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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 6I4\q.^qw  
    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! #U4F0BdA  
     
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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 L:j<c5  
    function z = zernfun(n,m,r,theta,nflag) O)*+="Rg  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. HGs $*  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 85:=4N%  
    %   and angular frequency M, evaluated at positions (R,THETA) on the DDP/DD;n}r  
    %   unit circle.  N is a vector of positive integers (including 0), and TH&U j1  
    %   M is a vector with the same number of elements as N.  Each element u(>^3PJ+  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) rk2j#>l$4  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, m@2QnA[ 4  
    %   and THETA is a vector of angles.  R and THETA must have the same '(f*2eE:  
    %   length.  The output Z is a matrix with one column for every (N,M) ,+DG2u  
    %   pair, and one row for every (R,THETA) pair. O7m(o:t x3  
    % ^R7lom.  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike QL&ZjSN  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), -`kW&I0  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral X ::JV7hu  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, wedbx00o  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized t7Iv?5]N  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. RQ'9m^  
    % 3 *"WG O5  
    %   The Zernike functions are an orthogonal basis on the unit circle. w !-gJmX>  
    %   They are used in disciplines such as astronomy, optics, and 2\MT;;ZTZ  
    %   optometry to describe functions on a circular domain. rNWw?_H-H(  
    % %9F([K  
    %   The following table lists the first 15 Zernike functions. |O\s|H  
    % (ylTp]~mR-  
    %       n    m    Zernike function           Normalization p Z|V 3  
    %       -------------------------------------------------- W.f/pu  
    %       0    0    1                                 1 30#s aGV  
    %       1    1    r * cos(theta)                    2 #uG%j  
    %       1   -1    r * sin(theta)                    2 XFHYQ2ME2  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) %+W{iu[|  
    %       2    0    (2*r^2 - 1)                    sqrt(3) UT~4x|b:O  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) WdH$JTk1  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) eCU:Q  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ifMRryN4  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) S"bg9o  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) o4F2%0gJ  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) &ZlVWK~v  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) l|JE#  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) NqazpB*  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) u^ +7hkk  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 58tARLDr  
    %       -------------------------------------------------- Ha0M)0Anv  
    % S}m)OmrmA  
    %   Example 1: taHJ ub  
    % %op**@4/t\  
    %       % Display the Zernike function Z(n=5,m=1) }I+E\ <  
    %       x = -1:0.01:1; ;40/yl3r3[  
    %       [X,Y] = meshgrid(x,x); Ct<udO  
    %       [theta,r] = cart2pol(X,Y); >reU#j  
    %       idx = r<=1; )np:lL$$  
    %       z = nan(size(X)); c \J:![x  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); #?U}&Bd  
    %       figure sQHv%]s 0  
    %       pcolor(x,x,z), shading interp F4-$~ v@  
    %       axis square, colorbar 8?#/o c  
    %       title('Zernike function Z_5^1(r,\theta)')  L2[($l  
    % YNyk1cE  
    %   Example 2: I#Y22&G1  
    % hP%M?MKC  
    %       % Display the first 10 Zernike functions ?|\ER#z  
    %       x = -1:0.01:1; W dK #ZOR  
    %       [X,Y] = meshgrid(x,x); Tj` ,Z5vy  
    %       [theta,r] = cart2pol(X,Y); .]Y$o^mf  
    %       idx = r<=1; B?gOHG*vd>  
    %       z = nan(size(X)); x*\Y)9Vgy  
    %       n = [0  1  1  2  2  2  3  3  3  3]; k<nZ+! M  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ~|D Ut   
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; A7Cm5>Y_S  
    %       y = zernfun(n,m,r(idx),theta(idx)); CAig ]=2'  
    %       figure('Units','normalized') Wa>}wA=v  
    %       for k = 1:10 T@H ^BGs  
    %           z(idx) = y(:,k); \_VA 50  
    %           subplot(4,7,Nplot(k)) ~k-y &<UR  
    %           pcolor(x,x,z), shading interp p}z<Fdu 0  
    %           set(gca,'XTick',[],'YTick',[]) b4%??"&<Y  
    %           axis square W s3)gvpPA  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}'])  L^/5ux  
    %       end }1L4 "}L.  
    % R3)~?X1n  
    %   See also ZERNPOL, ZERNFUN2. 3)t.p>VgO  
    a_^\=&?'  
    %   Paul Fricker 11/13/2006 TPQ%L@^ L+  
    c)6m$5]  
    Gt8M&S-;  
    % Check and prepare the inputs: :%_LpZ  
    % ----------------------------- RtkEGxw*^  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) '2A)}uR  
        error('zernfun:NMvectors','N and M must be vectors.') G/y5H;<9M  
    end P[G)sA_"  
    "b~+;<}Q  
    if length(n)~=length(m) ^&9zw\x;z  
        error('zernfun:NMlength','N and M must be the same length.') #X+JHl  
    end IEL%!RFG  
    ^lnK$i  
    n = n(:); 58}U^IW  
    m = m(:); XFVE>/H  
    if any(mod(n-m,2)) \S `:y?[Y  
        error('zernfun:NMmultiplesof2', ... /wGM#sFH  
              'All N and M must differ by multiples of 2 (including 0).') nK1Slg#U  
    end ANAVn@ [  
    XAD- 'i  
    if any(m>n) V@.Ior}w  
        error('zernfun:MlessthanN', ... gs^Xf;g vI  
              'Each M must be less than or equal to its corresponding N.') CCs%%U/=  
    end `f,/`''R  
    >4x(e\B  
    if any( r>1 | r<0 ) Y Vt% 0  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') (R,#a *CV  
    end B-RjMxX4>  
    "@^k)d$  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) @Myo'{3vF  
        error('zernfun:RTHvector','R and THETA must be vectors.') JMCKcZ%N  
    end |MTnH/|  
    )rIwqUgp6\  
    r = r(:); rET\n(AJ  
    theta = theta(:); }`@vF|2L  
    length_r = length(r); L8@f-Kk  
    if length_r~=length(theta) ^x]r`b  
        error('zernfun:RTHlength', ...  C#.->\  
              'The number of R- and THETA-values must be equal.') ~p6 V,Q  
    end %_H<:uGO%  
    ?d\N(s9F  
    % Check normalization: +zqn<<9  
    % -------------------- ]6,\r"  
    if nargin==5 && ischar(nflag) w?PkO p  
        isnorm = strcmpi(nflag,'norm'); J/`<!$<c  
        if ~isnorm L]|gZ&^  
            error('zernfun:normalization','Unrecognized normalization flag.') /aCc17>2V{  
        end )EPjAv  
    else u=*FI  
        isnorm = false; olB.*#gA  
    end ;$,U~0  
    G{~J|{t\yz  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tn\yI!a  
    % Compute the Zernike Polynomials LG9+GszX 2  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% oi7@s0@  
    *}qWj_RT  
    % Determine the required powers of r: b<[Or^X ]  
    % ----------------------------------- e-/&$Qq  
    m_abs = abs(m); LtF,kAIt7v  
    rpowers = []; 0@0w+&*"@  
    for j = 1:length(n) 6?gW-1mY  
        rpowers = [rpowers m_abs(j):2:n(j)]; dA}-]  
    end & GO}|W  
    rpowers = unique(rpowers); ] Jg&VXrH  
    _IHV7*u{;  
    % Pre-compute the values of r raised to the required powers, IxN9&xa  
    % and compile them in a matrix: q CC.^8  
    % ----------------------------- _#E0g'3  
    if rpowers(1)==0 un"Gozmt5  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); W &W5lArr  
        rpowern = cat(2,rpowern{:}); .bl/*s  
        rpowern = [ones(length_r,1) rpowern]; J9nX"Sb  
    else IJp-BTO{V  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);  #4NaL  
        rpowern = cat(2,rpowern{:}); 8mrUotjS  
    end [ZwjOi:)  
    VR8-&N  
    % Compute the values of the polynomials: pZ{+c  
    % -------------------------------------- ha<[b ue  
    y = zeros(length_r,length(n)); e;q!6%  
    for j = 1:length(n) 2eS~/Pq5=i  
        s = 0:(n(j)-m_abs(j))/2; `:fZ)$sY  
        pows = n(j):-2:m_abs(j); %)8}X>xq  
        for k = length(s):-1:1 {%5eMyF#  
            p = (1-2*mod(s(k),2))* ... LKB$,pR~1l  
                       prod(2:(n(j)-s(k)))/              ... 'W^YM@  
                       prod(2:s(k))/                     ... (UD@q>c  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... i v38p%Zm  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); epe)a  
            idx = (pows(k)==rpowers); l}|%5.5-  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 3AtGy'NTp  
        end 1z4OI6$Af  
         Yx%Hs5}8  
        if isnorm K&]G3W%V  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); N0Lw}@p  
        end 9d659i C  
    end Xza(k  
    % END: Compute the Zernike Polynomials ifQ*,+@fxR  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% kd(8I_i@  
    ORw,)l  
    % Compute the Zernike functions: DU'`ewLL7  
    % ------------------------------ lIS-4QX1  
    idx_pos = m>0; H[$"+&q  
    idx_neg = m<0; !>&o01i  
    nPl?K:(  
    z = y; C`9+6T  
    if any(idx_pos) ` p-cSxR_  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 9wwqcx)3(  
    end s~g *@K>+  
    if any(idx_neg) u'DRN,h+  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 0RLg:SV  
    end }B+C~@j  
    lvz7#f L~  
    % EOF zernfun
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag)  "Og7rl  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. pJ"qu,w  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated d#4**BM  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive [EXs  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Ckuh:bs  
    %   and THETA is a vector of angles.  R and THETA must have the same BLiF 5  
    %   length.  The output Z is a matrix with one column for every P-value, ]MitOkX  
    %   and one row for every (R,THETA) pair. [!#L6&:a8  
    % .jE{3^  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 9IfmW^0  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 0gr/<v  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) Yk Ki|k  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 {@{']Y  
    %   for all p. MaQqs=  
    % *H2r@)Y[~  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 {qJ1ko)$  
    %   Zernike functions (order N<=7).  In some disciplines it is ag[wdoj  
    %   traditional to label the first 36 functions using a single mode joAv{Tc  
    %   number P instead of separate numbers for the order N and azimuthal Zt{[ *~  
    %   frequency M. ,i`,Oy(BI  
    % rcG"o\g@+  
    %   Example: C XMLt  
    % FHg 9OI67  
    %       % Display the first 16 Zernike functions {]@= ijjf  
    %       x = -1:0.01:1; "e>;'%W  
    %       [X,Y] = meshgrid(x,x); O;jrCB  
    %       [theta,r] = cart2pol(X,Y); `e&Suyf4B  
    %       idx = r<=1; ~4Fvy'  
    %       p = 0:15; `kXs;T6&  
    %       z = nan(size(X)); +lcbi  
    %       y = zernfun2(p,r(idx),theta(idx)); S g![Lsj  
    %       figure('Units','normalized') -zeG1gr3  
    %       for k = 1:length(p) .|fH y  
    %           z(idx) = y(:,k); s-Tv8goNV  
    %           subplot(4,4,k) AH7}/Rc  
    %           pcolor(x,x,z), shading interp J<h $ wM  
    %           set(gca,'XTick',[],'YTick',[]) HBXOjr<,{  
    %           axis square 2eY_%Y0  
    %           title(['Z_{' num2str(p(k)) '}']) flbd0NB  
    %       end <<5(0#y#  
    % N5 6g+,w%)  
    %   See also ZERNPOL, ZERNFUN. iz PDd{[  
    Y]2A&0  
    %   Paul Fricker 11/13/2006 N<VJ(20y  
    ?NsW|w_  
    })Vi  
    % Check and prepare the inputs: xY(*.T9K  
    % ----------------------------- 0GCEqQy8  
    if min(size(p))~=1 xfe+n$~ c  
        error('zernfun2:Pvector','Input P must be vector.') &B1WtW  
    end [hv~o~q  
    0 /U{p,r6`  
    if any(p)>35 \Uq(Zga4)  
        error('zernfun2:P36', ... 33B]RGq  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... [waIi3Dv\  
               '(P = 0 to 35).']) "@0]G<H  
    end m&&m,6``P  
    . 3T3E X|G  
    % Get the order and frequency corresonding to the function number: hhc,uJ">!  
    % ---------------------------------------------------------------- VuZuS6~#J  
    p = p(:); ;iL#7NG-R  
    n = ceil((-3+sqrt(9+8*p))/2); =GMkR+<)  
    m = 2*p - n.*(n+2); F{;((VboN  
    RMu~l@  
    % Pass the inputs to the function ZERNFUN: 'I6i ,+D/q  
    % ---------------------------------------- R!gEwTk  
    switch nargin >U27];}y  
        case 3 y _k l:Ssa  
            z = zernfun(n,m,r,theta); $DaNbLV  
        case 4 Btn]}8K  
            z = zernfun(n,m,r,theta,nflag); Z,Dl` w  
        otherwise I:1C8*/  
            error('zernfun2:nargin','Incorrect number of inputs.') 1^JS Dd  
    end .Vvx,>>D  
    #?- wm  
    % 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) U0P~  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Y\g3h M  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of TJXT-\Vk  
    %   order N and frequency M, evaluated at R.  N is a vector of PtiOz :zV  
    %   positive integers (including 0), and M is a vector with the t&e{_|i#+  
    %   same number of elements as N.  Each element k of M must be a ZyFjFHe+  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) e^1Twz3z  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is &`2)V;t  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix m#\ dSl}  
    %   with one column for every (N,M) pair, and one row for every R.yvjPwJ  
    %   element in R. 8XE7]&)];  
    % z9Rp`z&`E  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- J)p l|I  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is -]M5wb2,  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 0{-q#/  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 V1B5w_^>h'  
    %   for all [n,m]. <=C!VVk4f  
    % "87:?v[[1  
    %   The radial Zernike polynomials are the radial portion of the ds[|   
    %   Zernike functions, which are an orthogonal basis on the unit aWF655Fs*  
    %   circle.  The series representation of the radial Zernike {kR#p %E]  
    %   polynomials is c(s.5p ^  
    % v"Es*-{B  
    %          (n-m)/2 bY~pc\V:`w  
    %            __ k~1?VQ+?M  
    %    m      \       s                                          n-2s 0oIe> r  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r .3Oap*X  
    %    n      s=0 PB\x3pV!}  
    % \z(gqkc 6  
    %   The following table shows the first 12 polynomials. S;`A{Mow  
    % 1#+S+g@#  
    %       n    m    Zernike polynomial    Normalization 49HZ2`Y  
    %       --------------------------------------------- 5VU2[ \  
    %       0    0    1                        sqrt(2) Q*~]h;6\{d  
    %       1    1    r                           2 r3UUlR/Do  
    %       2    0    2*r^2 - 1                sqrt(6) E$p+}sP(C  
    %       2    2    r^2                      sqrt(6) t;\Y{`  
    %       3    1    3*r^3 - 2*r              sqrt(8) sLxc(d'A  
    %       3    3    r^3                      sqrt(8) Q>i^s@0  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ##"HF  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) JDT`C2-Q  
    %       4    4    r^4                      sqrt(10) BLD gt~h#  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 9p(. A$  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ., 6-u  
    %       5    5    r^5                      sqrt(12) vUM4S26"NT  
    %       --------------------------------------------- XlR@pr6tw  
    % c\AfaK^KF  
    %   Example: cSV aI  
    % Jdj4\j u  
    %       % Display three example Zernike radial polynomials zy }$i?  
    %       r = 0:0.01:1; _xhax+,! ~  
    %       n = [3 2 5]; Uz]|N6`  
    %       m = [1 2 1]; HN|%9{VeB  
    %       z = zernpol(n,m,r); {R6ZKB  
    %       figure 97!;.f-  
    %       plot(r,z) /IMFO:c  
    %       grid on _I5Y"o  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') pFjK}J OF  
    % o?\?@H  
    %   See also ZERNFUN, ZERNFUN2. %1+4_g9  
    pYf-S?Y/V  
    % A note on the algorithm. fI|Nc  
    % ------------------------ $~T4hv :  
    % The radial Zernike polynomials are computed using the series EXqE~afm2  
    % representation shown in the Help section above. For many special 3(80:@|  
    % functions, direct evaluation using the series representation can 0<@@?G  
    % produce poor numerical results (floating point errors), because t*w/{|yO  
    % the summation often involves computing small differences between 92oFlEJ  
    % large successive terms in the series. (In such cases, the functions :d'8x  
    % are often evaluated using alternative methods such as recurrence }k.Z~1y  
    % relations: see the Legendre functions, for example). For the Zernike e+fN6v5pU  
    % polynomials, however, this problem does not arise, because the 7B66]3v  
    % polynomials are evaluated over the finite domain r = (0,1), and K]w'&Qm8W  
    % because the coefficients for a given polynomial are generally all ey$&;1x#5  
    % of similar magnitude. \qJXF|z<K  
    % G]&qx`TBK  
    % ZERNPOL has been written using a vectorized implementation: multiple 7 HYwLG:\~  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] uQKT  
    % values can be passed as inputs) for a vector of points R.  To achieve bV3|6]k^  
    % this vectorization most efficiently, the algorithm in ZERNPOL Cq~dp/V  
    % involves pre-determining all the powers p of R that are required to b@hqz!)l`  
    % compute the outputs, and then compiling the {R^p} into a single 88$8d>-  
    % matrix.  This avoids any redundant computation of the R^p, and pOoEI+t  
    % minimizes the sizes of certain intermediate variables. $/Uq0U  
    % H0vfUF53l  
    %   Paul Fricker 11/13/2006 67FWa   
    $6R-5oQ  
    I{=Qtnlb  
    % Check and prepare the inputs: +9sQZB# (  
    % ----------------------------- dioGAai'  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) e~"U @8xk~  
        error('zernpol:NMvectors','N and M must be vectors.') 1 [Bk%G@D&  
    end xr^LFn)  
     _;\_l  
    if length(n)~=length(m) ")p\q:z6  
        error('zernpol:NMlength','N and M must be the same length.') j8:\%|  
    end vKAN@HSYr  
    yyTnL 2Y9  
    n = n(:); S)"Jf?  
    m = m(:); Q^^niVz  
    length_n = length(n); g2Z`zQA7  
    XfIJ4ZM5  
    if any(mod(n-m,2)) 7D_=  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') m+z& Q  
    end 6[AL|d DK  
    /Z}}(6T  
    if any(m<0) t\O16O7S  
        error('zernpol:Mpositive','All M must be positive.')  &q*Aj17  
    end QIFgQ0{  
    rEz^  
    if any(m>n) '8kP.l  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') C\hM =%  
    end &_8 947  
    h 'nY3GrU  
    if any( r>1 | r<0 ) [0("Q;Ec[j  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') |CbikE}kL  
    end (S Yln>o  
    Bk{]g=DO  
    if ~any(size(r)==1) P16~Qj  
        error('zernpol:Rvector','R must be a vector.') SSzIih@u  
    end NDokSw-  
    Zx>=tx}  
    r = r(:); Q22 GIr  
    length_r = length(r); W[r>.7>?h  
    ?:9"X$XR  
    if nargin==4 sV*H`N')S  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); t sRdvFFq  
        if ~isnorm lH~[f  
            error('zernpol:normalization','Unrecognized normalization flag.') G=bCNn<  
        end I)HPO,7  
    else j![\& z  
        isnorm = false; z\4.Gm-  
    end b%c9oR's^  
    F_P~x(X  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fI|$K )K  
    % Compute the Zernike Polynomials dqcL]e  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  ZWm6eD  
    _,*r_D61S  
    % Determine the required powers of r: &BSn?  
    % ----------------------------------- h Xya*#n#  
    rpowers = []; *qpSXmOz  
    for j = 1:length(n) 7a}k  
        rpowers = [rpowers m(j):2:n(j)]; "$vRMpW:  
    end x.4m|f0;  
    rpowers = unique(rpowers); y8xE 6i  
    tpx2 IE  
    % Pre-compute the values of r raised to the required powers, \z)%$#I  
    % and compile them in a matrix: K:WDl;8 (d  
    % ----------------------------- sa8Vvzvo.  
    if rpowers(1)==0 pTuS*MYz  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); :rP=t ,  
        rpowern = cat(2,rpowern{:}); #lO Mm9  
        rpowern = [ones(length_r,1) rpowern]; I( Mm?9F  
    else z'7]h TA  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); TkF[x%o  
        rpowern = cat(2,rpowern{:}); l%=;  
    end ^=*;X;7  
    0tJ Z4(0  
    % Compute the values of the polynomials: Ew$C ;&9  
    % -------------------------------------- 1AFA=t:]p  
    z = zeros(length_r,length_n); 6wg^FD_Q  
    for j = 1:length_n \}G^\p6?M  
        s = 0:(n(j)-m(j))/2; uEx-]F  
        pows = n(j):-2:m(j); UGatWj  
        for k = length(s):-1:1 3iU=c&P  
            p = (1-2*mod(s(k),2))* ... hCo|HB  
                       prod(2:(n(j)-s(k)))/          ...  f)<6  
                       prod(2:s(k))/                 ... 0IWf!Sk ]  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... &,)&%Sg[  
                       prod(2:((n(j)+m(j))/2-s(k))); onV>.7sG  
            idx = (pows(k)==rpowers); (QiAisE  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 51.%;aY~z  
        end YHl;flv  
         bs1Rvx1:J%  
        if isnorm q0 \6F^;M  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ,iwp,=h=  
        end ABYcH]m  
    end OB}Ib]  
    EEL,^3KR  
    % 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)  | %Vh`HT  
    kZ3ThIk%  
    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*U4?M  
    C~iL3C b  
    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)