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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 x?k6ek  
    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! \%K6T)9  
     
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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 gG#M-2P  
    function z = zernfun(n,m,r,theta,nflag) f\ Qi()  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. z 6p.{M  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N rx%lL  
    %   and angular frequency M, evaluated at positions (R,THETA) on the FN G]  
    %   unit circle.  N is a vector of positive integers (including 0), and Z/%>/  
    %   M is a vector with the same number of elements as N.  Each element &n['#7 <(!  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) lLnD%*03  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, IF<jq\M  
    %   and THETA is a vector of angles.  R and THETA must have the same H=*;3gM,'  
    %   length.  The output Z is a matrix with one column for every (N,M) iZ&CE5+  
    %   pair, and one row for every (R,THETA) pair. -(Yq$5Zc&  
    % %/4ChKf!VR  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike |A"zxNeS"  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), emTqbO  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral }e1f kjWk  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ]c*&5c$  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized *S7<QyVh  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. UZqr6A(/H  
    % KZ&{Ya  
    %   The Zernike functions are an orthogonal basis on the unit circle. Fvg>>HVu  
    %   They are used in disciplines such as astronomy, optics, and h/5.>[VwDh  
    %   optometry to describe functions on a circular domain. >]FRHJo_  
    % li(g?|AD  
    %   The following table lists the first 15 Zernike functions. U4Il1| M&  
    % Zhf+u r  
    %       n    m    Zernike function           Normalization L_Z>*s&  
    %       -------------------------------------------------- 3b~k)t4R  
    %       0    0    1                                 1 y4+Km*am,W  
    %       1    1    r * cos(theta)                    2 :GK]"sNC  
    %       1   -1    r * sin(theta)                    2 Gq?JMq#  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) (V#5Cs,o:  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ?m0|>[j  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) [$$i1%c%Z<  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) \Gg6&:Ua  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Ubv<3syR'  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) n{aD4&  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) kyMWO*>|  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) z`XX[9$qm  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Rjt]^gb!*  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) `5:b=^'D /  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ibha`  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) O(#DaFJv  
    %       -------------------------------------------------- (} ?")$.  
    % 741Sd8  
    %   Example 1: w6aq/m"'  
    % IBZ_xU\2  
    %       % Display the Zernike function Z(n=5,m=1) 0F/o  
    %       x = -1:0.01:1; HW"@~-\  
    %       [X,Y] = meshgrid(x,x); @#rF8;  
    %       [theta,r] = cart2pol(X,Y); gAD,  
    %       idx = r<=1; kIrb;bZ+l  
    %       z = nan(size(X)); 2M@,g8O+B=  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); g[4pG`z  
    %       figure dD ?ZF6  
    %       pcolor(x,x,z), shading interp yH/m@#  
    %       axis square, colorbar XcL jUz?  
    %       title('Zernike function Z_5^1(r,\theta)') 5o2w)<d!  
    % j`7q7}  
    %   Example 2: OO#_ 0qK  
    % '*lVVeSiFw  
    %       % Display the first 10 Zernike functions ^ZuwUuuf  
    %       x = -1:0.01:1; "n'kv!?\  
    %       [X,Y] = meshgrid(x,x); }LeizbU  
    %       [theta,r] = cart2pol(X,Y); a]\l:r  
    %       idx = r<=1; OXp(rJ*bK  
    %       z = nan(size(X)); KDxqz$14 -  
    %       n = [0  1  1  2  2  2  3  3  3  3]; %W` }  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; n` M!K:Pq  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; $ra q,SP  
    %       y = zernfun(n,m,r(idx),theta(idx)); ~xCv_u^=  
    %       figure('Units','normalized') <x-7MU&  
    %       for k = 1:10 4 ))ZBq?  
    %           z(idx) = y(:,k); eI%9.Cx#I  
    %           subplot(4,7,Nplot(k)) ?sD4S   
    %           pcolor(x,x,z), shading interp 2fN2!OT  
    %           set(gca,'XTick',[],'YTick',[]) \:y oS>G  
    %           axis square %>Q[j`9y  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) \w#)uYK{i_  
    %       end XCvL`  
    % v9*31Jx  
    %   See also ZERNPOL, ZERNFUN2. ?*LVn~y  
    [8jIu&tJf  
    %   Paul Fricker 11/13/2006 4Dy|YH$>S  
    x/NjdK  
    i/|}#yw8A  
    % Check and prepare the inputs: sD#*W<  
    % ----------------------------- /Ixv{H)H  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) k~Z;S QyN  
        error('zernfun:NMvectors','N and M must be vectors.') qBF6LhR  
    end &$yxAqdab  
    Zz/ z7~{  
    if length(n)~=length(m) *(E]]8o  
        error('zernfun:NMlength','N and M must be the same length.') pF/s5z  
    end iVT)V>Up  
    &8\6%C  
    n = n(:); 2Y>#FEW/  
    m = m(:); L;h|Sk]{  
    if any(mod(n-m,2)) InA=ty]"_U  
        error('zernfun:NMmultiplesof2', ... Uz =OTM  
              'All N and M must differ by multiples of 2 (including 0).') ^|%u%UR  
    end *Za'^Z2  
    o3W@)|>  
    if any(m>n) twMDEw#VL  
        error('zernfun:MlessthanN', ... `l2h65\  
              'Each M must be less than or equal to its corresponding N.') MFeY}_d<  
    end otA'+4\  
    |_njN  
    if any( r>1 | r<0 ) (}m2}  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') XFSHl[uS1  
    end %0Ibi  
    # Rhtaq9  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) nvB< pSm  
        error('zernfun:RTHvector','R and THETA must be vectors.') LEuDDJ -  
    end U4=m>Ty  
    &6e A.  
    r = r(:); yXQ 28A  
    theta = theta(:); m<sCRWa-  
    length_r = length(r); } _=h]|6t  
    if length_r~=length(theta) ra;:  
        error('zernfun:RTHlength', ... <!=:{&d%  
              'The number of R- and THETA-values must be equal.') BdB9M8fM  
    end osn ,kD*  
    & LhQr-g  
    % Check normalization: vM?,#:5  
    % -------------------- mWF\h>]|.  
    if nargin==5 && ischar(nflag) O{x-9p  
        isnorm = strcmpi(nflag,'norm'); CC)Mws+2  
        if ~isnorm 7jw5'`;)"  
            error('zernfun:normalization','Unrecognized normalization flag.') @ 3rJ$6W  
        end f}EsS  
    else -H F1c  
        isnorm = false; D@%!|:  
    end b ,x$wP+  
    ;o158H$gz;  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% lWDSF]ZYV  
    % Compute the Zernike Polynomials N :OLN[  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Dtox/ ,"  
    fu iTy72  
    % Determine the required powers of r: rgo!t028^  
    % ----------------------------------- 87F]a3  
    m_abs = abs(m); 8=)9ZjfD  
    rpowers = []; >}B53.;.k  
    for j = 1:length(n) jx'hxC'3  
        rpowers = [rpowers m_abs(j):2:n(j)]; ^HU>fkSk  
    end WDI3*  
    rpowers = unique(rpowers); m2HO .ljc  
    Y()ZM  
    % Pre-compute the values of r raised to the required powers, Ej $.x6:  
    % and compile them in a matrix: \0K&2'  
    % ----------------------------- +TAyCxfmt  
    if rpowers(1)==0 )-m/(-  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); J| 1!4R~  
        rpowern = cat(2,rpowern{:}); NtmmPJ|5  
        rpowern = [ones(length_r,1) rpowern]; '|}H ,I{  
    else MP_/eC ;  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Wj8WT)cB  
        rpowern = cat(2,rpowern{:}); n\< uT1n  
    end 'Z|Czd8E  
    tkmzOc H  
    % Compute the values of the polynomials: =3nA5'UZ  
    % -------------------------------------- y Ni3@f  
    y = zeros(length_r,length(n)); v|dt[>G  
    for j = 1:length(n) *TrpW?]Y&  
        s = 0:(n(j)-m_abs(j))/2; >U.7>K V&  
        pows = n(j):-2:m_abs(j); 9rIv-&7'm  
        for k = length(s):-1:1 #7"";"{ z|  
            p = (1-2*mod(s(k),2))* ... N/[!$B0H@  
                       prod(2:(n(j)-s(k)))/              ... zDBm^ s  
                       prod(2:s(k))/                     ... a^Z=xlJ/uZ  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... e:K'e2  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); V+zn` \a  
            idx = (pows(k)==rpowers); WpOH1[ 8v  
            y(:,j) = y(:,j) + p*rpowern(:,idx); /z(d!0_q|v  
        end Q3'P<"u  
         !?sB=qo  
        if isnorm K"!U&`T  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); RRro.r,  
        end m {&lU@uL  
    end zcuz @  
    % END: Compute the Zernike Polynomials TEbIU8{Y  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `<#O8,7`  
    P`S'F_IN  
    % Compute the Zernike functions: C`uL 4r  
    % ------------------------------ FxT]*mo  
    idx_pos = m>0; k@pEs# a  
    idx_neg = m<0; t. HwX9  
    D&=+PAX  
    z = y; 2Ima15^+F  
    if any(idx_pos) L8oqlq( 9  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); qiq=v)  
    end 8w#4T:hsuN  
    if any(idx_neg) Bat@  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); a '?LC)^  
    end `)kxFD_bH  
    It VVI"-  
    % 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) m8eoD{  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. EpTc{  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated :GFK |  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive FE:} D ;$  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ,s 6lB0  
    %   and THETA is a vector of angles.  R and THETA must have the same #TV #*  
    %   length.  The output Z is a matrix with one column for every P-value, 9yu#G7  
    %   and one row for every (R,THETA) pair. -FrK'!\  
    % of >  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike o)Px d  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) mR|']^!SE  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) )`2ncb   
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ^f 0-w`D  
    %   for all p. .bRtK+}F#  
    % }*!_M3O  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Pj*]%V  
    %   Zernike functions (order N<=7).  In some disciplines it is :*R+ee,& -  
    %   traditional to label the first 36 functions using a single mode a/rQ@c>  
    %   number P instead of separate numbers for the order N and azimuthal bxWzm|  
    %   frequency M. +i}uRO  
    % uH7!)LE#  
    %   Example: rT[b ^l}  
    % #KuBEHr  
    %       % Display the first 16 Zernike functions #iRd2Qj%  
    %       x = -1:0.01:1; FuAs$;  
    %       [X,Y] = meshgrid(x,x); sp_19u  
    %       [theta,r] = cart2pol(X,Y); b/tc D r  
    %       idx = r<=1; j'Gezx^.<e  
    %       p = 0:15; >u|4490<0  
    %       z = nan(size(X)); @y6^/'  
    %       y = zernfun2(p,r(idx),theta(idx)); jN(c`Gb  
    %       figure('Units','normalized') 9lkl-b6xG  
    %       for k = 1:length(p) |Qpd<L  
    %           z(idx) = y(:,k); \K lY8\c[  
    %           subplot(4,4,k) :c(I-xif  
    %           pcolor(x,x,z), shading interp e?\hz\^  
    %           set(gca,'XTick',[],'YTick',[]) kt7Emb}  
    %           axis square '$4&q629d  
    %           title(['Z_{' num2str(p(k)) '}']) B->oTC`5  
    %       end {@'#|]4y.  
    % cnDF`7xrT  
    %   See also ZERNPOL, ZERNFUN. BFqM6_/J  
    @udc/J$  
    %   Paul Fricker 11/13/2006 1M?Sl?+j  
    zs'Jgm.v  
    <W^>:!?w  
    % Check and prepare the inputs: n:}'f- :T  
    % ----------------------------- +O8}twt@  
    if min(size(p))~=1 >#`{(^  
        error('zernfun2:Pvector','Input P must be vector.') gEmsPk,  
    end s -F3(mc(  
    B9`_~~^U5  
    if any(p)>35 =`!# V/=  
        error('zernfun2:P36', ... }_Y\6fcd  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... oJc7a z  
               '(P = 0 to 35).']) aCyn9Y$=  
    end {>Qs+]  
    f J%A_N}  
    % Get the order and frequency corresonding to the function number: qg*xdefQ%  
    % ---------------------------------------------------------------- ;Wn0-`_1,  
    p = p(:); aq9Ej]1b  
    n = ceil((-3+sqrt(9+8*p))/2); ApcE)mjpc  
    m = 2*p - n.*(n+2); I@1VX5  
    /1hcw|cfC  
    % Pass the inputs to the function ZERNFUN: #qEUGD`  
    % ---------------------------------------- Nig)!4CG  
    switch nargin Lp+?5DjLT  
        case 3 ph3dm\U.  
            z = zernfun(n,m,r,theta); JP]-a!5Ru  
        case 4 #-PUm0|  
            z = zernfun(n,m,r,theta,nflag); -(E-yC u  
        otherwise #BI6+rfv|  
            error('zernfun2:nargin','Incorrect number of inputs.') wFJ*2W:  
    end Gd|jE  
    `Tr !Gj_  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) 1+ V<-I@{  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. sw,p6T[  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of qITd.< k  
    %   order N and frequency M, evaluated at R.  N is a vector of "@itn  
    %   positive integers (including 0), and M is a vector with the jZ?^ |1  
    %   same number of elements as N.  Each element k of M must be a bFfDaO<k  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) |3gWH4M4**  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is s~I#K[[5  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix _9z+xl  
    %   with one column for every (N,M) pair, and one row for every 3e|,Z'4}4  
    %   element in R. :(4];Va  
    % eGI&4JgJ.  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- NOtwgZ-  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is &G7@lz@sK+  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to e!4Kl:  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 rU7t~DKS  
    %   for all [n,m]. ^ ;cJjl'=  
    % -n6T^vf  
    %   The radial Zernike polynomials are the radial portion of the I! ~3xZ  
    %   Zernike functions, which are an orthogonal basis on the unit B_0]$D0 ^  
    %   circle.  The series representation of the radial Zernike '>% c@C[  
    %   polynomials is :;o?d&C  
    % :a M@"#F  
    %          (n-m)/2 Aoy=gK  
    %            __ L$; gf_L  
    %    m      \       s                                          n-2s Jv D`RUh  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 'V*8'?  
    %    n      s=0 ^I!gteU;  
    % u6P U(f  
    %   The following table shows the first 12 polynomials. o6S`7uwJ*/  
    % 9`DY6qfly  
    %       n    m    Zernike polynomial    Normalization Z DnAzAR  
    %       --------------------------------------------- -V}ZbXJD  
    %       0    0    1                        sqrt(2) ,jMV # H[  
    %       1    1    r                           2 &0J/V>k  
    %       2    0    2*r^2 - 1                sqrt(6) P)hawH=  
    %       2    2    r^2                      sqrt(6) Jl89}Sf  
    %       3    1    3*r^3 - 2*r              sqrt(8) lziC.Dpa  
    %       3    3    r^3                      sqrt(8) * 9}~?#b  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) Wr.~Ns <  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) *q(HW  
    %       4    4    r^4                      sqrt(10) L9 H.DNA  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) /Pa<I^-#  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ZRCUM"R_  
    %       5    5    r^5                      sqrt(12) KnYHjJa  
    %       --------------------------------------------- U,nQnD"!t&  
    % `O}bPwa{>  
    %   Example: ,9/s`o  
    % Y ^uYc}  
    %       % Display three example Zernike radial polynomials 0"=}d y  
    %       r = 0:0.01:1; Rj,M|9Y)o  
    %       n = [3 2 5]; CV6W)B%Se  
    %       m = [1 2 1]; -!ERe@k(  
    %       z = zernpol(n,m,r); /M~rmIks  
    %       figure u9dL-Nr`  
    %       plot(r,z)  %L gfi  
    %       grid on OGU#%5"<  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') zy[|4Q(?  
    % s&qr2'F+z  
    %   See also ZERNFUN, ZERNFUN2. ,5Tw5<S  
    }ilX 2s?>  
    % A note on the algorithm. r#K"d  
    % ------------------------ {X>U`0P  
    % The radial Zernike polynomials are computed using the series 69/qH_Y  
    % representation shown in the Help section above. For many special .@]M'S^1  
    % functions, direct evaluation using the series representation can c)=UX_S!  
    % produce poor numerical results (floating point errors), because .;~K*GC  
    % the summation often involves computing small differences between ]lzOz<0q  
    % large successive terms in the series. (In such cases, the functions .A Z+|?d  
    % are often evaluated using alternative methods such as recurrence XY`2>7  
    % relations: see the Legendre functions, for example). For the Zernike (Zu V5|N  
    % polynomials, however, this problem does not arise, because the WjMP]ND#c  
    % polynomials are evaluated over the finite domain r = (0,1), and =6+j Po{F  
    % because the coefficients for a given polynomial are generally all +l_$}UN  
    % of similar magnitude. &0S/]E`_M  
    % M;qV% k  
    % ZERNPOL has been written using a vectorized implementation: multiple \Rvsy;7  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] b1qli5  
    % values can be passed as inputs) for a vector of points R.  To achieve zs@[!?A,  
    % this vectorization most efficiently, the algorithm in ZERNPOL Yg14aKZl  
    % involves pre-determining all the powers p of R that are required to ) XHcrm&  
    % compute the outputs, and then compiling the {R^p} into a single T2MX_rt#D  
    % matrix.  This avoids any redundant computation of the R^p, and t9 m],aH  
    % minimizes the sizes of certain intermediate variables. QYTwGThWR  
    % ^7~w yAr  
    %   Paul Fricker 11/13/2006 %epK-q9[  
    ._z[T@!9  
    :7Q, `W9  
    % Check and prepare the inputs: },LW@Z}  
    % ----------------------------- 7 }sj&  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) *npe]cC  
        error('zernpol:NMvectors','N and M must be vectors.') 7%4@*  
    end /I&b5Vp  
    r#[YBaCZJ  
    if length(n)~=length(m) @3b|jJyf  
        error('zernpol:NMlength','N and M must be the same length.') I3V>VLv  
    end i<Be)Y-'  
    /1q] D8  
    n = n(:); }ZWeb#\  
    m = m(:); #FV(a~  
    length_n = length(n); Ey "<hAF  
    *oX~z>aE  
    if any(mod(n-m,2)) >, }m=X8  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') "i_}\p.,X  
    end dO[w3\~  
    "ci<W_lx  
    if any(m<0) ?RD)a`y51  
        error('zernpol:Mpositive','All M must be positive.') 4x:Odt5  
    end cIgicp}U  
    Kv:ih=?  
    if any(m>n) [2,u:0"  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') \l leO|m  
    end qlg~W/  
    k]`3if5>  
    if any( r>1 | r<0 ) c};Qr@vpo  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') -h8!O+7 .  
    end *7I=vro  
    $V8B =k~  
    if ~any(size(r)==1) 8Q1){M9 '  
        error('zernpol:Rvector','R must be a vector.') z {J1pH_X  
    end ;G3{ e  
    y|X\f!  
    r = r(:); A4?_ 0:<  
    length_r = length(r); ]eq3cwR[|  
    ud-.R~f{e  
    if nargin==4 UmU=3et<Wj  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); Sk$ XC  
        if ~isnorm }r /L 9  
            error('zernpol:normalization','Unrecognized normalization flag.') BI=Ie?  
        end k>Qr 14F  
    else mHox  
        isnorm = false; .-N9\GlJ,d  
    end W3K"5E0ck  
    Q_#X*I  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }h>e=<  
    % Compute the Zernike Polynomials <..%@]+  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $/45*  
    b,G+=&6u  
    % Determine the required powers of r: MT#9x>  
    % ----------------------------------- ASaG }h  
    rpowers = []; ap9eQsC  
    for j = 1:length(n) N2tvP+Z6D  
        rpowers = [rpowers m(j):2:n(j)]; Xb3vvHdI  
    end VPg`vI$(X  
    rpowers = unique(rpowers); H$ xSl1>E  
    $zz=>BOk  
    % Pre-compute the values of r raised to the required powers, -ij1%#tz  
    % and compile them in a matrix: )11/BB\v  
    % ----------------------------- :"O=/p+*Us  
    if rpowers(1)==0 e= "/oo  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ce=6EYl  
        rpowern = cat(2,rpowern{:}); > KH4X:  
        rpowern = [ones(length_r,1) rpowern]; aM.l+D P  
    else qg6Hk:^r  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); XW]|Mv[M  
        rpowern = cat(2,rpowern{:}); uD:O[H-x  
    end HA"dw2 |  
    ~rY<y%K  
    % Compute the values of the polynomials: ~5}b$qL#`  
    % -------------------------------------- +I\ bs.84  
    z = zeros(length_r,length_n); W.:k E|a.g  
    for j = 1:length_n i$:CGUb  
        s = 0:(n(j)-m(j))/2; ZVIBmx  
        pows = n(j):-2:m(j); .#WF'  
        for k = length(s):-1:1 #0Oqw=F  
            p = (1-2*mod(s(k),2))* ... Qn%*kU0X  
                       prod(2:(n(j)-s(k)))/          ... #^#)OQq]  
                       prod(2:s(k))/                 ... Z*]n]eS  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... Cfz1\a&V{  
                       prod(2:((n(j)+m(j))/2-s(k))); 3+ i(fg_  
            idx = (pows(k)==rpowers); S8,+6+_7  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Cv$TNkP*  
        end 8@+YcN;->  
         vW)GUAF[  
        if isnorm tg8VFH2q.z  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); XcfTE m  
        end NKd@ Kp`,  
    end 2,QkktJLo  
    s^KxAw_IV  
    % 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)  g& "(- :  
    8Wj=|Ow-q  
    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。 w}.'Tebu  
    ;/V@N |$n  
    07年就写过这方面的计算程序了。
    2024年6月28-30日于上海组织线下成像光学设计培训,欢迎报名参加。请关注子在川上光学公众号。详细内容请咨询13661915143(同微信号)