| 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 fkKk/M>1 function z = zernfun(n,m,r,theta,nflag) ;t6)(d4z? %ZERNFUN Zernike functions of order N and frequency M on the unit circle. iO$Z?Dyg9 % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N olA 1,8 % and angular frequency M, evaluated at positions (R,THETA) on the UdGa#rcNW % unit circle. N is a vector of positive integers (including 0), and 1u `{yl*+? % M is a vector with the same number of elements as N. Each element 'xK ,|U % k of M must be a positive integer, with possible values M(k) = -N(k) ''p7!V? % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Gl am(V1 % and THETA is a vector of angles. R and THETA must have the same ZMp5d4y5 % length. The output Z is a matrix with one column for every (N,M) lftT55Tki % pair, and one row for every (R,THETA) pair. O@9<7@h+Nl % 76IjM4&a % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike P5?M"j0/^ % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), z4!Y9 % with delta(m,0) the Kronecker delta, is chosen so that the integral ,a6Oi=+>/U % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, d ,"L8 % and theta=0 to theta=2*pi) is unity. For the non-normalized Fu%D2%V$/ % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. )t?_3'W % 6gy;Xg % The Zernike functions are an orthogonal basis on the unit circle. xZ=6 % They are used in disciplines such as astronomy, optics, and rjaG{ i % optometry to describe functions on a circular domain. itU
P% % !3*:6 % The following table lists the first 15 Zernike functions. qI=j>x % 7my7|s[ % n m Zernike function Normalization PI-o)U$Ehv % -------------------------------------------------- sKCfI] % 0 0 1 1 ]ykMh % 1 1 r * cos(theta) 2 ABG>W>H-S % 1 -1 r * sin(theta) 2 R?Ys%~5 % 2 -2 r^2 * cos(2*theta) sqrt(6) (_ TKDx_ % 2 0 (2*r^2 - 1) sqrt(3) o[Gp *o\ % 2 2 r^2 * sin(2*theta) sqrt(6) 5f}GV0=n % 3 -3 r^3 * cos(3*theta) sqrt(8) c{(4s6D % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8)
26[. te9 % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) LX%UkfA9 % 3 3 r^3 * sin(3*theta) sqrt(8) T GuvyY % 4 -4 r^4 * cos(4*theta) sqrt(10) -~+Y0\%E % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) vyhxS .[9 % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 1RU+d.&D % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^MczumG[ % 4 4 r^4 * sin(4*theta) sqrt(10) +)<H,?/ % -------------------------------------------------- JI7.:k; % g[D`. % Example 1: X/AA8QV o % Jc9BZ`~i % % Display the Zernike function Z(n=5,m=1) eb*w$|y6" % x = -1:0.01:1; \L]T|]}( % [X,Y] = meshgrid(x,x); x6iT"\MO % [theta,r] = cart2pol(X,Y); _ry7[/) % idx = r<=1; Su>UXuNdE# % z = nan(size(X)); d{FD.eI0 % z(idx) = zernfun(5,1,r(idx),theta(idx)); tj<0q<is % figure U/j+\Kc~ % pcolor(x,x,z), shading interp z@tIC^s % axis square, colorbar F#>^S9Gml % title('Zernike function Z_5^1(r,\theta)') mA" 82" % :G/.h[\R| % Example 2: 'kE^oX_ % ^(.utO % % Display the first 10 Zernike functions *{%d{x}l % x = -1:0.01:1;
1k39KO@ % [X,Y] = meshgrid(x,x); 8 aC]" C % [theta,r] = cart2pol(X,Y); nep-?7x % idx = r<=1; Fq`wx % z = nan(size(X)); Y)KO*40c % n = [0 1 1 2 2 2 3 3 3 3]; iTpK:pX % m = [0 -1 1 -2 0 2 -3 -1 1 3]; \+I+Lrj% % Nplot = [4 10 12 16 18 20 22 24 26 28]; bqI| wGCA" % y = zernfun(n,m,r(idx),theta(idx)); ~'3hK4 % figure('Units','normalized') 3`^NaQ % for k = 1:10 .#wU+t> % z(idx) = y(:,k); MoP,a9p % subplot(4,7,Nplot(k)) n%%u0a% % pcolor(x,x,z), shading interp vkg."G:= % set(gca,'XTick',[],'YTick',[]) &-B&s.,kj % axis square 6~y7A<[^ % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) P;~P:qKd % end z<Y
>phc % P6([[mmG % See also ZERNPOL, ZERNFUN2. +ug[TV AOVoOd+6 % Paul Fricker 11/13/2006 {WYmO1 Z`97=:W oHj64fE9 % Check and prepare the inputs: \[ 5mBuk % ----------------------------- ypA)G /; if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) :1(UC}v error('zernfun:NMvectors','N and M must be vectors.') DUOSL end u*C"d1v= HZ.Jc"+M if length(n)~=length(m) /c9%|<O% error('zernfun:NMlength','N and M must be the same length.') "RG #e+ end Pln*?o R$xk cg2( n = n(:); .jps6{ m = m(:); ~hxo_& if any(mod(n-m,2)) v*.#LJEm error('zernfun:NMmultiplesof2', ... |Je+y;P7 'All N and M must differ by multiples of 2 (including 0).') 7IV:X
_y end d;c<" + my(yN| if any(m>n) KJLK]lf}d error('zernfun:MlessthanN', ... 4 fxD$%9 'Each M must be less than or equal to its corresponding N.') va_TC!{; end I-`qo7dQ_S -a(\(^NW if any( r>1 | r<0 ) &ivPY error('zernfun:Rlessthan1','All R must be between 0 and 1.') fX 41o# end <0hJo=6a8 GP/Gv if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) U~Ai'1?xz error('zernfun:RTHvector','R and THETA must be vectors.') N;BS;W5I end 0XNj!^& #:?MtVC r = r(:); H%\\-Z$# theta = theta(:); 8;r7ksE~ length_r = length(r); =*u:@T=d5 if length_r~=length(theta) l8_TeO error('zernfun:RTHlength', ... d{LQr}_o$$ 'The number of R- and THETA-values must be equal.') <(%cb.^c=N end W%k0_Y/5 [@_zsz,`L % Check normalization: Hx]{'? % -------------------- c*"TmDY if nargin==5 && ischar(nflag) _Di}={1[. isnorm = strcmpi(nflag,'norm'); vs)1Rm if ~isnorm }813.U error('zernfun:normalization','Unrecognized normalization flag.') `Y`QxU!d% end uPtHCP6 else sN
`NZyG isnorm = false; FSvtiNW< end W2a9P_ 1K ;i/ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q;kN+NK64 % Compute the Zernike Polynomials vG<JOxP %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [@qUQ,Ie 5 ^\f[} % Determine the required powers of r: rl,6ru % ----------------------------------- -;?5<>zZ m_abs = abs(m); M zLx2? rpowers = []; t*? CD.S for j = 1:length(n) h4GR:` rpowers = [rpowers m_abs(j):2:n(j)]; uT}Jw end S>]Jc$ rpowers = unique(rpowers); E!4Qc+. E&/D%}Wl % Pre-compute the values of r raised to the required powers, 3d{v5. C#X % and compile them in a matrix: .+H8c. % ----------------------------- _`JYA if rpowers(1)==0 !S/hH% C rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); <9S?wju4W' rpowern = cat(2,rpowern{:}); &B^vHH rpowern = [ones(length_r,1) rpowern]; +McKyEa else FRicHs n rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); %^l77:O rpowern = cat(2,rpowern{:}); Cl;B%5yl end =#i#IF42? 6NCa=9 % Compute the values of the polynomials: EX[X|"r % -------------------------------------- AcN~Q/xU y = zeros(length_r,length(n)); Musz+<] for j = 1:length(n) d0b--v/ s = 0:(n(j)-m_abs(j))/2; yD`{9'L
- pows = n(j):-2:m_abs(j); VZ*Q| for k = length(s):-1:1 6 qK0G$> p = (1-2*mod(s(k),2))* ... U4_< prod(2:(n(j)-s(k)))/ ... !}()mrIlP prod(2:s(k))/ ... qVFz-!6b prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Q^v8n1 prod(2:((n(j)+m_abs(j))/2-s(k))); XbJ=lH idx = (pows(k)==rpowers); rbnu:+! y(:,j) = y(:,j) + p*rpowern(:,idx); FeS6>/ end N1Y*IkW" [Bp[=\ if isnorm %HuQc^ y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); [&rW+/ end j>M
'nQ,;d end 2I:vie
% END: Compute the Zernike Polynomials twx8TQ9 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "w`f>]YLA IPcAE!h6zN % Compute the Zernike functions: BNg\;2r % ------------------------------ xZS idx_pos = m>0; yov:JnWo idx_neg = m<0; gv;=Yhw.c >ofS'mp z = y; !+ IxPn if any(idx_pos) 3Fh<%<= z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); {!B0&x end J=TbZL4y}4 if any(idx_neg) L.15EXAB z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 4aAr|!8|h! end =[`wyQe`_ E8>npDFv. % EOF zernfun
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