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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 /[20e1 w!  
    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! DD@)z0W  
     
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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 :M22P`:  
    function z = zernfun(n,m,r,theta,nflag) fg9?3x Z  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. N+CXOI=6x  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N W NwJM  
    %   and angular frequency M, evaluated at positions (R,THETA) on the *'9)H 0  
    %   unit circle.  N is a vector of positive integers (including 0), and 2E`~ qn  
    %   M is a vector with the same number of elements as N.  Each element 2PVx++*]C  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) |'V DI]p&  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ]Q6+e(:~ZH  
    %   and THETA is a vector of angles.  R and THETA must have the same 3[0w+{ (Q  
    %   length.  The output Z is a matrix with one column for every (N,M) _ yfdj[Ot`  
    %   pair, and one row for every (R,THETA) pair. Aautih@LX  
    % zVM4BT(  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike "wA0 LH_  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), {8^Gs^c c  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral V19e>  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, EKZ$Q4YE  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized xn(+G$m  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. D9qX->p  
    % *0%4l_i  
    %   The Zernike functions are an orthogonal basis on the unit circle. p+, 1Fi  
    %   They are used in disciplines such as astronomy, optics, and IK*oFo{C=K  
    %   optometry to describe functions on a circular domain. :8p&#M  
    % /635B*g  
    %   The following table lists the first 15 Zernike functions. t *{,Gk  
    % 9o-!ecx}  
    %       n    m    Zernike function           Normalization )46 0 Ed  
    %       -------------------------------------------------- \\=.6cg<K  
    %       0    0    1                                 1 UdT&cG  
    %       1    1    r * cos(theta)                    2 5^)?mA  
    %       1   -1    r * sin(theta)                    2 [f<"p[  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ds- yif6   
    %       2    0    (2*r^2 - 1)                    sqrt(3) [NYj.#,oR  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ?VFM ]hO  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ?22d},.  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) f?,-j>[.=f  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) TE3*ktB{N  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) pG/ NuImA  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) '@'B>7C#  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) BjM+0[HC  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) :/+>e IE  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) }l~]b3@qu  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) as>:\hjP##  
    %       -------------------------------------------------- 82lr4  
    % 5^\m`gS  
    %   Example 1:  cp$.,V  
    % \CcmePTN#x  
    %       % Display the Zernike function Z(n=5,m=1) IuNkfBe4m  
    %       x = -1:0.01:1; @4;&hP2Z:  
    %       [X,Y] = meshgrid(x,x); +H7y/#e+3  
    %       [theta,r] = cart2pol(X,Y); E]NY (1  
    %       idx = r<=1; {5>3;.  
    %       z = nan(size(X)); d-~vR(tU  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); vCj4;P g  
    %       figure 7'Lp8  
    %       pcolor(x,x,z), shading interp l1&5uwuF  
    %       axis square, colorbar ~%`EeJwT  
    %       title('Zernike function Z_5^1(r,\theta)') d+tj%7  
    % V|TA:&:7  
    %   Example 2: 'f 3HKn<L  
    % djUihcqA`  
    %       % Display the first 10 Zernike functions GE@uO J6H  
    %       x = -1:0.01:1; ;TtaH  
    %       [X,Y] = meshgrid(x,x); 5? Wg%@  
    %       [theta,r] = cart2pol(X,Y); ] GNh)  
    %       idx = r<=1; J==}QEhQ{  
    %       z = nan(size(X)); ) ]73S@P(=  
    %       n = [0  1  1  2  2  2  3  3  3  3];  ozU2  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; T)8p:}P!  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; L/BHexOB  
    %       y = zernfun(n,m,r(idx),theta(idx)); yr5NRs  
    %       figure('Units','normalized') 6z Ay)~  
    %       for k = 1:10 QO2Ut!Y  
    %           z(idx) = y(:,k); T8U[xu.>  
    %           subplot(4,7,Nplot(k)) gV|Y54}T  
    %           pcolor(x,x,z), shading interp H<,bq*@  
    %           set(gca,'XTick',[],'YTick',[]) M+0x;53nz  
    %           axis square $.a|ae|K  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) >PIPp7C  
    %       end Xtkw Z3  
    % u#FXW_-TK  
    %   See also ZERNPOL, ZERNFUN2. &3I$8v|!?  
    ilv_D~|  
    %   Paul Fricker 11/13/2006 ;u,rtEMy;  
    I0iY+@^5  
    ,ijW(95{k  
    % Check and prepare the inputs: `y2ljIWJ  
    % ----------------------------- U+} y %3l  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) wlrIgn%  
        error('zernfun:NMvectors','N and M must be vectors.') RJx{eck%  
    end G,]z (%  
    Wab.|\c  
    if length(n)~=length(m) t@)my[!  
        error('zernfun:NMlength','N and M must be the same length.') .a,(pq Jg  
    end 9<l-NU9 _  
    4:U0f;Fs  
    n = n(:); @bT3'K-4  
    m = m(:);  i S  
    if any(mod(n-m,2)) j7}lF?cJ2  
        error('zernfun:NMmultiplesof2', ... q!&B6]  
              'All N and M must differ by multiples of 2 (including 0).') V9T 4 +  
    end 4 [1k\  
    gLD{1-v  
    if any(m>n) ^X &)'H  
        error('zernfun:MlessthanN', ... "y$ qrN-  
              'Each M must be less than or equal to its corresponding N.') MqdB\OW&  
    end xl8#=qmCD  
    J)*8|E9P  
    if any( r>1 | r<0 ) nW GR5*e:  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') @Dj:4  
    end ufP Cx|x~  
    @+&'%1  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) /PqUXF  
        error('zernfun:RTHvector','R and THETA must be vectors.') W`x)=y]Z  
    end uoCGSXsi  
    PBrnzkoY  
    r = r(:); OR;&TbWF(R  
    theta = theta(:); /UHp [yod  
    length_r = length(r); ;& ~929  
    if length_r~=length(theta) [D[D`gpjA  
        error('zernfun:RTHlength', ... t#5:\U5r.  
              'The number of R- and THETA-values must be equal.') Lm|al.Z  
    end 6vobta^w  
    sJ~P:g  
    % Check normalization: l3p3tT3+  
    % -------------------- 3gc"_C\$  
    if nargin==5 && ischar(nflag) D0ruTS  
        isnorm = strcmpi(nflag,'norm'); wAh#   
        if ~isnorm Q#pnj thM  
            error('zernfun:normalization','Unrecognized normalization flag.') <KLg0L<W  
        end a5?A!k\2  
    else C3}Aq8$6  
        isnorm = false; G9Qe121m  
    end lw[<STpD;  
    =dGKF`tR  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% j"hASBTgp  
    % Compute the Zernike Polynomials TwFb%YM  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% azX`oU,l  
    9p`r7:  
    % Determine the required powers of r: B< hEx@  
    % ----------------------------------- {|6z+vR  
    m_abs = abs(m); s.:r;%a  
    rpowers = []; Wc|z7P~',%  
    for j = 1:length(n) 5UO k)rOf  
        rpowers = [rpowers m_abs(j):2:n(j)]; |I^y0Q:K  
    end G),db%,X2  
    rpowers = unique(rpowers); B 8{ uR  
    dy:d=Z  
    % Pre-compute the values of r raised to the required powers, /{X_ .fv<v  
    % and compile them in a matrix: Ae49n4J  
    % ----------------------------- {/ &B!zvl  
    if rpowers(1)==0 :Jl Di>B  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); UX_I6_&  
        rpowern = cat(2,rpowern{:}); uyT/Xzo3  
        rpowern = [ones(length_r,1) rpowern]; GN%(9N'W  
    else >^3zU   
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); FH*RU1Z  
        rpowern = cat(2,rpowern{:}); }bMWTT  
    end Mr* |9h  
    .pvxh|V  
    % Compute the values of the polynomials: uV~e|X "9s  
    % -------------------------------------- uTGcQs}  
    y = zeros(length_r,length(n)); H/J<Pd$p  
    for j = 1:length(n) K@r*;T  
        s = 0:(n(j)-m_abs(j))/2; Y6ben7j%-  
        pows = n(j):-2:m_abs(j); *#2Rvt*Ox  
        for k = length(s):-1:1 @^? XaU  
            p = (1-2*mod(s(k),2))* ... <AUWby,"  
                       prod(2:(n(j)-s(k)))/              ... ``9 GY  
                       prod(2:s(k))/                     ... gX,9Gh  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... >IY,be6>P  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Y=Hz;Ni  
            idx = (pows(k)==rpowers); 8i: [:Z  
            y(:,j) = y(:,j) + p*rpowern(:,idx); @!\K>G >9[  
        end z+3 9ee  
         r7I B{}>-  
        if isnorm %-j&e44  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); wPnybb{  
        end {oWsh)[x2  
    end "^%Z'ou  
    % END: Compute the Zernike Polynomials ]US[5)EL-  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1V%'.l9  
    \L[i9m|e  
    % Compute the Zernike functions: H06Bj(Y!  
    % ------------------------------ CLN+I'uX0  
    idx_pos = m>0; Nn#u%xvJt  
    idx_neg = m<0; 6vp0*ww  
    NHiq^ojk  
    z = y; =Od>;|]m  
    if any(idx_pos) Dg2uE8k  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); FC}oL"kk  
    end iV hJH4  
    if any(idx_neg) h^M^7S  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 7& 6Y  
    end HXks_ix )  
    ]}2Ztr)zZ  
    % 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) _;e\:7<m  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. "=|t~`  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated +?d}7zh  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive o&-L0]i|  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, dZ2`{@AYY  
    %   and THETA is a vector of angles.  R and THETA must have the same G6O/(8  
    %   length.  The output Z is a matrix with one column for every P-value, \G;CQV#{9  
    %   and one row for every (R,THETA) pair. [Ox(.  
    % % vS8?nG  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike \&q=@rJp(z  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) O&$0&dhc  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ?R6`qe_F  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 D1X{:#|  
    %   for all p. SS8ocGX  
    % 9]$`)wZ  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 v>-Y uS  
    %   Zernike functions (order N<=7).  In some disciplines it is p&3> `C  
    %   traditional to label the first 36 functions using a single mode 6Rz[?-mkLO  
    %   number P instead of separate numbers for the order N and azimuthal r nBOj#N  
    %   frequency M. R&So4},B  
    % DO^y;y>  
    %   Example: aRwnRii  
    % Ew4 g'A:H  
    %       % Display the first 16 Zernike functions C\Ayv)S #2  
    %       x = -1:0.01:1; Hj~O49%j&  
    %       [X,Y] = meshgrid(x,x); Lq0 4T0  
    %       [theta,r] = cart2pol(X,Y); Q}P-$X+/ n  
    %       idx = r<=1; /V^sJ($V$~  
    %       p = 0:15; e@jfIF0=}  
    %       z = nan(size(X)); JR1 *|u  
    %       y = zernfun2(p,r(idx),theta(idx)); nem@sB;v#  
    %       figure('Units','normalized') r_2b tpL^  
    %       for k = 1:length(p) !_^g8^>2(  
    %           z(idx) = y(:,k); xo~g78jm7,  
    %           subplot(4,4,k) u!1/B4!'O  
    %           pcolor(x,x,z), shading interp /`+7_=-  
    %           set(gca,'XTick',[],'YTick',[]) pFIecca w  
    %           axis square M#M?1(O/NE  
    %           title(['Z_{' num2str(p(k)) '}']) tWk{1IL  
    %       end ! F7:i  
    % `K?1L{p'4  
    %   See also ZERNPOL, ZERNFUN. 9X]f[^  
    V/bH^@,sA  
    %   Paul Fricker 11/13/2006 LK+felL  
    detLjlE  
    4<}A]BQVkJ  
    % Check and prepare the inputs: &jm[4'$ *z  
    % ----------------------------- =Ahw%`/&}]  
    if min(size(p))~=1 IVteF*8hU  
        error('zernfun2:Pvector','Input P must be vector.') iz`jDa Q|1  
    end p>p'.#M  
    -,GEv%6c  
    if any(p)>35 E5{n?e  
        error('zernfun2:P36', ... SDc" 4g`  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 3*WS"bt  
               '(P = 0 to 35).'])  :]c=pH  
    end x/I;nM Y  
    Yu3_=: <C  
    % Get the order and frequency corresonding to the function number: `d*b]2  
    % ---------------------------------------------------------------- e2Jp'93o'  
    p = p(:); btQet.  
    n = ceil((-3+sqrt(9+8*p))/2); j9xXKa5  
    m = 2*p - n.*(n+2); fn1pa@P  
    :[?!\m%0  
    % Pass the inputs to the function ZERNFUN: E@pFTvo  
    % ---------------------------------------- FpzP #;  
    switch nargin 3!Bj{;A  
        case 3 DHzkRCM  
            z = zernfun(n,m,r,theta); Wk[)+\WQ?  
        case 4 _,Q[2gQ5N  
            z = zernfun(n,m,r,theta,nflag); xG%*PNM0q  
        otherwise e?<D F.Md+  
            error('zernfun2:nargin','Incorrect number of inputs.') }17bV, t  
    end h5!d  
    -+P7:4/  
    % 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) A^g>fv  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. x@Vt[}e  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ~04[KG  
    %   order N and frequency M, evaluated at R.  N is a vector of `[@VxGy_  
    %   positive integers (including 0), and M is a vector with the 4NUN Ov`[{  
    %   same number of elements as N.  Each element k of M must be a d h?dO`  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) w, 7Cr  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is ue9h   
    %   a vector of numbers between 0 and 1.  The output Z is a matrix yoW> BX  
    %   with one column for every (N,M) pair, and one row for every @+t (xCv  
    %   element in R. 6ZEdihBei  
    % "Q?_ EEn  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 1p=&WM  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is I-{^[pp  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ! tr9(d  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 -t>Z 9  
    %   for all [n,m]. b[0S=e G  
    % yM|g|;U  
    %   The radial Zernike polynomials are the radial portion of the 9A<0zt  
    %   Zernike functions, which are an orthogonal basis on the unit C9pnU,[  
    %   circle.  The series representation of the radial Zernike - 3]|[  
    %   polynomials is ,-:a?#f>  
    % to51hjV  
    %          (n-m)/2 [DhEh@  
    %            __ `:gYXeR  
    %    m      \       s                                          n-2s OA5f}+  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r U1kh-8  :  
    %    n      s=0 yDuq6`R*  
    % 7@@<5&mN  
    %   The following table shows the first 12 polynomials. Z+,CL/  
    %  < GU  
    %       n    m    Zernike polynomial    Normalization E@mkm  
    %       --------------------------------------------- %HVD^. V  
    %       0    0    1                        sqrt(2) sL8>GtVo  
    %       1    1    r                           2 2_.CX(kI  
    %       2    0    2*r^2 - 1                sqrt(6) h[,XemwX  
    %       2    2    r^2                      sqrt(6) #@q1Ko!NZ  
    %       3    1    3*r^3 - 2*r              sqrt(8) <K,[sy&Qy  
    %       3    3    r^3                      sqrt(8) S2bexbp0o  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) w_!%'9m>  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) Z:TFOnJ  
    %       4    4    r^4                      sqrt(10) ,0,Oe=d  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) <dS5|||  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) v!KJ|c@m  
    %       5    5    r^5                      sqrt(12) GqMB^Ad  
    %       --------------------------------------------- 18rp; l{  
    % yH+c#w  
    %   Example: w O89&XZ<  
    % %2,/jhHL  
    %       % Display three example Zernike radial polynomials P]- #wz=S  
    %       r = 0:0.01:1; w#rVSSXQ3  
    %       n = [3 2 5]; Yq{jEatY{/  
    %       m = [1 2 1]; >-eS&rma  
    %       z = zernpol(n,m,r); IOS^|2:,  
    %       figure ;8uHRcdQ  
    %       plot(r,z) xjE7DCmA  
    %       grid on K,]woNxaw  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') <oQ6ZX  
    % +2El  
    %   See also ZERNFUN, ZERNFUN2. sX Z4U0 #  
    Gg}t-_M  
    % A note on the algorithm. 0a@c/ XGBp  
    % ------------------------ ,, 7.=#  
    % The radial Zernike polynomials are computed using the series ?o8a_9+  
    % representation shown in the Help section above. For many special shD+eHo$  
    % functions, direct evaluation using the series representation can yj'Cy8  
    % produce poor numerical results (floating point errors), because kM,@[V  
    % the summation often involves computing small differences between fmBkB8  
    % large successive terms in the series. (In such cases, the functions =8@RKG`>;  
    % are often evaluated using alternative methods such as recurrence -&$%|cyThQ  
    % relations: see the Legendre functions, for example). For the Zernike $.;iu2iyo  
    % polynomials, however, this problem does not arise, because the *0lt$F$~b  
    % polynomials are evaluated over the finite domain r = (0,1), and ig+k[`W  
    % because the coefficients for a given polynomial are generally all ~RAzFLt6x  
    % of similar magnitude. b13nE .  
    % !#C)99L"F  
    % ZERNPOL has been written using a vectorized implementation: multiple *XHj)DC;  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 4I z.fAw  
    % values can be passed as inputs) for a vector of points R.  To achieve y>4p~  
    % this vectorization most efficiently, the algorithm in ZERNPOL raSF3b/0  
    % involves pre-determining all the powers p of R that are required to W31LNysH!;  
    % compute the outputs, and then compiling the {R^p} into a single ^kc>m$HY  
    % matrix.  This avoids any redundant computation of the R^p, and uQO(?nCi  
    % minimizes the sizes of certain intermediate variables. .V7Y2!4TE  
    %  y/z9Ce*>  
    %   Paul Fricker 11/13/2006 1<;\6sg  
    j )<;g(  
    QziN]  
    % Check and prepare the inputs: jQO* oq}  
    % ----------------------------- K3j_C` Se  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) C3]\$  
        error('zernpol:NMvectors','N and M must be vectors.') E*Pz <  
    end tX+0 GLz  
    Q S5dP  
    if length(n)~=length(m) &t[z  
        error('zernpol:NMlength','N and M must be the same length.') ,G/\@x%  
    end pM1=U F  
    %g!yccD9  
    n = n(:); |~7+/VvI+  
    m = m(:); ?T tQZ  
    length_n = length(n); 3| GNi~  
    #8P#^v]H  
    if any(mod(n-m,2)) <)r,CiS  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') K(%dcUGDK>  
    end ^vYVl{$bT  
    EI[e+@J  
    if any(m<0) :(5]Z^  
        error('zernpol:Mpositive','All M must be positive.') d;;>4}XJ]  
    end b #o}=m  
    AGw1Pl8]K  
    if any(m>n) 1EKcD^U,  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') >1#DPU(g  
    end p ~,a=  
    dt`9RB$  
    if any( r>1 | r<0 ) )->-~E}p9  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') SS l8  
    end 23(B43zy  
    i{Y=!r5r  
    if ~any(size(r)==1) :DS2zA  
        error('zernpol:Rvector','R must be a vector.') [Q2S3szbt6  
    end @2x0V]AI  
    s!8J.hD'I  
    r = r(:); T4%i`<i  
    length_r = length(r);  }qgqb  
    z&>9 s)^-  
    if nargin==4 S!`4Bl  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); eXskwV+7  
        if ~isnorm \'\N"g`Fr  
            error('zernpol:normalization','Unrecognized normalization flag.') YyQf  
        end / K2.V@T  
    else | TQedC  
        isnorm = false; P#vv+]/  
    end @p9e:[  
    Zztt)/6*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ECmHy@(  
    % Compute the Zernike Polynomials a}[=_vb}K  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /-G qG)PX  
    DK#65H'  
    % Determine the required powers of r: ZNL;8sI?>  
    % ----------------------------------- 0-;DN:>  
    rpowers = []; mVc'%cPaw  
    for j = 1:length(n) zm;*:]S  
        rpowers = [rpowers m(j):2:n(j)]; !Vp,YN+yN  
    end Egjk^:@  
    rpowers = unique(rpowers); 7gZVg@   
    _D7HQ  
    % Pre-compute the values of r raised to the required powers, SoXX}<~E4  
    % and compile them in a matrix: T@d_ t  
    % ----------------------------- Mc#O+'](f  
    if rpowers(1)==0 = C$ @DNEc  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 5'{qEZs^QU  
        rpowern = cat(2,rpowern{:}); 1?e>x91  
        rpowern = [ones(length_r,1) rpowern]; c'TiWZP~  
    else `a/PIc"  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);  "df13U"  
        rpowern = cat(2,rpowern{:}); }Uqa8&  
    end MQbNWUi  
    }v'PY/d.  
    % Compute the values of the polynomials: Eezlx9b  
    % -------------------------------------- }LHT#{+ x  
    z = zeros(length_r,length_n); C>k;MvqO  
    for j = 1:length_n <x>k3bD  
        s = 0:(n(j)-m(j))/2; N18diP[C  
        pows = n(j):-2:m(j); .JD4gF2N  
        for k = length(s):-1:1 3-_U-:2"  
            p = (1-2*mod(s(k),2))* ... `DWi4y7  
                       prod(2:(n(j)-s(k)))/          ... O0=,&=i  
                       prod(2:s(k))/                 ... >2/wzsW  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... Nak'g/uP>  
                       prod(2:((n(j)+m(j))/2-s(k))); Q'Q72Fg  
            idx = (pows(k)==rpowers); Ls$g-k%c@Q  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ]\os`At  
        end vhE}{ED  
         HhY2`P8  
        if isnorm Hq"<vp  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); uz#eO|z@o  
        end kj<D4)  
    end tsSS31cv  
    1 ">d|oC  
    % 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)  ?_j6})2zY  
    59Q 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。 "XsY~  
    t\bxd`,  
    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)