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niuhelen 2011-03-12 18:40

ansys分析后面型数据如何进行zernike多项式拟合?

小弟不是学光学的,所以想请各位大侠指点啊!谢谢啦 4a-wGx#h  
就是我用ansys计算出了镜面的面型的数据,怎样可以得到zernike多项式的系数,然后用zemax各阶得到像差!谢谢啦! :%{7Q$Xv<  
phility 2011-03-12 22:31
可以用matlab编程,用zernike多项式进行波面拟合,求出zernike多项式的系数,拟合的算法有很多种,最简单的是最小二乘法,你可以查下相关资料,挺简单的
phility 2011-03-12 22:41
泽尼克多项式的前9项对应象差的
niuhelen 2011-03-12 23:00
非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 R 9(^CWs  
function z = zernfun(n,m,r,theta,nflag) |4vk@0L  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. /hQ!dU.+  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N <vs.Ucxx  
%   and angular frequency M, evaluated at positions (R,THETA) on the )1/O_N6C  
%   unit circle.  N is a vector of positive integers (including 0), and fJuJ#MX{:  
%   M is a vector with the same number of elements as N.  Each element }R^{<{KVJ  
%   k of M must be a positive integer, with possible values M(k) = -N(k) k:sh:G+=$d  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Y2Bu,/9^  
%   and THETA is a vector of angles.  R and THETA must have the same \GWC5R7Q0j  
%   length.  The output Z is a matrix with one column for every (N,M) ',f[y:v;  
%   pair, and one row for every (R,THETA) pair. Sc&_6} K  
% I,D=ixK  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike !SnpesTn  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Ax ^9J)C  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral K\G|q}E/1  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, m`Z4#_s2  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized g:HIiGN0Ic  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ]P.S5s'  
% y03l_E,  
%   The Zernike functions are an orthogonal basis on the unit circle. Ne%X:h  
%   They are used in disciplines such as astronomy, optics, and ~0L>l J  
%   optometry to describe functions on a circular domain. #]rw@c  
% VuGSP]$q  
%   The following table lists the first 15 Zernike functions. Uu ,Re  
% fw<'ygd  
%       n    m    Zernike function           Normalization BtspnVB ez  
%       -------------------------------------------------- xfb%bkr  
%       0    0    1                                 1 J{H475GqiT  
%       1    1    r * cos(theta)                    2 piU4%EO  
%       1   -1    r * sin(theta)                    2 ?S"xR0 *  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) V%))%?3x_  
%       2    0    (2*r^2 - 1)                    sqrt(3) ctf'/IZ5  
%       2    2    r^2 * sin(2*theta)             sqrt(6) ]BA8[2=m  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) 1*Z}M%  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) CXa$QSu>  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) /)~M cP3  
%       3    3    r^3 * sin(3*theta)             sqrt(8) \>+gZc]an  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) =3FXU{"Qi4  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 2l9_$evK~  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) p?Y1^/   
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) TWy1)30x  
%       4    4    r^4 * sin(4*theta)             sqrt(10) YPN|qn(  
%       -------------------------------------------------- S5j#&i  
% aD.A +es  
%   Example 1:  BzDS  
% >6Q-e$GS@  
%       % Display the Zernike function Z(n=5,m=1)  A/9 wr  
%       x = -1:0.01:1; dG1qrh9_-  
%       [X,Y] = meshgrid(x,x); nv|&|6?`oK  
%       [theta,r] = cart2pol(X,Y); N7|ctO  
%       idx = r<=1; W_?S^>?l/  
%       z = nan(size(X)); \eN}V  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); Ox58L>:0m  
%       figure uJi|@{V  
%       pcolor(x,x,z), shading interp U@H SU%H  
%       axis square, colorbar [K^RC;}nV^  
%       title('Zernike function Z_5^1(r,\theta)') ZW2U9  
% wuPx6hCl  
%   Example 2: VP[ J#TPU  
% {&xKS WNc  
%       % Display the first 10 Zernike functions X4jtti  
%       x = -1:0.01:1; s+aeP  
%       [X,Y] = meshgrid(x,x); ALhu\x>AY  
%       [theta,r] = cart2pol(X,Y); q?`bu:yS  
%       idx = r<=1; B7cXbUAQs  
%       z = nan(size(X)); *\emRI>  
%       n = [0  1  1  2  2  2  3  3  3  3]; ^X^4R1V)  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ?>2k>~xlQ  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; W}Z'zU?[  
%       y = zernfun(n,m,r(idx),theta(idx)); k5((@[  
%       figure('Units','normalized') b?y3m +V`  
%       for k = 1:10 )8yNqnD  
%           z(idx) = y(:,k); `U)~fu/\2M  
%           subplot(4,7,Nplot(k)) < }G7#xg  
%           pcolor(x,x,z), shading interp G"wy?  
%           set(gca,'XTick',[],'YTick',[]) ;as B@Q  
%           axis square <`BUk< uf#  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) moh7:g  
%       end DvU(rr\p  
% d&F8nBIM5  
%   See also ZERNPOL, ZERNFUN2. c'[l%4U8[  
*U8Pjb1  
%   Paul Fricker 11/13/2006 Q 1g@FsW&U  
4\3Z$%2^LZ  
Ve<l7U;  
% Check and prepare the inputs: t=5 K#SX}  
% ----------------------------- woQYP,  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) sT|8a  
    error('zernfun:NMvectors','N and M must be vectors.') 4;x{@Ln  
end SO9j/  
"d9"Md0k  
if length(n)~=length(m) Eb[*nWF=  
    error('zernfun:NMlength','N and M must be the same length.') K%O%#Kk  
end z.--"cF  
4Z,MqG>  
n = n(:); .hXxh)F  
m = m(:); k68\ _NUL  
if any(mod(n-m,2)) }/Pz1,/  
    error('zernfun:NMmultiplesof2', ... "1t%J7c_  
          'All N and M must differ by multiples of 2 (including 0).') wUv Zc  
end ng"R[/)In  
O)n"a\LD  
if any(m>n) ,dP-sD;<  
    error('zernfun:MlessthanN', ... P-.>vi^+  
          'Each M must be less than or equal to its corresponding N.') ycTX\.KV  
end 1Jjay#  
!K'j[cA^  
if any( r>1 | r<0 ) S{&,I2aO  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') To.CY^M  
end B|zJrz0q3  
akoK4!z  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 1YL6:5n  
    error('zernfun:RTHvector','R and THETA must be vectors.') !RN(/ &%y  
end fYBmW')  
{1Z8cV   
r = r(:); ~dg7c{o5  
theta = theta(:); Cz` !j  
length_r = length(r); j#hFx+S  
if length_r~=length(theta) Yi1lvB?m  
    error('zernfun:RTHlength', ... e0Zwhz,  
          'The number of R- and THETA-values must be equal.') Iy% fg',%  
end mII7p LbQ  
-{n2^vvF  
% Check normalization: qbrY5;U  
% -------------------- $dIu${lu  
if nargin==5 && ischar(nflag) j51Wod<[  
    isnorm = strcmpi(nflag,'norm'); %5Q5xw]w3  
    if ~isnorm LQ(z~M0B  
        error('zernfun:normalization','Unrecognized normalization flag.') Q8OA{EUtq  
    end e=e^;K4  
else /%fBkA#n  
    isnorm = false; o."k7fLB  
end Z<jio  
]zK'aod  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Y>W$n9d&G2  
% Compute the Zernike Polynomials IYAvO%~  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% qz[qjGdHg  
>U .  
% Determine the required powers of r: 2^RWGCEv  
% ----------------------------------- >ka*-8?  
m_abs = abs(m); 4IfOvAN%  
rpowers = []; `< _A#@  
for j = 1:length(n) vM G>Xb  
    rpowers = [rpowers m_abs(j):2:n(j)]; ts|dk%  
end DD5 S R  
rpowers = unique(rpowers); 3*INDD=  
"u^%~2  
% Pre-compute the values of r raised to the required powers, nwSujD  
% and compile them in a matrix:  KT'Ebb]  
% ----------------------------- |W $epOLg  
if rpowers(1)==0 {P/ sxh:e  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); _:"PBN9  
    rpowern = cat(2,rpowern{:}); !A_<(M<  
    rpowern = [ones(length_r,1) rpowern]; k_ d)  
else "wwAbU<  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 4PdJ  
    rpowern = cat(2,rpowern{:}); $r>$ u  
end DpA"5RV  
}MU}-6  
% Compute the values of the polynomials: 8d4:8}  
% -------------------------------------- a* 2*aH7  
y = zeros(length_r,length(n)); <=O/_Iu(  
for j = 1:length(n) i*ibx;s-  
    s = 0:(n(j)-m_abs(j))/2; [k<"@[8)  
    pows = n(j):-2:m_abs(j); o}^/K m+t  
    for k = length(s):-1:1 pX 4:WV  
        p = (1-2*mod(s(k),2))* ... -O&u;kh4g  
                   prod(2:(n(j)-s(k)))/              ... +`jI z'+  
                   prod(2:s(k))/                     ... VT@,RlB0  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... `3wzOMgJ  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); 3jeB\  
        idx = (pows(k)==rpowers); &>%R)?SZh  
        y(:,j) = y(:,j) + p*rpowern(:,idx); q!fdiv`  
    end _.}1 Y,Q  
     ko7*9`  
    if isnorm yLFZo"r  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 'J[ n}r  
    end ,q_'l?Pn  
end XEX ."y  
% END: Compute the Zernike Polynomials p*LG Y+  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }8lvi vR4  
5Yxs_t4  
% Compute the Zernike functions: owR`Z`^h)  
% ------------------------------ D6Q6yNE  
idx_pos = m>0; `qXCY^BH2  
idx_neg = m<0; KzgW+6*G  
Bm e_#  
z = y; 9sQ #v-+Yx  
if any(idx_pos) mK Ta.  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); !Py SYY  
end \jR('5DcB  
if any(idx_neg) k'6Poz+<  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); = n>aJ(=Pd  
end 9e aqq  
}piDg(D  
% EOF zernfun
niuhelen 2011-03-12 23:01
function z = zernfun2(p,r,theta,nflag) wO-](3A-8P  
%ZERNFUN2 Single-index Zernike functions on the unit circle. 'B"A*!" b  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated -O~ V4004  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive s:p6oEQ=J  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, \>7hT;Av=G  
%   and THETA is a vector of angles.  R and THETA must have the same RX"~m!26  
%   length.  The output Z is a matrix with one column for every P-value, ",O}{z  
%   and one row for every (R,THETA) pair. (>)f#t[9J  
% Lh_Q@>k  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike L/+KY_b:*  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) .dE2,9{Z  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) I\y=uC  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 N!=$6`d  
%   for all p. 'F .tOD  
% 5H{dLZ],  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ^Gt9.  
%   Zernike functions (order N<=7).  In some disciplines it is l)P~#G+C  
%   traditional to label the first 36 functions using a single mode H)5V \  
%   number P instead of separate numbers for the order N and azimuthal Nn:>c<[  
%   frequency M. l2.L h<G  
% /4xp?Lo:  
%   Example: 6xC$R q  
% sM  _m  
%       % Display the first 16 Zernike functions tqMOh R  
%       x = -1:0.01:1; "TQ3{=j{  
%       [X,Y] = meshgrid(x,x); _Pe,84Ro  
%       [theta,r] = cart2pol(X,Y); VNggDKS~K  
%       idx = r<=1; G&DL)ePu]m  
%       p = 0:15; wJ}9(>id*  
%       z = nan(size(X)); CHGV1X,  
%       y = zernfun2(p,r(idx),theta(idx)); j~#nJI5]  
%       figure('Units','normalized') ;*AK eI2  
%       for k = 1:length(p) ^6?)EM#  
%           z(idx) = y(:,k); I5$]{:L|9  
%           subplot(4,4,k) U%qE=u-  
%           pcolor(x,x,z), shading interp [m+):q^  
%           set(gca,'XTick',[],'YTick',[]) Y5&mJp\G  
%           axis square I<}<!.Bc!  
%           title(['Z_{' num2str(p(k)) '}']) M#'j7EMu  
%       end gQ8FjL6?  
% /t $J<bU  
%   See also ZERNPOL, ZERNFUN. v "Yo  
:,~]R,tJQ  
%   Paul Fricker 11/13/2006 o 00(\ -eb  
xkPH_+4i8  
R{0nk   
% Check and prepare the inputs: C\RJ){dk  
% ----------------------------- g/_j"Nn  
if min(size(p))~=1 O&vE 5%x  
    error('zernfun2:Pvector','Input P must be vector.') yr"BeTrS.  
end P-25]-  
fa:V8xa  
if any(p)>35 aZBS!X  
    error('zernfun2:P36', ... ))xyaYIZkk  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... -&UP[Mq  
           '(P = 0 to 35).']) {OBV+}#  
end wiE'6CM  
NJqjW  
% Get the order and frequency corresonding to the function number: 4IUdlb  
% ---------------------------------------------------------------- ob(S/t  
p = p(:); J6s@}@R1  
n = ceil((-3+sqrt(9+8*p))/2); B*c@w~E  
m = 2*p - n.*(n+2); Rg,]d u u?  
83[gV@LW0m  
% Pass the inputs to the function ZERNFUN: s(u,mtG  
% ---------------------------------------- U1,f$McZs  
switch nargin u.~`/O  
    case 3 [i 7^a/e  
        z = zernfun(n,m,r,theta); POl_chq  
    case 4 +"8-)'  
        z = zernfun(n,m,r,theta,nflag); c1>:|D7w  
    otherwise *41 2)zEy  
        error('zernfun2:nargin','Incorrect number of inputs.') EH2a  
end FQp@/H^  
8+=-!": ]  
% EOF zernfun2
niuhelen 2011-03-12 23:01
function z = zernpol(n,m,r,nflag) 873'=m&  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. K H&o`U(}  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of x } X1 O)  
%   order N and frequency M, evaluated at R.  N is a vector of )/4xR]  
%   positive integers (including 0), and M is a vector with the nbf w7u  
%   same number of elements as N.  Each element k of M must be a 6:$+"@ps  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) Q(0eq_X|6  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Uk0Fo(HY  
%   a vector of numbers between 0 and 1.  The output Z is a matrix ;mtv  
%   with one column for every (N,M) pair, and one row for every .R! /?eN  
%   element in R. bayDdR4T  
% ?]In@h-  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 23_\UTM}1  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 1*vt\,G  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to Du7DMo=l  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1  ~d\>f  
%   for all [n,m]. 4Y!_tZ>  
% JA(M'&q4  
%   The radial Zernike polynomials are the radial portion of the *$/!.e  
%   Zernike functions, which are an orthogonal basis on the unit oD?c]}3  
%   circle.  The series representation of the radial Zernike lAZn0EU  
%   polynomials is cE SSSH!m  
% lQ!)0F  
%          (n-m)/2 F4$N:J kl  
%            __ .Km6 (U  
%    m      \       s                                          n-2s GDBxciv  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 2 ,bLEhu  
%    n      s=0 ?VaAVxd29  
% tLc 9-  
%   The following table shows the first 12 polynomials. x}(p\Efx  
% GL- r;  
%       n    m    Zernike polynomial    Normalization ZlEH3-Zv  
%       --------------------------------------------- #lo1GoL\  
%       0    0    1                        sqrt(2) Sru}0M#M  
%       1    1    r                           2 stcbM  
%       2    0    2*r^2 - 1                sqrt(6) )cUFb:D*"  
%       2    2    r^2                      sqrt(6) Y-vLEIX=  
%       3    1    3*r^3 - 2*r              sqrt(8) *!p#1fE  
%       3    3    r^3                      sqrt(8) U) B^R  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) _A5.  
%       4    2    4*r^4 - 3*r^2            sqrt(10) kntM  
%       4    4    r^4                      sqrt(10) S{.G=O  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) NJ];Ck  
%       5    3    5*r^5 - 4*r^3            sqrt(12) N sNk  
%       5    5    r^5                      sqrt(12) 6B .x=  
%       --------------------------------------------- uFH ]w] X  
% 4,.B#: 8  
%   Example: J~,Ny_L  
% fH6mv0  
%       % Display three example Zernike radial polynomials mYRW/8+g  
%       r = 0:0.01:1; mO|YX/>  
%       n = [3 2 5]; fRT4,;  
%       m = [1 2 1]; CUG3C  
%       z = zernpol(n,m,r); y/d/#}\:  
%       figure i<ug("/  
%       plot(r,z) fO'"UI  
%       grid on 2zQ62t}  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') y^xEZD1X6-  
% vD@ =V#T  
%   See also ZERNFUN, ZERNFUN2. [n!5!/g>j  
!\RBOdw C  
% A note on the algorithm. z&x3":@u<  
% ------------------------ 3|qT.QR`Z  
% The radial Zernike polynomials are computed using the series &Bfgvws;  
% representation shown in the Help section above. For many special ;5TQH_g  
% functions, direct evaluation using the series representation can wx2 EMr   
% produce poor numerical results (floating point errors), because $~s|%>@  
% the summation often involves computing small differences between e <]^7pz  
% large successive terms in the series. (In such cases, the functions THcK,`lX@  
% are often evaluated using alternative methods such as recurrence DE659=Tq  
% relations: see the Legendre functions, for example). For the Zernike 52H'aHO1  
% polynomials, however, this problem does not arise, because the <[W41{  
% polynomials are evaluated over the finite domain r = (0,1), and WVa#nU^  
% because the coefficients for a given polynomial are generally all P g{/tM Y  
% of similar magnitude. qY^@^)b[  
% EJ&aT etQ  
% ZERNPOL has been written using a vectorized implementation: multiple Utv#E.VI  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] v0|[w2Q2  
% values can be passed as inputs) for a vector of points R.  To achieve 2qQG  
% this vectorization most efficiently, the algorithm in ZERNPOL ^xZo .P  
% involves pre-determining all the powers p of R that are required to Nu]& ?  
% compute the outputs, and then compiling the {R^p} into a single oI2YJ2?Je8  
% matrix.  This avoids any redundant computation of the R^p, and vSf ?o\O  
% minimizes the sizes of certain intermediate variables. 4x;vn8 yh  
% @s[Vtw%f  
%   Paul Fricker 11/13/2006 G{C27k>wa  
F[HMX4  
Za4X ;  
% Check and prepare the inputs: G,]%dZH e  
% ----------------------------- qC;1ND  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) NrTK+6 z  
    error('zernpol:NMvectors','N and M must be vectors.') )4@La&  
end tn@MOOP l  
}g@5%DI]  
if length(n)~=length(m) Frxim  
    error('zernpol:NMlength','N and M must be the same length.') Q;EQ8pL?"  
end U! xOJ  
ImnN&[Cu  
n = n(:); ;WG6|QgV?-  
m = m(:); oI/jGyY;  
length_n = length(n); Ny oRp  
P*M$^p  
if any(mod(n-m,2)) T+9#&  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') .&R j2d  
end sh|@X\EZO  
:% o32  
if any(m<0) ~ R:=zGDV  
    error('zernpol:Mpositive','All M must be positive.') $: %U`46%s  
end "h>B`S  
M1sR+e$"  
if any(m>n) 6n}5>GSF  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') &-s/F`  
end t(,_  
c(tX761qz  
if any( r>1 | r<0 ) ecjjCt2S  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') \4B2%H  
end `z?6.+C  
kS@6'5U  
if ~any(size(r)==1) +d=cI  
    error('zernpol:Rvector','R must be a vector.') *;]j#0  
end %%=PpKYtSD  
k'hJ@ 6eKS  
r = r(:); Uz&XqjS  
length_r = length(r); uBPxMwohR  
pO~lVM  
if nargin==4 Mr8r(LGY  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ^I4/{,Ev  
    if ~isnorm /1Q i9uit  
        error('zernpol:normalization','Unrecognized normalization flag.') p? q~.YY  
    end GdxMHnn=  
else RkW)B^#  
    isnorm = false; /<\B8^yQ  
end D02_ Jrg  
i5QG_^X&  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vU1*:3  
% Compute the Zernike Polynomials kN99(  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jZ;dY~fE  
j |N8"8"  
% Determine the required powers of r: 0d%p<c  
% ----------------------------------- +Je(]b @  
rpowers = []; cc"L> XoK  
for j = 1:length(n) [^U#ic>cT  
    rpowers = [rpowers m(j):2:n(j)]; =h#3D?b0n  
end o,#[Se*n  
rpowers = unique(rpowers); Ckelr  
ylEQeN  
% Pre-compute the values of r raised to the required powers, ) ag8]   
% and compile them in a matrix: Bx>)i8P7i0  
% ----------------------------- !E0zj9 [ R  
if rpowers(1)==0 7Z(F-B +j  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); bg8<}~zg  
    rpowern = cat(2,rpowern{:});  3e<FlH{  
    rpowern = [ones(length_r,1) rpowern]; LHt{y3l]  
else eTV%+  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); {APsi7HYBr  
    rpowern = cat(2,rpowern{:}); *TjolE~o  
end '_~X(izc  
_-vf<QO]  
% Compute the values of the polynomials: k]vrqjn Q  
% -------------------------------------- m}x&]">9  
z = zeros(length_r,length_n); HhmC+3w.7  
for j = 1:length_n f-V8/  
    s = 0:(n(j)-m(j))/2; ?Q~6\xA  
    pows = n(j):-2:m(j); R"au8f.  
    for k = length(s):-1:1 ON(H7  
        p = (1-2*mod(s(k),2))* ... .Q,IOCHk  
                   prod(2:(n(j)-s(k)))/          ... HlkG^:)  
                   prod(2:s(k))/                 ... :uAW  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... 9Yh0' <Z  
                   prod(2:((n(j)+m(j))/2-s(k))); k{fCU%  
        idx = (pows(k)==rpowers); ?D].Za^km  
        z(:,j) = z(:,j) + p*rpowern(:,idx); x<0-'EF/S  
    end !Cm<K*c"&E  
     PgKA>50a  
    if isnorm \LFRu  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); zS@"ITy  
    end 6z^Kg~a   
end f PoC yl  
CvCk#:@HM  
% EOF zernpol
niuhelen 2011-03-12 23:03
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!
li_xin_feng 2012-09-28 10:52
我也正在找啊
guapiqlh 2014-03-04 11:35
我也一直想了解这个多项式的应用,还没用过呢
phoenixzqy 2014-04-22 23:39
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)  .Od.lxz"mp  
k+>-?S,  
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。 FbB> Md;  
1FU(j*~:  
07年就写过这方面的计算程序了。
查看本帖完整版本: [-- ansys分析后面型数据如何进行zernike多项式拟合? --] [-- top --]

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