| 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
B +<i=w function z = zernfun(n,m,r,theta,nflag) ^w6~?'} %ZERNFUN Zernike functions of order N and frequency M on the unit circle. <F6LC_ % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N =?oYEO7 % and angular frequency M, evaluated at positions (R,THETA) on the %XiF7<A& % unit circle. N is a vector of positive integers (including 0), and m$!Ex}2 % M is a vector with the same number of elements as N. Each element kB3@;z: % k of M must be a positive integer, with possible values M(k) = -N(k) mh"9V5T % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ;{:bq`56f % and THETA is a vector of angles. R and THETA must have the same ?
e<D + % length. The output Z is a matrix with one column for every (N,M) T'${*NVn % pair, and one row for every (R,THETA) pair. RM6*c
. % aYrbB# % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike W~Ae&gcn# % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ipH'}~=ID % with delta(m,0) the Kronecker delta, is chosen so that the integral dQ`=CIr % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, I6lWB(H!u % and theta=0 to theta=2*pi) is unity. For the non-normalized 7I;A5f % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. $-p#4^dg % K bM1b % The Zernike functions are an orthogonal basis on the unit circle. (!"&c*
< % They are used in disciplines such as astronomy, optics, and {}DoRpq= % optometry to describe functions on a circular domain. a*bAf'= % 6X[Mn2wYW % The following table lists the first 15 Zernike functions. 6u[
B}%l % -W'T3_ % n m Zernike function Normalization :=e"D;5 % -------------------------------------------------- rJw
Ws % 0 0 1 1 bW?cb5C % 1 1 r * cos(theta) 2 b 67l\L % 1 -1 r * sin(theta) 2 ^udl&> % 2 -2 r^2 * cos(2*theta) sqrt(6) " gQJeMU % 2 0 (2*r^2 - 1) sqrt(3) {2=f,,|+f % 2 2 r^2 * sin(2*theta) sqrt(6) r9y(j
z % 3 -3 r^3 * cos(3*theta) sqrt(8) V8-*dE % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) u)9YRMl % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Y
wu
> k % 3 3 r^3 * sin(3*theta) sqrt(8) )=5,S~IT % 4 -4 r^4 * cos(4*theta) sqrt(10) ^j *H % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .Hm1ispq % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) [/GCy0jk % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Y@2v/O,\ % 4 4 r^4 * sin(4*theta) sqrt(10) =l+~}/7'Z % -------------------------------------------------- !.@F,wZvY % [|tlTk % Example 1: QUKv :; % RZbiiMC> % % Display the Zernike function Z(n=5,m=1) "pTU&He % x = -1:0.01:1; qj1Fj % [X,Y] = meshgrid(x,x); _qvzZ6 % [theta,r] = cart2pol(X,Y); c$b~?Mx % idx = r<=1; Bh5z4 % z = nan(size(X)); 'h3yxf}\ % z(idx) = zernfun(5,1,r(idx),theta(idx)); -n~%v0D8c % figure A#Ne07d % pcolor(x,x,z), shading interp YlJ_$Q[ % axis square, colorbar \kEC|O)8 % title('Zernike function Z_5^1(r,\theta)') qt`HP3J& % ]*TW%mY % Example 2: h42dk(B % nl+8C}=u % % Display the first 10 Zernike functions mIah[~G % x = -1:0.01:1; O*udV E> % [X,Y] = meshgrid(x,x); *_H^]wNJG % [theta,r] = cart2pol(X,Y); l9vJ] % idx = r<=1; ,&iZ*6=X?0 % z = nan(size(X)); n0%5mTUN % n = [0 1 1 2 2 2 3 3 3 3]; o|Kd\<rY % m = [0 -1 1 -2 0 2 -3 -1 1 3]; bu,xIT ^ % Nplot = [4 10 12 16 18 20 22 24 26 28]; M@<r8M]G % y = zernfun(n,m,r(idx),theta(idx)); Wo7`gf_ ( % figure('Units','normalized') oz&RNB.K % for k = 1:10 t-?#x
% z(idx) = y(:,k); *^i"q\n5( % subplot(4,7,Nplot(k)) P0ZY;/e5h % pcolor(x,x,z), shading interp 4MPR % set(gca,'XTick',[],'YTick',[]) 8Az|SJ< % axis square ]6@6g>f? % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ;uN&yj<}a % end 8 c5=Px2\ % Uc( z| % See also ZERNPOL, ZERNFUN2. nQ08(8 >Y=qSg>Ik % Paul Fricker 11/13/2006 9T%b#~?3P d5#z\E?? q]#j,}cN9 % Check and prepare the inputs: h.4FY< % ----------------------------- 4azqH;i if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) #+(@i|!ifo error('zernfun:NMvectors','N and M must be vectors.') =h,J!0Y end bA\(oD+: $%.,=~W7 if length(n)~=length(m) VYnB&3%DF error('zernfun:NMlength','N and M must be the same length.') NS){D7T end =F/ EzS zvR;Tl6] n = n(:); <6.?:Jj m = m(:); a^7QHYJ6 if any(mod(n-m,2)) =+w/t9I[ error('zernfun:NMmultiplesof2', ... ~WKWx.ul 'All N and M must differ by multiples of 2 (including 0).') FXh*!%"* end TFDzTD kJpr:4;@_ if any(m>n) 3hfv^H error('zernfun:MlessthanN', ... BMItHn]. 'Each M must be less than or equal to its corresponding N.') bJ^Jmb end 2?kVbF -FQc_k?VF if any( r>1 | r<0 ) ;^cMP1SH error('zernfun:Rlessthan1','All R must be between 0 and 1.') O:Wd
,3_ end 2Ws'3Jz X/FR e[R if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) uTNy{RBD+ error('zernfun:RTHvector','R and THETA must be vectors.') dpcU`$kt end X3HJ3F;== Uj^Y\w-@Z r = r(:); %e+{wU}w?2 theta = theta(:); py$i{v% length_r = length(r); ]-jaIvM if length_r~=length(theta) Mo]aB:a error('zernfun:RTHlength', ... [~ !9t9+~ 'The number of R- and THETA-values must be equal.') 00pe4^U end q@i.4>x ]0=THq\H % Check normalization: _7<G6q2( % -------------------- H/l,;/q]b
if nargin==5 && ischar(nflag) IwR=@Ne8 isnorm = strcmpi(nflag,'norm'); *1h@Jb34 if ~isnorm Kl]l[!c7$ error('zernfun:normalization','Unrecognized normalization flag.') )3^#CD end @ 1FWBH~ else 3`Dyrj#! isnorm = false; Z/LYTo$Bz end LBIEG_/m %'eaW %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .&.L@CRH % Compute the Zernike Polynomials Iv/h1j> H %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7%W@Hr,%F 2 ]}e4@{ % Determine the required powers of r: 2=$ F*B>9 % ----------------------------------- e}ivvs2 m_abs = abs(m); 4%7Oaf>9 rpowers = []; |WSmpuf for j = 1:length(n) vj"['6Xa rpowers = [rpowers m_abs(j):2:n(j)]; S2?)Sb` end QB*n
[(? rpowers = unique(rpowers); Y#FSU#a$< aT8A+=K6 % Pre-compute the values of r raised to the required powers,
pp()Hu3J % and compile them in a matrix: E//*bmww % ----------------------------- gF\a c%9 if rpowers(1)==0 4F+G;'JV rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); pIY3ft\ rpowern = cat(2,rpowern{:}); 1-PFM- rpowern = [ones(length_r,1) rpowern]; JC9OL.Ob else +f,I$&d.V rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); j#${L6 rpowern = cat(2,rpowern{:}); aZ}z/.b] end 1~vv<`- qot{#tk
d % Compute the values of the polynomials: xLw[
aYy4 % -------------------------------------- X
[;n149o y = zeros(length_r,length(n)); cq9d;~q for j = 1:length(n) Oyp)Wm;@ s = 0:(n(j)-m_abs(j))/2; c[EG
cY={ pows = n(j):-2:m_abs(j); *2Q x69` for k = length(s):-1:1 gXB&Sgjo p = (1-2*mod(s(k),2))* ... BG+X8t8\ prod(2:(n(j)-s(k)))/ ... cBU@853 prod(2:s(k))/ ... =<U'Jtu6' prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... \>+BvF prod(2:((n(j)+m_abs(j))/2-s(k))); `!.c_%m2 idx = (pows(k)==rpowers); ihIRB9 y(:,j) = y(:,j) + p*rpowern(:,idx); BXr._y, cr end m^4O jik <9`/Y"\ p if isnorm :U-yO 9!j y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); t_Ul;HVPS end M B,Z4 ^ end &sGLm~m# % END: Compute the Zernike Polynomials /_r{7Gq. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fw0Z- 9* kaV Ye)~ % Compute the Zernike functions: K555z+,'e % ------------------------------ +N!/>w]n idx_pos = m>0; >Yfo $S_ idx_neg = m<0; e_Q(l'f
DIh[% z = y; OgkbN` if any(idx_pos) cQh=Mri] z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); T7Yg^ -" end ,@t#)HV if any(idx_neg) }j,G)\g# z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ,tuZ_"?M end #4!6pMW(&7 RueL~$*6.~ % EOF zernfun
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