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<2U5 % 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 =QOtag1; % -------------------------------------------------- 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>} % -------------------------------------------------- !YsLx[+ % 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(1 YGuH 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 s03DL 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 !ge,]@/ wb2N$Ew= if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) xojy[c# error('zernfun:RTHvector','R and THETA must be vectors.') u|<Z};a end udX4SBq-pC +j_Vs+0 r = r(:); <1.].A@b* theta = theta(:); s/0-DHd length_r = length(r); B<
P H7 if length_r~=length(theta) 2/RK
pl & error('zernfun:RTHlength', ... j9XY%4. 'The number of R- and THETA-values must be equal.') ,GS8Gu end I&3L1rl3{*
81V,yq] % Check normalization: t(VG#} % -------------------- >Y?B(I2e if nargin==5 && ischar(nflag) e2*0NT^R isnorm = strcmpi(nflag,'norm'); ptQr8[FA if ~isnorm )I9AF,K error('zernfun:normalization','Unrecognized normalization flag.') UTc$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)]; \v44 Vmfz end K~z*P0g* rpowers = unique(rpowers); 'Sppm;? 5s8S;Pb]< | |