| 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 P+h&tXZn8 function z = zernfun(n,m,r,theta,nflag) 4fswx@l %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ) /'s&
D % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ri
~2t3gg % and angular frequency M, evaluated at positions (R,THETA) on the /+msrrpD % unit circle. N is a vector of positive integers (including 0), and TZg7BLfy % M is a vector with the same number of elements as N. Each element KG$2u:n % k of M must be a positive integer, with possible values M(k) = -N(k) ZD(gYNi % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, .EO1{2= % and THETA is a vector of angles. R and THETA must have the same 9K!='u` % length. The output Z is a matrix with one column for every (N,M) r8rR _M{P % pair, and one row for every (R,THETA) pair. HenJlo % >.|gmo>b % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike hLRQ) % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), xJCpWU3wM % with delta(m,0) the Kronecker delta, is chosen so that the integral >Fz$DKr[ % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, #ZA
YP % and theta=0 to theta=2*pi) is unity. For the non-normalized }T,uw8?f! % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. hh9{md\ % [@6iStRg7 % The Zernike functions are an orthogonal basis on the unit circle. 3QpTO, % They are used in disciplines such as astronomy, optics, and V_!i KEU % optometry to describe functions on a circular domain. 8*wI^*Q % e=2D^G#qE % The following table lists the first 15 Zernike functions. bd4q/w4q % 0N.*c % n m Zernike function Normalization YVT^}7# % -------------------------------------------------- o9i\[Ul % 0 0 1 1 OjZ@_V: % 1 1 r * cos(theta) 2 JFZ p^{ % 1 -1 r * sin(theta) 2 i weP3u## % 2 -2 r^2 * cos(2*theta) sqrt(6) W=!f % 2 0 (2*r^2 - 1) sqrt(3) #82B`y<<y/ % 2 2 r^2 * sin(2*theta) sqrt(6) c+JlM1p@ % 3 -3 r^3 * cos(3*theta) sqrt(8) !T*izMX} % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Lmb<)YY % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 2xX7dl(cC % 3 3 r^3 * sin(3*theta) sqrt(8) PO&`rr % 4 -4 r^4 * cos(4*theta) sqrt(10) scdT/|(U$ % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) r`2& o % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) =R05H2hs % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) amRtFrc| % 4 4 r^4 * sin(4*theta) sqrt(10) #XsqTK_nk % -------------------------------------------------- )n.peZ % o0 Ae*Y0 % Example 1: ~J|0G6H % yFSL7`p+ % % Display the Zernike function Z(n=5,m=1) KjadX&JD % x = -1:0.01:1; 6{M.S}.^ % [X,Y] = meshgrid(x,x); B !XT:.+ % [theta,r] = cart2pol(X,Y); L$g;^@j % idx = r<=1; _3hEYeh % z = nan(size(X)); b7-a0zaN % z(idx) = zernfun(5,1,r(idx),theta(idx)); 6XP>p$- % figure J?&9ofj& % pcolor(x,x,z), shading interp DsoF4&>g[B % axis square, colorbar mS0W@# |K % title('Zernike function Z_5^1(r,\theta)') + '`RJ,K+[ % @:63OLlrG % Example 2: 1 !sYd@iD@ % M0|z^2 % % Display the first 10 Zernike functions qVfOf\x.e % x = -1:0.01:1; T4[eBO % [X,Y] = meshgrid(x,x); \21!NPXH2 % [theta,r] = cart2pol(X,Y); WI%,m~ % idx = r<=1; CV k8MA % z = nan(size(X)); !Ej<J&e % n = [0 1 1 2 2 2 3 3 3 3]; YW*ti|u|w % m = [0 -1 1 -2 0 2 -3 -1 1 3]; y3x_B@}BY % Nplot = [4 10 12 16 18 20 22 24 26 28]; q45n.A6a % y = zernfun(n,m,r(idx),theta(idx)); 344- ~i* % figure('Units','normalized') W+QI
D/ % for k = 1:10 zIu1oF4[ % z(idx) = y(:,k); fA8 ,wy|> % subplot(4,7,Nplot(k)) !59q@Mya[ % pcolor(x,x,z), shading interp ?IK[]=! % set(gca,'XTick',[],'YTick',[]) K&/W cuP& % axis square y=t
-/*K % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) !>M: G:K % end L(.5:&Y=` % ]]+"`t,- % See also ZERNPOL, ZERNFUN2. 2'D2>^os >">-4L17m % Paul Fricker 11/13/2006 .L}ar7 +U[A.^t ujaaO6oZ7 % Check and prepare the inputs: E11"uWk` % ----------------------------- oZQu&O' if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Lr`yl$6 error('zernfun:NMvectors','N and M must be vectors.') Yv>% 5` end 1'ZBtX~A 1c]GS&(RP if length(n)~=length(m) zJPzI{-w| error('zernfun:NMlength','N and M must be the same length.') !^y'G0
end #jQITS7 SO|$X n = n(:); G3q\Z`|3h m = m(:); 0L'h5i>H) if any(mod(n-m,2)) "bJW yUb error('zernfun:NMmultiplesof2', ... 4v;/"4)' 'All N and M must differ by multiples of 2 (including 0).') WHL@]^E@m end *t63c.S jVr:O` if any(m>n) OF}vY0oiw? error('zernfun:MlessthanN', ... \]zHM.E1 'Each M must be less than or equal to its corresponding N.') $. Ih- end V
V<Zl 'Je;3"@ if any( r>1 | r<0 ) rAgb<D@,H error('zernfun:Rlessthan1','All R must be between 0 and 1.') 5~v({R. end k/>k&^? hDCR>G if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ^]K_k7`I error('zernfun:RTHvector','R and THETA must be vectors.') yN9/'c~ end }}<^fM }5EvBEv-) r = r(:); *5u0`k^j theta = theta(:); /@:I\&{f'9 length_r = length(r); dW6sA65<Y if length_r~=length(theta) Hi#hf"V error('zernfun:RTHlength', ... arm26YA-, 'The number of R- and THETA-values must be equal.') d-y8c end W;Ct[Y8m 12.|E d*72 % Check normalization: v#TU7v?~ % -------------------- 6YNd;,it>p if nargin==5 && ischar(nflag) %AaZc=a[c isnorm = strcmpi(nflag,'norm'); 9J*.'Y if ~isnorm W|4:3c4 error('zernfun:normalization','Unrecognized normalization flag.') EJrP{GH end \<TWy&2& else qf;x~1efC4 isnorm = false; )vn{?Ulj end G8}k9?26( @P@?KZ..v! %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Phr+L9Eog % Compute the Zernike Polynomials wt]onve}% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;X , A|m$( !cW6dc^ % Determine the required powers of r: $i1$nc8 % ----------------------------------- }Y:V&4DW m_abs = abs(m); W^k95%zBM rpowers = []; {\hjKP for j = 1:length(n) h/k00hD60 rpowers = [rpowers m_abs(j):2:n(j)]; 3?5JY;}h>" end DHQS7%)f` rpowers = unique(rpowers); fN&@y$ FF #T"y0Y % Pre-compute the values of r raised to the required powers, 3$G &~A{ % and compile them in a matrix: H\RejGR % ----------------------------- jl9hFubwW if rpowers(1)==0 VkFMr8@| rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); >e>%AMzo[ rpowern = cat(2,rpowern{:}); >jz9o9?8 rpowern = [ones(length_r,1) rpowern]; >e^bq/' else &n9&k
Em rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ^p)#;$6b rpowern = cat(2,rpowern{:}); z;DNl#|!L end Wz%H?m:g# |P@N}P@ % Compute the values of the polynomials: ,<k%'a!B
% -------------------------------------- z^vfha y = zeros(length_r,length(n)); ox*1F+Xri for j = 1:length(n) dIW@L s = 0:(n(j)-m_abs(j))/2; hi`[ pows = n(j):-2:m_abs(j); xpX<iT>5u for k = length(s):-1:1 o%7-<\qS p = (1-2*mod(s(k),2))* ... D6-R>"} prod(2:(n(j)-s(k)))/ ... >
a;iX.K prod(2:s(k))/ ... `*6|2 prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... /% g+|C prod(2:((n(j)+m_abs(j))/2-s(k))); R:4@a ':H idx = (pows(k)==rpowers); 60;_^v y(:,j) = y(:,j) + p*rpowern(:,idx); LTxP@pr end %_."JT$v{ OClG dFJ| if isnorm hC[=e`j y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Om^(CAp end s]]lB018O\ end 63'm
@oZ % END: Compute the Zernike Polynomials ; [G: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% HjIIhl?UY G9NI`]k % Compute the Zernike functions: ?7}ybw3t] % ------------------------------ v4<W57oH idx_pos = m>0; ^s6}[LDW>@ idx_neg = m<0; ;plBo%EBV Z#.1p'3qm1 z = y; ^D<CoxG if any(idx_pos) /{f"0]-RA z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); " i:[|7 end `6)(Fk--" if any(idx_neg) GF6 o z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); !NH(EWER end -'Ay(h 0N^+d,Xt. % EOF zernfun
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