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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 $:vS_#  
    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! ^Q!A4 qOQ  
     
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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 U8<C4  
    function z = zernfun(n,m,r,theta,nflag) `9|Uu#x  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. }8`>n4  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N K4OiKYq  
    %   and angular frequency M, evaluated at positions (R,THETA) on the j%8 1q  
    %   unit circle.  N is a vector of positive integers (including 0), and LQ||7>{eX  
    %   M is a vector with the same number of elements as N.  Each element `9acR>00$  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) !=6\70lJ  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, +Y>oNX1KN  
    %   and THETA is a vector of angles.  R and THETA must have the same ?5j~"  
    %   length.  The output Z is a matrix with one column for every (N,M) :_o^oi7G  
    %   pair, and one row for every (R,THETA) pair. 0*AXd=)"*  
    % | vxmgX)  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ]q&NO(:kbq  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Y6(= cm  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral A5sz[k  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ^szi[Cj  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized KD- -w(4  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. w`gT]Rn  
    % Bz>5OuOVS\  
    %   The Zernike functions are an orthogonal basis on the unit circle. dKa2_|k'  
    %   They are used in disciplines such as astronomy, optics, and *dsI>4%m  
    %   optometry to describe functions on a circular domain. ff00s+  
    % & +yo PF  
    %   The following table lists the first 15 Zernike functions. |ZOdfr4uW  
    % Au:R]7   
    %       n    m    Zernike function           Normalization ^S!;snhn  
    %       -------------------------------------------------- aF>&X-2  
    %       0    0    1                                 1 F#.ph?W  
    %       1    1    r * cos(theta)                    2 8uA!Vrp3  
    %       1   -1    r * sin(theta)                    2 T*'WS!z  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) g~76c.u-  
    %       2    0    (2*r^2 - 1)                    sqrt(3) z8xBq%97us  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ! w;/J^  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) r Cb#E}  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) A>_,tt  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) K'f2 S  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) YoWXHg!U  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Ns5P,[pBOZ  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Fe.90)  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) aDu[iaZ  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) "CZv5)  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) )g KC}_h=  
    %       -------------------------------------------------- 1pjx8*!B  
    % _z9~\N/@[  
    %   Example 1: S27s Rxfr  
    % ,RP9v*  
    %       % Display the Zernike function Z(n=5,m=1) :@-.whj  
    %       x = -1:0.01:1; kU.@HJ[@j  
    %       [X,Y] = meshgrid(x,x); .bj:tmz  
    %       [theta,r] = cart2pol(X,Y); &2I8!Ia  
    %       idx = r<=1; {uJ"%  
    %       z = nan(size(X)); Ty7)j]b"zl  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); l+X\>,  
    %       figure s^Xs*T@~h  
    %       pcolor(x,x,z), shading interp Z$zX%w  
    %       axis square, colorbar r`< x@,  
    %       title('Zernike function Z_5^1(r,\theta)') 0f_A"K  
    % xC}'"``s  
    %   Example 2: U} w@,6  
    % $9:  @M.  
    %       % Display the first 10 Zernike functions D|^N9lDaQ  
    %       x = -1:0.01:1; m;L 3c(r.  
    %       [X,Y] = meshgrid(x,x); n~tb z"&  
    %       [theta,r] = cart2pol(X,Y); ukRmjHbLf  
    %       idx = r<=1; t D4-Llj6  
    %       z = nan(size(X)); >Psq" Xj  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ($W%&(:/  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; [jrfh>v  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; MH0wpHz  
    %       y = zernfun(n,m,r(idx),theta(idx)); v5U'ky :  
    %       figure('Units','normalized') i'\-Y]?[  
    %       for k = 1:10 .tQ(q=#  
    %           z(idx) = y(:,k); S\!vDtD@  
    %           subplot(4,7,Nplot(k)) VN'\c3;  
    %           pcolor(x,x,z), shading interp KVUub'k  
    %           set(gca,'XTick',[],'YTick',[]) < RtyW  
    %           axis square YHMJ5IM@.  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) {7;QZk(  
    %       end MU\Pggs  
    % 1kR. .p<"  
    %   See also ZERNPOL, ZERNFUN2. AWssDbh/[  
    %s^1de  
    %   Paul Fricker 11/13/2006 ;zV<63tW  
    3i'01z  
    WWo"De@  
    % Check and prepare the inputs: B<n[yiJ}  
    % ----------------------------- 5(E&jKn&  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) L 4Z+8*  
        error('zernfun:NMvectors','N and M must be vectors.') (U_HX2f  
    end ]lqZ9rO  
    rS8\Vf]F  
    if length(n)~=length(m) 62y:i  
        error('zernfun:NMlength','N and M must be the same length.') jzBW'8  
    end xq=!1>  
    {<-wm-]mo  
    n = n(:); RDjw|V  
    m = m(:); Z:es7<#y  
    if any(mod(n-m,2)) }^j8<  
        error('zernfun:NMmultiplesof2', ... e4tC[6;  
              'All N and M must differ by multiples of 2 (including 0).') sLXM$SMBh  
    end zmL VFGnS  
    po,U e>n/  
    if any(m>n) \7pEn  
        error('zernfun:MlessthanN', ... `H$=hr  
              'Each M must be less than or equal to its corresponding N.') z%iPk'^  
    end rm$dv%q  
    aw~h03R_Z  
    if any( r>1 | r<0 ) ^S?f"''y3  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') x%HxM~&  
    end Gf:dN_e6.  
    5`gVziS!S  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ;[[6[i  
        error('zernfun:RTHvector','R and THETA must be vectors.') #g0N/  
    end 11kyrv  
    cMnN} '  
    r = r(:); C=v+e%)x@  
    theta = theta(:); "Z;({a$v  
    length_r = length(r); 5MKM;6cA&p  
    if length_r~=length(theta) 4;r,U{uR  
        error('zernfun:RTHlength', ... "@/pQoLy  
              'The number of R- and THETA-values must be equal.') =&qH%S6  
    end ~TeOl|!lE+  
    0a#v}w^ *  
    % Check normalization: (E&M[hH+  
    % -------------------- S]~5iO_bst  
    if nargin==5 && ischar(nflag) q9{)nU  
        isnorm = strcmpi(nflag,'norm'); /!A"[Tyt  
        if ~isnorm !.q 9:|oc  
            error('zernfun:normalization','Unrecognized normalization flag.') j(]O$""  
        end 4z26a  
    else ^cSfkBh  
        isnorm = false; &zJ*afi)  
    end IYXN}M.=  
    WBkx!{\z  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (Z[c7  
    % Compute the Zernike Polynomials u%E8&T8,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s/s&d pT*  
    -1d*zySL  
    % Determine the required powers of r: c00rq ~<K  
    % ----------------------------------- D %)L "5C  
    m_abs = abs(m); m)"(S  
    rpowers = []; B8n[ E  
    for j = 1:length(n) NH}o`x/  
        rpowers = [rpowers m_abs(j):2:n(j)]; \[.qN  
    end %"fO^KA.h]  
    rpowers = unique(rpowers); _KxR~k^  
    )oz2V9X{  
    % Pre-compute the values of r raised to the required powers, $Cfp1#  
    % and compile them in a matrix: Kg"eS`-  
    % ----------------------------- J'7;+.s(  
    if rpowers(1)==0 VP^Yf_  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); B@0#*I Rm  
        rpowern = cat(2,rpowern{:}); % XZ&(  
        rpowern = [ones(length_r,1) rpowern]; ztX$kX:_m  
    else |9IOZ>H9  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 92A9gY  
        rpowern = cat(2,rpowern{:}); knph549  
    end ~u2f`67{  
    alHA&YC{K  
    % Compute the values of the polynomials: -T{2R:\{  
    % -------------------------------------- j>:N0:  
    y = zeros(length_r,length(n)); 5;p|iT  
    for j = 1:length(n) |3!)  
        s = 0:(n(j)-m_abs(j))/2; Pmd[2/][  
        pows = n(j):-2:m_abs(j); j 4=iHnE;  
        for k = length(s):-1:1 Ddg!1SF  
            p = (1-2*mod(s(k),2))* ... Wkjp:`(-$r  
                       prod(2:(n(j)-s(k)))/              ... aGi`(|shW  
                       prod(2:s(k))/                     ...  JJ}DYv  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... H)gc"aRe;Y  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ZAN~TG<n  
            idx = (pows(k)==rpowers); %X %zK1  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Cb+$|Kg/"b  
        end NW`.7'aWT  
         2gZp O9  
        if isnorm QSa#}vCp*  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Rk#'^ }  
        end Y:,C_^$w;  
    end GWPBP-)0  
    % END: Compute the Zernike Polynomials c!7WRHJE_a  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1 Ga3[ g  
    }8aqSD<:  
    % Compute the Zernike functions: zb!1o0, J  
    % ------------------------------ _0'X!1"  
    idx_pos = m>0; un-%p#  
    idx_neg = m<0; uyB2   
    &,jUaC5I  
    z = y; 2z;3NUL$n  
    if any(idx_pos) 7]T(=gg /  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ux(~+<k  
    end M kJBKS  
    if any(idx_neg) =d^hiR!GN  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); GU2TQx{V  
    end tJ>>cFx  
    ,-E'059  
    % 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) Er j{_i?R?  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. qwj7CIc(  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated nf"#F@dk  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Wd)\r.pJ  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, a4~B  
    %   and THETA is a vector of angles.  R and THETA must have the same y _"V=:  
    %   length.  The output Z is a matrix with one column for every P-value, M NwY   
    %   and one row for every (R,THETA) pair. <%uEWb)  
    % JP6 Noia  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike wW\@^5  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 54>0Dv??H  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) } (-9d  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 9]IZ3 fQX  
    %   for all p. \l /}` w  
    % FauASu,A  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Fd<Ouyxqe  
    %   Zernike functions (order N<=7).  In some disciplines it is {X(nn.GpC  
    %   traditional to label the first 36 functions using a single mode i]k)wr(  
    %   number P instead of separate numbers for the order N and azimuthal LS<+V+o2%  
    %   frequency M. L k nK  
    % oydP}X  
    %   Example: ,>6a)2xh  
    % Evm3Sm!S  
    %       % Display the first 16 Zernike functions `IwZVz  
    %       x = -1:0.01:1; ]YhQQH1> ]  
    %       [X,Y] = meshgrid(x,x); EDgtn)1  
    %       [theta,r] = cart2pol(X,Y); Y"8@\73(R  
    %       idx = r<=1; 2ak]&ll+h  
    %       p = 0:15; }'x)e  
    %       z = nan(size(X)); $aJay]F  
    %       y = zernfun2(p,r(idx),theta(idx)); ff.k1%wr^  
    %       figure('Units','normalized') Q34u>VkdQI  
    %       for k = 1:length(p) !vu-`u~86  
    %           z(idx) = y(:,k); xk>cdgt  
    %           subplot(4,4,k) dyn)KDS  
    %           pcolor(x,x,z), shading interp h?n?3x!(  
    %           set(gca,'XTick',[],'YTick',[]) E<3xv;v8r  
    %           axis square |Vz)!M  
    %           title(['Z_{' num2str(p(k)) '}']) O[MFp  
    %       end }?mSMqnB  
    % 3<$Ek3X  
    %   See also ZERNPOL, ZERNFUN. z3S"1L7  
    t.;._'  
    %   Paul Fricker 11/13/2006 M]{~T7n-  
    8ly)G  
    06AgY0\  
    % Check and prepare the inputs: sd%)g<t  
    % ----------------------------- COHBju fmR  
    if min(size(p))~=1 A8mc+ Bf(  
        error('zernfun2:Pvector','Input P must be vector.') ]m 3cm  
    end de W1>yh^_  
    u,8)M' UU  
    if any(p)>35 ;AOLbmb)H4  
        error('zernfun2:P36', ... jnJ*e-AW  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... SceHdx(]  
               '(P = 0 to 35).']) y-.{){uaD  
    end (y!bvp[" m  
    s;oe Qa}TB  
    % Get the order and frequency corresonding to the function number: w"[T  
    % ---------------------------------------------------------------- Sq,>^|v4&e  
    p = p(:); s1cu5eCt  
    n = ceil((-3+sqrt(9+8*p))/2); t6+W  
    m = 2*p - n.*(n+2); xP_%d,  
    y'^U4# (  
    % Pass the inputs to the function ZERNFUN: rMIX{K)'f  
    % ---------------------------------------- l@GJcCufE  
    switch nargin W3UxFs]$  
        case 3 3)W_^6>bM  
            z = zernfun(n,m,r,theta); V^Z5i]zT  
        case 4 #OM'2@  
            z = zernfun(n,m,r,theta,nflag); Q+Q"JU  
        otherwise *\'t$se+  
            error('zernfun2:nargin','Incorrect number of inputs.') z~`X4Segw  
    end $6UU58>n  
    n^{h@u  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) IYq#|^)5+  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. VS ECD;u4c  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ,h1\PT9ULY  
    %   order N and frequency M, evaluated at R.  N is a vector of p({@t=L3g  
    %   positive integers (including 0), and M is a vector with the dO2?&f  
    %   same number of elements as N.  Each element k of M must be a cA 4?[F  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) r3' DXP  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is lbt8S.fx  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix dDl+  
    %   with one column for every (N,M) pair, and one row for every h9m|f|cH  
    %   element in R. ;0m J4G  
    % 9Cd/SlNV2  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- lq53 xT  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is `. JW_F)1  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to fgL"\d}  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 N}VoO0I  
    %   for all [n,m]. x*F- d2D  
    % /y{fDCC  
    %   The radial Zernike polynomials are the radial portion of the ~cp=B>*(  
    %   Zernike functions, which are an orthogonal basis on the unit ,8Q0AkG  
    %   circle.  The series representation of the radial Zernike B=]L%~xL$  
    %   polynomials is +pT;; 9  
    % JXkx!X_{  
    %          (n-m)/2 k]:`<`/I_  
    %            __ qh6b;ae\x  
    %    m      \       s                                          n-2s ku*k+4rz  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r [g@qZ5I.  
    %    n      s=0 Gev\bQa  
    % ~.^:?yCA  
    %   The following table shows the first 12 polynomials. 3O*iv{-&  
    % ZhCz]z~tj6  
    %       n    m    Zernike polynomial    Normalization mz1m^p)~{  
    %       --------------------------------------------- 'MYKAnZ-i  
    %       0    0    1                        sqrt(2) <Tgubv+J  
    %       1    1    r                           2 XN t` 4$L  
    %       2    0    2*r^2 - 1                sqrt(6) -eV*I >G  
    %       2    2    r^2                      sqrt(6) Ygg+=@].@  
    %       3    1    3*r^3 - 2*r              sqrt(8) (T2HUmkQ6  
    %       3    3    r^3                      sqrt(8) ) C~#W  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 9=iMP~?xF  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 7^rT-f07  
    %       4    4    r^4                      sqrt(10) & ;5f/  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Oz\J+  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Y'P^]Q=}_#  
    %       5    5    r^5                      sqrt(12) L=Aj+  
    %       --------------------------------------------- ] g9SUFM  
    % "&D0Sd@[?  
    %   Example: Gl{'a1  
    % YG*<jKcX  
    %       % Display three example Zernike radial polynomials n)a/pO_  
    %       r = 0:0.01:1; )ZLj2H<  
    %       n = [3 2 5]; GBg  
    %       m = [1 2 1]; a0JMLLa [I  
    %       z = zernpol(n,m,r); 34)l3UI~  
    %       figure #gWok'ZcR  
    %       plot(r,z) J:uFQWxZ   
    %       grid on <<qzZ+u  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') $Nvox<d0  
    % F3!6}u\F  
    %   See also ZERNFUN, ZERNFUN2. |]q{ qsy  
    [W[awGf  
    % A note on the algorithm. *dB3Gu{ +  
    % ------------------------ En-=z`j G  
    % The radial Zernike polynomials are computed using the series J Z@sk2  
    % representation shown in the Help section above. For many special `3[W~Cq  
    % functions, direct evaluation using the series representation can h-Ks:pcR  
    % produce poor numerical results (floating point errors), because ueW/i  
    % the summation often involves computing small differences between obbg# ,  
    % large successive terms in the series. (In such cases, the functions |iSwG=&  
    % are often evaluated using alternative methods such as recurrence :G9d,B7*  
    % relations: see the Legendre functions, for example). For the Zernike {Gfsiz6  
    % polynomials, however, this problem does not arise, because the L*Ffic  
    % polynomials are evaluated over the finite domain r = (0,1), and #+" D?  
    % because the coefficients for a given polynomial are generally all g] IPNW^n  
    % of similar magnitude. )knK'H(  
    % WQw11uMt@q  
    % ZERNPOL has been written using a vectorized implementation: multiple 0.!vp?  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] eUa:@cA  
    % values can be passed as inputs) for a vector of points R.  To achieve ~Odclrs  
    % this vectorization most efficiently, the algorithm in ZERNPOL hP[/xe  
    % involves pre-determining all the powers p of R that are required to ;gJAxVD<  
    % compute the outputs, and then compiling the {R^p} into a single C)qG<PW.!  
    % matrix.  This avoids any redundant computation of the R^p, and S9b=?? M)  
    % minimizes the sizes of certain intermediate variables. OHngpe4  
    % {KTZSs $n  
    %   Paul Fricker 11/13/2006 A;/,</  
    b4KNIP7E  
    J~ @W":v  
    % Check and prepare the inputs: {RsdI=%  
    % ----------------------------- 7S= ]@*  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Bz,Xg-k+  
        error('zernpol:NMvectors','N and M must be vectors.') ) cOBP}j+  
    end ,HE{&p2y  
    (i<\n`h1K  
    if length(n)~=length(m) tnb'\}Vn  
        error('zernpol:NMlength','N and M must be the same length.') :%fnJg(  
    end N-p||u  
    KxJDAP  
    n = n(:); 54]UfmT%I  
    m = m(:); _!vuDv%  
    length_n = length(n); "0>AefFd#  
    aJs! bx>K  
    if any(mod(n-m,2)) h^H)p`[Gme  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') '|%\QWuZ  
    end ,4,./wIq  
    L`w_Q2{sv  
    if any(m<0) l~1l~Gx_&n  
        error('zernpol:Mpositive','All M must be positive.') Fv^>^txh  
    end .q 4FGPWz  
    uXGAcUx(  
    if any(m>n) &xC5Mecb*  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') -ebyW#  
    end dZd]p8  
    k1D|Cpnp  
    if any( r>1 | r<0 ) `apCu  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 8X\":l:  
    end R C!~eJG!  
    \%W"KLP  
    if ~any(size(r)==1) _4lKd`  
        error('zernpol:Rvector','R must be a vector.') /dR:\ffz2  
    end m$'ZiS5  
    ``h* A  
    r = r(:); 2tp95E`(O  
    length_r = length(r); eN  TKX  
    >/-Bg:  
    if nargin==4 c5eimA%`  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 2) Q/cH\g  
        if ~isnorm M"OCwBT U  
            error('zernpol:normalization','Unrecognized normalization flag.') k#5Qwxu`  
        end nG| NRp  
    else Q,o"[ &Gp  
        isnorm = false; %$R]NL|  
    end p" Di;3!y!  
    SUoUXh^!w  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #.@D}7y5  
    % Compute the Zernike Polynomials Q"GZh.m  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [-=y*lx %g  
    u>03l(X6f  
    % Determine the required powers of r: [@$t35t~  
    % ----------------------------------- pc](  
    rpowers = []; s%l^zA(  
    for j = 1:length(n) A9y3B^\*  
        rpowers = [rpowers m(j):2:n(j)]; ~5~Cpu2v7  
    end Bh q]h  
    rpowers = unique(rpowers); ~2 J!I^ J  
    ? C6t Yd  
    % Pre-compute the values of r raised to the required powers, [jKhC<t}  
    % and compile them in a matrix: y>JSo9[@  
    % ----------------------------- 7Y1FFw |  
    if rpowers(1)==0 KA9v?_@{F  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); h}GzQry1  
        rpowern = cat(2,rpowern{:}); T5TA kEVl  
        rpowern = [ones(length_r,1) rpowern]; v==/tr)  
    else 2Ni {fC?  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); OGnuBK  
        rpowern = cat(2,rpowern{:}); GaOM|F'>  
    end rn-CQ2{?  
    r )f+j@KF  
    % Compute the values of the polynomials: f] kG%JEK  
    % -------------------------------------- {60U6n  
    z = zeros(length_r,length_n); f;a55%3c  
    for j = 1:length_n c"S{5xh0&  
        s = 0:(n(j)-m(j))/2; iq`caoi  
        pows = n(j):-2:m(j); ys} I~MK-  
        for k = length(s):-1:1 6tBe,'*  
            p = (1-2*mod(s(k),2))* ... N?mQ50o~C  
                       prod(2:(n(j)-s(k)))/          ... /G!M\teeF  
                       prod(2:s(k))/                 ... jtQ}  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ,\ zx4 *  
                       prod(2:((n(j)+m(j))/2-s(k))); 0-I L@Di`F  
            idx = (pows(k)==rpowers); I73=PfS:m  
            z(:,j) = z(:,j) + p*rpowern(:,idx); $36.*s m  
        end WKONK;U+7  
         :Mnl1;oh  
        if isnorm / #D R|  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); !TPKD  
        end [|APMMYK1  
    end 78t:ge eX  
    y3@5~4+  
    % 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)  [9G=x[  
    %<fs \J^k  
    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。 AG><5 }  
    U9jdb9 |  
    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)