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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 ~_F<"40  
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! r h c&#JS  
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 Ie[DTy  
function z = zernfun(n,m,r,theta,nflag) #0:rBKm,  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. I_Omv{&u  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ]m :Y|,:6  
%   and angular frequency M, evaluated at positions (R,THETA) on the 'A,)PZL9i  
%   unit circle.  N is a vector of positive integers (including 0), and $q##Tys  
%   M is a vector with the same number of elements as N.  Each element 6@VgLa,  
%   k of M must be a positive integer, with possible values M(k) = -N(k) e0M'\'J  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, B"rO  
%   and THETA is a vector of angles.  R and THETA must have the same Ki>XLX,er=  
%   length.  The output Z is a matrix with one column for every (N,M) (sSGJS'X  
%   pair, and one row for every (R,THETA) pair. K 8W99:v  
% &W}6Xg(  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike X G E.*aI  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), y"|gC!V}  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral %R<xe.X  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, &Z(6i}f,Gp  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized lf\^!E:  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. *mkVk7]c  
% !ou;yE&<,  
%   The Zernike functions are an orthogonal basis on the unit circle. Nj.;mr<  
%   They are used in disciplines such as astronomy, optics, and w8bvqTQ  
%   optometry to describe functions on a circular domain. /@1pm/>ZaN  
% LvMA('4  
%   The following table lists the first 15 Zernike functions. {TvB3QOsj  
% mRy0zN>?  
%       n    m    Zernike function           Normalization !j& #R%D  
%       -------------------------------------------------- F#~*j  
%       0    0    1                                 1 VHG}'r9KC%  
%       1    1    r * cos(theta)                    2 >=86*U~  
%       1   -1    r * sin(theta)                    2 <b?$-Rx  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) S4pEBbV^n  
%       2    0    (2*r^2 - 1)                    sqrt(3) Q)=2%X  
%       2    2    r^2 * sin(2*theta)             sqrt(6) y@r0"cvz9  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) (o^?i2)g  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) $$XeCPs 0  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) rl x6a@MiD  
%       3    3    r^3 * sin(3*theta)             sqrt(8) {$V2L4  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) [{: l?  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) dU2:H}  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) k\r^GB  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) C#B|^A_  
%       4    4    r^4 * sin(4*theta)             sqrt(10) eCiI=HcW;  
%       -------------------------------------------------- j^/=.cD|  
% $}fY B/  
%   Example 1: ltKMvGEF  
% %/d1x  
%       % Display the Zernike function Z(n=5,m=1) ,20l` :  
%       x = -1:0.01:1; f@k.4aS  
%       [X,Y] = meshgrid(x,x); ^b4i9n,t1  
%       [theta,r] = cart2pol(X,Y); Tv9\` F[  
%       idx = r<=1; tO?-@Qf/9<  
%       z = nan(size(X)); {q^UWv?1  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); dK4w$~j{k  
%       figure q8HnPXV  
%       pcolor(x,x,z), shading interp F:~@e(  
%       axis square, colorbar DG}s`'  
%       title('Zernike function Z_5^1(r,\theta)') y8Rq2jI;(e  
% Y ` Z,52  
%   Example 2: nX\mCO4T  
% mW~*GD~r  
%       % Display the first 10 Zernike functions +|TXKhm{  
%       x = -1:0.01:1; !L<z(dV|(  
%       [X,Y] = meshgrid(x,x); 5vLA)Al3  
%       [theta,r] = cart2pol(X,Y); qt6@]Y  
%       idx = r<=1; /(?s\}O  
%       z = nan(size(X)); ";/ogFi  
%       n = [0  1  1  2  2  2  3  3  3  3]; nFWiS~(#sW  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; =MvB9gx@r  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; qC5IV}9`  
%       y = zernfun(n,m,r(idx),theta(idx)); )Cat$)I#,  
%       figure('Units','normalized') C{+JrHV%h  
%       for k = 1:10 $R+rB;=a!  
%           z(idx) = y(:,k); ?6HnN0A)  
%           subplot(4,7,Nplot(k)) 4$81ilBcL  
%           pcolor(x,x,z), shading interp :i|]iXEI"  
%           set(gca,'XTick',[],'YTick',[]) (g&@E(@]?  
%           axis square Z:^ S-h  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ~SmFDg$/m  
%       end 0{I-x^FI  
% Xq<_r^  
%   See also ZERNPOL, ZERNFUN2. +~=j3U  
SbQ:vAE*ho  
%   Paul Fricker 11/13/2006 y.s\MWvv>u  
-w f>N:  
m4yWhUi(o  
% Check and prepare the inputs: 9 Q*:II  
% ----------------------------- i52JY&N  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Z>l<.T"t'  
    error('zernfun:NMvectors','N and M must be vectors.') ZAn9A>5_  
end .&`apQD}  
"{trK?-8%  
if length(n)~=length(m) u \<APn  
    error('zernfun:NMlength','N and M must be the same length.') A8o)^T(vJ  
end eNO[ikm  
uvw1 _j?  
n = n(:); OJhMM-  
m = m(:); Z[0/x.pp$  
if any(mod(n-m,2)) `m #i|8  
    error('zernfun:NMmultiplesof2', ... ~N7;. 3 7  
          'All N and M must differ by multiples of 2 (including 0).') PS=q):R|  
end V F b  
E}THG=6  
if any(m>n) w9mAeGyE  
    error('zernfun:MlessthanN', ... 7 toIbC#  
          'Each M must be less than or equal to its corresponding N.') g++-v HD  
end C\OZs%]At  
e}P@7e  h  
if any( r>1 | r<0 ) yk(r R  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') Z9*@w`x^u  
end )vpYVr-  
TtH!5{$s  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) }`!-WY  
    error('zernfun:RTHvector','R and THETA must be vectors.') lR9uD9Dr  
end I?Z"YR+MQ  
TP~1-(M)}  
r = r(:); DrbjklcUU  
theta = theta(:); jw 5 U-zi  
length_r = length(r); P,xJVo\  
if length_r~=length(theta) iq^;csyKb  
    error('zernfun:RTHlength', ... B(5>H2  
          'The number of R- and THETA-values must be equal.') .:GOKyr(~  
end Hs_7oy|P  
b'z\|jY  
% Check normalization: SLUQFoz}  
% -------------------- E@#<p-@~  
if nargin==5 && ischar(nflag) wh2E$b(-  
    isnorm = strcmpi(nflag,'norm'); 0\s&;@xKk  
    if ~isnorm -M_>]ubG  
        error('zernfun:normalization','Unrecognized normalization flag.') x9S9%JG :  
    end #!%zf{(C+  
else _OS,zZ0  
    isnorm = false; SU6Aq?`@  
end 6j+X@|2^  
W-Vc6cq  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?STO#<a  
% Compute the Zernike Polynomials lV$#>2Hh5  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {E7STLQ_%  
F%af05L[  
% Determine the required powers of r: B#35)QI  
% ----------------------------------- -YmIRocx  
m_abs = abs(m); {,Rlq  
rpowers = []; [1ClZ~f  
for j = 1:length(n) &\Lu}t7Ru  
    rpowers = [rpowers m_abs(j):2:n(j)]; !IB}&m  
end q)KOI` A  
rpowers = unique(rpowers); }$r]\v  
4HX;9HPHE<  
% Pre-compute the values of r raised to the required powers, ry@p  
% and compile them in a matrix: kHhku!CH  
% ----------------------------- rLA-q||  
if rpowers(1)==0 N:S2X+}(  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); N 7Y X  
    rpowern = cat(2,rpowern{:}); [9:";JSl"Y  
    rpowern = [ones(length_r,1) rpowern]; q">}3`k  
else bd% M.,  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); +c, ^KHW  
    rpowern = cat(2,rpowern{:}); `&xdSH  
end 9zrTf%m F  
q^n LC6q  
% Compute the values of the polynomials: \Sv|yQUT  
% -------------------------------------- @$ lX%p>  
y = zeros(length_r,length(n)); 0>]&9'cn  
for j = 1:length(n) 5,V*aP  
    s = 0:(n(j)-m_abs(j))/2; &GvSgdttv  
    pows = n(j):-2:m_abs(j); H@~tJ\L  
    for k = length(s):-1:1 fX6pW%Q'6  
        p = (1-2*mod(s(k),2))* ... JG1q5j##]b  
                   prod(2:(n(j)-s(k)))/              ... qPWf=s7!  
                   prod(2:s(k))/                     ... [p}~M-$V8Y  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... -1`}|t;  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); >!{8)ti  
        idx = (pows(k)==rpowers); Ggst s  
        y(:,j) = y(:,j) + p*rpowern(:,idx); bAuiMw7!  
    end bn5O2  
     pSIXv%1J  
    if isnorm pGy k61  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); VGu(HB8n#  
    end ]KXyi;n2  
end U5@B7v1  
% END: Compute the Zernike Polynomials Bss *-K]  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jO|`aUY Tf  
8*&73cp  
% Compute the Zernike functions: ua &uR7  
% ------------------------------ #F2DEo^0  
idx_pos = m>0; QZa^Cng~  
idx_neg = m<0; ?hR0 MnP  
AN[pjC<  
z = y; b9 li   
if any(idx_pos) "wKJ8  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); I,,SR"  
end g}K/ba'  
if any(idx_neg) gm8Jx hL  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); .h4Z\R`  
end E?|NYu#I6  
@,6*yyO  
% EOF zernfun
niuhelen 2011-03-12 23:01
function z = zernfun2(p,r,theta,nflag) M(C}2.20  
%ZERNFUN2 Single-index Zernike functions on the unit circle. @Gw]cm  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated )J+rt^4|  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive ,1JQjsR   
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, yv>uzb`N  
%   and THETA is a vector of angles.  R and THETA must have the same {TMng&  
%   length.  The output Z is a matrix with one column for every P-value, ) .' + {  
%   and one row for every (R,THETA) pair. u(? U[pe[  
% 0oBAJP  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike B,Tv9(sv  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) wgvCgr<  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) _ D8 zKp  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ?:l:fS0:{  
%   for all p. CL-?Mi=Uc  
% -~4kh]7%  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 -4F}I3I  
%   Zernike functions (order N<=7).  In some disciplines it is U7f o4y1}  
%   traditional to label the first 36 functions using a single mode NE) w$>0M  
%   number P instead of separate numbers for the order N and azimuthal :J2^Y4l2  
%   frequency M. XH/!A`ZK  
% VsK8:[Al  
%   Example: Jcs /i  
% erTly2-SJ  
%       % Display the first 16 Zernike functions CD^@*jH9"  
%       x = -1:0.01:1; -y?ve od#  
%       [X,Y] = meshgrid(x,x); xUa9>=JU{  
%       [theta,r] = cart2pol(X,Y); iXXaB +w  
%       idx = r<=1; *UN*&DmF  
%       p = 0:15; HL>l.IG?  
%       z = nan(size(X)); een62-`  
%       y = zernfun2(p,r(idx),theta(idx)); 6Iz!_  
%       figure('Units','normalized') Lk^bzW>f  
%       for k = 1:length(p) {CVZ7tU7]  
%           z(idx) = y(:,k); kXUJlLod  
%           subplot(4,4,k) wGIRRM !b  
%           pcolor(x,x,z), shading interp k][{4~z  
%           set(gca,'XTick',[],'YTick',[]) (r}StR+  
%           axis square Zc&pJP+M'U  
%           title(['Z_{' num2str(p(k)) '}']) z\K %  
%       end ]2iIk=r$  
% 1L\r:mx3  
%   See also ZERNPOL, ZERNFUN. 78?{;iNv  
q9InO]s&~=  
%   Paul Fricker 11/13/2006 Ip8:~Fl]  
p_apVm\t_  
4U'sBaY!K  
% Check and prepare the inputs: fJ\ u8  
% ----------------------------- Z7e"4w A  
if min(size(p))~=1 iF.f*3-NJB  
    error('zernfun2:Pvector','Input P must be vector.') Cgf4E{\U!  
end 7.n\a@I/  
92b}N|u  
if any(p)>35 Tpkt'|8  
    error('zernfun2:P36', ... .uN(44^+x  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Cyg(~7]  
           '(P = 0 to 35).']) 3k8. 5W  
end 'C:i5?zh(q  
"l#"c{ee{  
% Get the order and frequency corresonding to the function number: 8hyX He  
% ---------------------------------------------------------------- &rG]]IO  
p = p(:); MBQ|*}+;  
n = ceil((-3+sqrt(9+8*p))/2); -ntQqHs  
m = 2*p - n.*(n+2); />>KCmc  
R7FI{ A  
% Pass the inputs to the function ZERNFUN: WBzPSnS2  
% ---------------------------------------- F)/4#[  
switch nargin -ni@+Dy  
    case 3 9]/:B8k  
        z = zernfun(n,m,r,theta); evVxzU&  
    case 4 rO/Sj<0^  
        z = zernfun(n,m,r,theta,nflag); G2|G}#E  
    otherwise #D >:'ezm  
        error('zernfun2:nargin','Incorrect number of inputs.') 6W;`}'ap  
end M%SNq|Lo  
u{l4O1k/c  
% EOF zernfun2
niuhelen 2011-03-12 23:01
function z = zernpol(n,m,r,nflag) }XCh>LvX  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. cVmF'g  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of C}9|e?R[Rz  
%   order N and frequency M, evaluated at R.  N is a vector of \3F)M`g  
%   positive integers (including 0), and M is a vector with the sY]pszjT  
%   same number of elements as N.  Each element k of M must be a 7.kH="@  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) BcQw-<veu  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is mFd|JbW  
%   a vector of numbers between 0 and 1.  The output Z is a matrix vP%:\u:{  
%   with one column for every (N,M) pair, and one row for every aH, NS   
%   element in R. -7 Kstc-  
% l)EtK&er(}  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- eqWs(`  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 2xBh  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to dLR[<@E  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 KT*"Sbh  
%   for all [n,m]. '3E25BsL  
% Pb59RE:7V  
%   The radial Zernike polynomials are the radial portion of the @?0))@kPc3  
%   Zernike functions, which are an orthogonal basis on the unit ,E+\SBQS_  
%   circle.  The series representation of the radial Zernike A43[i@o  
%   polynomials is FEV Ya#S  
% m?;/H  
%          (n-m)/2 /}5)[9GC  
%            __ -Gm}i8;  
%    m      \       s                                          n-2s N Zwi3  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r CXi:?6OG  
%    n      s=0 b,(<74!#8  
% DKl\N~{F  
%   The following table shows the first 12 polynomials. [Qqss8a  
% ]XYD2fR2qA  
%       n    m    Zernike polynomial    Normalization aq(i^d  
%       --------------------------------------------- K_qA[n  
%       0    0    1                        sqrt(2) }5B\:*yW  
%       1    1    r                           2 B? XK;*])  
%       2    0    2*r^2 - 1                sqrt(6) "!D,9AkZS  
%       2    2    r^2                      sqrt(6) "VMb1Zhf  
%       3    1    3*r^3 - 2*r              sqrt(8) ykxAm\O  
%       3    3    r^3                      sqrt(8) (!Fu5m=<8  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 1Z_ H% (  
%       4    2    4*r^4 - 3*r^2            sqrt(10) |"eC0u  
%       4    4    r^4                      sqrt(10) SUsdX[byb  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ?|n@ %'  
%       5    3    5*r^5 - 4*r^3            sqrt(12) SkU9iW(k  
%       5    5    r^5                      sqrt(12) Y{X%C\  
%       --------------------------------------------- 0P;LH3sx  
% gaE8\JSr  
%   Example: #"[EVF0%1D  
% sAF="uB  
%       % Display three example Zernike radial polynomials D\DwBZ>  
%       r = 0:0.01:1; U.N& ~S  
%       n = [3 2 5]; V7 hO}  
%       m = [1 2 1]; :y_] JL;w  
%       z = zernpol(n,m,r); Lu4>C2{  
%       figure 5BM6Pnle  
%       plot(r,z) f{{J_""?&  
%       grid on `SS[[FT$>  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') !H6X%hlk  
% 0J;Qpi!u2v  
%   See also ZERNFUN, ZERNFUN2. GB)< 5I  
'GLpSWL+*  
% A note on the algorithm. gMU%.%p2  
% ------------------------ CH5>u  
% The radial Zernike polynomials are computed using the series H9WXp&  
% representation shown in the Help section above. For many special >g{b'Xx  
% functions, direct evaluation using the series representation can TW1#'G_#  
% produce poor numerical results (floating point errors), because pK0@H"$8  
% the summation often involves computing small differences between zbvV:9N  
% large successive terms in the series. (In such cases, the functions C87 9eeJ  
% are often evaluated using alternative methods such as recurrence +Y>oNX1KN  
% relations: see the Legendre functions, for example). For the Zernike [|(=15;  
% polynomials, however, this problem does not arise, because the 8<=sUO  
% polynomials are evaluated over the finite domain r = (0,1), and JX#0<U|L  
% because the coefficients for a given polynomial are generally all \\`(x:\  
% of similar magnitude. ylQ9Su>o  
% VWqZ`X  
% ZERNPOL has been written using a vectorized implementation: multiple yI%q3lB}^  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] XS.*CB_m_  
% values can be passed as inputs) for a vector of points R.  To achieve f#gV>.P;h\  
% this vectorization most efficiently, the algorithm in ZERNPOL w`gT]Rn  
% involves pre-determining all the powers p of R that are required to @\a~5CLN  
% compute the outputs, and then compiling the {R^p} into a single }R_Rw:W  
% matrix.  This avoids any redundant computation of the R^p, and XaMsIyhI  
% minimizes the sizes of certain intermediate variables. +R;s< pZ^  
% 6S0Gjekr  
%   Paul Fricker 11/13/2006 6wZ)GLW[  
^S!;snhn  
aF>&X-2  
% Check and prepare the inputs: ; {v2s;  
% ----------------------------- "ZFH_5<  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) goJ'z|))  
    error('zernpol:NMvectors','N and M must be vectors.') bo@, B  
end Wmx3@]<  
s3 VD6xi7  
if length(n)~=length(m) @\W-=YKLg  
    error('zernpol:NMlength','N and M must be the same length.') Tf('iZ2+  
end YoWXHg!U  
Ns5P,[pBOZ  
n = n(:); eL{$=Um  
m = m(:); aS~~*UHW  
length_n = length(n); i>dFpJ  
Z{?T1 =n  
if any(mod(n-m,2))  "+Sq}WR  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ]kF1~kXBe  
end Ei=rBi  
vb}c)w dp?  
if any(m<0) Fqh./@o  
    error('zernpol:Mpositive','All M must be positive.') e&!8UYP  
end J|<C;[du>  
nC)"% Sa  
if any(m>n) 4 =T_h`  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') (^E5y,H<g  
end W{~ y< `D  
 2IGU{&s  
if any( r>1 | r<0 ) w(K|0|t  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') g1(5QWb  
end Hx!eCTO:*  
k$pND,Ws  
if ~any(size(r)==1) N7YCg  
    error('zernpol:Rvector','R must be a vector.') '%o^#gJp  
end z#GSt ZT  
.K`n;lVs  
r = r(:); /vu]ch  
length_r = length(r); >qmNT/  
w^,Xa  
if nargin==4 [70 5[  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); Y/T-q<ag8  
    if ~isnorm )<qL8#["U  
        error('zernpol:normalization','Unrecognized normalization flag.') Z'Pe%}3  
    end hTmJ ~m'J  
else yB 'C9wEH  
    isnorm = false; {L [   
end O\)Kg2  
^YB2E*  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5fYWuc9}z  
% Compute the Zernike Polynomials g yhy0  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% m9+?>/R  
B]6Lbp"oo  
% Determine the required powers of r: ,t:P  
% ----------------------------------- p(K ^Zc  
rpowers = []; ,+I]\ZeO  
for j = 1:length(n) n$9Xj@  +  
    rpowers = [rpowers m(j):2:n(j)]; uX]]wj-R3  
end ]'w5s dP  
rpowers = unique(rpowers); S4-jFD)U  
H4j1yD(d  
% Pre-compute the values of r raised to the required powers, g,W34*7=Q  
% and compile them in a matrix: Of-xGo YZ  
% ----------------------------- =6'bGC%c  
if rpowers(1)==0 rS8\Vf]F  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Upcx@zJ  
    rpowern = cat(2,rpowern{:}); "fUNrhCx  
    rpowern = [ones(length_r,1) rpowern]; 6a_U[-a9;  
else 0Ub'=`]5a  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ]yL+lv  
    rpowern = cat(2,rpowern{:}); O'{kNr{u  
end `AvK=]  
LsaX HI/?b  
% Compute the values of the polynomials: zmL VFGnS  
% -------------------------------------- )jg*u}u 0  
z = zeros(length_r,length_n); NL;sn"  
for j = 1:length_n P#`M8k  
    s = 0:(n(j)-m(j))/2; o4Cq  /K  
    pows = n(j):-2:m(j); _VTpfeL@n  
    for k = length(s):-1:1 `%"x'B`mM  
        p = (1-2*mod(s(k),2))* ... \okv}x^L=Z  
                   prod(2:(n(j)-s(k)))/          ... \NEk B&^n  
                   prod(2:s(k))/                 ... 'J5F+, \Ka  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... @+Sr~:K  
                   prod(2:((n(j)+m(j))/2-s(k))); MNb9~kM  
        idx = (pows(k)==rpowers); CDsl)  
        z(:,j) = z(:,j) + p*rpowern(:,idx); T?6<1nU)  
    end _ qwf3Q@  
     1~3dX[&  
    if isnorm O:pg+o&  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 4;r,U{uR  
    end "@/pQoLy  
end TW70z]B  
cAot+N+9|]  
% 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)  8F`8=L NO  
.Y?]r6CC/  
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。 K%mR=u#%&  
C58o="L3S  
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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