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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 E._/PB  
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! `?vI_>md'!  
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 P6 mDwR  
function z = zernfun(n,m,r,theta,nflag) ] iiB|xT  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle.  ev(E  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 1.z !u%2  
%   and angular frequency M, evaluated at positions (R,THETA) on the f.U.(  
%   unit circle.  N is a vector of positive integers (including 0), and l65Qk2<YC  
%   M is a vector with the same number of elements as N.  Each element xh!aB6m8R  
%   k of M must be a positive integer, with possible values M(k) = -N(k) 4yRX{Bl|  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, S >\\n^SbT  
%   and THETA is a vector of angles.  R and THETA must have the same x/#.%Ga#T  
%   length.  The output Z is a matrix with one column for every (N,M) M0uC0\' #P  
%   pair, and one row for every (R,THETA) pair. \'Ca%j  
% lKy4Nry9  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike m\J" P'=  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), U,^jN|v  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral wEbO|S+K1  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ]4&B*]j  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized coc :$Sr%  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ' ui`EL%  
% (I(k$g[>  
%   The Zernike functions are an orthogonal basis on the unit circle. ~=pyA#VVJ"  
%   They are used in disciplines such as astronomy, optics, and %J|xPp)  
%   optometry to describe functions on a circular domain. +Ram%"Zwh  
% wHhIa3_v  
%   The following table lists the first 15 Zernike functions. /)xQ# yfX  
% U\",!S~<  
%       n    m    Zernike function           Normalization ;i;;{j@$i  
%       -------------------------------------------------- yg@}j   
%       0    0    1                                 1 M% FKg/  
%       1    1    r * cos(theta)                    2 x\6i(k-  
%       1   -1    r * sin(theta)                    2 m_>~e}2'A  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) 0'tm.,  
%       2    0    (2*r^2 - 1)                    sqrt(3) 05vu{>  
%       2    2    r^2 * sin(2*theta)             sqrt(6) m?D k(DJ  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) \G &q[8F\  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Qx !! Ttd{  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) V@1K  
%       3    3    r^3 * sin(3*theta)             sqrt(8) oJ)v6"j  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) . aqP=  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Zl`sY5{1  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) I'16-  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) uB uwE6  
%       4    4    r^4 * sin(4*theta)             sqrt(10) {_*$X  
%       -------------------------------------------------- ZlsdO.G  
% l j*J|%~  
%   Example 1: d9uT*5f  
% Y@M l}43  
%       % Display the Zernike function Z(n=5,m=1) {:d9q  
%       x = -1:0.01:1; N&+DhKw  
%       [X,Y] = meshgrid(x,x); e.^Y4(  
%       [theta,r] = cart2pol(X,Y); nXF|AeAco  
%       idx = r<=1; "t)|N dZm  
%       z = nan(size(X)); {V9}W<  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); 9k>=y n  
%       figure wa4(tM2  
%       pcolor(x,x,z), shading interp /2PsC*y  
%       axis square, colorbar SB`"%6  
%       title('Zernike function Z_5^1(r,\theta)') n`)wD~mk  
% s|=.L&"   
%   Example 2: auT$-Ki8  
% t#[u X?  
%       % Display the first 10 Zernike functions az ?2  
%       x = -1:0.01:1; iVGc\6+'  
%       [X,Y] = meshgrid(x,x); 4FgY!k  
%       [theta,r] = cart2pol(X,Y); p~THliwd  
%       idx = r<=1; XZ8;Ow=  
%       z = nan(size(X)); L]HYk}oD.  
%       n = [0  1  1  2  2  2  3  3  3  3]; 0Ku%9wh-  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Ev;ocb,  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; ZM%z"hO9R  
%       y = zernfun(n,m,r(idx),theta(idx)); R]{AJ"p  
%       figure('Units','normalized') uua1_# a  
%       for k = 1:10 B>&eciY  
%           z(idx) = y(:,k); ku}I; k |  
%           subplot(4,7,Nplot(k)) hq^@t6!C\m  
%           pcolor(x,x,z), shading interp P>t[35/1  
%           set(gca,'XTick',[],'YTick',[]) S*1Km&  
%           axis square )pXw 3Fo  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) #D{//P|;  
%       end ly] n2RK  
% ;hLne0|)}  
%   See also ZERNPOL, ZERNFUN2. ~:%rg H  
k99ANW  
%   Paul Fricker 11/13/2006 yxa~R z/  
&`'gO 9  
mJ|7Jc  
% Check and prepare the inputs: rn?:utP  
% ----------------------------- o[!g,Gmoh  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) _8'FI_E3  
    error('zernfun:NMvectors','N and M must be vectors.') Q`}n; DV  
end E*Q><UU  
K>`7f]?H*e  
if length(n)~=length(m) #?z 1cgCg  
    error('zernfun:NMlength','N and M must be the same length.') ,e ~@  
end jbqhNsTNK  
1+^L,-k!  
n = n(:); :>[;XT<  
m = m(:); ?_F,HhQ  
if any(mod(n-m,2)) TvWhy`RQ  
    error('zernfun:NMmultiplesof2', ... <Z c:  
          'All N and M must differ by multiples of 2 (including 0).') ?)cNe:KY  
end Ir*,fyl  
G1"=}Wt`  
if any(m>n) {[4Y(l1  
    error('zernfun:MlessthanN', ... 66%#$WH#  
          'Each M must be less than or equal to its corresponding N.') U!-|.N,  
end ?6 "F.\ O@  
jz$)*Kdi*  
if any( r>1 | r<0 ) GGs3r;(t  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') oWpy ^=D_  
end 8<t?o'9I  
k:w\4Oqd  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) r^?%N3  
    error('zernfun:RTHvector','R and THETA must be vectors.') OwQ 9y<v  
end E$FXs~a  
yVxR||e  
r = r(:); >=2nAv/(  
theta = theta(:); gT R:9E:B  
length_r = length(r); Wv"[,5 Z13  
if length_r~=length(theta) PL8eM]XS  
    error('zernfun:RTHlength', ... ,F J9C3  
          'The number of R- and THETA-values must be equal.') (o\:rLZu  
end %rT XT  
,h1r6&MEY  
% Check normalization: +MQf2|--  
% -------------------- R9yK"  
if nargin==5 && ischar(nflag) P$@5&/]  
    isnorm = strcmpi(nflag,'norm'); t9PS5O ;  
    if ~isnorm 2D MH@U2  
        error('zernfun:normalization','Unrecognized normalization flag.') Lvc*L6  
    end 1C=}4^Pu  
else f$k#\=2%  
    isnorm = false; eR8qO"%2:  
end WZCX&ui  
H#G~b""mY  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EG0NikT?  
% Compute the Zernike Polynomials ;X\>oV3#  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {61NLF\0H  
oa`,|dA"  
% Determine the required powers of r: $<]y.nr|CX  
% ----------------------------------- QHsS|\u  
m_abs = abs(m); @yB!?x  
rpowers = []; k-T_,1l{  
for j = 1:length(n) jo<[|ZD  
    rpowers = [rpowers m_abs(j):2:n(j)]; ~?6V-m{>#  
end o)?"P;UhJX  
rpowers = unique(rpowers); 5gV8=Ml"V  
.d9VV&  
% Pre-compute the values of r raised to the required powers, i[^?24~ c  
% and compile them in a matrix: DSy,#yA  
% ----------------------------- [8SW0wsk  
if rpowers(1)==0 G_[|N>  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); eF}Q8]da  
    rpowern = cat(2,rpowern{:}); lV$U!v: b  
    rpowern = [ones(length_r,1) rpowern]; d Z"bc]z{  
else Os8]iNvW\  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); #0L :h ?L  
    rpowern = cat(2,rpowern{:}); 7esG$sVj(  
end w])bQ7)  
OV>T}Fq  
% Compute the values of the polynomials: E] t:_v  
% -------------------------------------- t22BO@gt74  
y = zeros(length_r,length(n)); KE16BjX@  
for j = 1:length(n) xvjHGgWSxc  
    s = 0:(n(j)-m_abs(j))/2; Cz?N[dhh  
    pows = n(j):-2:m_abs(j);  X\ \\RCp  
    for k = length(s):-1:1 JO7IzD\  
        p = (1-2*mod(s(k),2))* ... z8>KY/c  
                   prod(2:(n(j)-s(k)))/              ... {*t'h?b  
                   prod(2:s(k))/                     ... ED"5y  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 9|Jmj @9  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); ERGDo=j  
        idx = (pows(k)==rpowers); =t&B8+6  
        y(:,j) = y(:,j) + p*rpowern(:,idx); $|6Le; K  
    end [CRy>hfV  
     w>H!H6Q  
    if isnorm GMgsM6.R  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); '|4/aHU  
    end Lvv`_  
end ui G7  
% END: Compute the Zernike Polynomials D}cq_|mmn[  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <s+=v!  
`W?aq]4x5  
% Compute the Zernike functions: ^67P(h  
% ------------------------------ ]v94U b   
idx_pos = m>0; IDE@{Dy  
idx_neg = m<0; %'4dg k  
^bL.|vB  
z = y; 9eGM6qW\_  
if any(idx_pos) l%fnGe` _  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); wm*`  
end '4dnC2a]  
if any(idx_neg) og. dYs7W4  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); z O$SL8U  
end *[W!ng  
yn!LJT[~2  
% EOF zernfun
niuhelen 2011-03-12 23:01
function z = zernfun2(p,r,theta,nflag) AZl|; y  
%ZERNFUN2 Single-index Zernike functions on the unit circle. 4^70r9hV9  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated I.y|AQB  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive X-oou'4<  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, o0s+ roiD  
%   and THETA is a vector of angles.  R and THETA must have the same rI; e!EW  
%   length.  The output Z is a matrix with one column for every P-value, b%,5B  
%   and one row for every (R,THETA) pair. Jev@IORN\  
% .__X[Mzth3  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 1/gY]ghL  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) j*W]^uT,  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ;13lu1  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 W1r-uR  
%   for all p. }4_izKS  
% kc~Z1  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 q+gqa<kM  
%   Zernike functions (order N<=7).  In some disciplines it is wGnjuIR  
%   traditional to label the first 36 functions using a single mode Ik;~u8j1e  
%   number P instead of separate numbers for the order N and azimuthal Zn #ri 8S  
%   frequency M. i1  SP  
% "St,4 b  
%   Example: s'~_pP  
% lP9a*>=a  
%       % Display the first 16 Zernike functions S 5m1~fz  
%       x = -1:0.01:1; 6<]&T lS]  
%       [X,Y] = meshgrid(x,x); .t"s>jq 1  
%       [theta,r] = cart2pol(X,Y); vuDp_p*]S  
%       idx = r<=1; 6sQ"go$}  
%       p = 0:15; ms;Lu- UR  
%       z = nan(size(X)); fcJ#\-+E  
%       y = zernfun2(p,r(idx),theta(idx)); d<7xSRC   
%       figure('Units','normalized') "~d)$]+  
%       for k = 1:length(p) zFB$^)v"<  
%           z(idx) = y(:,k); YfxZ<  
%           subplot(4,4,k) |P5?0{  
%           pcolor(x,x,z), shading interp 68 -I2@&  
%           set(gca,'XTick',[],'YTick',[]) ;EF s2-{K  
%           axis square *RxbqB-  
%           title(['Z_{' num2str(p(k)) '}']) : ]CZS  
%       end =B<g_9d4  
% LsV!Sd  
%   See also ZERNPOL, ZERNFUN. QdC>fy  
zq{L:.#ha  
%   Paul Fricker 11/13/2006 N^mY/`2  
G;fP  
Q*mPU=<  
% Check and prepare the inputs: P-^Z7^o-bX  
% ----------------------------- c?<FMb3]  
if min(size(p))~=1 4~G9._  
    error('zernfun2:Pvector','Input P must be vector.') :kY][_  
end {T:2+iS9:  
Gt6$@ji4u  
if any(p)>35 $ZQPf  
    error('zernfun2:P36', ... ;"joebZ/  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... \8_&@uLm  
           '(P = 0 to 35).']) dx Mz!  
end Fo;xA  
1C.<@IZ  
% Get the order and frequency corresonding to the function number: KS(s<ip|  
% ----------------------------------------------------------------  g<UjB  
p = p(:); m:p1O3[R  
n = ceil((-3+sqrt(9+8*p))/2); Wv(VV[?/&  
m = 2*p - n.*(n+2); i/)Uj-*G)  
n/^wzG  
% Pass the inputs to the function ZERNFUN: s|EP/=9i  
% ---------------------------------------- \i`/k(  
switch nargin @xO< ~  
    case 3 93o}vy->  
        z = zernfun(n,m,r,theta); AO8`ItNZdT  
    case 4 \n:'>:0X!  
        z = zernfun(n,m,r,theta,nflag); s)ZL`S?</  
    otherwise 4\Q ?4ZX  
        error('zernfun2:nargin','Incorrect number of inputs.') #G9S[J=xe  
end ]'T-6  
T-|z18|!  
% EOF zernfun2
niuhelen 2011-03-12 23:01
function z = zernpol(n,m,r,nflag) 4V<s"  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. iSIj ?.  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ]_F%{8|  
%   order N and frequency M, evaluated at R.  N is a vector of lm]4zs /A  
%   positive integers (including 0), and M is a vector with the qjUQ2d  
%   same number of elements as N.  Each element k of M must be a &IOChQ`8P  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) t55 '  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is c5]^jUB6  
%   a vector of numbers between 0 and 1.  The output Z is a matrix a{Tv#P*!  
%   with one column for every (N,M) pair, and one row for every =AZ>2P  
%   element in R. =3/||b4c  
% hQ8/-#LO_  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- C4Bh#C  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is g4I(uEJk  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to rf]]I#C7  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 mv<z%y?Oj  
%   for all [n,m]. 8BLtTpu  
% ^aXyho  
%   The radial Zernike polynomials are the radial portion of the H$;K(,'  
%   Zernike functions, which are an orthogonal basis on the unit oDM}h +  
%   circle.  The series representation of the radial Zernike HtmJIH:  
%   polynomials is dbE $T  
% [T~O%ly7x&  
%          (n-m)/2 )Hl;9  
%            __ ,Iwri\  
%    m      \       s                                          n-2s M<g>z6   
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r #N@sJyI N  
%    n      s=0 65s|gfu/  
% VEo>uR  
%   The following table shows the first 12 polynomials. 4J-)+C/edx  
% cKFzn+  
%       n    m    Zernike polynomial    Normalization N<Ti]G  
%       --------------------------------------------- {$O.@#'  
%       0    0    1                        sqrt(2) ykX}T6T  
%       1    1    r                           2 Mq6.!j  
%       2    0    2*r^2 - 1                sqrt(6) }|{yd03 +  
%       2    2    r^2                      sqrt(6) m3P%E8<Q#  
%       3    1    3*r^3 - 2*r              sqrt(8) lj'c0k8  
%       3    3    r^3                      sqrt(8) &{/ `Q ,  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) m0XK?;\V  
%       4    2    4*r^4 - 3*r^2            sqrt(10) kM]?  
%       4    4    r^4                      sqrt(10) Xk(c2s&  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) SDDs}mV  
%       5    3    5*r^5 - 4*r^3            sqrt(12) 7cY_=X-?Y  
%       5    5    r^5                      sqrt(12) EJb"/oLla  
%       --------------------------------------------- sAC1Pda  
% VO#]IXaP  
%   Example: OUnt?[U\  
% >L?/Ph%d  
%       % Display three example Zernike radial polynomials SYeCz(H>d  
%       r = 0:0.01:1; WLv( K_3Y  
%       n = [3 2 5]; Wi hQj  
%       m = [1 2 1]; 2EycFjO  
%       z = zernpol(n,m,r); mNWmp_c,1  
%       figure < yE(p  
%       plot(r,z) #CHsH{d  
%       grid on I zM=?,`  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') g9'50<|J  
% M2%<4(UwI  
%   See also ZERNFUN, ZERNFUN2. j#3}nJB%#i  
n M,m#"AI  
% A note on the algorithm. YJ. 'Yc  
% ------------------------ :7@"EW  
% The radial Zernike polynomials are computed using the series .e"De-u  
% representation shown in the Help section above. For many special O&E1(M|*>  
% functions, direct evaluation using the series representation can qYGnebn@\  
% produce poor numerical results (floating point errors), because %* vYX0W"  
% the summation often involves computing small differences between Uk6Y6mU V  
% large successive terms in the series. (In such cases, the functions ZQsE07  
% are often evaluated using alternative methods such as recurrence 8/ZJkI  
% relations: see the Legendre functions, for example). For the Zernike VKS:d!}3E  
% polynomials, however, this problem does not arise, because the "Yq-s$yBi  
% polynomials are evaluated over the finite domain r = (0,1), and CZxQz  
% because the coefficients for a given polynomial are generally all ]J5[ZVz  
% of similar magnitude. Ep.,2H  
% e7>)Z  
% ZERNPOL has been written using a vectorized implementation: multiple /vD5C  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M]  ^qy$M>  
% values can be passed as inputs) for a vector of points R.  To achieve Ajs<a(,6  
% this vectorization most efficiently, the algorithm in ZERNPOL m/cbRuPWgP  
% involves pre-determining all the powers p of R that are required to )PB&w%J  
% compute the outputs, and then compiling the {R^p} into a single $6Nm`[V  
% matrix.  This avoids any redundant computation of the R^p, and Czid"Ih-  
% minimizes the sizes of certain intermediate variables. E)NH6 ~  
% 9Rb-QI  
%   Paul Fricker 11/13/2006 lVARe3#  
q !EJs:AS  
Rr|VGtg  
% Check and prepare the inputs: N@58R9P<p  
% ----------------------------- @HMt}zD  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 0A9x9l9Wd  
    error('zernpol:NMvectors','N and M must be vectors.') ;7K5Bo  
end bxqXFy/I  
w`c9_V  
if length(n)~=length(m) J=Ak+  J  
    error('zernpol:NMlength','N and M must be the same length.') |.*),t3 (w  
end h 34|v=8d  
z%`Tf&UL  
n = n(:); X>wB=z5PXK  
m = m(:); E`=y9r* Z  
length_n = length(n); ryW1OV6?_0  
/Fk LZm  
if any(mod(n-m,2)) agM.-MK  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') n-}:D<\7  
end jE&kN$.7j  
>XSe  
if any(m<0) Olj]A]v}  
    error('zernpol:Mpositive','All M must be positive.') dV<M$+;s]  
end ,B5Ptf#  
k#c BBrY  
if any(m>n) Y)hLu:P]  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') oZA|IF8U0  
end H{\tQ->(2  
]i8K )/  
if any( r>1 | r<0 ) PxfeU2^{0  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') fh b&_T  
end {,]BqFXv  
?" {+m  
if ~any(size(r)==1) m{JiF-=u  
    error('zernpol:Rvector','R must be a vector.') P qagep d  
end HsR#dp+s~  
prlB9,3|C  
r = r(:); oeDsJ6;  
length_r = length(r); ,au64sH  
1 _fFbb"  
if nargin==4 xv$^%(Ujp  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); tmRD$O%:  
    if ~isnorm e&OMW ,7  
        error('zernpol:normalization','Unrecognized normalization flag.') V~Tjz%<  
    end "f!*%SR: 1  
else |Mo# +{~c  
    isnorm = false; p? iJ'K  
end $\81WsL '  
?Ih24>:D  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pvI(hjMYPk  
% Compute the Zernike Polynomials y z9`1R2c  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v0"|J3  
Kzo{L  
% Determine the required powers of r: f#eTi&w  
% ----------------------------------- L9Fx Lw41  
rpowers = []; uqTOEHH7  
for j = 1:length(n) eb9qg.9Z  
    rpowers = [rpowers m(j):2:n(j)]; gW/H#T,  
end oxO}m7 ULH  
rpowers = unique(rpowers); nXi6Q+YI  
"{z9 L+  
% Pre-compute the values of r raised to the required powers, 1G.+)*:3  
% and compile them in a matrix: nz+o8L,  
% ----------------------------- g+ 2SB5 2D  
if rpowers(1)==0 M?[lpH3  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); P%ZWm=lg  
    rpowern = cat(2,rpowern{:}); ?z:xQ*#X  
    rpowern = [ones(length_r,1) rpowern]; }^`{YD  
else Rn(|  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); LTzf&TZbx5  
    rpowern = cat(2,rpowern{:}); \M>AN Z}  
end U?$v 1||  
_& KaI }O  
% Compute the values of the polynomials: _JR4 PKtx  
% -------------------------------------- ,Yi =s;E  
z = zeros(length_r,length_n); ik8e  
for j = 1:length_n Y] P}7GZ  
    s = 0:(n(j)-m(j))/2; OR?8F5o?p  
    pows = n(j):-2:m(j); })mez[UmZ  
    for k = length(s):-1:1 }PED#Uv  
        p = (1-2*mod(s(k),2))* ... h$ZF[Xbfe  
                   prod(2:(n(j)-s(k)))/          ... Jc:G7}j6  
                   prod(2:s(k))/                 ... .-RWlUe;,  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... jDXmre?  
                   prod(2:((n(j)+m(j))/2-s(k))); 6@l:(-(j2A  
        idx = (pows(k)==rpowers); i w m7M  
        z(:,j) = z(:,j) + p*rpowern(:,idx); p%3';7W\  
    end nF=Ig-NX^  
     /f# rN_4  
    if isnorm ?[m5|ty#  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); OVj,qL)  
    end k?J}-+Bm[|  
end X,y$!2QI  
NoKYHN^*w  
% 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)  #cmj?y()  
( E0be.  
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。 =UYc~VUYnT  
Rq\.RR](  
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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