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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 :/5G Hfyj  
    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! t>N2K-8Qh  
     
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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 3 i>NKS  
    function z = zernfun(n,m,r,theta,nflag) +'93%/:  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. cc`u{F9  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N W vu 1?  
    %   and angular frequency M, evaluated at positions (R,THETA) on the @f|~$$k=  
    %   unit circle.  N is a vector of positive integers (including 0), and ( [a$Z2m  
    %   M is a vector with the same number of elements as N.  Each element 8|\ -(:v  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) G;wh).jG5  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, G~u$BV'  
    %   and THETA is a vector of angles.  R and THETA must have the same [hot,\+f  
    %   length.  The output Z is a matrix with one column for every (N,M) "'Gq4<&y  
    %   pair, and one row for every (R,THETA) pair. VE*`J i  
    % `# !>}/m  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 1~E4]Ef:W  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), kxyOe[7 S  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ,+h<qBsV@  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, CXz9bhn<4  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Z<AZO ^  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. %q;y74  
    % <liprUFsn  
    %   The Zernike functions are an orthogonal basis on the unit circle. d^tY?*n  
    %   They are used in disciplines such as astronomy, optics, and W]bytsl  
    %   optometry to describe functions on a circular domain. 7 u Q +]d  
    % jg%mWiKwK7  
    %   The following table lists the first 15 Zernike functions. ABp8PD  
    % ^e_uprZWm  
    %       n    m    Zernike function           Normalization :iE`=( o  
    %       -------------------------------------------------- 1lA? 5:  
    %       0    0    1                                 1 L_:~{jV  
    %       1    1    r * cos(theta)                    2 T:K}mLSg  
    %       1   -1    r * sin(theta)                    2 uhaHY`w  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) `<T4 En  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ~^'t70 :D  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ? ][/hL@[  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ^j pQfDe6  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) J%q)6&  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Mkt_pr  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) NaQ~iY?  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Wq1OYZ,  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) V^n=@CZT9C  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) +5R8mbD!  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) vF@|cTRR)  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) -cn`D2RP  
    %       -------------------------------------------------- 5__B M5|  
    % 7rGp^  
    %   Example 1: b rDyjh  
    % vGJw/ij'X  
    %       % Display the Zernike function Z(n=5,m=1) XS&;8 PO  
    %       x = -1:0.01:1; ,|zwY~l t5  
    %       [X,Y] = meshgrid(x,x); /9D mK%d  
    %       [theta,r] = cart2pol(X,Y); }LEasj  
    %       idx = r<=1; d:]ZFk_*  
    %       z = nan(size(X)); r t)[}+ox  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ?wIEXKI  
    %       figure &i`(y>\  
    %       pcolor(x,x,z), shading interp #!yX2lR  
    %       axis square, colorbar n1R{[\ >1  
    %       title('Zernike function Z_5^1(r,\theta)') :y{@=E=XSC  
    % 0R]'HA>  
    %   Example 2: y6G6wk;  
    % c5Kc iTD^  
    %       % Display the first 10 Zernike functions ,]9p&xu  
    %       x = -1:0.01:1; 23bTCp.d  
    %       [X,Y] = meshgrid(x,x); fv* $=m  
    %       [theta,r] = cart2pol(X,Y); rT4qx2u  
    %       idx = r<=1; pf yJL?_%  
    %       z = nan(size(X)); w; f LnEz_  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 28}L.>5k  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3];  aqi]5,  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; :8+x&zn  
    %       y = zernfun(n,m,r(idx),theta(idx)); 7}&vEc@w&  
    %       figure('Units','normalized') wI F'|"  
    %       for k = 1:10 }^n"t>Z8  
    %           z(idx) = y(:,k); brqmi<*9"[  
    %           subplot(4,7,Nplot(k)) =6fJUy^M\  
    %           pcolor(x,x,z), shading interp *J4 \KU  
    %           set(gca,'XTick',[],'YTick',[]) =|^R<#%/  
    %           axis square ?c fFJl  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 0NvicZ7VR  
    %       end @lh]? |*[  
    % bQ0+Y?,+/  
    %   See also ZERNPOL, ZERNFUN2. ^ Vc(oa&;  
    a?W<<9]  
    %   Paul Fricker 11/13/2006 +J42pSxzoo  
    JRtDjZ4>  
    "%rU1/@#  
    % Check and prepare the inputs: 1u }2}c|  
    % ----------------------------- }tH_YF}u  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7< ?Aou  
        error('zernfun:NMvectors','N and M must be vectors.') Te^_gdf  
    end >ca`0gu  
     [cfXcl  
    if length(n)~=length(m) =%[vHQ\%  
        error('zernfun:NMlength','N and M must be the same length.') $JK,9G[Vu  
    end P}!pmg6V  
    bl|)/)6o  
    n = n(:); TD!c+ ${w  
    m = m(:); 7Mh!@Rd_V  
    if any(mod(n-m,2)) JY D\VaW  
        error('zernfun:NMmultiplesof2', ... Orlf5 {P  
              'All N and M must differ by multiples of 2 (including 0).') m='_ O+ $  
    end ,LU|WXRB  
    a3 t||@v!  
    if any(m>n) 2>^jMln  
        error('zernfun:MlessthanN', ... h{ EnS5~  
              'Each M must be less than or equal to its corresponding N.') 3X`N~_+  
    end +\cG{n*  
    '|yBz1uL  
    if any( r>1 | r<0 ) P@Pe5H"o  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 4%SA%]a L1  
    end Z =*h9,MY  
    `TDS 4Y  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) "haJwV6-  
        error('zernfun:RTHvector','R and THETA must be vectors.') u6*0% Km  
    end LdX'V]ITh  
    =MmAnjo  
    r = r(:); 0\o5+  
    theta = theta(:); 92/_!P>  
    length_r = length(r); FeZGPxc~  
    if length_r~=length(theta) W)odaab7  
        error('zernfun:RTHlength', ... >H]|R }h  
              'The number of R- and THETA-values must be equal.') 1#vi]CX  
    end v 5&8C  
    <;!#+|L/  
    % Check normalization: _xo;[rEw8  
    % -------------------- ?r.U5}PBI  
    if nargin==5 && ischar(nflag) ]_! . xx>  
        isnorm = strcmpi(nflag,'norm'); ev5m(wR  
        if ~isnorm RJD(c#r$  
            error('zernfun:normalization','Unrecognized normalization flag.') ,Q+.kAh !G  
        end }};AV)}J  
    else }FkF1?C  
        isnorm = false; *Ud P1?Y  
    end !z+'mF?V+X  
    mqQC`Aqx:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ot~buf'|  
    % Compute the Zernike Polynomials 6{[ uCxxl  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~HUO$*U4<  
    zi+NQOhR  
    % Determine the required powers of r: G,@ Jo[e  
    % ----------------------------------- xGw|@d  
    m_abs = abs(m); SJ&+"S&  
    rpowers = []; AaDMX,  
    for j = 1:length(n) (U|WP%IM'  
        rpowers = [rpowers m_abs(j):2:n(j)]; AZbFj-^4  
    end G;^,T/q47  
    rpowers = unique(rpowers); xL!@$;J  
    @F!oRm5  
    % Pre-compute the values of r raised to the required powers, *#o2b-[V  
    % and compile them in a matrix: >q1rdq  
    % ----------------------------- EzXi*/  
    if rpowers(1)==0 yOm#c>X  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); N/8B@}@n  
        rpowern = cat(2,rpowern{:}); h"ATRr^  
        rpowern = [ones(length_r,1) rpowern]; "0?" E\  
    else T $o;PJc  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); n,b6|Y0  
        rpowern = cat(2,rpowern{:});  75T+6 u  
    end f/^T:F6  
    i [2bz+Z?  
    % Compute the values of the polynomials: P,K^ oz}  
    % -------------------------------------- $gaGaB  
    y = zeros(length_r,length(n)); 6'1Lu1w  
    for j = 1:length(n) xHuw ?4  
        s = 0:(n(j)-m_abs(j))/2; nM H:7[x3  
        pows = n(j):-2:m_abs(j); q.d qr<  
        for k = length(s):-1:1 cwI3ANV  
            p = (1-2*mod(s(k),2))* ... Lz`_&&6  
                       prod(2:(n(j)-s(k)))/              ... 1<pb=H  
                       prod(2:s(k))/                     ...  y7.oy"  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... dwUs[v   
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Y]+KsiOL  
            idx = (pows(k)==rpowers); gq&jNj7V  
            y(:,j) = y(:,j) + p*rpowern(:,idx); K5(:0Q.5y  
        end Qa,$_ ,E  
         p 8lm1;  
        if isnorm y:R+;91  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); (Cbm*VL  
        end mC!^`y)  
    end v=?/c-J*  
    % END: Compute the Zernike Polynomials (6X{ &  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 23P7%\  
    uB+ :sX-L  
    % Compute the Zernike functions: LTnbBh*mc  
    % ------------------------------ )W!\D/C+  
    idx_pos = m>0; @6{F4  
    idx_neg = m<0; hU 5_ dV  
    [yDOv Q[  
    z = y; 88A,ll%  
    if any(idx_pos) nF=[m; ~  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); E#c9n%E\sz  
    end #++D|oE  
    if any(idx_neg) kQO5sX$;  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 6MelN^\[7  
    end B8?j"AF  
    .}iRe}=  
    % 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) I\('b9"*  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. MN_1^T5  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated +D M,+{}  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive [.nkNda5)v  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, j`_tb   
    %   and THETA is a vector of angles.  R and THETA must have the same Kj#h9e  
    %   length.  The output Z is a matrix with one column for every P-value,  +Q+!#  
    %   and one row for every (R,THETA) pair. kf_*=ER  
    % 5)p!}hWs  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike H=*2A!O[_  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) N`7+] T  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) &IkHP/  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 r9~IR  
    %   for all p. /1?{,Das=  
    % #kAk d-QY6  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 pm:#@sl  
    %   Zernike functions (order N<=7).  In some disciplines it is g\@zQ^O?  
    %   traditional to label the first 36 functions using a single mode M&0U@ r-  
    %   number P instead of separate numbers for the order N and azimuthal 0|]qW cD  
    %   frequency M. uo7[T*<Q  
    % $%5vJiuk  
    %   Example: Ss'Dto35Q  
    % 9zaSA,}  
    %       % Display the first 16 Zernike functions 6bBNC2K$-  
    %       x = -1:0.01:1; p I@!2c:}  
    %       [X,Y] = meshgrid(x,x); ?nD]p!  
    %       [theta,r] = cart2pol(X,Y); u\=yY.   
    %       idx = r<=1; z O6Sl[)  
    %       p = 0:15; %`]fZr A]#  
    %       z = nan(size(X)); h]k1vp)Q y  
    %       y = zernfun2(p,r(idx),theta(idx)); #Z}YQ $g  
    %       figure('Units','normalized') 4J$dG l#f  
    %       for k = 1:length(p) <Mf(2`T  
    %           z(idx) = y(:,k); k~qZ^9QB~  
    %           subplot(4,4,k) ,ydn]0SS  
    %           pcolor(x,x,z), shading interp ePTN^#|W  
    %           set(gca,'XTick',[],'YTick',[]) Ol*|J  
    %           axis square # 0/,teJ k  
    %           title(['Z_{' num2str(p(k)) '}']) 5>rjL ;  
    %       end S|T*-?|  
    % BU=Ta$#BZ  
    %   See also ZERNPOL, ZERNFUN. -m Sf`1l0  
    6KKQ)DNu_  
    %   Paul Fricker 11/13/2006 +}NQ |y V  
    DK(8Ml:k  
    -7A2@g  
    % Check and prepare the inputs: PAv<J<d  
    % ----------------------------- y_{v&AGmgm  
    if min(size(p))~=1 n;~6'f xe  
        error('zernfun2:Pvector','Input P must be vector.') tdn|mX#  
    end TU?$yNE  
    4_o+gG%HaM  
    if any(p)>35 : W0;U  
        error('zernfun2:P36', ... \w2X.2b.F  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... }1P v6L(o)  
               '(P = 0 to 35).']) 2yA+zJ 46B  
    end =p#:v  
    ybpU?n  
    % Get the order and frequency corresonding to the function number: HkyN$1s  
    % ---------------------------------------------------------------- _"- ,ia[D  
    p = p(:); / 2h6  
    n = ceil((-3+sqrt(9+8*p))/2); %QX"oRMn0  
    m = 2*p - n.*(n+2); opqf)C  
    qQCds}<w  
    % Pass the inputs to the function ZERNFUN: qEnmms1  
    % ---------------------------------------- -d6PXf5  
    switch nargin pNc4o@-  
        case 3 }62Q{>`  
            z = zernfun(n,m,r,theta); #,rP1#?  
        case 4 p *GAs C  
            z = zernfun(n,m,r,theta,nflag); ~}s0~j~  
        otherwise vXibg  
            error('zernfun2:nargin','Incorrect number of inputs.') ,~7+r#q7  
    end BmCBC,j<v>  
    Fzn#>`qG  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) ]d,#PF  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. `-<m#HF:)d  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ;V)94YT  
    %   order N and frequency M, evaluated at R.  N is a vector of 6>- Gi  
    %   positive integers (including 0), and M is a vector with the =N{-lyr)  
    %   same number of elements as N.  Each element k of M must be a f9J]-#Iif  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) LQ4F/[1}  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is rcG-V f@  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix O]\eMM&  
    %   with one column for every (N,M) pair, and one row for every Axx{G~n![  
    %   element in R. 4mF=A$Q_/  
    % kl9<l*  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- (JeRJ4  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 7' TXR[   
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ]\oE}7K%r  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 T!gq Z  
    %   for all [n,m]. > V%3w7  
    % ?=\_U  
    %   The radial Zernike polynomials are the radial portion of the v*EErQML8b  
    %   Zernike functions, which are an orthogonal basis on the unit r2>y !Q?  
    %   circle.  The series representation of the radial Zernike  Fs1ms)  
    %   polynomials is QNCG^ub  
    % 7\JA8mm  
    %          (n-m)/2 X>VxE/  
    %            __ `jH0FJQ  
    %    m      \       s                                          n-2s )lB*] n`Z]  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r wvg>SfV,e  
    %    n      s=0 s h^&3}  
    % ut;KphvSH  
    %   The following table shows the first 12 polynomials. dG'5: ,n/  
    % Qv:J#uVw?O  
    %       n    m    Zernike polynomial    Normalization y{1|@?ii  
    %       --------------------------------------------- cLCzLNyKl  
    %       0    0    1                        sqrt(2) 1F }mlyS  
    %       1    1    r                           2 Nyo,6 AA  
    %       2    0    2*r^2 - 1                sqrt(6) &|) (lX  
    %       2    2    r^2                      sqrt(6) `PvGfmYOl  
    %       3    1    3*r^3 - 2*r              sqrt(8) 7(bE;(4  
    %       3    3    r^3                      sqrt(8) fIcra  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ' C|yUsBC  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) .%N*g[J  
    %       4    4    r^4                      sqrt(10) ' 8bT9  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 0qMf6  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) .=-K7.X.)  
    %       5    5    r^5                      sqrt(12) Vch!&8xii  
    %       --------------------------------------------- \.jT"Z~  
    % E>6:59+  
    %   Example: a]|k w4  
    % KmlpB  
    %       % Display three example Zernike radial polynomials IOi6' 1l  
    %       r = 0:0.01:1; >QM$ NIf@  
    %       n = [3 2 5]; kVb8$Sp  
    %       m = [1 2 1]; OM 5h>\9  
    %       z = zernpol(n,m,r); "Crm\UI6  
    %       figure Qr l>A*  
    %       plot(r,z) eA(c{  
    %       grid on gAgP("  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') S[W|=(f9  
    % 5UHxB"`C  
    %   See also ZERNFUN, ZERNFUN2. Nm]\0m0p-  
    ;qT7BUh(%  
    % A note on the algorithm. gKGM|0u|r  
    % ------------------------ O%(k$ fvM  
    % The radial Zernike polynomials are computed using the series ="*8ja-K  
    % representation shown in the Help section above. For many special ^zr]#`@G  
    % functions, direct evaluation using the series representation can 7`f',ZK%  
    % produce poor numerical results (floating point errors), because 4?{e?5)  
    % the summation often involves computing small differences between E64d6z^7u  
    % large successive terms in the series. (In such cases, the functions *T'>-nm]  
    % are often evaluated using alternative methods such as recurrence saaN$tU7  
    % relations: see the Legendre functions, for example). For the Zernike /N&)r wc  
    % polynomials, however, this problem does not arise, because the <C9_5C e~  
    % polynomials are evaluated over the finite domain r = (0,1), and Hc{0O7  
    % because the coefficients for a given polynomial are generally all h .Iscr^~  
    % of similar magnitude. ?"[h P=3J  
    % yy6?16@  
    % ZERNPOL has been written using a vectorized implementation: multiple ard<T}|N  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] SlZ>N$E  
    % values can be passed as inputs) for a vector of points R.  To achieve 4X &\/X  
    % this vectorization most efficiently, the algorithm in ZERNPOL H]7;O M/g  
    % involves pre-determining all the powers p of R that are required to *.DTcV  
    % compute the outputs, and then compiling the {R^p} into a single &zYo   
    % matrix.  This avoids any redundant computation of the R^p, and c{u~=24;%#  
    % minimizes the sizes of certain intermediate variables. z@0*QZ.y 1  
    % v*7lJNN.  
    %   Paul Fricker 11/13/2006 e/;chMCq  
    OxraaN`  
    ~D)!zQkD  
    % Check and prepare the inputs: ?>W4*8 (  
    % ----------------------------- V nv9 <=R  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) -B-nTS`  
        error('zernpol:NMvectors','N and M must be vectors.') bVc;XZwI  
    end 0/."R ;  
    I~$LIdzw  
    if length(n)~=length(m) }'KVi=qnHb  
        error('zernpol:NMlength','N and M must be the same length.') VzR (O B  
    end YolO-5  
    0pS|t/h0  
    n = n(:); c2z%|\q  
    m = m(:); XACbDKyS  
    length_n = length(n); xPY/J#X$  
    _Z|s!~wdz  
    if any(mod(n-m,2)) JxP=[>I  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ?[K+Ym+  
    end ? 5qo>W<7  
    uLsGb=m%b  
    if any(m<0) >Udb*76 D  
        error('zernpol:Mpositive','All M must be positive.') *@q+A1P7@  
    end d))(hk:  
    lGI5  
    if any(m>n) o?f7_8fG  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') xP.B,1\X  
    end +b.qzgH>r  
    Vs~^r>  
    if any( r>1 | r<0 ) B8^tIq  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 5O Ob(  
    end p1Q[c0NMK  
    0}YR=  
    if ~any(size(r)==1) "-4V48ci  
        error('zernpol:Rvector','R must be a vector.') 6rEt!v #K[  
    end @+ VvZc2Y  
    ?=]`X=g 6  
    r = r(:); sD$ \!7:b  
    length_r = length(r); 3,bA&c3  
    FX"%  
    if nargin==4 8x jJ  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); q!k  F  
        if ~isnorm U RDb  
            error('zernpol:normalization','Unrecognized normalization flag.') w| x=^  
        end S<f&?\wK=v  
    else AC=cz!3iB  
        isnorm = false; I?v)>| |Q  
    end oh`I$  
    aC>r5b#:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X6 ~y+ R  
    % Compute the Zernike Polynomials +(5H$O{h  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V; 1r  
    Wxg,y{(`  
    % Determine the required powers of r: z12[vN  
    % ----------------------------------- WIXzxI<)  
    rpowers = []; I]>-~_  
    for j = 1:length(n) hA?j"y0?  
        rpowers = [rpowers m(j):2:n(j)]; VuwBnQ.2k  
    end $=$I^hV  
    rpowers = unique(rpowers); !%NxSJ  
    p2PD';"  
    % Pre-compute the values of r raised to the required powers, .s|n}{D_i  
    % and compile them in a matrix: 6g!#"=ls;  
    % ----------------------------- FpE83}@".w  
    if rpowers(1)==0 9Ps:]Kp!vN  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); #6_?7 (X  
        rpowern = cat(2,rpowern{:}); 3O<<XXar  
        rpowern = [ones(length_r,1) rpowern]; /a6\G.C5  
    else ~)D2U:"^xm  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); z.eqOPW  
        rpowern = cat(2,rpowern{:}); \~Zj](#  
    end B8 -/ C\  
    bK; -Xcm  
    % Compute the values of the polynomials: 7|T<dfQk  
    % -------------------------------------- 1j6ZSE/*|  
    z = zeros(length_r,length_n); q|om^:n.  
    for j = 1:length_n Kwfrh?  
        s = 0:(n(j)-m(j))/2; iwCnW7:  
        pows = n(j):-2:m(j); H|T:_*5  
        for k = length(s):-1:1 _< 69d  
            p = (1-2*mod(s(k),2))* ...  2 Ua_7  
                       prod(2:(n(j)-s(k)))/          ... *(L4rK\2  
                       prod(2:s(k))/                 ... h|dVVCsN  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... GY<Y,  
                       prod(2:((n(j)+m(j))/2-s(k))); y<k-dbr  
            idx = (pows(k)==rpowers); =ALy.^J=  
            z(:,j) = z(:,j) + p*rpowern(:,idx); eg) =^b  
        end :D-d`OyjG>  
         B+ GPTQSTb  
        if isnorm IoJkM-^H&)  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); >Fz_]z   
        end J @"wJEF  
    end )rz4IfE  
    w@.E}%bwq  
    % 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)  * Od_Cl  
    D}SRr,4v  
    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。 " nq4!  
    N~;=*)_VH  
    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)