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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 Pi,86?  
    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! pE0@m-p  
     
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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 9w1`_r[J  
    function z = zernfun(n,m,r,theta,nflag) Oz "_KMz  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. A+fXt`YNM  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N r*FAUb`bG  
    %   and angular frequency M, evaluated at positions (R,THETA) on the j|[>f  
    %   unit circle.  N is a vector of positive integers (including 0), and \"Qa)1 |  
    %   M is a vector with the same number of elements as N.  Each element f %q ?  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) $wl_  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, V%`\x\Xat  
    %   and THETA is a vector of angles.  R and THETA must have the same 3XncEdy_  
    %   length.  The output Z is a matrix with one column for every (N,M) 2cZgG^  
    %   pair, and one row for every (R,THETA) pair. i7&ay\+@  
    % 3]7j, 1^  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike @jZ1WHS_a  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Ak3V< =gx  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral C[><m2T  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Nkn2\ w  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized FyChH7  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. dChMjaix  
    % jFI`CA6P  
    %   The Zernike functions are an orthogonal basis on the unit circle. D23 c/8K  
    %   They are used in disciplines such as astronomy, optics, and SXNde@% {  
    %   optometry to describe functions on a circular domain. '<6DLtZl  
    % on1B~?*D  
    %   The following table lists the first 15 Zernike functions. I`x[1%y2 F  
    % IUD@Kf]S  
    %       n    m    Zernike function           Normalization `1lGAKv  
    %       -------------------------------------------------- sdN1BV2  
    %       0    0    1                                 1 n-OQCz9Xl  
    %       1    1    r * cos(theta)                    2 ,Z8)DC=  
    %       1   -1    r * sin(theta)                    2 ROO@EQ#`Z  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) TrQUhmS/!  
    %       2    0    (2*r^2 - 1)                    sqrt(3) T5dnj&N ]  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) [7,q@>:CS  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) NFqGbA|  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) L08lkq,  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 7s Gf_`Z  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) N_l_^yD  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) F4IU2_CnPD  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) <driD'=F  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) B'bOK`p  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) [* |+ it+!  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) "kjSg7m*:  
    %       -------------------------------------------------- p@oz[017/J  
    % @]Ac >&  
    %   Example 1: z:&/O&?  
    % D J7U6{KLq  
    %       % Display the Zernike function Z(n=5,m=1) T`GiM%R;g  
    %       x = -1:0.01:1; y<c7RK]  
    %       [X,Y] = meshgrid(x,x); !x")uYf  
    %       [theta,r] = cart2pol(X,Y); ^zfs8]QSf  
    %       idx = r<=1; ~_ wSB[z  
    %       z = nan(size(X)); 7j88^59  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); {+EnJ"  
    %       figure ?}(B8^  
    %       pcolor(x,x,z), shading interp RNt9Qdr4y  
    %       axis square, colorbar 3u< ntx ><  
    %       title('Zernike function Z_5^1(r,\theta)') S F da?>  
    % fm!\**Q1  
    %   Example 2: `v)ZOw9&  
    % F45-M[z  
    %       % Display the first 10 Zernike functions 20I/En  
    %       x = -1:0.01:1; pnXwE-c_  
    %       [X,Y] = meshgrid(x,x); jsP+,brO  
    %       [theta,r] = cart2pol(X,Y); 1\g r ;b  
    %       idx = r<=1; r;+a%?P  
    %       z = nan(size(X)); (O& HCT|  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 8is QL  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; R*2F)e\|  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ex66GJQe1  
    %       y = zernfun(n,m,r(idx),theta(idx)); {BgJ=0g?  
    %       figure('Units','normalized') 8\jsGN.$JZ  
    %       for k = 1:10 #7KR`H  
    %           z(idx) = y(:,k);  R*r"};  
    %           subplot(4,7,Nplot(k)) %"WhD'*z}  
    %           pcolor(x,x,z), shading interp [X*u`J  
    %           set(gca,'XTick',[],'YTick',[]) T1'\!6_5  
    %           axis square p1W6s0L  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Y~?Z'uR  
    %       end $EzWUt  
    % PKQ.gPu6*@  
    %   See also ZERNPOL, ZERNFUN2. <(H<*Xf9  
    ^F&j;8U  
    %   Paul Fricker 11/13/2006 ~YByyJG   
    - FJLM  
    gdq6jz  
    % Check and prepare the inputs: CTxP3a9]  
    % ----------------------------- 2jxIr-a1G  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) G,<l}(tEG  
        error('zernfun:NMvectors','N and M must be vectors.') :zNNtv iA  
    end :}-?X\|\  
    mu5r4W47  
    if length(n)~=length(m) O0 $V+fE  
        error('zernfun:NMlength','N and M must be the same length.') q([{WZ:6Oq  
    end >?6HUUQ  
    }6=? zs}  
    n = n(:); #%w)w R3  
    m = m(:); Z] x6np  
    if any(mod(n-m,2)) D4uAwmc  
        error('zernfun:NMmultiplesof2', ... ]&dPY[~,/i  
              'All N and M must differ by multiples of 2 (including 0).') 6Nt/>[  
    end %?Q&a ]  
    ^a Q&.q  
    if any(m>n) N Hn #c3o  
        error('zernfun:MlessthanN', ... {s@ 0<!  
              'Each M must be less than or equal to its corresponding N.') SpYmgL?wJ  
    end K}2G4*8S_G  
    Zxozhmg  
    if any( r>1 | r<0 ) b*/Mco 9O  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') `zB bB^\`W  
    end GLX{EG9Z  
    X(\L1N  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) WW2hwB (  
        error('zernfun:RTHvector','R and THETA must be vectors.') )lz~Rt;1i  
    end 7[!dm_  
    B9%%jEH*  
    r = r(:); yH>C7M7 t  
    theta = theta(:); `uZMln @  
    length_r = length(r); $15H_X*!  
    if length_r~=length(theta) R[)bGl6#  
        error('zernfun:RTHlength', ... ?%Ww3cU+J  
              'The number of R- and THETA-values must be equal.') UEhFId  
    end c{KJNH%7  
    (E,Ibz2G:e  
    % Check normalization: 6 jm@`pYbE  
    % -------------------- !l Egta[Ql  
    if nargin==5 && ischar(nflag) |I29m`  
        isnorm = strcmpi(nflag,'norm'); +r9neS.l  
        if ~isnorm E.+%b;Eqe  
            error('zernfun:normalization','Unrecognized normalization flag.') T7Y}v,+-  
        end w=a$]`  
    else WuFBt=%  
        isnorm = false; _:WNk(  
    end #TC}paIpj  
     ST0TWE'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Jb^{o+s53  
    % Compute the Zernike Polynomials ^!0z+M:>^  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M ?AX:0  
    vxLr034  
    % Determine the required powers of r: 8n-Xt7z  
    % -----------------------------------  K+XUC  
    m_abs = abs(m); 9%"`9j~H>  
    rpowers = []; fOME&$=O  
    for j = 1:length(n) 3D1y^I  
        rpowers = [rpowers m_abs(j):2:n(j)]; ~8"oH5  
    end <RZqs  
    rpowers = unique(rpowers); dv+ZxP%g  
    9q 2 vT^  
    % Pre-compute the values of r raised to the required powers, o4J@M{xb_  
    % and compile them in a matrix: -sZb+2tDa  
    % ----------------------------- aM(#J7;  
    if rpowers(1)==0 ~PpDrJ; Va  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); &43c/T Sb  
        rpowern = cat(2,rpowern{:}); +6 =lN[b  
        rpowern = [ones(length_r,1) rpowern]; T93st<F=R  
    else YOj&1ymBZ  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); odC"#Rb  
        rpowern = cat(2,rpowern{:}); jD}h`(bE  
    end B]: |;d  
    9V[}#(f$  
    % Compute the values of the polynomials: i3Bpim.  
    % -------------------------------------- ",J&UTUh  
    y = zeros(length_r,length(n)); N)AlQ'Lwx  
    for j = 1:length(n) &;)B qqXc  
        s = 0:(n(j)-m_abs(j))/2; `JpFqZ'58  
        pows = n(j):-2:m_abs(j); ey,f igjd.  
        for k = length(s):-1:1 ,UNk]vd  
            p = (1-2*mod(s(k),2))* ... LExm#T`  
                       prod(2:(n(j)-s(k)))/              ... Lo#G. s|  
                       prod(2:s(k))/                     ... R=<::2_Y96  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 0"T/a1S7bl  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); dJQK|/  
            idx = (pows(k)==rpowers); ViMl{3  
            y(:,j) = y(:,j) + p*rpowern(:,idx); "DfjUk  
        end >]ZE<.  
         ]'M B3@T  
        if isnorm HLG5SS7  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); .`5|NUhN  
        end nqo1+OR  
    end $I>]61l%  
    % END: Compute the Zernike Polynomials `O%nDry  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% cL ~WDW/  
    6)ln,{  
    % Compute the Zernike functions: xW*Lceb  
    % ------------------------------ kWVk^ ,  
    idx_pos = m>0; 0Xw>_#Y/xS  
    idx_neg = m<0; .UQ|k,,t  
    cNxxX!P/  
    z = y; ge.>#1f}  
    if any(idx_pos) Z"_8 l3  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); -N wic|  
    end VPuR4 p.  
    if any(idx_neg) REE .8_  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); <tZZ]Y]  
    end DB-79U%W  
    X&LJ"ahK  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) d=%NFCIV  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 5X+`aB  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated wv."  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive UhL1Y NF_  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, T$%QK?B  
    %   and THETA is a vector of angles.  R and THETA must have the same amC)t8L?  
    %   length.  The output Z is a matrix with one column for every P-value, U&u63 56  
    %   and one row for every (R,THETA) pair. gj @9(dk%  
    % LO)!Fj4|  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike `N.:3]B t  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) P-gjSE|yh  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) G$V=\60a-  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 tlUh8os  
    %   for all p. XX%K_p`&Z  
    % 43B0ynagN  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 E z}1Xse  
    %   Zernike functions (order N<=7).  In some disciplines it is JZ`h+fAt  
    %   traditional to label the first 36 functions using a single mode @0 P4pt;(  
    %   number P instead of separate numbers for the order N and azimuthal ox&? `DO  
    %   frequency M. 9?O8j1F  
    % G"J nQ  
    %   Example: @W/k}<07  
    % cl `Wl/Q#  
    %       % Display the first 16 Zernike functions ou~$XZ7oi  
    %       x = -1:0.01:1; eT3!"+p-F  
    %       [X,Y] = meshgrid(x,x); WA43}CyAe  
    %       [theta,r] = cart2pol(X,Y); abUO3 Y{  
    %       idx = r<=1; {\0V$#q   
    %       p = 0:15; 4jTO:aPh_  
    %       z = nan(size(X)); W-#DEU 7_  
    %       y = zernfun2(p,r(idx),theta(idx)); ;#9?3O s  
    %       figure('Units','normalized') ?Ce=h+l  
    %       for k = 1:length(p) vbeE}7 *2  
    %           z(idx) = y(:,k); d[,Rgdd@I  
    %           subplot(4,4,k) =dA] nM  
    %           pcolor(x,x,z), shading interp K=JDl-#!  
    %           set(gca,'XTick',[],'YTick',[]) L +Uq4S^  
    %           axis square `MPR-"Z6  
    %           title(['Z_{' num2str(p(k)) '}']) ]-{ fr+  
    %       end -_5Dk'R#`  
    % a dr\l5pWQ  
    %   See also ZERNPOL, ZERNFUN. Oj_]`  
    /tj]^QspS  
    %   Paul Fricker 11/13/2006 T[1iZ  
    [niFJI sc  
    &(Fm@ksh\  
    % Check and prepare the inputs: Qw2-Vv4!"  
    % ----------------------------- QCZ88 \jX[  
    if min(size(p))~=1 3\j`g  
        error('zernfun2:Pvector','Input P must be vector.') $RY-yKmi  
    end lkTA"8d  
    "0jwCX Cu  
    if any(p)>35 m=@xZw<  
        error('zernfun2:P36', ... ;k`51=Wi  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... U?f-/@fc  
               '(P = 0 to 35).']) EniV-Uj\D  
    end Lr:n  
    grgs r_)[  
    % Get the order and frequency corresonding to the function number: J(d+EjC  
    % ---------------------------------------------------------------- 3 5.&!4}  
    p = p(:); \&kj#)JYA  
    n = ceil((-3+sqrt(9+8*p))/2); uGC5XX^  
    m = 2*p - n.*(n+2); )GVTa4}p  
    (BPp2^  
    % Pass the inputs to the function ZERNFUN: WJONk_WAc  
    % ---------------------------------------- l3F$5n  
    switch nargin K)>F03=uE  
        case 3 F<p`)?  
            z = zernfun(n,m,r,theta); Ro9:kEG$  
        case 4 'a+^= c  
            z = zernfun(n,m,r,theta,nflag); vA_,TS#Bo  
        otherwise ^P~,bO&H.Z  
            error('zernfun2:nargin','Incorrect number of inputs.') K\%\p$ZD  
    end tkHmH/'7  
    _ncBq;j{  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) P-2DBNB7  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. ;E 9o%f:o  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of z4BU}`;b3t  
    %   order N and frequency M, evaluated at R.  N is a vector of 6 ~0kb_td  
    %   positive integers (including 0), and M is a vector with the I ]o|mjvs  
    %   same number of elements as N.  Each element k of M must be a \\:%++}J  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ,5W u  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is bR"4:b>K  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix a_}C*+D  
    %   with one column for every (N,M) pair, and one row for every dpq(=s`s  
    %   element in R. PRiE2Di2S  
    % "$(D7yFO  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ^"|q~2  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is V1M|p!  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to .-HM{6J  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 azIhp{rH w  
    %   for all [n,m]. $Q#n'#c  
    % 7c(j1:Ku-  
    %   The radial Zernike polynomials are the radial portion of the FNraof @Oy  
    %   Zernike functions, which are an orthogonal basis on the unit 4Us,DS_/  
    %   circle.  The series representation of the radial Zernike H[?S*/n,<  
    %   polynomials is W*S4gPGM  
    % n.9k5r@  
    %          (n-m)/2 SW}Rkr\e  
    %            __ HN`qMGW^  
    %    m      \       s                                          n-2s DTA$,1JuD  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r `]l` t"x  
    %    n      s=0  YMv}]  
    % 2Cy,#X%j>  
    %   The following table shows the first 12 polynomials.  o%j?}J7y  
    % 7WSP0Xyz  
    %       n    m    Zernike polynomial    Normalization p+?`ru  
    %       --------------------------------------------- x[TLlV:{  
    %       0    0    1                        sqrt(2) 3s%DF,  
    %       1    1    r                           2 I$sXbM;z=  
    %       2    0    2*r^2 - 1                sqrt(6) | X1axRO  
    %       2    2    r^2                      sqrt(6) >%`SXB& 9  
    %       3    1    3*r^3 - 2*r              sqrt(8) RYvdfj.ij  
    %       3    3    r^3                      sqrt(8) .zdaY, U  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ~ :{mKc  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) X(E`cH |  
    %       4    4    r^4                      sqrt(10) L;*7p9  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) w+ ')wyB  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Z>g&%3j  
    %       5    5    r^5                      sqrt(12) Xa>'DO2  
    %       --------------------------------------------- HFjSM~  
    % T$0)un  
    %   Example:  d<xi/  
    % *A48shfO  
    %       % Display three example Zernike radial polynomials dr"$@  
    %       r = 0:0.01:1; ?;UR9f|!  
    %       n = [3 2 5]; P4yUm(@  
    %       m = [1 2 1]; g?xXX /Qe  
    %       z = zernpol(n,m,r); S)ipkuj X  
    %       figure O{ 3X`xAf  
    %       plot(r,z) %k?/pRv$>  
    %       grid on #E Bd g  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ^zT=qB l  
    % 7P2(q  
    %   See also ZERNFUN, ZERNFUN2. _oa*E2VN  
    RgH 6l2  
    % A note on the algorithm. go=xx.WJ  
    % ------------------------ #w5%^ HwO  
    % The radial Zernike polynomials are computed using the series sbVEA  
    % representation shown in the Help section above. For many special pLF,rOb  
    % functions, direct evaluation using the series representation can /0k'w%V{n  
    % produce poor numerical results (floating point errors), because Sx~mc_ekY  
    % the summation often involves computing small differences between %e25Z .Se$  
    % large successive terms in the series. (In such cases, the functions +<@1)qZ(E  
    % are often evaluated using alternative methods such as recurrence U9PI#TX &O  
    % relations: see the Legendre functions, for example). For the Zernike ~eP~c"L  
    % polynomials, however, this problem does not arise, because the U:7w8$_  
    % polynomials are evaluated over the finite domain r = (0,1), and ;,]4A{|  
    % because the coefficients for a given polynomial are generally all KY< $+/B!  
    % of similar magnitude. JDa_;bqL  
    % on8$Kc  
    % ZERNPOL has been written using a vectorized implementation: multiple 1jKj' 7/K  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] OB=bRLd.IR  
    % values can be passed as inputs) for a vector of points R.  To achieve CTg79 ITYk  
    % this vectorization most efficiently, the algorithm in ZERNPOL *uK!w(;2  
    % involves pre-determining all the powers p of R that are required to =ePwGm1:c  
    % compute the outputs, and then compiling the {R^p} into a single !Y|xu07  
    % matrix.  This avoids any redundant computation of the R^p, and .%J<zqk-  
    % minimizes the sizes of certain intermediate variables. X xwcvE  
    % a 01s'9Be  
    %   Paul Fricker 11/13/2006 |*ZM{$  
    ]==7P;_-  
    9k62_]w@6  
    % Check and prepare the inputs: < <0[PJ  
    % ----------------------------- Z?[ R;V1j  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 3zfpFgD!  
        error('zernpol:NMvectors','N and M must be vectors.') @Kt!uKrI  
    end 1xkk5\3]  
    m7A3i<6p  
    if length(n)~=length(m) U . <c#S  
        error('zernpol:NMlength','N and M must be the same length.') %sZ3Gpi  
    end 8* m,#   
    O:,Gmft+  
    n = n(:); t vW0 W  
    m = m(:); JY:Fu  
    length_n = length(n); ".AW   
    rKOa9M  
    if any(mod(n-m,2)) JB5%\   
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') )2d1@]6#  
    end )9/iH(  
    5xUZeLj  
    if any(m<0) P^q!Pye  
        error('zernpol:Mpositive','All M must be positive.') iV@\v0k  
    end 75P!`9bE  
    uJ1oo| sn  
    if any(m>n) Pu!C,7vUQ  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') K",Xe>  
    end ESIeZhXVH  
    {=!BzNMj  
    if any( r>1 | r<0 ) 8&+u+@H  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 1z,P"?Q  
    end $_HyE%F#  
    qgWsf-di=  
    if ~any(size(r)==1)  bFA lC  
        error('zernpol:Rvector','R must be a vector.') eA(FWO  
    end :zfMRg  
    9zBt a  
    r = r(:); A2PeI"y  
    length_r = length(r); ei]Q<vT6  
    !qH)ttW  
    if nargin==4 )P+<=8@a  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); Yu[MNX ;G  
        if ~isnorm ]E.FBGT  
            error('zernpol:normalization','Unrecognized normalization flag.') X,8Zn06M  
        end nhm#_3!6A  
    else H-m).^  
        isnorm = false; 1]8Hpd  
    end 51AA,"2[_  
    %vG;'_gM B  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% YWANBM(v+  
    % Compute the Zernike Polynomials X2np.9hie  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :q2RgZE  
    n-wOLH  
    % Determine the required powers of r: ZWB3R  
    % ----------------------------------- 5ry[Lgg  
    rpowers = []; ;'!h(H  
    for j = 1:length(n) p[!9objU  
        rpowers = [rpowers m(j):2:n(j)]; $['`H)z  
    end /Vv)00  
    rpowers = unique(rpowers); Mp J3*$Dr  
    #aP;a-Q|k  
    % Pre-compute the values of r raised to the required powers, O15~\8#'  
    % and compile them in a matrix: b!EqYT  
    % ----------------------------- 3)^ 2X  
    if rpowers(1)==0  $iH  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); m[N&UM#  
        rpowern = cat(2,rpowern{:}); 4!M0)Nix  
        rpowern = [ones(length_r,1) rpowern]; K_X(j$2Xc  
    else UG]5Dxk  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); q}"HxMJ  
        rpowern = cat(2,rpowern{:}); uE#i3( J  
    end z3fU|*_c  
    jlD3SF~2  
    % Compute the values of the polynomials: u<+RA  
    % -------------------------------------- `V[ hE r|  
    z = zeros(length_r,length_n); X5Y. o&  
    for j = 1:length_n LXc;`]  
        s = 0:(n(j)-m(j))/2; q6 4bP4K  
        pows = n(j):-2:m(j); ?/Aql_?3  
        for k = length(s):-1:1 T(kG"dz   
            p = (1-2*mod(s(k),2))* ... (Y?}'?  
                       prod(2:(n(j)-s(k)))/          ... M!PK3  
                       prod(2:s(k))/                 ... fAT M?  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... qyBo|AQ5  
                       prod(2:((n(j)+m(j))/2-s(k))); ~~,#<g[  
            idx = (pows(k)==rpowers); -K5u5l}  
            z(:,j) = z(:,j) + p*rpowern(:,idx); .m%ygoO  
        end f(-3d*g  
         \AD|;tA\vE  
        if isnorm R"71)ob4  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); $! UEpQ  
        end **n109R  
    end rAu@`H?  
    l_ &T)Ei  
    % 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)  - HiRXB  
    2\J-7o=P  
    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。 ];8S<KiS~  
    :M _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)