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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 V0s,f .a  
    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! k06xz#pL  
     
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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  dhZ Zb  
    function z = zernfun(n,m,r,theta,nflag) oDz*~{BHg  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 'G<}U343=8  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N /X@7ju;   
    %   and angular frequency M, evaluated at positions (R,THETA) on the ('T4Db  
    %   unit circle.  N is a vector of positive integers (including 0), and l8er$8S}  
    %   M is a vector with the same number of elements as N.  Each element jo<>Hc{g>  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ri"?, }(  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, wTHK=n\i  
    %   and THETA is a vector of angles.  R and THETA must have the same {EOn r1  
    %   length.  The output Z is a matrix with one column for every (N,M) qo6 1O\qm  
    %   pair, and one row for every (R,THETA) pair. sk~za  
    % U&,r4>V@h>  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ^uC"dfH  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), `@4 2jG}*  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Sc%aJ1  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, )!N2'Ld  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized y=-{Q  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. tceIA8d6  
    % W"W@WG9X0  
    %   The Zernike functions are an orthogonal basis on the unit circle. BHF{-z  
    %   They are used in disciplines such as astronomy, optics, and \H,V 9!B  
    %   optometry to describe functions on a circular domain. w/qQ(]n8  
    % h~,x7]w6  
    %   The following table lists the first 15 Zernike functions. B1x'5S;Bq  
    % Z"l`e0 {  
    %       n    m    Zernike function           Normalization Tq9,c#}&  
    %       -------------------------------------------------- :|?~B%-p[  
    %       0    0    1                                 1 ;n3uV`\  
    %       1    1    r * cos(theta)                    2 <dq,y>  
    %       1   -1    r * sin(theta)                    2 UN,<6D3\b  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) +F1]M2p]  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 0\V\qAk  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) eA~J4k_  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) }UyzM y,  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) p#ZMABlE,P  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) TvQWdX=  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Z|]l"W*w  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) F;cI0kP=>  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Iu)L3_+  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) (jp1; #P!  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) " 7l jc  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) p6<E=5RRd1  
    %       -------------------------------------------------- Hi9 G^Q  
    % B(S5+Y  
    %   Example 1: sqm%iyC=q  
    % RA*_&Ll&!C  
    %       % Display the Zernike function Z(n=5,m=1) 9`ri J4zl  
    %       x = -1:0.01:1; PFImqojHd  
    %       [X,Y] = meshgrid(x,x); 2z.k)Qx!Z  
    %       [theta,r] = cart2pol(X,Y); 0|],d?-h  
    %       idx = r<=1; +9<,3IJe6  
    %       z = nan(size(X)); &>d:ewM\  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); (1j(* ?2  
    %       figure ;s}-X_O<  
    %       pcolor(x,x,z), shading interp d/0/$Bz}P  
    %       axis square, colorbar pKO T  Qf  
    %       title('Zernike function Z_5^1(r,\theta)') C!aX45eg  
    % <wIp$F.  
    %   Example 2: qg_>`Bv"a  
    % S#dyRTmI  
    %       % Display the first 10 Zernike functions !1ie:z>s  
    %       x = -1:0.01:1; tEi@p;Z>  
    %       [X,Y] = meshgrid(x,x); !mw{T D  
    %       [theta,r] = cart2pol(X,Y); 1G e)p4  
    %       idx = r<=1; <[ g$N4  
    %       z = nan(size(X)); +=n x|:no  
    %       n = [0  1  1  2  2  2  3  3  3  3]; UQC'(>.}  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; rXHHD#\oF  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ,gFL Wb`B'  
    %       y = zernfun(n,m,r(idx),theta(idx)); \GjXsR*b5  
    %       figure('Units','normalized') ~G|{q VO7A  
    %       for k = 1:10 ~NNaLl  
    %           z(idx) = y(:,k); &5kjjQ*HB  
    %           subplot(4,7,Nplot(k)) 5n|MA  
    %           pcolor(x,x,z), shading interp J@u!S~&r  
    %           set(gca,'XTick',[],'YTick',[]) |Fh`.iT%c  
    %           axis square @B>%B EC  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) puf;"c6e'  
    %       end = y,yQO  
    % d\1:1ucV  
    %   See also ZERNPOL, ZERNFUN2. IkE'_F  
    x|~D(zo  
    %   Paul Fricker 11/13/2006 &?`d8\z  
    -r6(=A  
    a9mr-`<  
    % Check and prepare the inputs: MJ*oeI!.=  
    % ----------------------------- ?kT~)k  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) x~3>1Wr#M  
        error('zernfun:NMvectors','N and M must be vectors.') &9jUf:gJ0  
    end 2WbZ>^:Nsk  
    he#Tr'j  
    if length(n)~=length(m) ~' PS|  
        error('zernfun:NMlength','N and M must be the same length.') tyG nG0GK  
    end *aSRKY  
    _If@#WnoyA  
    n = n(:); hg86#jq%  
    m = m(:); \8C*O{w  
    if any(mod(n-m,2)) -Z\UYt  
        error('zernfun:NMmultiplesof2', ... 0SGczgg  
              'All N and M must differ by multiples of 2 (including 0).') ( .6tz  
    end 9X^-)G>  
    ' /@!"IXz  
    if any(m>n) G`3vH,  
        error('zernfun:MlessthanN', ... =t>`< T|(  
              'Each M must be less than or equal to its corresponding N.') )}zA,FOA*  
    end {?h6*>-^Z  
     onS{  
    if any( r>1 | r<0 ) P[J qJi/H  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') LeRh (a`=$  
    end wTJMq`sY_  
    `P)64So-1  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) {F{[!.  
        error('zernfun:RTHvector','R and THETA must be vectors.') n(F<  
    end A=2nj  
    |[n|=ORI'  
    r = r(:); Tl0+Bq  
    theta = theta(:); !Z9ikn4A  
    length_r = length(r); 2Dwt4V  
    if length_r~=length(theta) Nr*ibtz|D  
        error('zernfun:RTHlength', ... ">4[+'  
              'The number of R- and THETA-values must be equal.') S)AE   
    end N?u2,h-  
    *b7 ^s,?  
    % Check normalization: <?`e9o  
    % -------------------- S+\Mt+o  
    if nargin==5 && ischar(nflag) f*R_\  
        isnorm = strcmpi(nflag,'norm'); n6-!@RYr  
        if ~isnorm &hM,b!R|  
            error('zernfun:normalization','Unrecognized normalization flag.') $K>d\{@+7  
        end `&&6-/  
    else b ffml  
        isnorm = false; *^$N $t/2  
    end HpgN$$\@  
    7E84@V[\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?2bE=|  
    % Compute the Zernike Polynomials oCru5F  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )~o`QM+  
    ysP/@;jC  
    % Determine the required powers of r: @5nkI$>3z  
    % ----------------------------------- "9Fv!*<-W  
    m_abs = abs(m); Z;> aW;Wt  
    rpowers = []; I7-PF?  
    for j = 1:length(n) jzOMjz~:)  
        rpowers = [rpowers m_abs(j):2:n(j)]; ;U:o'9^9T  
    end M`g Kt (3  
    rpowers = unique(rpowers); '&L   
    j2&OYg  
    % Pre-compute the values of r raised to the required powers, I>(z)"1  
    % and compile them in a matrix: sC*E;7gT,  
    % ----------------------------- oFx gR9  
    if rpowers(1)==0 @X / =.  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); fJN9+l  
        rpowern = cat(2,rpowern{:}); 7Bb@9M?i  
        rpowern = [ones(length_r,1) rpowern];  x+j/v5  
    else mjJlXA  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); c\?/^xr'!}  
        rpowern = cat(2,rpowern{:}); Y&:\s8C  
    end U";Rp&\3;  
    mwiPvwHrg  
    % Compute the values of the polynomials: 0~I) /T  
    % -------------------------------------- hCx#Heh  
    y = zeros(length_r,length(n)); IaZAP  
    for j = 1:length(n) !c;p4B)  
        s = 0:(n(j)-m_abs(j))/2; (6_/n&mF  
        pows = n(j):-2:m_abs(j); 5Szo5  
        for k = length(s):-1:1 k/f_@8  
            p = (1-2*mod(s(k),2))* ... _rWXcK3cjr  
                       prod(2:(n(j)-s(k)))/              ... wB 0WR  
                       prod(2:s(k))/                     ... P6Ol+SI#m  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... J'oz P^N  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 7PPsEU:rf  
            idx = (pows(k)==rpowers); S%%qn  
            y(:,j) = y(:,j) + p*rpowern(:,idx); W;j)ux7jMY  
        end bJu,R-f  
         A}+r;Y8[h  
        if isnorm T%b^|="@  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); )FiU1E  
        end Z-=7QK.\{  
    end yOm6HA``hT  
    % END: Compute the Zernike Polynomials HAOrwJFqU  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% m<;" 1<k  
    LA(JA  
    % Compute the Zernike functions: 206jeH9  
    % ------------------------------ Xrs~ove1V  
    idx_pos = m>0; O? <_,-.  
    idx_neg = m<0; W8/6  
    nK; rEL  
    z = y; K*D]\/;^  
    if any(idx_pos) r/w@Dh]{_  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); X%qR6mMfT7  
    end %Y[/Ucdm  
    if any(idx_neg) lY8Qy2k|  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Hw3 ES  
    end ~w% +y  
    !,WRXE&j  
    % 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 k4i%hGk  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. m$j;FKz+|  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated Vi~+C@96  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive `{[C4]Ew/  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, -B! TA0=oJ  
    %   and THETA is a vector of angles.  R and THETA must have the same TW? MS em  
    %   length.  The output Z is a matrix with one column for every P-value, JG$J,!.\  
    %   and one row for every (R,THETA) pair. KPrxw }P  
    % l$@lk?dc  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 5,fzB~$TX(  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ;hp; Rd  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) tV%\Jk),  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ~UFsiVpL  
    %   for all p. wYM{x!D  
    % Hc3/`.nt  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 iIRigW  
    %   Zernike functions (order N<=7).  In some disciplines it is !y0 O['7  
    %   traditional to label the first 36 functions using a single mode !I$RE?7eY  
    %   number P instead of separate numbers for the order N and azimuthal RGOwm~a  
    %   frequency M. T!$HVHh&,}  
    % =l{KYv  
    %   Example: @1X1E 2:  
    % lsf?R'1  
    %       % Display the first 16 Zernike functions Z k_&Kw|  
    %       x = -1:0.01:1; a2n#T,kq&  
    %       [X,Y] = meshgrid(x,x); 2sq<"TlQXI  
    %       [theta,r] = cart2pol(X,Y); breVTY7 S  
    %       idx = r<=1; 6f1Y:qK'@  
    %       p = 0:15; cVi CWc2  
    %       z = nan(size(X)); j(N9%/4u  
    %       y = zernfun2(p,r(idx),theta(idx)); Q4 S8NqE  
    %       figure('Units','normalized') QJ'C?hn  
    %       for k = 1:length(p) Cl=ExpX/O  
    %           z(idx) = y(:,k); ;bmd<1  
    %           subplot(4,4,k) bBL"F!.  
    %           pcolor(x,x,z), shading interp 1Tkz!  
    %           set(gca,'XTick',[],'YTick',[]) B 8,{jwB  
    %           axis square )Qp?LECrt  
    %           title(['Z_{' num2str(p(k)) '}']) w=5qth7  
    %       end w?"l4.E%  
    % 3 Q;l*xu  
    %   See also ZERNPOL, ZERNFUN. efm<bJB2  
    ^\;5O(9  
    %   Paul Fricker 11/13/2006 7 |A,GH  
    |&.)_+w  
    ~{{:-XkVB  
    % Check and prepare the inputs: Qmn5-yiw1d  
    % ----------------------------- %hh8\5l.:  
    if min(size(p))~=1 $Vh82Id^  
        error('zernfun2:Pvector','Input P must be vector.') L x&ZWF$  
    end iddT.   
    nz+KA\iW  
    if any(p)>35 6cvm\ opH  
        error('zernfun2:P36', ... n9yxZu   
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ~b/>TKn+  
               '(P = 0 to 35).']) 8X5XwFf}  
    end I Cs1=  
     -W ,b*U  
    % Get the order and frequency corresonding to the function number: 1lM0pl6M  
    % ---------------------------------------------------------------- Uyh#g^r  
    p = p(:); sa($3`d  
    n = ceil((-3+sqrt(9+8*p))/2); dE~ns ,+  
    m = 2*p - n.*(n+2); u""= 9>0  
    X"sN~Q.0  
    % Pass the inputs to the function ZERNFUN: H'.d'OE:I  
    % ---------------------------------------- E'}$'n?:  
    switch nargin H?m2|.  
        case 3 -1:asM7  
            z = zernfun(n,m,r,theta); # ,Y}  
        case 4 Z:{Z&HQC  
            z = zernfun(n,m,r,theta,nflag); W*2SlS7  
        otherwise Pa*yo:U'h  
            error('zernfun2:nargin','Incorrect number of inputs.') ~Q0}>m,S  
    end [ 0Sd +{Q  
    Q2o:wXvj  
    % 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) E4Sp^,  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. &}oDSD H^,  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of |N*>K a;  
    %   order N and frequency M, evaluated at R.  N is a vector of N78Ev7PN  
    %   positive integers (including 0), and M is a vector with the K"D9.%7  
    %   same number of elements as N.  Each element k of M must be a !PgYn  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) d@<XR~);  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is C,E 5/XW  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix udB}`<Q  
    %   with one column for every (N,M) pair, and one row for every F {[Q  
    %   element in R. )`)cB)s  
    % o7 kGZ  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- .IqS}Rh  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is q/Q*1  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to p=zjJ~DVd  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ??F{Gli"C`  
    %   for all [n,m]. c09uCito  
    % q#Bdq8  
    %   The radial Zernike polynomials are the radial portion of the xc!"?&\*  
    %   Zernike functions, which are an orthogonal basis on the unit ;tHF$1!J  
    %   circle.  The series representation of the radial Zernike /1Eg6hf9B  
    %   polynomials is C$P3&k#W  
    % w/&#UsEIr  
    %          (n-m)/2 )9*WmFc+#  
    %            __ Vrnx# j-U  
    %    m      \       s                                          n-2s B>R6j}rh'k  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r |Qm 7x[i  
    %    n      s=0 ?h {&  
    % <X: 9y  
    %   The following table shows the first 12 polynomials. ^71sIf;+  
    % vm(% u!_P  
    %       n    m    Zernike polynomial    Normalization :G!Kaa,r  
    %       --------------------------------------------- }} IvZG&  
    %       0    0    1                        sqrt(2) P6MT[  
    %       1    1    r                           2 I*X| pRD  
    %       2    0    2*r^2 - 1                sqrt(6) xd* kNY  
    %       2    2    r^2                      sqrt(6) @A:Xct  
    %       3    1    3*r^3 - 2*r              sqrt(8) <+6)E@Y  
    %       3    3    r^3                      sqrt(8) rIXAn4,dTv  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) WPPmh~:  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) Eq|_> f@@8  
    %       4    4    r^4                      sqrt(10) Z@1rs#  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 9N9;EY-U  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) t ({:TQ  
    %       5    5    r^5                      sqrt(12) :5ji.g* 0  
    %       --------------------------------------------- N(D_*% 96  
    % ~($h9* \  
    %   Example: n04Zji(F@  
    % /vBpRm  
    %       % Display three example Zernike radial polynomials RJ0w3T]7  
    %       r = 0:0.01:1; @6\8&(|  
    %       n = [3 2 5]; c(o8uWn  
    %       m = [1 2 1]; *b> ~L  
    %       z = zernpol(n,m,r); lO:[^l?F  
    %       figure <@oK ^ja  
    %       plot(r,z) 5R qkAC  
    %       grid on LNe- ]3wB  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') K(hqDif*6  
    % 'E6)6N  
    %   See also ZERNFUN, ZERNFUN2. E}~ GXG  
    ^)X^Pcx  
    % A note on the algorithm. 0%v p'v  
    % ------------------------ GR/ p%Y(  
    % The radial Zernike polynomials are computed using the series &QvWT+]c'0  
    % representation shown in the Help section above. For many special 9=:!XkT.  
    % functions, direct evaluation using the series representation can {4 *ob@w*  
    % produce poor numerical results (floating point errors), because qPWYY  
    % the summation often involves computing small differences between @;pTQ 5 I  
    % large successive terms in the series. (In such cases, the functions g,\<fY+ 4  
    % are often evaluated using alternative methods such as recurrence ?L'ijzP  
    % relations: see the Legendre functions, for example). For the Zernike uA,K}sNRZ  
    % polynomials, however, this problem does not arise, because the }y'KS:Jb  
    % polynomials are evaluated over the finite domain r = (0,1), and euQ d  
    % because the coefficients for a given polynomial are generally all h"j{B  
    % of similar magnitude. tlc&Wx  
    % {eS!cZJ  
    % ZERNPOL has been written using a vectorized implementation: multiple 7,Nd[ oL*7  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 41$7P[M;  
    % values can be passed as inputs) for a vector of points R.  To achieve 68d(6?OgW  
    % this vectorization most efficiently, the algorithm in ZERNPOL p5E|0p  
    % involves pre-determining all the powers p of R that are required to LvB-%@n  
    % compute the outputs, and then compiling the {R^p} into a single eQA89 :j,  
    % matrix.  This avoids any redundant computation of the R^p, and wuI+$?  
    % minimizes the sizes of certain intermediate variables. hmQD-E{Ab  
    % *Z Aue.  
    %   Paul Fricker 11/13/2006 D.X%wJ8  
    _.zW[;84b  
    BJ1txdxvS  
    % Check and prepare the inputs: >AJtoJ=j  
    % ----------------------------- iN<Tn8-YH6  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) dYW19$W n  
        error('zernpol:NMvectors','N and M must be vectors.') V 9][a  
    end ob-y {x,R  
    yPKeatH]  
    if length(n)~=length(m)  ^~?VD  
        error('zernpol:NMlength','N and M must be the same length.') .pK_j~}P  
    end c1Xt$[_  
    .(`#q@73  
    n = n(:); &?v^xAr?B  
    m = m(:); Y ~xcJH  
    length_n = length(n); uee2WGD  
    S+7>Y? B!  
    if any(mod(n-m,2)) !Hxx6/  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') v~9PS2  
    end [*Wq6n  
    :k#Y|(  
    if any(m<0) @ITJ}e4  
        error('zernpol:Mpositive','All M must be positive.') C&D!TR!K  
    end {O[a +r.n  
    ,_D`0B6o  
    if any(m>n) [YLaR r  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') ,aU_bve  
    end 3t)07(x_B  
    eE '\h  
    if any( r>1 | r<0 ) ^/U-(4O05*  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') b[%sKl  
    end @/g%l1$`  
    amK"Z<V F  
    if ~any(size(r)==1) /z.Y<xOc  
        error('zernpol:Rvector','R must be a vector.') nZ0- Kb  
    end n>" 0y^v  
    1.6yi];6  
    r = r(:); IXDj;~GF  
    length_r = length(r); nRzD[ 3I  
    oYG9i=lZ  
    if nargin==4 kFg@|#0v9  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); N`h,2!(j  
        if ~isnorm ZBUEg7c  
            error('zernpol:normalization','Unrecognized normalization flag.')  olB?"M=H  
        end |@`F !bnLr  
    else &!SdO<agZ  
        isnorm = false; j'R{llZW  
    end _0 Qp[l-  
    R?V s8?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% e$u=>=jV]  
    % Compute the Zernike Polynomials &Op_!]8`U  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% U-&dn%Sq  
    6vAq&Y{JB'  
    % Determine the required powers of r: 0K<y }  
    % ----------------------------------- mnh>gl!l  
    rpowers = []; >x]b"@Hkw  
    for j = 1:length(n) 3#<b!Yz  
        rpowers = [rpowers m(j):2:n(j)]; ^K. d|z  
    end % P .(L  
    rpowers = unique(rpowers); <=[,_P6|  
    0}tf*M+a  
    % Pre-compute the values of r raised to the required powers, <&^P1x<x  
    % and compile them in a matrix: +L03. rf  
    % ----------------------------- `K5Lp>=R  
    if rpowers(1)==0 E%8Op{zv_  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); a|?&  
        rpowern = cat(2,rpowern{:}); ]/g&y5RG  
        rpowern = [ones(length_r,1) rpowern]; lQ(I/[qVd  
    else "*UN\VV+s  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); RdaAS{>Sk  
        rpowern = cat(2,rpowern{:}); DLggR3K_\  
    end *'[8FZ|dQ  
    Zq1Z rwPF  
    % Compute the values of the polynomials: @`t#Bi9  
    % -------------------------------------- HEh,Cf7`'  
    z = zeros(length_r,length_n); @D1}).  
    for j = 1:length_n goBl~fqy0  
        s = 0:(n(j)-m(j))/2; r&!Ebe-  
        pows = n(j):-2:m(j); u-qwG/$E  
        for k = length(s):-1:1 mW EaUi)Zz  
            p = (1-2*mod(s(k),2))* ... R<(kiD\?]  
                       prod(2:(n(j)-s(k)))/          ... ~C M%WvS  
                       prod(2:s(k))/                 ... Uao8#<CkvJ  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... $.HZz  
                       prod(2:((n(j)+m(j))/2-s(k)));  rG[iEY  
            idx = (pows(k)==rpowers); X% JQ_Z  
            z(:,j) = z(:,j) + p*rpowern(:,idx); d?[gd(O  
        end 0APh=Alq  
         ^V6cx2M  
        if isnorm 5\!t!FL_  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); GO&~)Vh&7  
        end "- 2HKs  
    end .h c-uaL  
    nUb0R~wr$G  
    % 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)  /v-:ca)7mI  
    _q z^|J  
    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。 |y$8!*S~(  
    ;6655C  
    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)