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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 &M|rRd~*  
    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! r_+Vb*|Y  
     
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    离线phility
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    只看该作者 1楼 发表于: 2011-03-12
    可以用matlab编程,用zernike多项式进行波面拟合,求出zernike多项式的系数,拟合的算法有很多种,最简单的是最小二乘法,你可以查下相关资料,挺简单的
    离线phility
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    只看该作者 2楼 发表于: 2011-03-12
    泽尼克多项式的前9项对应象差的
    离线niuhelen
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    只看该作者 3楼 发表于: 2011-03-12
    回 2楼(phility) 的帖子
    非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 ^p(aZj3k  
    function z = zernfun(n,m,r,theta,nflag) Rxdj}xy  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. FWu:5fBZY  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ;?u cC@  
    %   and angular frequency M, evaluated at positions (R,THETA) on the y],op G6  
    %   unit circle.  N is a vector of positive integers (including 0), and |mMsU,*gB  
    %   M is a vector with the same number of elements as N.  Each element =mLp g4  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) &en2t=a  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, }"+"nf5h  
    %   and THETA is a vector of angles.  R and THETA must have the same xY?p(>(  
    %   length.  The output Z is a matrix with one column for every (N,M) g7323m1=  
    %   pair, and one row for every (R,THETA) pair. (A=PDjP!  
    % _1)n_P4  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike "]jN'N(.  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 7=G6ao7  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral & &CrF~  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, u.q3~~[=  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ="]lN  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. A< .5=E,/  
    % 2<*"@Vj  
    %   The Zernike functions are an orthogonal basis on the unit circle. TeuZVy8a  
    %   They are used in disciplines such as astronomy, optics, and t,LK92?  
    %   optometry to describe functions on a circular domain. qJF'KHyU{l  
    % R:n|1]*f3X  
    %   The following table lists the first 15 Zernike functions. yW?-Z[  
    % 4U\>TFO  
    %       n    m    Zernike function           Normalization %UdE2D'bC  
    %       -------------------------------------------------- Mx w-f4j  
    %       0    0    1                                 1 +6>2= ,?Z  
    %       1    1    r * cos(theta)                    2 'bRf>=  
    %       1   -1    r * sin(theta)                    2 $m ;p@#n  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) AAfhh5i  
    %       2    0    (2*r^2 - 1)                    sqrt(3) kKRu]0J~[  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) '{0O!y[H6  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Pg.JI:>2Ku  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) @|;[ ;:h@  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) M#Z^8(  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) j)G%I y[`  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) G[e,7jev  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) pS-o*!\C.  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) w}6~t\9D  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) o~Hq&C"^}  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) d_(;sW"I  
    %       -------------------------------------------------- K?M~x&Q  
    % XR8`,qH>  
    %   Example 1: = mQY%l  
    % o[wiQ9Tl  
    %       % Display the Zernike function Z(n=5,m=1) Q`K^>L1  
    %       x = -1:0.01:1; fFVQu\  
    %       [X,Y] = meshgrid(x,x); 7h(  
    %       [theta,r] = cart2pol(X,Y); cq]0|\Vz  
    %       idx = r<=1; E9k%:&]vd  
    %       z = nan(size(X)); [Cd#<Te3  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); dH0>lV  
    %       figure  e+#Oj  
    %       pcolor(x,x,z), shading interp &:C[ nq  
    %       axis square, colorbar b i-Am/9  
    %       title('Zernike function Z_5^1(r,\theta)') ^xk4HF   
    % A##Q>|>)  
    %   Example 2: pqvl,G5  
    % sAO/yG  
    %       % Display the first 10 Zernike functions U(+QrC:  
    %       x = -1:0.01:1; M`#g>~bI#R  
    %       [X,Y] = meshgrid(x,x); zxs)o}8icO  
    %       [theta,r] = cart2pol(X,Y); 9*JxP%8T~X  
    %       idx = r<=1; 6(\-aH'Ol  
    %       z = nan(size(X)); xP9R d/xa|  
    %       n = [0  1  1  2  2  2  3  3  3  3]; wmK;0 )|H  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; zZ9Ei-Q  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; dC4`xUv  
    %       y = zernfun(n,m,r(idx),theta(idx)); I|bX;l  
    %       figure('Units','normalized') r#j3O}(n  
    %       for k = 1:10 )y!gApNs"  
    %           z(idx) = y(:,k); ?l[#d7IB  
    %           subplot(4,7,Nplot(k)) 1IgTJ" \  
    %           pcolor(x,x,z), shading interp b+RU <qR  
    %           set(gca,'XTick',[],'YTick',[]) ]ml'd  
    %           axis square /QlzWson  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ?3LV$S)U  
    %       end  j AoI`J  
    % y]i} j,e0L  
    %   See also ZERNPOL, ZERNFUN2. %26HB w=JF  
    [vBP,_Tjx  
    %   Paul Fricker 11/13/2006 V/\`:  
    -hF!_);{  
    @G=:@;  
    % Check and prepare the inputs: zb~;<:<  
    % ----------------------------- CyVi{"aF3  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) @VND}{j  
        error('zernfun:NMvectors','N and M must be vectors.') bQ?Vh@j(M  
    end d]_].D$  
    w4^ $@GtN  
    if length(n)~=length(m) yWN'va1+$  
        error('zernfun:NMlength','N and M must be the same length.') ~s?y[yy6i  
    end L`:V]p  
    /a$Zzs&xs  
    n = n(:); &RS)U72  
    m = m(:); <|3F('Q"  
    if any(mod(n-m,2)) 0|hOoO]?q&  
        error('zernfun:NMmultiplesof2', ... $Zi {1w  
              'All N and M must differ by multiples of 2 (including 0).') F_}y[Yn^  
    end IAmMO[9H  
    e'v_eD T^  
    if any(m>n) !t)uRJ   
        error('zernfun:MlessthanN', ... X)TZ  S  
              'Each M must be less than or equal to its corresponding N.') I#F, Mb>:  
    end oY\;KPz  
    :E|+[}|  
    if any( r>1 | r<0 ) *|+$7j  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') a[=B?Bd  
    end Vn^8nS  
    0!c/4^  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) VzM (u _)  
        error('zernfun:RTHvector','R and THETA must be vectors.') ~2NT Xp  
    end 'o8,XBv-  
     pXNH  
    r = r(:); N>j*{]OY+{  
    theta = theta(:); MqWM!v-M  
    length_r = length(r); : T4ap_Ycq  
    if length_r~=length(theta) FGo)] U  
        error('zernfun:RTHlength', ... grd fR`3  
              'The number of R- and THETA-values must be equal.') nwDW<J{f|U  
    end Ko0T[TNkh  
    e 7Sg-NWV  
    % Check normalization: ~a>3,v -  
    % -------------------- fhHTp_u)2  
    if nargin==5 && ischar(nflag) mL@7,GD  
        isnorm = strcmpi(nflag,'norm'); *:chN' <  
        if ~isnorm PB:r+[91  
            error('zernfun:normalization','Unrecognized normalization flag.') r_V^sX  
        end {X\FS   
    else V2 }.X+u&<  
        isnorm = false; TU2MG VYy  
    end 57N<OQWf  
    1(VskFtZF  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B an" H~  
    % Compute the Zernike Polynomials 8?o{{ay  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% lb)i0`AN+  
    !6+V  
    % Determine the required powers of r: UXS+GAWU  
    % ----------------------------------- i?F~]8  
    m_abs = abs(m); m`,h nDp  
    rpowers = []; xAf?E%_pi  
    for j = 1:length(n) B/EGaYH  
        rpowers = [rpowers m_abs(j):2:n(j)]; %C >Win)g  
    end yA<\?Ps  
    rpowers = unique(rpowers); T,4REbm^  
    ]"vpCL  
    % Pre-compute the values of r raised to the required powers, 1i|5ii*vc  
    % and compile them in a matrix: VBu6,6  
    % ----------------------------- [4}U*\/>C  
    if rpowers(1)==0 L<N=,~  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); XCO;t_%  
        rpowern = cat(2,rpowern{:}); gn${@y?  
        rpowern = [ones(length_r,1) rpowern]; 74~ %4  
    else ,Ct1)%   
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); wIQt f|ZI>  
        rpowern = cat(2,rpowern{:}); .ffb*gZ4  
    end PkdL] !:  
    eOd'i{f@F  
    % Compute the values of the polynomials: Ar$ Am  
    % -------------------------------------- u,Cf4H*xS  
    y = zeros(length_r,length(n)); OmECvL'Z  
    for j = 1:length(n) l9$"zEC  
        s = 0:(n(j)-m_abs(j))/2; L q;=UE  
        pows = n(j):-2:m_abs(j); iC<qWq|S_m  
        for k = length(s):-1:1 ~w$ ^`e!]  
            p = (1-2*mod(s(k),2))* ... gs=(h*  
                       prod(2:(n(j)-s(k)))/              ... 2o`L^^  
                       prod(2:s(k))/                     ... AhSN'gWpbF  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 4"%LgV`  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); $C&E3 'O  
            idx = (pows(k)==rpowers); hQbz}x  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ?xCWg.#l4V  
        end <a%RKjQvT  
         O>2i)M-h9x  
        if isnorm ,y*|f0&"~  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Ne2eBmY}(  
        end -xU4s  
    end B P0*`TY  
    % END: Compute the Zernike Polynomials ~fF;GtP  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 69{q*qCW  
    HY7#z2L  
    % Compute the Zernike functions: ^/$bd4,z  
    % ------------------------------ |`ZW(} ~  
    idx_pos = m>0; XXPpj< c  
    idx_neg = m<0; S8)6@ECC  
    zM|Y X<  
    z = y; ,9~2#[|lq  
    if any(idx_pos) +T]D\];D  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Vqxxm&^P  
    end ~myY-nEY  
    if any(idx_neg) 5'[b:YC  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); p6W|4_a?  
    end XlU`jv+  
    45tQ$jr`1  
    % 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) oD&axNk  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. RD0=\!w*5  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated <K zEn+  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive |'N)HH>;  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, q jmlwVw  
    %   and THETA is a vector of angles.  R and THETA must have the same ,\=,,1_  
    %   length.  The output Z is a matrix with one column for every P-value, MI\35~JAN  
    %   and one row for every (R,THETA) pair. QNm8`1  
    % R*r;`x  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike BXB ZX@jVk  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) fu $<*Sa2  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) .FpeVjR''  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 8a3h)R  
    %   for all p. EK$3T5e  
    % /V$U%0  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 @PAT|6  
    %   Zernike functions (order N<=7).  In some disciplines it is ;30SnR/  
    %   traditional to label the first 36 functions using a single mode 'nSo0cyQ  
    %   number P instead of separate numbers for the order N and azimuthal h0)Dj( C  
    %   frequency M. i-gN< 8\v  
    % V3<H8pL  
    %   Example: JBt2R=  
    % ~Y/o9x0  
    %       % Display the first 16 Zernike functions g}n-H4LI  
    %       x = -1:0.01:1; T?HW=v_a  
    %       [X,Y] = meshgrid(x,x); xSy`VuSl  
    %       [theta,r] = cart2pol(X,Y); :B  9>  
    %       idx = r<=1; 6,a:s:$>}R  
    %       p = 0:15; +Fa!<txn  
    %       z = nan(size(X)); (]N- HN]v  
    %       y = zernfun2(p,r(idx),theta(idx)); M2lvD&  
    %       figure('Units','normalized') jiqE^j3;  
    %       for k = 1:length(p) ZGj ^,?a  
    %           z(idx) = y(:,k); d=d*:<Zx  
    %           subplot(4,4,k) h1 pEC  
    %           pcolor(x,x,z), shading interp _kXq0~  
    %           set(gca,'XTick',[],'YTick',[]) '|^x[8^  
    %           axis square k{ ~0BK  
    %           title(['Z_{' num2str(p(k)) '}']) ]I'dnd3e  
    %       end Cd2A&RB  
    % +o-jMvK9  
    %   See also ZERNPOL, ZERNFUN. i8->3uB  
    'M!M$<j  
    %   Paul Fricker 11/13/2006 T7~H|%  
    <NXJ&xs-+  
    dJ>~  
    % Check and prepare the inputs: )9i$ 1"a(  
    % ----------------------------- <\EJ:  
    if min(size(p))~=1 1V.oR`&2E  
        error('zernfun2:Pvector','Input P must be vector.') @8*lqV2  
    end <NWq0 3:&  
    f9D01R fo  
    if any(p)>35 c*.-mS~Z`  
        error('zernfun2:P36', ... LS]0p#  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... sm"s2Ci=}  
               '(P = 0 to 35).']) je8 5G`{DC  
    end L Iz<fB  
    |p.|zH  
    % Get the order and frequency corresonding to the function number: &&g02>gE  
    % ---------------------------------------------------------------- hjD%=Ri0Z  
    p = p(:); uH]oHh!}j  
    n = ceil((-3+sqrt(9+8*p))/2); +}R#mco5K  
    m = 2*p - n.*(n+2); KX J7\}  
    Xz`0nU  
    % Pass the inputs to the function ZERNFUN: \{ve6`7Rn  
    % ---------------------------------------- lHAWZyO  
    switch nargin LhRd0  
        case 3 m~0Kos%^*b  
            z = zernfun(n,m,r,theta); 1xI  
        case 4 R*/s#*gmL  
            z = zernfun(n,m,r,theta,nflag); Z@(KZ|  
        otherwise EpH_v`  
            error('zernfun2:nargin','Incorrect number of inputs.') 3$wK*xK  
    end RdBIbm  
    J6Vx7  
    % 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) Wy%FF\D.Y  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. P ^ 4 @  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Z?xRSi2~7  
    %   order N and frequency M, evaluated at R.  N is a vector of 0` .5gxm  
    %   positive integers (including 0), and M is a vector with the $,yAOaa  
    %   same number of elements as N.  Each element k of M must be a u"4 B5D  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 5vg="@O K  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is /'VbV8%  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 9:P]{}  
    %   with one column for every (N,M) pair, and one row for every |c^?tR<  
    %   element in R. AJm$(3?/D  
    % [dAQrou6P  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- sM+~x<}0  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is '%82pZ,?  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to a>x6n3{  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 s&Y"a,|Z  
    %   for all [n,m]. n}I?.r@e  
    % _OC@J*4.  
    %   The radial Zernike polynomials are the radial portion of the ?xrOhA9  
    %   Zernike functions, which are an orthogonal basis on the unit SnR2o3r-Of  
    %   circle.  The series representation of the radial Zernike 4Y$\QZO  
    %   polynomials is a ydNSgu  
    % G:p85k `  
    %          (n-m)/2 &7oL2 Wf  
    %            __ +FAj30  
    %    m      \       s                                          n-2s lPI~5N8  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r vENf3;o0  
    %    n      s=0 r0 )ne|&Hp  
    % X\/M(byn  
    %   The following table shows the first 12 polynomials. r *]pL<  
    % 9 @xl{S-  
    %       n    m    Zernike polynomial    Normalization !nCq8~#  
    %       --------------------------------------------- N@L{9ak1  
    %       0    0    1                        sqrt(2) |/vJ+aKq  
    %       1    1    r                           2 E^zfI9R  
    %       2    0    2*r^2 - 1                sqrt(6) naW!b&:  
    %       2    2    r^2                      sqrt(6) y?3.W  
    %       3    1    3*r^3 - 2*r              sqrt(8) //_H _ue$  
    %       3    3    r^3                      sqrt(8) " X8jpg  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) {eV8h}KIl  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) KVJ, a  
    %       4    4    r^4                      sqrt(10) q+y\pdhdO  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 9&5<ZC-D  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) f+Sb> $  
    %       5    5    r^5                      sqrt(12) }&t>j[  
    %       --------------------------------------------- UhpJGO  
    % ?UZt30|1  
    %   Example: \1Xk[%  
    % !~Uj 'w  
    %       % Display three example Zernike radial polynomials BUJ\[/  
    %       r = 0:0.01:1; 8v4 o+w P  
    %       n = [3 2 5]; yB2h/~+  
    %       m = [1 2 1]; acR|X@ \3  
    %       z = zernpol(n,m,r); b1Kt SRLV  
    %       figure CMaph  
    %       plot(r,z) {PcJuRTHB  
    %       grid on {^ b2nOMv  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') Ch_rV+  
    % `(sb  
    %   See also ZERNFUN, ZERNFUN2. [/UchU]DT  
    jb -kg</A  
    % A note on the algorithm. WXl+w7jr  
    % ------------------------ w8UuwFG?<  
    % The radial Zernike polynomials are computed using the series M .b8 -`V  
    % representation shown in the Help section above. For many special `x#Ud)g  
    % functions, direct evaluation using the series representation can A913*O: \  
    % produce poor numerical results (floating point errors), because n%s%i-[5B  
    % the summation often involves computing small differences between !qj[$x-ns  
    % large successive terms in the series. (In such cases, the functions >s f g`4  
    % are often evaluated using alternative methods such as recurrence {P]C>  
    % relations: see the Legendre functions, for example). For the Zernike 6 :] N%  
    % polynomials, however, this problem does not arise, because the X,7y|tb  
    % polynomials are evaluated over the finite domain r = (0,1), and &)%+DUV|  
    % because the coefficients for a given polynomial are generally all S{rltT-  
    % of similar magnitude. o@Oz a  
    % DPTk5o[  
    % ZERNPOL has been written using a vectorized implementation: multiple _`|1B$@x  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] K>@yk9)vi  
    % values can be passed as inputs) for a vector of points R.  To achieve \ ;npdFy  
    % this vectorization most efficiently, the algorithm in ZERNPOL xzm]v9k&  
    % involves pre-determining all the powers p of R that are required to Nr4:Gih  
    % compute the outputs, and then compiling the {R^p} into a single "6$V1B0KW  
    % matrix.  This avoids any redundant computation of the R^p, and bf"'xn9  
    % minimizes the sizes of certain intermediate variables. `}}:9d  
    % <U@N ^#  
    %   Paul Fricker 11/13/2006 CWn\K R  
    N Qk aW)  
    N1c=cZDV  
    % Check and prepare the inputs: B7C3r9wj  
    % ----------------------------- qd*}d)!  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) -n|bi cP  
        error('zernpol:NMvectors','N and M must be vectors.') &9p!J(C  
    end /o9T [ ^\  
    3@> F-N  
    if length(n)~=length(m) ,#A(I#wL~  
        error('zernpol:NMlength','N and M must be the same length.') W)In.?>]W  
    end /\I6j;$z  
    ?Z7`TnG$uf  
    n = n(:); fu/8r%:h  
    m = m(:); /kW Z 8Z  
    length_n = length(n); r ]DiB:.  
    Gk,Bx1y  
    if any(mod(n-m,2)) U(#<D7}  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') H}^'  
    end lkn|>U[  
    a1U|eLmUb  
    if any(m<0) e1m?g&[  
        error('zernpol:Mpositive','All M must be positive.') !eLj + 0  
    end B0fOAP1  
     +$dJA  
    if any(m>n) J D\tt-  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') RP4/:sO  
    end O VV@  
    BU O5g8m{  
    if any( r>1 | r<0 ) =q4}(  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') *'< AwG&  
    end T*z]<0E]  
    (#Y~z',I  
    if ~any(size(r)==1) RgJ@J/p"  
        error('zernpol:Rvector','R must be a vector.') [ GLH8R  
    end )g0lI  
    f%Q{}fC{*  
    r = r(:); X z+%Ym  
    length_r = length(r); ) r2Y@+.FN  
    RZOK+!H:  
    if nargin==4 ;[v!#+yml  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); +)Z]<O  
        if ~isnorm D*XrK0#Z`  
            error('zernpol:normalization','Unrecognized normalization flag.') :QE5 7 .  
        end oaY_6  
    else NB-dlv1  
        isnorm = false; sVLvnX,  
    end BQ@7^E[  
    1Ms[$$b$  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t&r.Kf9Z\  
    % Compute the Zernike Polynomials "HMEoZ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "s2_X+4oY  
    ( ;FxKm<P@  
    % Determine the required powers of r: Z*,e<zNQ  
    % ----------------------------------- K`4rUEf}V"  
    rpowers = []; p@ <Q?  
    for j = 1:length(n) <Vat@e  
        rpowers = [rpowers m(j):2:n(j)]; 2w?q7N%  
    end vwCQvt  
    rpowers = unique(rpowers); )`u17 {  
    (`x_MTLL  
    % Pre-compute the values of r raised to the required powers, ZoW1Cc&p  
    % and compile them in a matrix: $%<{zWQm  
    % ----------------------------- i! .]U@{k  
    if rpowers(1)==0 cG|fau<G  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @= -(H<0  
        rpowern = cat(2,rpowern{:}); LCF}Y{  
        rpowern = [ones(length_r,1) rpowern]; s\*L5{kiSl  
    else 9^gYy&+>6]  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); pwFp<O"  
        rpowern = cat(2,rpowern{:}); q t"D!S_  
    end =7Ln&tZ  
    ?w3RqF@}  
    % Compute the values of the polynomials: XlmX3RU  
    % -------------------------------------- /]0qI  
    z = zeros(length_r,length_n); w:&" "'E  
    for j = 1:length_n e~r/!B5X  
        s = 0:(n(j)-m(j))/2; ^7gGtz2  
        pows = n(j):-2:m(j); w^yb`\$  
        for k = length(s):-1:1 X Xque-  
            p = (1-2*mod(s(k),2))* ... -IPo/?}  
                       prod(2:(n(j)-s(k)))/          ... wi(Y=?=  
                       prod(2:s(k))/                 ... ( }-*irSsj  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... !Sc"V.o @!  
                       prod(2:((n(j)+m(j))/2-s(k))); =|3BkmO  
            idx = (pows(k)==rpowers); wQ_4_W  
            z(:,j) = z(:,j) + p*rpowern(:,idx); c[dSO(=  
        end : 4ryi&Y  
         ~Y 6'sM|  
        if isnorm 0w?da~  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); tKbxC>w  
        end d&AG~,&d|  
    end l|CM/(99-  
    DfXkLOGik  
    % 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)  N_G&nw  
    *mV?_4!,f7  
    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。 ?* ~4~ZE E  
    [XPAI["  
    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)