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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 <v[,A8Q  
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! m&|?mTo>m  
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 '1r:z, o|  
function z = zernfun(n,m,r,theta,nflag) 9[6*FAFJPP  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. =UNzjmP503  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N m2<sVTN`^  
%   and angular frequency M, evaluated at positions (R,THETA) on the HcQ{ok9u  
%   unit circle.  N is a vector of positive integers (including 0), and 4U>  
%   M is a vector with the same number of elements as N.  Each element uu=e~K  
%   k of M must be a positive integer, with possible values M(k) = -N(k) zc,fJM  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, O2Rv^la  
%   and THETA is a vector of angles.  R and THETA must have the same Dw6Q2Gnv  
%   length.  The output Z is a matrix with one column for every (N,M) XRj<2U 5  
%   pair, and one row for every (R,THETA) pair. cKK 1$x  
% pU`Q[HOs  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike )BS./zD*[<  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ga~rllm;i  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral ?exV:OKLb  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ]eP&r?B  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized 3Xf}vdgdM$  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. MWsBZJRr  
% vVZ@/D6w  
%   The Zernike functions are an orthogonal basis on the unit circle. x8gUP  
%   They are used in disciplines such as astronomy, optics, and o& $Fc8bH  
%   optometry to describe functions on a circular domain. OYNs1yB  
% C8@SuJ  
%   The following table lists the first 15 Zernike functions. M_UhFY='  
% i+T$&$b  
%       n    m    Zernike function           Normalization =Q Otag1;  
%       -------------------------------------------------- IM)\-O\Wd  
%       0    0    1                                 1 NBE)DL  
%       1    1    r * cos(theta)                    2 RNp3lXf O  
%       1   -1    r * sin(theta)                    2 >,A:zbs&  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) FrB}2  
%       2    0    (2*r^2 - 1)                    sqrt(3) hU+sg~E  
%       2    2    r^2 * sin(2*theta)             sqrt(6) Z]":xl\7  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) m_Z%[@L  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) p?=rQte([  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) `gD'q5.z;3  
%       3    3    r^3 * sin(3*theta)             sqrt(8) US0)^TKrj  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) ezCsbV;. [  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) x:"_B  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) SpjL\ p0  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )DfmO  
%       4    4    r^4 * sin(4*theta)             sqrt(10) ?7/n s>}  
%       -------------------------------------------------- !YsL x[+  
% b 9F=}.4  
%   Example 1: D*DCMMp=0  
% XNf%vC>  
%       % Display the Zernike function Z(n=5,m=1) mn?< Zz  
%       x = -1:0.01:1; =MRg  
%       [X,Y] = meshgrid(x,x); Pc`d@q  
%       [theta,r] = cart2pol(X,Y); Pfe&wA't  
%       idx = r<=1; AnfJyltS  
%       z = nan(size(X)); rH$0h2  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); H)G ^ Y1  
%       figure [F*t2 -ta  
%       pcolor(x,x,z), shading interp uRh`qnL  
%       axis square, colorbar ePa1 @dI  
%       title('Zernike function Z_5^1(r,\theta)') (p-a;.Twj  
% sr sDnf  
%   Example 2: z#SBt`c  
% k2" Z:\?z  
%       % Display the first 10 Zernike functions 4[9~g=y>  
%       x = -1:0.01:1; T`Sp!  
%       [X,Y] = meshgrid(x,x);  }aRV)F  
%       [theta,r] = cart2pol(X,Y); q4|TwRx~  
%       idx = r<=1; 8sx\b  
%       z = nan(size(X)); +D*b!5[  
%       n = [0  1  1  2  2  2  3  3  3  3]; @]Aul9.h  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; cx&jnF#$  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; KFkKr>S :  
%       y = zernfun(n,m,r(idx),theta(idx)); 5<<e_n.2q  
%       figure('Units','normalized') 6cb;iA  
%       for k = 1:10 1%G<gbHpI  
%           z(idx) = y(:,k); nMNAn}~*M  
%           subplot(4,7,Nplot(k)) Y]0oF_ :7  
%           pcolor(x,x,z), shading interp 'bN\bbR  
%           set(gca,'XTick',[],'YTick',[]) Xl.h&x0? 8  
%           axis square (?72 vCc  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) $C##S@  
%       end xYtY}?!"  
% tMad 2,:  
%   See also ZERNPOL, ZERNFUN2. &$.Vi&{.  
& %ej=O  
%   Paul Fricker 11/13/2006 #9,!IW]l  
DzkE*vR  
- (VV  
% Check and prepare the inputs: j/5>zS  
% ----------------------------- 1c(1YGuH  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) @sO*O4os>  
    error('zernfun:NMvectors','N and M must be vectors.') JR<#el  
end &kB[jz_[A  
(9CB&LZ(+E  
if length(n)~=length(m) !:,d^L!bh  
    error('zernfun:NMlength','N and M must be the same length.') S)p{4`p%  
end R4"["T+L`  
|Vs|&0  
n = n(:); |xG|HJm,  
m = m(:); 9t(B{S  
if any(mod(n-m,2)) C0[Rf.*  
    error('zernfun:NMmultiplesof2', ... 5r.\maW  
          'All N and M must differ by multiples of 2 (including 0).') y@ J\h8_  
end hV;Tm7I2  
ps[TiW{q;  
if any(m>n) Q*+@"tk<  
    error('zernfun:MlessthanN', ... .L0pS.=LT  
          'Each M must be less than or equal to its corresponding N.') L01R.3Z+  
end s03 DL  
E1_FK1*V;  
if any( r>1 | r<0 ) -@b&qi7&S  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') e,>L&9] ZI  
end !g e,]@/  
wb 2N$Ew=  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) xojy[c#  
    error('zernfun:RTHvector','R and THETA must be vectors.') u|<Z};a  
end udX4SBq-pC  
+j_Vs+0  
r = r(:); <1.].A@b*  
theta = theta(:); s/0-DHd  
length_r = length(r); B< P H7  
if length_r~=length(theta) 2/RK pl &  
    error('zernfun:RTHlength', ... j 9XY%4.  
          'The number of R- and THETA-values must be equal.') ,G S8Gu  
end I&3L1rl3{*  
81V,yq]  
% Check normalization: t(VG#}  
% -------------------- >Y?B(I2e  
if nargin==5 && ischar(nflag) e2*0NT^R  
    isnorm = strcmpi(nflag,'norm'); ptQr8[FA  
    if ~isnorm )I9AF,K  
        error('zernfun:normalization','Unrecognized normalization flag.') UTc$zc7  
    end &NZN_%  
else VG*BAFs  
    isnorm = false; /xJ,nwp7  
end 1eZ">,F6<  
S;M'qwN  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fu}NH \{  
% Compute the Zernike Polynomials a8rsF  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Bs =V-0  
[Tby+pC  
% Determine the required powers of r: daE/v.a4|  
% ----------------------------------- l %=yT6  
m_abs = abs(m); p%+ 0^]v1  
rpowers = []; E^zgYkZO  
for j = 1:length(n) ,RKBGOz?f  
    rpowers = [rpowers m_abs(j):2:n(j)]; \ v44Vmfz  
end K~z*P 0g*  
rpowers = unique(rpowers); 'Sppm;?  
5s8S;Pb]<  
% Pre-compute the values of r raised to the required powers, M(HU^?B{'  
% and compile them in a matrix: /p~"?9b[ i  
% ----------------------------- h#@l'Cye  
if rpowers(1)==0 ( t#w@<  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 91r9RG>  
    rpowern = cat(2,rpowern{:}); Z2)f$ c  
    rpowern = [ones(length_r,1) rpowern]; TC?kuQI  
else h>sz@\{  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); l " pCxA  
    rpowern = cat(2,rpowern{:}); 9vWKyzMi  
end sqi~j(&\1  
y A?>v'K  
% Compute the values of the polynomials: g_G?gO  
% -------------------------------------- #QXv[%k  
y = zeros(length_r,length(n)); bYQ h{q  
for j = 1:length(n) qGuz`&i  
    s = 0:(n(j)-m_abs(j))/2; O_K@\<;~  
    pows = n(j):-2:m_abs(j); 0a QtJ0e16  
    for k = length(s):-1:1 k(C?6Gfj  
        p = (1-2*mod(s(k),2))* ... z}.!q{Q  
                   prod(2:(n(j)-s(k)))/              ... j|FGb:  
                   prod(2:s(k))/                     ... Msn)jh  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... "Ol;0>$  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); TP{lt6wws(  
        idx = (pows(k)==rpowers); MG>g?s'!  
        y(:,j) = y(:,j) + p*rpowern(:,idx); Kv:UQdnU[  
    end ;!, ]}2w*X  
     bR?-B>EB  
    if isnorm QtJe){(z+  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); E"!9WF(2t5  
    end (9';zw   
end b<_*~af  
% END: Compute the Zernike Polynomials 1)?^N`xF  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\_I% yF  
Z{+h~?63  
% Compute the Zernike functions: {.bLh 0  
% ------------------------------ 9<kKno  
idx_pos = m>0; M$1+,[^f  
idx_neg = m<0; AJ2Xq*fk  
8H./@~_ =  
z = y; Fly@"W4a  
if any(idx_pos) _Ta9rDSP]  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); fpM 4q  
end =}\]i*  
if any(idx_neg) cNw<k&w6F  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); [T"oqO4%]  
end $qD8vu )|j  
`=uCp^ +v  
% EOF zernfun
niuhelen 2011-03-12 23:01
function z = zernfun2(p,r,theta,nflag) -yg9ug  
%ZERNFUN2 Single-index Zernike functions on the unit circle. ^4Tr @g#]"  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated tH 5f;mY,  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive ~Cks)mJs  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, \4K8*`$  
%   and THETA is a vector of angles.  R and THETA must have the same T=VVK6Lc:  
%   length.  The output Z is a matrix with one column for every P-value, EYGJDv(S  
%   and one row for every (R,THETA) pair. sa#=#0yg  
% YM3oqS D  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ]j*o&6cQf  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) (loUO;S=  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) pTGq4v@6x  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 \b.2f+;3  
%   for all p. 3=t}py7M  
% uWx/V+w  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 o4Fh`?d}  
%   Zernike functions (order N<=7).  In some disciplines it is lADi  
%   traditional to label the first 36 functions using a single mode )pr pG !  
%   number P instead of separate numbers for the order N and azimuthal Y4@~NCU/  
%   frequency M. TT .EQv5  
% ~W5 fJd0  
%   Example: J2aA"BhdC"  
% ]!YzbvoR  
%       % Display the first 16 Zernike functions [+{ ot   
%       x = -1:0.01:1; bT[Q:#GL  
%       [X,Y] = meshgrid(x,x); ;TmwIZ  
%       [theta,r] = cart2pol(X,Y); +/\.%S/  
%       idx = r<=1; 5y=X?hF~)  
%       p = 0:15; Ip8 Ap$  
%       z = nan(size(X)); &_" 3~:N8k  
%       y = zernfun2(p,r(idx),theta(idx)); F!pUfF,&  
%       figure('Units','normalized') b44H2A .  
%       for k = 1:length(p) Rr%]/%  
%           z(idx) = y(:,k); kG?tgO?*  
%           subplot(4,4,k) ,>{4*PM(  
%           pcolor(x,x,z), shading interp m\1*/6oV  
%           set(gca,'XTick',[],'YTick',[]) SjlkKulMF  
%           axis square }5Y.N7F  
%           title(['Z_{' num2str(p(k)) '}']) M*t@Q|$:  
%       end eqeVz`  
% >%#J8  
%   See also ZERNPOL, ZERNFUN. J'@ I!Jc  
bGK&W;Myk  
%   Paul Fricker 11/13/2006 &\0LR?Nh  
r+m8#uR  
K/MIDH  
% Check and prepare the inputs: =C`v+NPM)|  
% ----------------------------- \VtCkb  
if min(size(p))~=1 C!qW:H  
    error('zernfun2:Pvector','Input P must be vector.') M/UJb1<  
end 'QCvN b6  
. s? ''/(  
if any(p)>35 =b`>ggw#  
    error('zernfun2:P36', ... 0>Mm |x*5  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... D3XQ>T[*q  
           '(P = 0 to 35).']) XHN?pVZ7  
end >#!n"i;  
Fi7pq2  
% Get the order and frequency corresonding to the function number: Lb2Bu>  
% ---------------------------------------------------------------- Z]9 )1&  
p = p(:); v]VIUVd  
n = ceil((-3+sqrt(9+8*p))/2); 4RTEXoXs  
m = 2*p - n.*(n+2); IH>+P]+3"3  
xFg=Tyq:  
% Pass the inputs to the function ZERNFUN: 9oc[}k-M  
% ---------------------------------------- diTzolY7  
switch nargin `awk@  
    case 3 j1/J9F'  
        z = zernfun(n,m,r,theta); OmU.9PDg-  
    case 4 v+Mt/8  
        z = zernfun(n,m,r,theta,nflag); xg3G  
    otherwise 0Fbq/63  
        error('zernfun2:nargin','Incorrect number of inputs.') kx'6FkZPIr  
end $Q47>/CUc^  
bzUc;&WDz  
% EOF zernfun2
niuhelen 2011-03-12 23:01
function z = zernpol(n,m,r,nflag) _ZR2?y-M  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. M.|hnGX N  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of #wC4$y<>  
%   order N and frequency M, evaluated at R.  N is a vector of |W#^L`!G  
%   positive integers (including 0), and M is a vector with the oxGOn('  
%   same number of elements as N.  Each element k of M must be a apw8wL2  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) ENqJ9%sk7  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 2H]&3kM3X  
%   a vector of numbers between 0 and 1.  The output Z is a matrix Zqx5I~  
%   with one column for every (N,M) pair, and one row for every Dhef|E<  
%   element in R. VaQ}XM  
% ;| \Ojuf  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- C #TS  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is >@rp]xx  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ]^ j)4us  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 i(U*<1y  
%   for all [n,m]. 2RM0ca _F  
% Mb$&~!  
%   The radial Zernike polynomials are the radial portion of the XqJ@NgsY  
%   Zernike functions, which are an orthogonal basis on the unit s \kkD *  
%   circle.  The series representation of the radial Zernike B&.XGo)  
%   polynomials is u 4)i7  
% lW>bX C  
%          (n-m)/2 4|Z3;;%+  
%            __ ,&l>^w/  
%    m      \       s                                          n-2s '<XG@L  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r L\n_q6n  
%    n      s=0 r--"JO%2  
% ;itz` 9T  
%   The following table shows the first 12 polynomials. jfgAI7;b  
% y;Dw%m  
%       n    m    Zernike polynomial    Normalization >TtkG|/U-T  
%       --------------------------------------------- n{UB^-}5  
%       0    0    1                        sqrt(2) eb+[=nmP  
%       1    1    r                           2 L {\B9b2  
%       2    0    2*r^2 - 1                sqrt(6) eqjl$QWPJS  
%       2    2    r^2                      sqrt(6) [>6:xGSe9X  
%       3    1    3*r^3 - 2*r              sqrt(8) ~BZA_w"`1  
%       3    3    r^3                      sqrt(8) nk6xavQji  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) WH'[~O  
%       4    2    4*r^4 - 3*r^2            sqrt(10) fv`%w  
%       4    4    r^4                      sqrt(10) c()F%e:n  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Zkxt>%20~  
%       5    3    5*r^5 - 4*r^3            sqrt(12) 0! !pNK%(  
%       5    5    r^5                      sqrt(12) iyj&O"  
%       --------------------------------------------- v?Y9z!M  
% neOR/]  
%   Example: mtJI#P  
% tR2IjvmsX  
%       % Display three example Zernike radial polynomials nc l-VN  
%       r = 0:0.01:1; i<&2Ffvq  
%       n = [3 2 5]; E#_}y}7JY  
%       m = [1 2 1]; 4Jo:^JV  
%       z = zernpol(n,m,r); qFvtqv2  
%       figure "4L' 2w+  
%       plot(r,z) Af*^u|#  
%       grid on #ljfcQm  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') -bKli<C  
% FtE%<QHt  
%   See also ZERNFUN, ZERNFUN2. J^1w& 40  
{]|};E[}m  
% A note on the algorithm. i}M&1E  
% ------------------------ V&NOp  
% The radial Zernike polynomials are computed using the series 5v>(xl  
% representation shown in the Help section above. For many special ,D~C40f  
% functions, direct evaluation using the series representation can })s s.  
% produce poor numerical results (floating point errors), because SRj|XCd  
% the summation often involves computing small differences between {$Fg+~   
% large successive terms in the series. (In such cases, the functions 3!ulBiMh  
% are often evaluated using alternative methods such as recurrence _RjM .  
% relations: see the Legendre functions, for example). For the Zernike K3 "co1]u  
% polynomials, however, this problem does not arise, because the cH"M8gP#  
% polynomials are evaluated over the finite domain r = (0,1), and ly6?jVJ  
% because the coefficients for a given polynomial are generally all Vk>aU3\c  
% of similar magnitude. o),i2  
% ~@L$}Eu  
% ZERNPOL has been written using a vectorized implementation: multiple >#c]rk:  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] ,?i#NN5p  
% values can be passed as inputs) for a vector of points R.  To achieve ^=Up U B  
% this vectorization most efficiently, the algorithm in ZERNPOL zneK)C8&q3  
% involves pre-determining all the powers p of R that are required to {f)",#  
% compute the outputs, and then compiling the {R^p} into a single sx(yG9  
% matrix.  This avoids any redundant computation of the R^p, and Z/56JYt!~  
% minimizes the sizes of certain intermediate variables. /koNcpJ  
% /1Rm^s)2z  
%   Paul Fricker 11/13/2006 y]M/oH  
q4(&.Al\@  
E%jOJA  
% Check and prepare the inputs: vZ$uD,@;.  
% ----------------------------- ~])\xC  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) rN} {v}n  
    error('zernpol:NMvectors','N and M must be vectors.') F]SexP4:A  
end !^G+@~U  
}q27M  
if length(n)~=length(m) $eRxCX?b2  
    error('zernpol:NMlength','N and M must be the same length.') *F~"4g  
end 3vmLftZE}  
c?b?x 6 2  
n = n(:); K'n^, t  
m = m(:); (a]'}c$X9`  
length_n = length(n); >MS}7Hk\  
w doA>a?q  
if any(mod(n-m,2)) pk(<],0]X  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') A^%z;( 0p  
end op&,&  
=4+UX*&i?.  
if any(m<0) )!p=0&z@{  
    error('zernpol:Mpositive','All M must be positive.') ]#))#-&1  
end '-gk))u>)  
%+FM$xyJ  
if any(m>n) yBht4"\Al  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') uoaF(F-  
end pg*'2AT  
d<(1^Rto  
if any( r>1 | r<0 ) S #&HB  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') D@5&xd_@4  
end tCtR(mG=A  
7^as~5'&-  
if ~any(size(r)==1) #qm<4]9 1  
    error('zernpol:Rvector','R must be a vector.') Yca9G?^\v  
end W{ @lt}  
Vg6?a  
r = r(:); q.~.1 '`!  
length_r = length(r); 8p>%}LX/  
 CG$S?  
if nargin==4 v?n`kw  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); _(J- MCY\  
    if ~isnorm M+)%gnq`u  
        error('zernpol:normalization','Unrecognized normalization flag.') 1:q55!b  
    end ?2_u/x  
else 0!_D M^3  
    isnorm = false; ^*%p]r  
end m!N_TOl-^  
f1hi\p0q  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Rb\=\  
% Compute the Zernike Polynomials 1\kOjF)l  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B58H7NH ;G  
SECL(@0(^  
% Determine the required powers of r: {foF[M  
% ----------------------------------- ~ E>D0o  
rpowers = []; a5L#c=  
for j = 1:length(n) REnRpp$  
    rpowers = [rpowers m(j):2:n(j)]; ~e,  
end g4RkkoZ>)  
rpowers = unique(rpowers); C<6u}czA  
bN<c5  
% Pre-compute the values of r raised to the required powers, eV1O#FLbi  
% and compile them in a matrix: Qj[4gN?}=  
% ----------------------------- %jKR\f G  
if rpowers(1)==0 <s]K~ Vo  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); V|#B=W  
    rpowern = cat(2,rpowern{:}); (RWZ [-;)  
    rpowern = [ones(length_r,1) rpowern]; =lr*zeHLC  
else NT= ?@uxD  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 5#$E4k:YV  
    rpowern = cat(2,rpowern{:}); [$8*(d"F'  
end %w/o#*j<;  
W4|1wd}.t  
% Compute the values of the polynomials: Ud`V"X  
% -------------------------------------- ZV_mP'1*  
z = zeros(length_r,length_n); E J q=MP  
for j = 1:length_n .Z'CqBr[:  
    s = 0:(n(j)-m(j))/2; }@!d(U*  
    pows = n(j):-2:m(j); `: i|y  
    for k = length(s):-1:1 Drk9F"J  
        p = (1-2*mod(s(k),2))* ... ZJ=-cE2n  
                   prod(2:(n(j)-s(k)))/          ... qECc[)B  
                   prod(2:s(k))/                 ... 4kxy7] W  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... XRJ<1w:  
                   prod(2:((n(j)+m(j))/2-s(k))); R 4E0avt  
        idx = (pows(k)==rpowers); j05ahquI  
        z(:,j) = z(:,j) + p*rpowern(:,idx); vb{&T<  
    end $J=9$.4"  
     HR.S.(t[_  
    if isnorm [q9TTJ@2  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 1PjSa4  
    end rAn''X6H  
end jR }h3!  
W\N-~9UA  
% 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)  kp xd+w  
Ct$e`H!;  
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。 +)L 'qbCSM  
l'B`f)  
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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