^  
    end kSpy-bVn  
    &RHZ7T  
    if any(p)>35 Yyr qO^9m  
        error('zernfun2:P36', ... \8Hs[H!  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... |iA8aHFU  
               '(P = 0 to 35).']) &7 K=  
    end ri`;   
    'ln o#  
    % Get the order and frequency corresonding to the function number: *N |ak =  
    % ---------------------------------------------------------------- Kqz+:E8D  
    p = p(:); U2Tw_  
    n = ceil((-3+sqrt(9+8*p))/2); j8G$,~v  
    m = 2*p - n.*(n+2); iG ,z3/~v  
    ]$,3vYBf  
    % Pass the inputs to the function ZERNFUN: *Fg)`M3g  
    % ---------------------------------------- nWes,K6T  
    switch nargin 1I awi?73  
        case 3 I&6M{,rnM  
            z = zernfun(n,m,r,theta); !,^y!+,Qy  
        case 4 &qzy?/i8  
            z = zernfun(n,m,r,theta,nflag); %Z3B9  
        otherwise SsEpuEn  
            error('zernfun2:nargin','Incorrect number of inputs.') K))P 2ss  
    end OQIr"  
    (!PsK:wc  
    % EOF zernfun2
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) 0 @um  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. #+N_wIP4  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of }U(bMo@;  
    %   order N and frequency M, evaluated at R.  N is a vector of Na4O( d`  
    %   positive integers (including 0), and M is a vector with the FUXJy{n6"2  
    %   same number of elements as N.  Each element k of M must be a BIS.,  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) )vUS).;S`  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is +w7U7" xQ  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 2rW9ja  
    %   with one column for every (N,M) pair, and one row for every dW2 2v!  
    %   element in R. }Q*J!OH  
    % U)M&AYb  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- nLOK1@,4  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is  ^We}i  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to kl[(!"p  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 3:G$Y: #P  
    %   for all [n,m]. %#o@c  
    % VC0Tqk  
    %   The radial Zernike polynomials are the radial portion of the d'&OEGb<  
    %   Zernike functions, which are an orthogonal basis on the unit Io"3wL)2  
    %   circle.  The series representation of the radial Zernike kBLFK3i  
    %   polynomials is lU%oU&P/"S  
    % +'Y?K]zbt  
    %          (n-m)/2 P*B @it  
    %            __ }]#z0'Aqsu  
    %    m      \       s                                          n-2s Cn{v\Q~.4  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r jo0XF]  
    %    n      s=0 6"+9$nFyW  
    % ^eyVEN  
    %   The following table shows the first 12 polynomials. ]R>NmjAI  
    % 7_P33l8y  
    %       n    m    Zernike polynomial    Normalization # S/n3  
    %       --------------------------------------------- 3~7!=s\v  
    %       0    0    1                        sqrt(2) :iLRCK3 C  
    %       1    1    r                           2 "G~!J\  
    %       2    0    2*r^2 - 1                sqrt(6) Trs2M+r)  
    %       2    2    r^2                      sqrt(6) /qJCp![X  
    %       3    1    3*r^3 - 2*r              sqrt(8) )HC/J-  
    %       3    3    r^3                      sqrt(8) O$;#GpR  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) }!{R;,5/n  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) c~~4eia)  
    %       4    4    r^4                      sqrt(10) =+UtA f<n  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) S-}c_zbl;  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) !h4A7KBYG  
    %       5    5    r^5                      sqrt(12) k \qFWFR  
    %       --------------------------------------------- #rF`Hk:  
    % X(eW+,H  
    %   Example: |OAM;@jH  
    % mj?Gc  
    %       % Display three example Zernike radial polynomials /g. c( -#]  
    %       r = 0:0.01:1; 7V8k =  
    %       n = [3 2 5]; ^"p . 3Hy  
    %       m = [1 2 1]; zwU[!i)  
    %       z = zernpol(n,m,r); #R:&Irh  
    %       figure #;~HoOK*#  
    %       plot(r,z) ^"D^D`$@  
    %       grid on (CRx'R  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') _k26(rdI@-  
    % UimofFmI%  
    %   See also ZERNFUN, ZERNFUN2. Su7N?X!  
    6pSTw\/6  
    % A note on the algorithm. Y2XxfZ j  
    % ------------------------ 2"?DaX  
    % The radial Zernike polynomials are computed using the series 2C}Yvfm4  
    % representation shown in the Help section above. For many special qbD 7\%  
    % functions, direct evaluation using the series representation can $pAJ$0=sw  
    % produce poor numerical results (floating point errors), because GC7WRA  
    % the summation often involves computing small differences between A-:k4] {%P  
    % large successive terms in the series. (In such cases, the functions lq"X_M$  
    % are often evaluated using alternative methods such as recurrence Ky"F L   
    % relations: see the Legendre functions, for example). For the Zernike =f y|Dm74  
    % polynomials, however, this problem does not arise, because the * 30K}&T  
    % polynomials are evaluated over the finite domain r = (0,1), and Q&vdBO/  
    % because the coefficients for a given polynomial are generally all J<+ f7L  
    % of similar magnitude. ?RS:I%bL  
    % z`t~N  
    % ZERNPOL has been written using a vectorized implementation: multiple {pH#zs4Y  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] YPI,u7-  
    % values can be passed as inputs) for a vector of points R.  To achieve xx>h J!  
    % this vectorization most efficiently, the algorithm in ZERNPOL ]"HaE-`%  
    % involves pre-determining all the powers p of R that are required to wpYk`L r  
    % compute the outputs, and then compiling the {R^p} into a single ,>rvl P  
    % matrix.  This avoids any redundant computation of the R^p, and ;veD?|  
    % minimizes the sizes of certain intermediate variables. 5v)bs\x6  
    % m N}szW,  
    %   Paul Fricker 11/13/2006 j\IdB:}j  
    nOL.%  
    5 | ,b  
    % Check and prepare the inputs: qE[YZ(/f0&  
    % ----------------------------- PIa!N Py  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) V=*^C+6s  
        error('zernpol:NMvectors','N and M must be vectors.') M Zz21H  
    end Znb7OF^#"  
    |xcI~ X7Q  
    if length(n)~=length(m) GW;%~qH[,  
        error('zernpol:NMlength','N and M must be the same length.') PjEJ C@n  
    end G2kU_  
    [Cv./hEQi  
    n = n(:); rX?ZUw?u&  
    m = m(:); B8T$<  
    length_n = length(n); ; $80}TY '  
    $~.YB\3  
    if any(mod(n-m,2)) 9D1WUUa  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') |K Rt$t  
    end C$6FI `J  
    T9Q3I  
    if any(m<0) aqI"4v]~b  
        error('zernpol:Mpositive','All M must be positive.') T8z?_ *k  
    end w'(/dr  
    \(P?=] -  
    if any(m>n) B??07j  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') &;d N:F;  
    end %r(WS_%K|  
    I* C~w  
    if any( r>1 | r<0 ) `R8&(kQ  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') K#wA ;  
    end R*D<M3  
    )Q =>7%ZA  
    if ~any(size(r)==1) 4G(7V:  
        error('zernpol:Rvector','R must be a vector.') F =e9o*z  
    end q%d G>!  
    #mu L-V  
    r = r(:); O+=%Mz(l  
    length_r = length(r); lfc&#G i3  
    k(dakFaC^  
    if nargin==4 hvw9i7#  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ~< bpdI0  
        if ~isnorm nFEJO&1+  
            error('zernpol:normalization','Unrecognized normalization flag.') EYq?NL='  
        end edp I?  
    else zg<-%r'$  
        isnorm = false; Q p>b  
    end wL?Up>fr  
    ja_8n["z  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8J(j}</>a  
    % Compute the Zernike Polynomials MMFwT(l<1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \QK@wgu  
    w I_@  
    % Determine the required powers of r: p5fr}#en  
    % ----------------------------------- Res U5Ce~  
    rpowers = []; ux&"TkEp  
    for j = 1:length(n) F$?Ab\#B  
        rpowers = [rpowers m(j):2:n(j)]; TBBnsj6e  
    end ;AEfU^[  
    rpowers = unique(rpowers); 0!|d .jZI  
    !RmVb}m  
    % Pre-compute the values of r raised to the required powers, njy2pDC@  
    % and compile them in a matrix: Iy9hBAg\y  
    % ----------------------------- |qUGB.Q  
    if rpowers(1)==0 Y7}>yC/GY  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); _AX 9 Mu]  
        rpowern = cat(2,rpowern{:}); ^}=)jLS  
        rpowern = [ones(length_r,1) rpowern]; sW]^YT>?  
    else >S +}  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); FbE/x$;~O  
        rpowern = cat(2,rpowern{:}); m;OvOc,  
    end d+JK")$9C  
    2!/Kt O)i^  
    % Compute the values of the polynomials: +NPL.b|  
    % -------------------------------------- Lj1l ]OD  
    z = zeros(length_r,length_n); S 5S\zTPIf  
    for j = 1:length_n k6Kc{kY  
        s = 0:(n(j)-m(j))/2; ^Pn|Q'{/p  
        pows = n(j):-2:m(j); EMmgX*iu@  
        for k = length(s):-1:1 *DF3juf~  
            p = (1-2*mod(s(k),2))* ... Y P2VSK2Q  
                       prod(2:(n(j)-s(k)))/          ... lYx_8x2  
                       prod(2:s(k))/                 ... 03 @a G  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... pr0X7 #_E5  
                       prod(2:((n(j)+m(j))/2-s(k))); 7]h%?W !  
            idx = (pows(k)==rpowers); y *i&p4Y*  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Pp8S\%z~h  
        end +vh|m5"7I7  
         S>yiD`v  
        if isnorm n$/|r  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); &K9;GZS?  
        end v"bWVc~H  
    end 7*5B  
    jdxHWkQ   
    % EOF zernpol
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    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
    发帖
    59
    光币
    0
    光券
    0
    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
    发帖
    856
    光币
    846
    光券
    0
    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
    发帖
    4352
    光币
    5478
    光券
    1
    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)  U]Q 5};FK  
    ^gVQ6=z%  
    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。 b:(+d"S  
    7w73,r/D8A  
    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)