切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11149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 2xm?,p`  
    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! QNBzc {XB  
     
    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 JJO"\^,;~  
    function z = zernfun(n,m,r,theta,nflag) {QJ`.6Kt  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 0eIR)#j*  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N %vzpp\t  
    %   and angular frequency M, evaluated at positions (R,THETA) on the D':A-E  
    %   unit circle.  N is a vector of positive integers (including 0), and ~A( Pa-  
    %   M is a vector with the same number of elements as N.  Each element ^.7xu/T  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ]5CFL$_Q{  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, umYdr'p!v  
    %   and THETA is a vector of angles.  R and THETA must have the same c0~'5Mlp  
    %   length.  The output Z is a matrix with one column for every (N,M) VI{1SIhfa  
    %   pair, and one row for every (R,THETA) pair. P'';F}NwfX  
    % 6ZJQ '9f  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike qKXn=J/0tA  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), I-I5^s  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ([A;~ p;n  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, _\zf XHp  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized !LA#c'  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. lRq!|.C  
    % mbK$Wp#  
    %   The Zernike functions are an orthogonal basis on the unit circle. LgYzGlJp  
    %   They are used in disciplines such as astronomy, optics, and UgJHSl  
    %   optometry to describe functions on a circular domain. t!$/r]XM h  
    % 'AU!xG6OQ  
    %   The following table lists the first 15 Zernike functions. 8h=XQf6k0  
    % BH1To&ol  
    %       n    m    Zernike function           Normalization {zcjTJ=Zt8  
    %       -------------------------------------------------- #;)7~69  
    %       0    0    1                                 1  Qy%/+9L  
    %       1    1    r * cos(theta)                    2 ;DOz92X94  
    %       1   -1    r * sin(theta)                    2 VrG|/2  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) :1I,:L  
    %       2    0    (2*r^2 - 1)                    sqrt(3) K`sm  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) +( d2hSIF  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) !~#31kL&  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) l%O-c}X  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ueOvBFgZ  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) _e W*  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ? "gy`oCv  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) r_",E=e  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) )_ y{^kn3^  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) $i hI Hl6'  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) R.7" ZG  
    %       -------------------------------------------------- L r,$98Dy  
    % >_".  
    %   Example 1: 0qv)'[O  
    % l#Tm`br  
    %       % Display the Zernike function Z(n=5,m=1) KRQ/wuv  
    %       x = -1:0.01:1; )8_0d)  
    %       [X,Y] = meshgrid(x,x); ,DjZDw  
    %       [theta,r] = cart2pol(X,Y); 0WFZx Ad"  
    %       idx = r<=1; n.)-aRu[  
    %       z = nan(size(X)); E_z@\z MB  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); BsAglem  
    %       figure q&.!*rPD  
    %       pcolor(x,x,z), shading interp F^ f]*MhT"  
    %       axis square, colorbar ETIf x)B-  
    %       title('Zernike function Z_5^1(r,\theta)') mMR[(  
    % !dGgLU_  
    %   Example 2: ` mi!"pmw  
    % la-+ `  
    %       % Display the first 10 Zernike functions tPUQ"S  
    %       x = -1:0.01:1; >&TktQO_T  
    %       [X,Y] = meshgrid(x,x); }5gQZ'ys'  
    %       [theta,r] = cart2pol(X,Y); -%A6eRShk  
    %       idx = r<=1; ,/KHKLY7  
    %       z = nan(size(X)); z<ek?0?yS  
    %       n = [0  1  1  2  2  2  3  3  3  3]; CNwhH)*  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; FR&RIFy  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; `4o;Lz~  
    %       y = zernfun(n,m,r(idx),theta(idx)); Vo\d&}Q  
    %       figure('Units','normalized') * PZ=$>r  
    %       for k = 1:10 ZE9*i}r  
    %           z(idx) = y(:,k); yP@= x!$  
    %           subplot(4,7,Nplot(k)) _tjH=Ff$  
    %           pcolor(x,x,z), shading interp /xmd]XM=_  
    %           set(gca,'XTick',[],'YTick',[]) o)$sZ{` ="  
    %           axis square i|<*EXB"  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) $6_J` 7  
    %       end 3K'3Xp@A  
    % GV9"8M Z6  
    %   See also ZERNPOL, ZERNFUN2. 2`z+_DA  
    1F=x~FMvY  
    %   Paul Fricker 11/13/2006 r"n)I$  
    3RD Q{&J:  
    9(C Ke,  
    % Check and prepare the inputs: {3;4=R3  
    % ----------------------------- 71~V*  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Mfgd;FsX#  
        error('zernfun:NMvectors','N and M must be vectors.') m?csake.Me  
    end XhS<GF%  
    @a~K#Bvlm  
    if length(n)~=length(m) R(:q^?  
        error('zernfun:NMlength','N and M must be the same length.') F2u{Wzr_@  
    end 1.uyu  
    -Oo$\=d  
    n = n(:); {{O1C ~  
    m = m(:); {U4%aoBd8  
    if any(mod(n-m,2)) /q>"">  
        error('zernfun:NMmultiplesof2', ... 0$UE|yDs>  
              'All N and M must differ by multiples of 2 (including 0).') JeO(sj$e  
    end =.uE(L`]NA  
    v(af aN  
    if any(m>n) rR7}SEa  
        error('zernfun:MlessthanN', ... <mpkkCl,  
              'Each M must be less than or equal to its corresponding N.') D3_,2  
    end A5z`3T;1  
    eX=W+&lj  
    if any( r>1 | r<0 ) DukCXyB*l  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') S]<Hx_[}  
    end 4WNWn#M  
    ;}r#08I  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) O|8p #  
        error('zernfun:RTHvector','R and THETA must be vectors.') z-()7WY  
    end O*30|[  
    $FD0MrB_+  
    r = r(:); M[X& Q  
    theta = theta(:); ua2SW(C@  
    length_r = length(r); x1TB (^aX  
    if length_r~=length(theta) S3 &L  
        error('zernfun:RTHlength', ... E*CY/F I_  
              'The number of R- and THETA-values must be equal.') \s,ZE6dQ  
    end wp} PQw:  
    .~Td /o7  
    % Check normalization: r;9F@/  
    % -------------------- ba ,2.|  
    if nargin==5 && ischar(nflag) &u.t5m7(  
        isnorm = strcmpi(nflag,'norm'); :V8 \^  
        if ~isnorm q),yY]5  
            error('zernfun:normalization','Unrecognized normalization flag.') H1N%uk=kV  
        end r=u>TA$  
    else M[SWMVN{  
        isnorm = false;  aj1Zi3h  
    end ^f@EDG8  
    hMDy;oQ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% j134iVF%  
    % Compute the Zernike Polynomials |E|d"_Ma  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _%Jqyc"-  
    uP<tP:  
    % Determine the required powers of r: ,zO!`|I  
    % ----------------------------------- b1_HDC(  
    m_abs = abs(m); 8n NRn[oS  
    rpowers = []; ?oP<sGp  
    for j = 1:length(n) iFpJ /L  
        rpowers = [rpowers m_abs(j):2:n(j)]; D/{hLp{  
    end (oxe'\  
    rpowers = unique(rpowers); .I<#i9Le  
    ]H%y7kH8  
    % Pre-compute the values of r raised to the required powers, EE-jU<>|  
    % and compile them in a matrix: R0 AVAUG  
    % ----------------------------- :IvKxOv  
    if rpowers(1)==0 BlMc<k  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); dy`K5lC@  
        rpowern = cat(2,rpowern{:}); >}Fe9Y.o  
        rpowern = [ones(length_r,1) rpowern]; g"^<LX-  
    else oF8#gn_  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); m&cVda/  
        rpowern = cat(2,rpowern{:}); HvLvSy1U  
    end d%8hWlffz  
    rISg`-  
    % Compute the values of the polynomials: 6]1cy&SG  
    % -------------------------------------- U TC|8  
    y = zeros(length_r,length(n));  1ti+ Q0~  
    for j = 1:length(n) CM|?;PBuv  
        s = 0:(n(j)-m_abs(j))/2; >+LFu?y  
        pows = n(j):-2:m_abs(j); IXc"gO  
        for k = length(s):-1:1 :>+}|(v  
            p = (1-2*mod(s(k),2))* ... aOIE9wO  
                       prod(2:(n(j)-s(k)))/              ... }\?UmuolQ  
                       prod(2:s(k))/                     ... @Ge\odfF:  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... *#\da]"{  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); tUaDwIu#  
            idx = (pows(k)==rpowers); 5R"iF+p4  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 2M1}`H\  
        end ;Hk{bz(  
         R9xhO!   
        if isnorm __O@w.  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); DSf  
        end P;G Rk6  
    end D;*P'%_Z  
    % END: Compute the Zernike Polynomials mW- 4  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% gE;r;#Jt4  
    qp;eBa  
    % Compute the Zernike functions: SoC3)iqv/  
    % ------------------------------ FX}kH]  
    idx_pos = m>0; K8,Q^!5]"  
    idx_neg = m<0; bh V.uBH  
    Hwiw:lPq`E  
    z = y; ,}?x!3  
    if any(idx_pos) '~{bq'7`m  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); V'alzw7#  
    end J B[n]|  
    if any(idx_neg) dX^ ^ @7  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); I5Vp%mCY  
    end 8725ET t  
    ->_rSjnM{  
    % 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) VCf/EkC  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. JO[7_*s  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated `|&#=hl~  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive V)<Jj  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, J> Z.2  
    %   and THETA is a vector of angles.  R and THETA must have the same h$`zuz  
    %   length.  The output Z is a matrix with one column for every P-value, XSOSy2:  
    %   and one row for every (R,THETA) pair. e2F{}N  
    % + PAb+E|,  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike "@ 1+l&  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 4 x|yzUx  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) T@H<Fm_  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 X5_T?  
    %   for all p. X iW~? *Z  
    % RwyX,|  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 X^o0t^  
    %   Zernike functions (order N<=7).  In some disciplines it is 2pQ29  
    %   traditional to label the first 36 functions using a single mode xwSi.~.  
    %   number P instead of separate numbers for the order N and azimuthal o{[w6^D7  
    %   frequency M. 4(nwi[1Y  
    % }z,f8Yz  
    %   Example: %^KNY ;E  
    % o$q})!  
    %       % Display the first 16 Zernike functions }j`#s  
    %       x = -1:0.01:1; P!xN]or]u  
    %       [X,Y] = meshgrid(x,x); $Cnv]1%  
    %       [theta,r] = cart2pol(X,Y); y?P4EVknM3  
    %       idx = r<=1; )i/x%^ca$  
    %       p = 0:15; }kZ)|/]kn  
    %       z = nan(size(X)); GtLn h~)  
    %       y = zernfun2(p,r(idx),theta(idx)); !-AK@`i.  
    %       figure('Units','normalized') EBMZ7b-7  
    %       for k = 1:length(p) }Gf9.ACQ  
    %           z(idx) = y(:,k); nq%GLUH   
    %           subplot(4,4,k) Q@(tyW+8U@  
    %           pcolor(x,x,z), shading interp sD=iHO Am  
    %           set(gca,'XTick',[],'YTick',[]) 5c ($~EFr  
    %           axis square FE'F@aS\  
    %           title(['Z_{' num2str(p(k)) '}']) 1fMl8[!JLu  
    %       end 1ir~WFP  
    % 4{6XZ_J1  
    %   See also ZERNPOL, ZERNFUN. Mwtd<7<!A  
    rO[ Zx'a  
    %   Paul Fricker 11/13/2006 wl5+VC*l0  
    W&=F<n`  
    HDHC9E6  
    % Check and prepare the inputs: irooFR[L9  
    % ----------------------------- \AY*x=PF  
    if min(size(p))~=1 I?OnEw  
        error('zernfun2:Pvector','Input P must be vector.') HDQH7Bs  
    end 'U*Kb  
    VMl)_M:'  
    if any(p)>35 AQgagE^  
        error('zernfun2:P36', ... M0K+Vz=  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Qm@v}pD  
               '(P = 0 to 35).']) 1X-fiQJe  
    end yL #2|t(  
    (W'3Zv'f  
    % Get the order and frequency corresonding to the function number: |Ye%HpTTv  
    % ---------------------------------------------------------------- >5MHn@  
    p = p(:);  2p;N|V  
    n = ceil((-3+sqrt(9+8*p))/2); w$$vR   
    m = 2*p - n.*(n+2); ^3lEfI<pBm  
    |PutTcjQ  
    % Pass the inputs to the function ZERNFUN: N VBWF  
    % ---------------------------------------- s#>``E!  
    switch nargin F.$NYr/|y  
        case 3 glUf. :]  
            z = zernfun(n,m,r,theta); u(C?\HaH  
        case 4 JW9U&Bj{  
            z = zernfun(n,m,r,theta,nflag); ;@s'JSPt  
        otherwise d)'J:  
            error('zernfun2:nargin','Incorrect number of inputs.') l'FNp  
    end 7q@>d(xho  
    f0ME$:2  
    % 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) vI0::ah/  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. R6E.C!EI  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of dZ{yNh.]  
    %   order N and frequency M, evaluated at R.  N is a vector of j7v?NY  
    %   positive integers (including 0), and M is a vector with the {N`<TH PP  
    %   same number of elements as N.  Each element k of M must be a L%5g]=  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) sHf.xc  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is @ZtDjxN &  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix I1fUV72  
    %   with one column for every (N,M) pair, and one row for every s*UO!bHa  
    %   element in R. !fK9YW(Im  
    % 99u9L)  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- +kZW:t!-  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is C?fa-i0l^  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to u ioBI d  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 D9-D%R,  
    %   for all [n,m]. qcR"i+b  
    % ~[3B<^e  
    %   The radial Zernike polynomials are the radial portion of the bqSp4TI  
    %   Zernike functions, which are an orthogonal basis on the unit ?)mM]2%%  
    %   circle.  The series representation of the radial Zernike ,-.a! a  
    %   polynomials is d!#qBn$*[  
    % x$;kA}gy  
    %          (n-m)/2 A i5|N  
    %            __ 6rg?0\A<  
    %    m      \       s                                          n-2s  KSB{Z TE  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r EjFK zx  
    %    n      s=0 >'e(|P4  
    % =.yKl*WV{  
    %   The following table shows the first 12 polynomials.  "?(N  
    % g!.k>  
    %       n    m    Zernike polynomial    Normalization uBqZ62{G  
    %       --------------------------------------------- sEm064  
    %       0    0    1                        sqrt(2) ?h7(,39^>  
    %       1    1    r                           2 E'wJ+X9 +  
    %       2    0    2*r^2 - 1                sqrt(6) e{fm7Cc)D  
    %       2    2    r^2                      sqrt(6) 1PnWgu  
    %       3    1    3*r^3 - 2*r              sqrt(8) \&. ]!!Q  
    %       3    3    r^3                      sqrt(8) $G .ws  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 7<7 /NZ<I  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) %VmHw~xyF:  
    %       4    4    r^4                      sqrt(10) s6.#uT7h  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) cr"AK"TQ  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) {v~.zRW%]r  
    %       5    5    r^5                      sqrt(12) C3z#A3&J  
    %       --------------------------------------------- kSq1Q#Bxq  
    % 7qT>wCVT  
    %   Example: e9@7GaL`"S  
    % i! DO  
    %       % Display three example Zernike radial polynomials c]!Yb-  
    %       r = 0:0.01:1; N;.}g*_+}  
    %       n = [3 2 5]; ZA Xw=O5  
    %       m = [1 2 1]; 4;.y>~z  
    %       z = zernpol(n,m,r); ~.L\f%<  
    %       figure p`}'-A|@  
    %       plot(r,z) :qL1jnR^  
    %       grid on - }2AXP2q  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 6im!v<1Qx  
    % ( S=RFd  
    %   See also ZERNFUN, ZERNFUN2. R1lC_G]  
    41Htsj  
    % A note on the algorithm. +?[,{WtV  
    % ------------------------ }vspjplk^  
    % The radial Zernike polynomials are computed using the series C=uYX"  
    % representation shown in the Help section above. For many special k7\ ,N o}  
    % functions, direct evaluation using the series representation can afNqK~  
    % produce poor numerical results (floating point errors), because *D6X&Hg&5  
    % the summation often involves computing small differences between 7GVI={ b  
    % large successive terms in the series. (In such cases, the functions /Xo8 kC  
    % are often evaluated using alternative methods such as recurrence 1@L|EFa  
    % relations: see the Legendre functions, for example). For the Zernike `R+I(Cb  
    % polynomials, however, this problem does not arise, because the @.SuHd  
    % polynomials are evaluated over the finite domain r = (0,1), and Kfl#78$d  
    % because the coefficients for a given polynomial are generally all .,$<waGD  
    % of similar magnitude. \n`)>-  
    % @ky<5r*JU(  
    % ZERNPOL has been written using a vectorized implementation: multiple X cDu&6Dy  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 32KL~32Y  
    % values can be passed as inputs) for a vector of points R.  To achieve |NoTwK  
    % this vectorization most efficiently, the algorithm in ZERNPOL l6O8:XI  
    % involves pre-determining all the powers p of R that are required to MzudCMF  
    % compute the outputs, and then compiling the {R^p} into a single W{z{AxS  
    % matrix.  This avoids any redundant computation of the R^p, and '|JBA.s|  
    % minimizes the sizes of certain intermediate variables. %pk'YA{M)q  
    % j|/4V  
    %   Paul Fricker 11/13/2006 *dw6>G0U  
    svTKt%6X  
    4T<4Rb[  
    % Check and prepare the inputs: ;"N4Yflz  
    % ----------------------------- q+}KAk|]V  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ;ZVT[gi*  
        error('zernpol:NMvectors','N and M must be vectors.') p,'Z{7HG  
    end HX&G  k  
    1#m'u5L  
    if length(n)~=length(m) iF#|Z$g-(  
        error('zernpol:NMlength','N and M must be the same length.') .\6q\7Ej  
    end 6+s10?  
    VvSe`E*  
    n = n(:); U:1cbD7|3  
    m = m(:); *~>} *  
    length_n = length(n); vz1yH%~E  
    CfMCc:8mL  
    if any(mod(n-m,2)) ~aZy52H_#.  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') vdt":  
    end _b)=ERBbCo  
    pd Fa]  
    if any(m<0) m:  
        error('zernpol:Mpositive','All M must be positive.') -)Zp"  
    end 1;8%\r[|5^  
    pSC\[%K  
    if any(m>n) $t{;- DpNB  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') )5NjwLs  
    end >nqCUhS   
    {k"t`uo_  
    if any( r>1 | r<0 ) 4[VW~x07  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') :Ou[LF.O  
    end 2^;zj0]Rt  
    )A1u uW (  
    if ~any(size(r)==1) )4tOTi[  
        error('zernpol:Rvector','R must be a vector.') G3wkqd  
    end Nm.G,6<J  
    |3{"ANmm'  
    r = r(:); ^S%xaA9  
    length_r = length(r); %p t^?  
    r\."=l  
    if nargin==4 618k-  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); BJNZH#"  
        if ~isnorm MX )mm^A  
            error('zernpol:normalization','Unrecognized normalization flag.') bK69Rb@\A  
        end !-cK@>.pE  
    else m*f"Y"B.1I  
        isnorm = false; T?+%3z}8  
    end D<wz%*  
    @yj$  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~cL)0/j}  
    % Compute the Zernike Polynomials lh`ZEvt  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% qe<xH#6  
    AdgZau[Y6  
    % Determine the required powers of r: RE%25t|  
    % ----------------------------------- 5>!I6[{  
    rpowers = []; S\dG>F>S  
    for j = 1:length(n) !(8) '<t9  
        rpowers = [rpowers m(j):2:n(j)]; ;#XF.l,u  
    end F(DM$5z[  
    rpowers = unique(rpowers); >*]dB|2  
    Tf{lH9ca$  
    % Pre-compute the values of r raised to the required powers, X@pcL{T!  
    % and compile them in a matrix: ?[#4WH-G  
    % ----------------------------- 4L_AhX7  
    if rpowers(1)==0 k@ So l6  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); uGU-MC *  
        rpowern = cat(2,rpowern{:}); #\ l#f8(l  
        rpowern = [ones(length_r,1) rpowern]; dh-?_|"  
    else u/.# zn@9h  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); E@J}(76VS  
        rpowern = cat(2,rpowern{:}); 3S=$ng  
    end E0*62OI~O  
    k!0vpps  
    % Compute the values of the polynomials: @>q4hYF  
    % -------------------------------------- .Mxt F\  
    z = zeros(length_r,length_n); 8'-E>+L   
    for j = 1:length_n "BA&  
        s = 0:(n(j)-m(j))/2; fi  
        pows = n(j):-2:m(j); Xk?Y  
        for k = length(s):-1:1 5h [<!f=  
            p = (1-2*mod(s(k),2))* ... ^ ~kfo|  
                       prod(2:(n(j)-s(k)))/          ... RHu4cK!5  
                       prod(2:s(k))/                 ... orZwm9#].  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... )CoJ9PO7  
                       prod(2:((n(j)+m(j))/2-s(k))); >>T,M@s-:  
            idx = (pows(k)==rpowers); _Rk>yJD7s  
            z(:,j) = z(:,j) + p*rpowern(:,idx); RV>n Op}R  
        end MZ:Ty,pw:O  
         },%, v2}  
        if isnorm Ij?Qs{V  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 1B`JvNtd  
        end bo &QKK  
    end T!1Np'12zF  
    nn8uFISb  
    % EOF zernpol
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
    发帖
    59
    光币
    0
    光券
    0
    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
    发帖
    856
    光币
    846
    光券
    0
    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
    发帖
    4352
    光币
    5478
    光券
    1
    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  #OVf2  "  
    l:H}Y3_I  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 JJ$q*  
    sy;_%,}N  
    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)