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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 7}VqXUwabx  
    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! At\(/Z y  
     
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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 *Y(59J2  
    function z = zernfun(n,m,r,theta,nflag) +fk*c[FG  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. Jb"FY:/Qv+  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N =R=V  
    %   and angular frequency M, evaluated at positions (R,THETA) on the x/O;8^b  
    %   unit circle.  N is a vector of positive integers (including 0), and |E >h*Y  
    %   M is a vector with the same number of elements as N.  Each element K}CgFBk  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 6X@z(EEL  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, hH`x*:Qja  
    %   and THETA is a vector of angles.  R and THETA must have the same <2)AbI+3  
    %   length.  The output Z is a matrix with one column for every (N,M) <'4Wne.z!  
    %   pair, and one row for every (R,THETA) pair. @l CG)Ix<  
    % I:jIChT  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike YcA. Bn|as  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ^i8,9T'=  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral G0 EXgq8  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, "\@J0 |ppb  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized U(f@zGV  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. lBfthLBa  
    % \>5sW8P]H`  
    %   The Zernike functions are an orthogonal basis on the unit circle. 9Q1%+zjjMq  
    %   They are used in disciplines such as astronomy, optics, and ?V2P]|  
    %   optometry to describe functions on a circular domain. 0i\>(o  
    % Z)|~  
    %   The following table lists the first 15 Zernike functions. :Vxt2@p{  
    % sa+ JN^[X  
    %       n    m    Zernike function           Normalization 3?B1oIHQ  
    %       -------------------------------------------------- ^(TCUY~f&  
    %       0    0    1                                 1 lW c[Q1  
    %       1    1    r * cos(theta)                    2 )(]rUJ~+~A  
    %       1   -1    r * sin(theta)                    2 pl>b 6 |  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) c \??kQH  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ,?yjsJd.  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ;((t|  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) $}(Z]z}O;  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) {LiJ=Ebt  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 1#x5 o2n  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) p-"C^=l  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 9\Gk)0  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) _9=87u0  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) (LK@w9)i;  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (/uN+   
    %       4    4    r^4 * sin(4*theta)             sqrt(10) J~K O#`  
    %       -------------------------------------------------- OFr"RGW"  
    % 9C \}bT  
    %   Example 1: $?F_Qsy{d  
    % &n | <NF  
    %       % Display the Zernike function Z(n=5,m=1) C+/EPPi  
    %       x = -1:0.01:1; Lz1KDXr`)+  
    %       [X,Y] = meshgrid(x,x); +}m`$B}mJ  
    %       [theta,r] = cart2pol(X,Y); fL| 9/sojz  
    %       idx = r<=1; <zqIq9}r  
    %       z = nan(size(X)); !!L'{beF  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); {qHQ_ _Bl  
    %       figure \Yj_U'2"i  
    %       pcolor(x,x,z), shading interp UhJS=YvT  
    %       axis square, colorbar (72%au  
    %       title('Zernike function Z_5^1(r,\theta)') ?xwi2<zz  
    % oPs asa  
    %   Example 2: iY`[dsT  
    % \'=svJ   
    %       % Display the first 10 Zernike functions =A5i84y.2u  
    %       x = -1:0.01:1; _8$xsj4_  
    %       [X,Y] = meshgrid(x,x); U`) " ;WN  
    %       [theta,r] = cart2pol(X,Y); ]A[}:E 5}  
    %       idx = r<=1; .~I:Hcf/  
    %       z = nan(size(X)); Srw`vql{(  
    %       n = [0  1  1  2  2  2  3  3  3  3]; `}t5`:#k  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; (;g/wb:  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; |m\7/&@<  
    %       y = zernfun(n,m,r(idx),theta(idx)); kR1 12J9P  
    %       figure('Units','normalized') {KSLB8gtL  
    %       for k = 1:10 x(>XM:|  
    %           z(idx) = y(:,k); B[mZQ&Gz`a  
    %           subplot(4,7,Nplot(k)) 5q4wREh  
    %           pcolor(x,x,z), shading interp .Od@i$E>&  
    %           set(gca,'XTick',[],'YTick',[]) <>KQ8:  
    %           axis square u L v  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) L"0dB.  
    %       end lre(]oBXA  
    % nEUH;z  
    %   See also ZERNPOL, ZERNFUN2. 0Bgj.?l  
    6 [bQ'Ir^8  
    %   Paul Fricker 11/13/2006 |9i[*]  
    6a9:P@tY  
    R{H8@JLD  
    % Check and prepare the inputs: Y, Lpv|  
    % ----------------------------- @=g{4(zR ^  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) yz3=#  
        error('zernfun:NMvectors','N and M must be vectors.') 7&etnQJ{  
    end V,zFHXO  
    , MqoX-+  
    if length(n)~=length(m) ;|\j][A  
        error('zernfun:NMlength','N and M must be the same length.') hH$9GL{H  
    end vx$DKQK@l\  
    bOYM-\ {y  
    n = n(:); 0f_`;{  
    m = m(:); EFU)0IAL[  
    if any(mod(n-m,2)) @@3 NSKA  
        error('zernfun:NMmultiplesof2', ... ) F -8  
              'All N and M must differ by multiples of 2 (including 0).') tw 3zw`o:  
    end ?1|\(W#  
    MYJMZ3qBi  
    if any(m>n) bWp)'mx5u  
        error('zernfun:MlessthanN', ... ',+Zqog92  
              'Each M must be less than or equal to its corresponding N.') \u6.*w5TI  
    end asQ^33g z  
    "\lO Op^-  
    if any( r>1 | r<0 ) Bvj  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') _^?_Vb  
    end >C{8}Lg-.  
    Ya jAz5N  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) VeEa17g&  
        error('zernfun:RTHvector','R and THETA must be vectors.') lP4s"8E`h  
    end c8zok `\P_  
    @G>e Cj  
    r = r(:); 5%K|dYv^^  
    theta = theta(:); d=\TC'd"{  
    length_r = length(r); Z6So5r%wZ  
    if length_r~=length(theta) CZ^ ,bad  
        error('zernfun:RTHlength', ...  `uDOIl  
              'The number of R- and THETA-values must be equal.') B$OV^iwxK  
    end <v\$r2C*  
    0}` -<(  
    % Check normalization: CG35\b;Q  
    % -------------------- H7drDw  
    if nargin==5 && ischar(nflag) S]}}r)  
        isnorm = strcmpi(nflag,'norm'); Q"!GdKM  
        if ~isnorm ES(qu]CjI  
            error('zernfun:normalization','Unrecognized normalization flag.') I~HA ad,k  
        end E&"V~  
    else gLFSZ  
        isnorm = false; [k%u$  
    end Tqs|2at<t  
    &\ad.O/Q  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b'4}=Xpn  
    % Compute the Zernike Polynomials ;i [;%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wrJ" (:VZ  
    L6jwJwD  
    % Determine the required powers of r: . Y!dO@$:  
    % ----------------------------------- A&ceuu  
    m_abs = abs(m); |<8Fa%!HHc  
    rpowers = []; YJDJj x  
    for j = 1:length(n) 6B b+f"  
        rpowers = [rpowers m_abs(j):2:n(j)]; RA){\~@wC  
    end }t|i1{%_  
    rpowers = unique(rpowers); T' Jl,)"  
    Gy6x.GX  
    % Pre-compute the values of r raised to the required powers, 4qd( a)NdY  
    % and compile them in a matrix: LF{8hC[  
    % ----------------------------- !4z vkJO  
    if rpowers(1)==0 (6 RWI#  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @bAu R  
        rpowern = cat(2,rpowern{:});  e?o/H  
        rpowern = [ones(length_r,1) rpowern]; &- My[t  
    else }:s.m8LC5n  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); s|[qq7  
        rpowern = cat(2,rpowern{:}); 1bDXv, nD  
    end k O.iJcZg  
    V HLNJnA  
    % Compute the values of the polynomials: n-GoG(s..b  
    % -------------------------------------- I2) 2'j,B  
    y = zeros(length_r,length(n)); |WT]s B0Eq  
    for j = 1:length(n) u{sb^cmy  
        s = 0:(n(j)-m_abs(j))/2; tu ;Pm4q7  
        pows = n(j):-2:m_abs(j); 0hXx31JN N  
        for k = length(s):-1:1 W]>%*n  
            p = (1-2*mod(s(k),2))* ... (7$BF~s:,  
                       prod(2:(n(j)-s(k)))/              ... #oR@!?  
                       prod(2:s(k))/                     ... .rX,*|1x  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Bq-}BN?pz  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ]{t!J^Xn  
            idx = (pows(k)==rpowers); :+?r nb)N  
            y(:,j) = y(:,j) + p*rpowern(:,idx); /* "pylm  
        end {=U*!`D  
         fMM%,/b{  
        if isnorm PH^Gjm  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); }Q6o#oZ  
        end : Hu {MN\  
    end #D ]CuSi  
    % END: Compute the Zernike Polynomials )tS;gn  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% U?5G%o(q  
    >4+KEK  
    % Compute the Zernike functions: o?IrDQ2gmh  
    % ------------------------------ )4 ,U  
    idx_pos = m>0; e:rbyzf#  
    idx_neg = m<0; 5e?<x>e  
    ##alzC  
    z = y; Cm"S=gV  
    if any(idx_pos) Qf'g2 \  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); `z7,HJ.0c  
    end i;juwc^n}  
    if any(idx_neg) Pl2eDv-y  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); a#9pN?~  
    end y(^\]-fE  
    cHOC>|  
    % 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) *D9H3M[o#  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. <num!@2D  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated }WBHuVcZG  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive >6)|># Wi  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, q[/pE7FL  
    %   and THETA is a vector of angles.  R and THETA must have the same $~%h4  
    %   length.  The output Z is a matrix with one column for every P-value, ,g,Hb\_R)  
    %   and one row for every (R,THETA) pair. $2-_j)+  
    % V\l@_%D[(v  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike sc6NON#  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) l/\D0\x2  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) :)&vf<JL  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 6GqC]rd*:  
    %   for all p. 8Kk41=  
    % ibG>|hV  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 w8 `1'*HG  
    %   Zernike functions (order N<=7).  In some disciplines it is `46.!  
    %   traditional to label the first 36 functions using a single mode Q( e  
    %   number P instead of separate numbers for the order N and azimuthal sH > zsc  
    %   frequency M. HIC!:|  
    % I`i"*z  
    %   Example: M.:JT31>1  
    % SQ/HZ  
    %       % Display the first 16 Zernike functions ZE~zs~z|  
    %       x = -1:0.01:1; vbZ!NO!H  
    %       [X,Y] = meshgrid(x,x); 18Ju]U  
    %       [theta,r] = cart2pol(X,Y); "^;h'  
    %       idx = r<=1; NSH4 @x  
    %       p = 0:15; *-{|m1P  
    %       z = nan(size(X)); Nd{U|k3pL  
    %       y = zernfun2(p,r(idx),theta(idx));  X>P|-n#  
    %       figure('Units','normalized') gU NWM^n  
    %       for k = 1:length(p) 0r8Wv,7Bo  
    %           z(idx) = y(:,k); NK(_ &.F  
    %           subplot(4,4,k) )S/=5Uc  
    %           pcolor(x,x,z), shading interp -|>T? t'K  
    %           set(gca,'XTick',[],'YTick',[]) #N'9 w .  
    %           axis square %O<8H7e)V  
    %           title(['Z_{' num2str(p(k)) '}']) ?,8+1"|$A]  
    %       end U~;tk@  
    % ^H{YLO  
    %   See also ZERNPOL, ZERNFUN. 9 %i\)  
    3JkdPh  
    %   Paul Fricker 11/13/2006 k}NM]9EAE  
    Hrph>v  
    bS954d/  
    % Check and prepare the inputs: $ ]#WC\Hv  
    % ----------------------------- 4RYH^9;>K  
    if min(size(p))~=1 `(=Kp=b  
        error('zernfun2:Pvector','Input P must be vector.') BMuEfa^  
    end cCNRv$IO\  
    D|Wlq~IpQ  
    if any(p)>35 zNdkwj p+  
        error('zernfun2:P36', ... ~id:Rh>o  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ;ko6igx)+  
               '(P = 0 to 35).']) oD1k7Gq1  
    end $(]nl%<Q  
    IF\ @uo`  
    % Get the order and frequency corresonding to the function number: 0y?;o*&U\  
    % ---------------------------------------------------------------- :Jv5Flxl  
    p = p(:); /kg#i&bP~  
    n = ceil((-3+sqrt(9+8*p))/2); Hbd>sS  
    m = 2*p - n.*(n+2); ]dI^ S  
    js@L%1r#L  
    % Pass the inputs to the function ZERNFUN: +@?'dw  
    % ---------------------------------------- v+W'0ymbnV  
    switch nargin f.+1Ubq!5  
        case 3 |%mZ|,[  
            z = zernfun(n,m,r,theta); Lhe&  
        case 4 lw.[qP  
            z = zernfun(n,m,r,theta,nflag); aekke//y  
        otherwise wWiYxBeN  
            error('zernfun2:nargin','Incorrect number of inputs.') 4OpzGZ4+  
    end M*kE |q/K  
    ]+W){W=ai  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) c {%mi  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. D5snaGss9a  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of XGH:'^o_  
    %   order N and frequency M, evaluated at R.  N is a vector of HbsNF~;  
    %   positive integers (including 0), and M is a vector with the jqc}mI\#  
    %   same number of elements as N.  Each element k of M must be a nW7: ]  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) j9h fW'  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is UGy3 B)  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 9n{Y6I x:  
    %   with one column for every (N,M) pair, and one row for every ]0ErT9  
    %   element in R. Vb\g49\o/  
    % OB$Jv<C@  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 3zD#V3 =  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is :VZS7$5  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to "T5oUy&i  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 9$ ;5J  
    %   for all [n,m]. Af]zv~uM  
    % 4=Ru{ewRV  
    %   The radial Zernike polynomials are the radial portion of the A%Ka)UU+n  
    %   Zernike functions, which are an orthogonal basis on the unit O& Sk}^  
    %   circle.  The series representation of the radial Zernike d\]KG(T  
    %   polynomials is SYA~I-OYc  
    % A+* lV*@0  
    %          (n-m)/2 vu\W5M  
    %            __ * $fM}6}  
    %    m      \       s                                          n-2s }%/mPbd#  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ofQs /  
    %    n      s=0 ZF[W<Q  
    % vHcl7=)Q  
    %   The following table shows the first 12 polynomials. bHnKtaK4c  
    % /-[vC$B"  
    %       n    m    Zernike polynomial    Normalization uhfK\.3  
    %       --------------------------------------------- D5P-$1KPt  
    %       0    0    1                        sqrt(2) h$!YKfhq}  
    %       1    1    r                           2 mnK<5KLg1  
    %       2    0    2*r^2 - 1                sqrt(6) )LFbz#;Y  
    %       2    2    r^2                      sqrt(6) 3Z9Yzv)A  
    %       3    1    3*r^3 - 2*r              sqrt(8) A&t8C8,  
    %       3    3    r^3                      sqrt(8) Za|iU`e\  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) M1-tRF  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ;7:} iKU  
    %       4    4    r^4                      sqrt(10) xB Wl|j  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) :5%98V>02  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) L{%L*z9J  
    %       5    5    r^5                      sqrt(12) }={@_g#  
    %       --------------------------------------------- '2lzMc>wvP  
    % yC\UT ~j/  
    %   Example: n!/0yR2S  
    % xn2nh@;  
    %       % Display three example Zernike radial polynomials pS+w4gW  
    %       r = 0:0.01:1; O~V^]   
    %       n = [3 2 5]; =M;F&;\8  
    %       m = [1 2 1]; B.5+!z&7  
    %       z = zernpol(n,m,r); JT0j2_*Rr  
    %       figure }? / Blr  
    %       plot(r,z) >2{Y5__+e  
    %       grid on z Fm`e:td  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') mc?IM(t  
    % GuR^L@+ -.  
    %   See also ZERNFUN, ZERNFUN2. 1!MJ+?Jl  
    7wx=#  
    % A note on the algorithm.  (yP1}?  
    % ------------------------ OXrm!'  
    % The radial Zernike polynomials are computed using the series |hika`35K  
    % representation shown in the Help section above. For many special YXWDbr:JX  
    % functions, direct evaluation using the series representation can 3=uhy|f! /  
    % produce poor numerical results (floating point errors), because i6_}  
    % the summation often involves computing small differences between -fA=&$V  
    % large successive terms in the series. (In such cases, the functions 90W= v*  
    % are often evaluated using alternative methods such as recurrence K^fs #7  
    % relations: see the Legendre functions, for example). For the Zernike 6}E>B{Y  
    % polynomials, however, this problem does not arise, because the .yy*[56X  
    % polynomials are evaluated over the finite domain r = (0,1), and =fRS UtX  
    % because the coefficients for a given polynomial are generally all ,:(s=J N+  
    % of similar magnitude. {UP[iw$~  
    % qK.(w Fx  
    % ZERNPOL has been written using a vectorized implementation: multiple g8MW6Y  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] '/8/M{`s  
    % values can be passed as inputs) for a vector of points R.  To achieve b&[".ibN1  
    % this vectorization most efficiently, the algorithm in ZERNPOL hc q&`Gun  
    % involves pre-determining all the powers p of R that are required to 59)w+AW  
    % compute the outputs, and then compiling the {R^p} into a single USyc D`  
    % matrix.  This avoids any redundant computation of the R^p, and " 7^nRJy  
    % minimizes the sizes of certain intermediate variables. S3%2T  
    % yk4 @@kHW  
    %   Paul Fricker 11/13/2006 1}\p:`  
    G%bv<_R  
    %RV81H9B  
    % Check and prepare the inputs: #8et91qw  
    % ----------------------------- w~6UOA8}  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) )  6a,8t  
        error('zernpol:NMvectors','N and M must be vectors.') 5wVJ.B~s  
    end Hdew5Xn(:  
    %evb.h)  
    if length(n)~=length(m) D{B?2}X  
        error('zernpol:NMlength','N and M must be the same length.') *`+zf7-f  
    end G"F O%3&|  
    %9>w|%+;U+  
    n = n(:); ,A`|jF  
    m = m(:); 95'+8*YCY  
    length_n = length(n); =8 @DYz'  
    aU(.LC  
    if any(mod(n-m,2)) P'8RaO&d  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 6m?<"y8]  
    end !lfE7|\p  
    iG54 +]  
    if any(m<0) Qg/FFn^Kg*  
        error('zernpol:Mpositive','All M must be positive.') M6p\QKi  
    end s_y8+BJaV  
    htbE Q NW  
    if any(m>n)  ?P +Uv  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') <w^u^)iLy1  
    end 9o>D Uc  
    %mmV#vwp  
    if any( r>1 | r<0 ) ]?(kaNQ "D  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') &YY`XEG59O  
    end rB".!b  
    vgy.fP"@  
    if ~any(size(r)==1) D-{*3?x  
        error('zernpol:Rvector','R must be a vector.') j#p;XI  
    end m)L50ot:/  
    ZJ%NZAxy  
    r = r(:); 2|bt"y-5r  
    length_r = length(r); < ?B3^z$  
    ;'{7wr|9  
    if nargin==4 5.VPK 338A  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); m'}`+#C%)  
        if ~isnorm 'zm5wqrkAd  
            error('zernpol:normalization','Unrecognized normalization flag.') :^kAFLU  
        end 7vZO;FGtG  
    else Dazm8_x  
        isnorm = false; ;w. la  
    end v Z]j%c@  
    H[.)&7M\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9 3)fC  
    % Compute the Zernike Polynomials =%~- M  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !+3&%vQ)  
    EmT_T 3v  
    % Determine the required powers of r: |j0_^:2r=  
    % ----------------------------------- gamB]FPZ  
    rpowers = []; yP3I^>AZ3  
    for j = 1:length(n) ;l!<A  
        rpowers = [rpowers m(j):2:n(j)]; zp8x/,gwF  
    end }o:LwxNO  
    rpowers = unique(rpowers); cVx SO`jZw  
    AwG0E `SU  
    % Pre-compute the values of r raised to the required powers, v K{2  
    % and compile them in a matrix: .9x* YS  
    % ----------------------------- K*5gb^Ul  
    if rpowers(1)==0 zlEI_th:~  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); yQ/O[(  
        rpowern = cat(2,rpowern{:}); V7U*09 0*5  
        rpowern = [ones(length_r,1) rpowern]; GkU_01C  
    else q`p0ul,n  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 8WV5'cX  
        rpowern = cat(2,rpowern{:}); ]8%E'd  
    end 2+y wy^  
    }i^M<A O  
    % Compute the values of the polynomials: c!\T 0XtT  
    % -------------------------------------- BGi'UL,  
    z = zeros(length_r,length_n); J#C4A]A  
    for j = 1:length_n NTq_"`JjZ  
        s = 0:(n(j)-m(j))/2; <J%Z?3@ T  
        pows = n(j):-2:m(j); #EUT"^:d  
        for k = length(s):-1:1 wA$?e}  
            p = (1-2*mod(s(k),2))* ... r4P%.YO+X  
                       prod(2:(n(j)-s(k)))/          ... T&[6  
                       prod(2:s(k))/                 ... L@O>;zp;  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ;N.dzH2yA  
                       prod(2:((n(j)+m(j))/2-s(k))); H^kOwmSzh  
            idx = (pows(k)==rpowers); VB905%  
            z(:,j) = z(:,j) + p*rpowern(:,idx); jo&j<3i  
        end f 4pIF"U9>  
         %pjY^tM/  
        if isnorm F1BvDplQ>G  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); fUf 1G{4  
        end IN3-ZNx  
    end N<SW $ o  
    >[wxZ5))  
    % 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`'Iew }  
    4/\Ynb.L  
    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。 xrY >Or  
    ettBque  
    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)