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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 /db?ltb  
    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! Bx&` $lW  
     
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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 )sV# b  
    function z = zernfun(n,m,r,theta,nflag) <;=Y4$y[  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. (X>y)V  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N uTl:u  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 9Biw!%a  
    %   unit circle.  N is a vector of positive integers (including 0), and ~|uCZ.;o  
    %   M is a vector with the same number of elements as N.  Each element c4-&I"z  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) J~_p2TZJ\3  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 2M&4]d  
    %   and THETA is a vector of angles.  R and THETA must have the same x *qef_Hu  
    %   length.  The output Z is a matrix with one column for every (N,M) b,Z& P|  
    %   pair, and one row for every (R,THETA) pair. =\XAD+  
    % U~H'c p  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 21o_9=[^  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), \^#1~Kx  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral izC>-  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 2#(7,o}Y5  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized WN?T*bz2  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. dlT\VWMha(  
    % tjd"05"@:  
    %   The Zernike functions are an orthogonal basis on the unit circle. q#p)E=$  
    %   They are used in disciplines such as astronomy, optics, and )F~>  
    %   optometry to describe functions on a circular domain. ~HYP:6f  
    % Q?"[zX1  
    %   The following table lists the first 15 Zernike functions. |iwTzlt*#  
    % Bw_Ih|y,w  
    %       n    m    Zernike function           Normalization 25ayYO%PTc  
    %       -------------------------------------------------- -:~`g*3#  
    %       0    0    1                                 1 8m1zL[.8g  
    %       1    1    r * cos(theta)                    2 &R5M&IwL  
    %       1   -1    r * sin(theta)                    2 dt \O7Rjw8  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) vlPE8U=  
    %       2    0    (2*r^2 - 1)                    sqrt(3) $U8ap4EXM  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 9~; Ju^b  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) l?R_wu,Q  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) aDOH3Ri0K!  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) J<BdIKCma  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) +.N;h-'  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) W@ Z=1y  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) }cPV_^{  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) >bZ#  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #KK(Z \;  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) yBe/UFp+  
    %       -------------------------------------------------- =#V11j  
    % O#EBR<CuK  
    %   Example 1: \6'A^cE/PX  
    % xw-q)u  
    %       % Display the Zernike function Z(n=5,m=1) RdDcMZ  
    %       x = -1:0.01:1; ZbrE m  
    %       [X,Y] = meshgrid(x,x); = ]@xXVf/  
    %       [theta,r] = cart2pol(X,Y); |M?HdxPa  
    %       idx = r<=1; # _7c>gn  
    %       z = nan(size(X));  ~Afs  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); q#a21~S<  
    %       figure X,N@`  
    %       pcolor(x,x,z), shading interp UA9LI<Y  
    %       axis square, colorbar \\lC"Z#J`  
    %       title('Zernike function Z_5^1(r,\theta)') YHA[PF   
    % (s3%1OC[  
    %   Example 2: }dHiW:J>  
    % C\; 8l}t  
    %       % Display the first 10 Zernike functions {S}@P~H =  
    %       x = -1:0.01:1; q kKABow  
    %       [X,Y] = meshgrid(x,x); Sy'>JHx  
    %       [theta,r] = cart2pol(X,Y); E\zhxiI  
    %       idx = r<=1; </=PN1=A  
    %       z = nan(size(X)); UZ!hk*PF  
    %       n = [0  1  1  2  2  2  3  3  3  3]; %OtW\T=u  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; { &'TA  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Imw x~eo  
    %       y = zernfun(n,m,r(idx),theta(idx)); iN*>Z(b"  
    %       figure('Units','normalized') kW~F*  
    %       for k = 1:10 sZH7 EK  
    %           z(idx) = y(:,k); 10J*S[n1  
    %           subplot(4,7,Nplot(k)) 0/6&2  
    %           pcolor(x,x,z), shading interp uqUo4z5T  
    %           set(gca,'XTick',[],'YTick',[]) v wyDY%B"n  
    %           axis square s z\RmX  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) =c,gK8C  
    %       end [5VUcXGt*\  
    % yq}{6IyZ^  
    %   See also ZERNPOL, ZERNFUN2. k:TfE6JZ  
    TUaK:*x*  
    %   Paul Fricker 11/13/2006 7&3URglsL"  
    o.5j@ dr  
    l0&8vhw8k  
    % Check and prepare the inputs: i njmP9ed  
    % ----------------------------- ie(7m| .  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) y]l"u=$Tr{  
        error('zernfun:NMvectors','N and M must be vectors.') 752wK|o0|;  
    end bIArAS9%  
    wuzz%9;@B  
    if length(n)~=length(m) *r`Yz}  
        error('zernfun:NMlength','N and M must be the same length.') 9^^#I ~-  
    end $dP)8_Z2  
    g#4gGhI  
    n = n(:); #CPPdU$  
    m = m(:); aAri  
    if any(mod(n-m,2)) 7fay:_  
        error('zernfun:NMmultiplesof2', ... @__;RVQ  
              'All N and M must differ by multiples of 2 (including 0).') Hl;p>>n  
    end L:M9|/  
    k&/ )g3(N(  
    if any(m>n) 'j_H{kQy  
        error('zernfun:MlessthanN', ... {^W,e ^:  
              'Each M must be less than or equal to its corresponding N.') [kOA+\v  
    end F}]_/cY7B  
    `t1$Ew<  
    if any( r>1 | r<0 ) pxxFm~"d  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') L"iyjL<M  
    end ql~{`qoD~  
    QYgN39gp  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) XKX,7  
        error('zernfun:RTHvector','R and THETA must be vectors.') Pm^N0L9?q  
    end i)L:VkN  
    CFm1c1%Hg  
    r = r(:); 5|CiwQg|,p  
    theta = theta(:); (AG  
    length_r = length(r); ;_/q>DR>,3  
    if length_r~=length(theta) b 0b9#9x  
        error('zernfun:RTHlength', ... kI 3zYD^:  
              'The number of R- and THETA-values must be equal.') Jyci}CU3\Q  
    end A_Iu*pz^^  
    E`fssd~  
    % Check normalization: g/,Bx!'8p  
    % -------------------- i=UTc1  
    if nargin==5 && ischar(nflag) WKl'  
        isnorm = strcmpi(nflag,'norm'); RQCQGa^cP  
        if ~isnorm hIQ[:f  
            error('zernfun:normalization','Unrecognized normalization flag.') h.$__Gs  
        end %hbLT{w  
    else 4E-A@FR  
        isnorm = false; =>0M3 Qh{  
    end I'9s=~VfY,  
    4)HWPX  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% loUwR z  
    % Compute the Zernike Polynomials SP*JleQN  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% h ^h-pd  
    +;*(a3Gp  
    % Determine the required powers of r: 0BB @E(*  
    % ----------------------------------- BZ+ mO  
    m_abs = abs(m); r!$NZ2I  
    rpowers = []; 7~ese+\smG  
    for j = 1:length(n) G;HlII9x[  
        rpowers = [rpowers m_abs(j):2:n(j)]; Ik5jwfz  
    end z|]oM#Gt  
    rpowers = unique(rpowers); y3nm!tjyM  
    @B'8SLoP  
    % Pre-compute the values of r raised to the required powers, 4A/,X>W61  
    % and compile them in a matrix: 2^ bpH%  
    % ----------------------------- NhK(HTsvK  
    if rpowers(1)==0 As'M3 9*V  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 4@]xn  
        rpowern = cat(2,rpowern{:}); c =N]! ,MO  
        rpowern = [ones(length_r,1) rpowern]; *_<*bhR<  
    else V2s}<uG  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); sRyw\v-=P  
        rpowern = cat(2,rpowern{:}); {,f!'i&b@  
    end rrY{Jf9>  
    +Bq}>  
    % Compute the values of the polynomials: mU+FQX  
    % -------------------------------------- 12d}#G<q-  
    y = zeros(length_r,length(n)); 0"^oTmQN  
    for j = 1:length(n) j t`p<gI  
        s = 0:(n(j)-m_abs(j))/2; TFC!u 0Y"$  
        pows = n(j):-2:m_abs(j); n E,gQHw  
        for k = length(s):-1:1 @CaD8%j{  
            p = (1-2*mod(s(k),2))* ... C*s0r;  
                       prod(2:(n(j)-s(k)))/              ... UiK+c30FU  
                       prod(2:s(k))/                     ... -hVv  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... c,+(FQ9  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); c_z/At;4  
            idx = (pows(k)==rpowers); ~6:LUM  
            y(:,j) = y(:,j) + p*rpowern(:,idx); e}R2J `7  
        end ^wO_b'@v  
         ?St=7a(D  
        if isnorm E7yf[/it  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); A:.IBctsd  
        end {rb-DB-/5M  
    end G{f`K^  
    % END: Compute the Zernike Polynomials :%uyy5AZ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .hQ3A"  
    @r/Id{pCI  
    % Compute the Zernike functions: *K}z@a_  
    % ------------------------------ ll(e,9.D  
    idx_pos = m>0; 7/&C;"  
    idx_neg = m<0; nG},v%  
    b>bgUDq  
    z = y; Z9"{f)T  
    if any(idx_pos) V|3yZ8lE  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); urT/+deR  
    end -; us12SZ  
    if any(idx_neg) AU\xNF3  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); AJ>BF.>  
    end #0?"J)  
    W>?f^C!+m  
    % EOF zernfun
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) yD3bl%uZ  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. GhG%>U#&a  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated M.xhVgFf)  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive J.bF v/R  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, P\q<d  
    %   and THETA is a vector of angles.  R and THETA must have the same lBlSNDs  
    %   length.  The output Z is a matrix with one column for every P-value, u[V4OU}%  
    %   and one row for every (R,THETA) pair. 3{Na ZIk  
    % ;g: UE  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike +,&m7L  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) }g>dn  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) Q^a&qYK  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 LyuSZa]  
    %   for all p. +=ZWau   
    % qs6Nb'JvQR  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Y|%anTP  
    %   Zernike functions (order N<=7).  In some disciplines it is hFsA_x+L;  
    %   traditional to label the first 36 functions using a single mode j>$=SMc  
    %   number P instead of separate numbers for the order N and azimuthal vF9*tK'   
    %   frequency M. =@xN(] (  
    % 9GMH*=3[=  
    %   Example: %^a]J"Ydi8  
    % DC?21[60  
    %       % Display the first 16 Zernike functions iEiu%T>  
    %       x = -1:0.01:1; x r-;,W  
    %       [X,Y] = meshgrid(x,x); TqS2!/jp  
    %       [theta,r] = cart2pol(X,Y); Y brx%  
    %       idx = r<=1; =d go!k  
    %       p = 0:15; [kPD`be2#  
    %       z = nan(size(X)); |7svA<<[  
    %       y = zernfun2(p,r(idx),theta(idx)); >EBZ$X  
    %       figure('Units','normalized') ;\<""Yj@l  
    %       for k = 1:length(p) p&i. )/  
    %           z(idx) = y(:,k); nGq]$h  
    %           subplot(4,4,k) d>lt  
    %           pcolor(x,x,z), shading interp G;2R]H#p  
    %           set(gca,'XTick',[],'YTick',[]) wVx,JL5Jr  
    %           axis square XOu+&wOu  
    %           title(['Z_{' num2str(p(k)) '}']) J?._/RL8-  
    %       end 1pd 9s8CA  
    % Ap11b|v  
    %   See also ZERNPOL, ZERNFUN. <e;jW K  
    8 ne/=N|,  
    %   Paul Fricker 11/13/2006 ;$vLq&(}  
    nAIH`L"X  
    !cRfZ  
    % Check and prepare the inputs: )D:9R)m  
    % ----------------------------- z'7#"D  
    if min(size(p))~=1 n4^~gT%b5]  
        error('zernfun2:Pvector','Input P must be vector.') Ee{`Y0  
    end Wu4ot0SZ  
    tS?a){^:c  
    if any(p)>35 j*tk(o}qG  
        error('zernfun2:P36', ... 8V6=i'GK  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... A/!<kp{S  
               '(P = 0 to 35).']) inYM+o!Ub  
    end 2Oyy`k  
    #X#8ynt  
    % Get the order and frequency corresonding to the function number: 2$M,*Dnr  
    % ---------------------------------------------------------------- #":: ' ?,  
    p = p(:); ITVQLQ  
    n = ceil((-3+sqrt(9+8*p))/2); Tak t_N  
    m = 2*p - n.*(n+2); },rav]  
    zm3-C%:Bw  
    % Pass the inputs to the function ZERNFUN: #S%4?   
    % ---------------------------------------- 0Y7$d`  
    switch nargin kB8 Mi  
        case 3 @!e~G'j%VD  
            z = zernfun(n,m,r,theta); os[ZIHph  
        case 4 E(_ KN[}S  
            z = zernfun(n,m,r,theta,nflag); @cZ\*,T  
        otherwise VKy5=2&  
            error('zernfun2:nargin','Incorrect number of inputs.') auRY|j  
    end /?Vdqci  
    Z(p*Z,?u  
    % 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) f3G1r5x  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. #1[z;Mk0  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of X]!D;7^  
    %   order N and frequency M, evaluated at R.  N is a vector of W .U+.hR  
    %   positive integers (including 0), and M is a vector with the }z wX  
    %   same number of elements as N.  Each element k of M must be a ys%zlbj[  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) m2(E>raV6  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is :k~dj C  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix _8^0!,j  
    %   with one column for every (N,M) pair, and one row for every qp>N^)>  
    %   element in R. Wwhgo.Wx  
    % o&z!6"S<  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- d`Oe_<  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is !MoOKW  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to hU" F;4p  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 mw4JQ\  
    %   for all [n,m]. *g_w I%l  
    % IE;\7 r+h  
    %   The radial Zernike polynomials are the radial portion of the  <B )   
    %   Zernike functions, which are an orthogonal basis on the unit 89Z#|#uM5  
    %   circle.  The series representation of the radial Zernike |gv{z"  
    %   polynomials is DtI$9`~  
    % u]E.iXp  
    %          (n-m)/2 pDn&V(  
    %            __ r P'AJDuq  
    %    m      \       s                                          n-2s V&*D~Jq  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r zsVcXBz  
    %    n      s=0 |9IC/C!HC  
    % ]JdJe6`Mc  
    %   The following table shows the first 12 polynomials. 'Jydu   
    % vk'rA{x  
    %       n    m    Zernike polynomial    Normalization L^FcS\r;  
    %       ---------------------------------------------  ?Vc0)  
    %       0    0    1                        sqrt(2) 9i=B  
    %       1    1    r                           2 g1{/ 5{XI  
    %       2    0    2*r^2 - 1                sqrt(6) "|t!7hC  
    %       2    2    r^2                      sqrt(6) GoIQ>n  
    %       3    1    3*r^3 - 2*r              sqrt(8) [b/o$zR  
    %       3    3    r^3                      sqrt(8) *5D3vB*S  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) f*m[|0qI<X  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) _TUm$#@Y`  
    %       4    4    r^4                      sqrt(10) Y: psZ  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ?pG/m%[  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 3vW4<:Lgy  
    %       5    5    r^5                      sqrt(12) yMxTfR  
    %       --------------------------------------------- {eQWO.C{  
    % %;|0  
    %   Example: dJD(\a>r.u  
    % P :k+ y$  
    %       % Display three example Zernike radial polynomials SK R1E];4  
    %       r = 0:0.01:1; LZ<[ll#C  
    %       n = [3 2 5]; {@.Vh]  
    %       m = [1 2 1]; opp!0:jS*  
    %       z = zernpol(n,m,r); q3h'l,  
    %       figure (3;@^S4&w  
    %       plot(r,z) BStk&b  
    %       grid on K_ke2{4Jm  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') |V|+lx'sc  
    % N3%*7{X 9  
    %   See also ZERNFUN, ZERNFUN2. 3FNT|QF  
    %8r/oS  
    % A note on the algorithm. vFQ,5n;fF  
    % ------------------------ )`+YCCa6F  
    % The radial Zernike polynomials are computed using the series |"]PCb)!  
    % representation shown in the Help section above. For many special >jTp6tu,  
    % functions, direct evaluation using the series representation can E[g*O5  
    % produce poor numerical results (floating point errors), because u0 y 1  
    % the summation often involves computing small differences between PsnGXcj  
    % large successive terms in the series. (In such cases, the functions +Qj(B@ i  
    % are often evaluated using alternative methods such as recurrence )9L/sKz  
    % relations: see the Legendre functions, for example). For the Zernike lzhqcL"  
    % polynomials, however, this problem does not arise, because the L0NA*C   
    % polynomials are evaluated over the finite domain r = (0,1), and p&Ed\aQ%z;  
    % because the coefficients for a given polynomial are generally all 3BQ!qO17^d  
    % of similar magnitude. Q(Gl{#b  
    % ig+4S[L~n  
    % ZERNPOL has been written using a vectorized implementation: multiple cWLqU  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ,}SCa'PB  
    % values can be passed as inputs) for a vector of points R.  To achieve M.[rLJZ4  
    % this vectorization most efficiently, the algorithm in ZERNPOL T!|=El>  
    % involves pre-determining all the powers p of R that are required to ig!7BxM)<h  
    % compute the outputs, and then compiling the {R^p} into a single Z?G&.# :  
    % matrix.  This avoids any redundant computation of the R^p, and _Z Y\,_  
    % minimizes the sizes of certain intermediate variables. Um.qRZ?  
    % ,wAz^cK|  
    %   Paul Fricker 11/13/2006 Z`Z5sj 4{  
    F0lOlS   
    9`B$V##-L  
    % Check and prepare the inputs: YY5!_k  
    % ----------------------------- D Ml?o:l  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) <q2?S  
        error('zernpol:NMvectors','N and M must be vectors.') z-T{~{q  
    end pv,45z0  
    Jkt4@h2Q}  
    if length(n)~=length(m) s60 TxB  
        error('zernpol:NMlength','N and M must be the same length.') Y.6SOu5$]  
    end ~bK9R 0|<  
    {XCf-{a]~  
    n = n(:); >3.X?  
    m = m(:); g(E"4M@t!  
    length_n = length(n); 9Ul(GI(  
    jp2Q 9Z  
    if any(mod(n-m,2)) B&?sF" Y  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') {*m?Kc7k  
    end $ &UZy|9  
    PkuTg";  
    if any(m<0) 60>.ul2  
        error('zernpol:Mpositive','All M must be positive.') /j2H A^GT  
    end ??&<k   
    3G}AH E4  
    if any(m>n) aM3gRp51cj  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') p9eRZVy/  
    end 3L5r*fa  
    zZ-\a[F  
    if any( r>1 | r<0 ) k@mVxnC  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') TFQ!7'xk)  
    end j]rz] k  
    DGESba\2+  
    if ~any(size(r)==1) |I;$M;'r&  
        error('zernpol:Rvector','R must be a vector.') V@-)\RZm  
    end =n(3o$r(  
    C#0Qd%  
    r = r(:); s#9Ui#[=h  
    length_r = length(r); #'baPqdO  
    5s{j = .O  
    if nargin==4 (qM j-l  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); !D^c3d  
        if ~isnorm Dh4 6o|P  
            error('zernpol:normalization','Unrecognized normalization flag.') 2/ rt@{V(  
        end yY).mxRN  
    else _l`e#XbG  
        isnorm = false; OX]V) QHVZ  
    end >o,^b\  
    R"v 3!P  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% o`S ?  
    % Compute the Zernike Polynomials R\3VB NX.g  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *jq7X  
    "UFs~S|e  
    % Determine the required powers of r: Io`P,l:  
    % ----------------------------------- ZD/jX_!t  
    rpowers = []; -_OS%ARa  
    for j = 1:length(n) &C<yfRDu  
        rpowers = [rpowers m(j):2:n(j)]; jEdtJ EPa  
    end #SVNHpx  
    rpowers = unique(rpowers); g^Yl TB  
    qFX~[h8i+  
    % Pre-compute the values of r raised to the required powers, K kW;-{c  
    % and compile them in a matrix: YUU-D(  
    % ----------------------------- Z6C=T;w  
    if rpowers(1)==0 m0w;8uF2UV  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); )m3Uar  
        rpowern = cat(2,rpowern{:}); 8LkP)]4^sO  
        rpowern = [ones(length_r,1) rpowern]; wBj-m  
    else .jw}JJ  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 6DIZ@oi  
        rpowern = cat(2,rpowern{:}); f>o,N{|  
    end #hfuH=&oh  
    ^[E' 1$D  
    % Compute the values of the polynomials: ) /vhclkb  
    % -------------------------------------- d$ACDX2  
    z = zeros(length_r,length_n); 0-Y:v(|.  
    for j = 1:length_n ^)!F9h+  
        s = 0:(n(j)-m(j))/2; 1F'1>Bu~  
        pows = n(j):-2:m(j); IzOYduJ.  
        for k = length(s):-1:1 j1q[2'  
            p = (1-2*mod(s(k),2))* ... Gl %3XdU  
                       prod(2:(n(j)-s(k)))/          ... '7Nr8D4L  
                       prod(2:s(k))/                 ... 5wao1sd#  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... B5V_e!*5F*  
                       prod(2:((n(j)+m(j))/2-s(k))); 7M_U2cd|TD  
            idx = (pows(k)==rpowers); $0oO &)*  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 8(g:HR*;  
        end 8b.u'r174  
          MTER(L  
        if isnorm 0kQPJWF  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); c !ZM  
        end YYEJph@06q  
    end SnlyUP~P  
    6Tw#^;q-  
    % 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)  {bO|409>W  
    e]=!"nJ+  
    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。 [-&L8Un  
    Y(\T- bI  
    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)