| jssylttc |
2012-04-23 19:23 |
如何从zernike矩中提取出zernike系数啊
下面这个函数大家都不会陌生,计算zernike函数值的,并根据此可以还原出图像来, &Ivf!Bgm{Z 我输入10阶的n、m,r,theta为38025*1向量,最后得到的z是29525*10阶的矩阵, ,m[#<}xXA 这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? `]4tJJy$ 那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? |h-e+Wh1 j5!pS xOC NX8.
\Pf# kE854Ej !|~yf3 function z = zernfun(n,m,r,theta,nflag) k-I U}|Xz %ZERNFUN Zernike functions of order N and frequency M on the unit circle. =3|5=ZU034 % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Q.g44> % and angular frequency M, evaluated at positions (R,THETA) on the Ntlbn&lc;D % unit circle. N is a vector of positive integers (including 0), and nK6(0?/ % M is a vector with the same number of elements as N. Each element o#hFK'&~ % k of M must be a positive integer, with possible values M(k) = -N(k) l*b0uF % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, FJ2~SKWT % and THETA is a vector of angles. R and THETA must have the same 1pM>-"a8j % length. The output Z is a matrix with one column for every (N,M) ZVDi;
% pair, and one row for every (R,THETA) pair. FW:V<{f % lyw)4;wt\ % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike I]Ws
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), #:8V<rc^ % with delta(m,0) the Kronecker delta, is chosen so that the integral SI/3Dz[ % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, o'=i$Eb % and theta=0 to theta=2*pi) is unity. For the non-normalized <~
?LU^ % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. )j}v3@EM5 % kiu#THF % The Zernike functions are an orthogonal basis on the unit circle. z6Zd/mt~x % They are used in disciplines such as astronomy, optics, and '5\?l:z % optometry to describe functions on a circular domain. Ygc.0VKMR % a0PClbf2. % The following table lists the first 15 Zernike functions. j`QXl % ZJV;&[$[ % n m Zernike function Normalization +r$VrNVs % -------------------------------------------------- /! M%9gu % 0 0 1 1 Q+p9^_r % 1 1 r * cos(theta) 2 QeQxz1 % 1 -1 r * sin(theta) 2 }[: i!t.m % 2 -2 r^2 * cos(2*theta) sqrt(6) D<lV WP % 2 0 (2*r^2 - 1) sqrt(3) o$Z]qhq % 2 2 r^2 * sin(2*theta) sqrt(6) wUH:l % 3 -3 r^3 * cos(3*theta) sqrt(8) $?y\3GX % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) wLKC6@
W % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) y<m}dW6[\ % 3 3 r^3 * sin(3*theta) sqrt(8) qv8B$}F U % 4 -4 r^4 * cos(4*theta) sqrt(10) YbJB.;qK % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 3^]Kd % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) @Uqcym. % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +oiuulA % 4 4 r^4 * sin(4*theta) sqrt(10) 9^@#Ua % -------------------------------------------------- KNSMx<GP % (S8hr,%n % Example 1: %Vhj<gN % @gi / 1 cq % % Display the Zernike function Z(n=5,m=1) &X)^G# % x = -1:0.01:1; &G_XgQsg{ % [X,Y] = meshgrid(x,x); "LM[WcDX % [theta,r] = cart2pol(X,Y); h%] D[g % idx = r<=1; UxvT|~" % z = nan(size(X)); `t2Y IwOK % z(idx) = zernfun(5,1,r(idx),theta(idx)); )|
F O> % figure C:RA( % pcolor(x,x,z), shading interp v dPb-z4 % axis square, colorbar 2[1lwV % title('Zernike function Z_5^1(r,\theta)') 07>Iq8<mu % O7yIFqI=/ % Example 2: yK w.69. % Dj{=Y`Tw % % Display the first 10 Zernike functions ^F87gow%`B % x = -1:0.01:1; 7w?N-Q$y % [X,Y] = meshgrid(x,x); 4d%QJ7y % [theta,r] = cart2pol(X,Y); oA4<AJ2 % idx = r<=1; 4)d"}j % z = nan(size(X)); FLQ>,=O % n = [0 1 1 2 2 2 3 3 3 3]; lz(}N7SLa % m = [0 -1 1 -2 0 2 -3 -1 1 3]; A5,(P$@k % Nplot = [4 10 12 16 18 20 22 24 26 28]; :</KgR0I % y = zernfun(n,m,r(idx),theta(idx)); M!
uE#| % figure('Units','normalized') T*rz#O % for k = 1:10 '19kP. % z(idx) = y(:,k); "BvDLe': % subplot(4,7,Nplot(k)) #{zF~/Qq % pcolor(x,x,z), shading interp `}#n#C) % set(gca,'XTick',[],'YTick',[]) 0(+dXzcwM % axis square b\C1qM4 % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) xvW# ~T] % end ~Z5Wwp]a % Gy!bPVe % See also ZERNPOL, ZERNFUN2. 8)="Ee .|GnTC q 0[1/#0$ % Paul Fricker 11/13/2006 Rzxkz 0f"la=6 =]P|!$!}0 ?A~a}bFZ dwVo"_Yr % Check and prepare the inputs: E*83N@i % ----------------------------- >C -N0H if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) I GB) error('zernfun:NMvectors','N and M must be vectors.') >7 qZ\# end '|nAGkA zI'c 'X1, Y|aaZ|+ if length(n)~=length(m) VX e7b error('zernfun:NMlength','N and M must be the same length.') :@J.!dokF end HQ^:5XH )`}4rD^b ig4mj47wJ n = n(:); 'V}4_3#q m = m(:); 1p(9hVA if any(mod(n-m,2)) L!^^3vn error('zernfun:NMmultiplesof2', ... #A^(1 'All N and M must differ by multiples of 2 (including 0).') 0-8'.C1v end E*8).'S%k !6eF8T Pi=B\=gs if any(m>n) xN44>3# error('zernfun:MlessthanN', ... =5#sB* 'Each M must be less than or equal to its corresponding N.') o*xft6U end @T~~aQFk G/&Wc2k YBQ{/"v%| if any( r>1 | r<0 ) 5Pd^Sew error('zernfun:Rlessthan1','All R must be between 0 and 1.') z=Khbh end AmBLZ<f; DTCOhUIV <[tU.nh if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 9^^:Y3j error('zernfun:RTHvector','R and THETA must be vectors.') YI),yj end AHn
Yfxv_ *w6(nG'M{ Hs(U|BXU r = r(:); ^4{"h theta = theta(:); -Ps kUl' length_r = length(r); -h{| u{t if length_r~=length(theta) io*iA<@Gx error('zernfun:RTHlength', ... LRv[,]b 'The number of R- and THETA-values must be equal.') fk(h*L|sI end X!f` !tZ:{ >npFg@A IeGVLC % Check normalization: O
718s\# % -------------------- `h$^=84 if nargin==5 && ischar(nflag) NSQ#\:3:S isnorm = strcmpi(nflag,'norm'); KNqs=:i if ~isnorm d 8M8O3 error('zernfun:normalization','Unrecognized normalization flag.') +XL|bdK end Ogp@! else +%LR1+/%b isnorm = false; 0-Vx!( end RV_+-m{] sjwD x0(7= GA2kg7 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,=`iQl3(y/ % Compute the Zernike Polynomials wak'L5GQE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f}VIkx]X" Y%@a~| NnqAr , % Determine the required powers of r: /@+[D{_Fw % ----------------------------------- aui3Mq#f m_abs = abs(m); '*n2<y rpowers = []; #]lK! : for j = 1:length(n) 6Wc'5t3 rpowers = [rpowers m_abs(j):2:n(j)]; 3nX={72<b end xc_-1u4a9 rpowers = unique(rpowers); </9@RO =x}p>#o,J 4pZ=CB+j % Pre-compute the values of r raised to the required powers, i|QL6e*0 % and compile them in a matrix: ZMmf!cKY:' % ----------------------------- _?a.S8LxJZ if rpowers(1)==0 U$ 22 r b rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); !r`/vQ# rpowern = cat(2,rpowern{:}); v}BXH4 &Y rpowern = [ones(length_r,1) rpowern]; FmC
[u else {V5eHn9/Q' rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); =Bb/Y`Q rpowern = cat(2,rpowern{:}); {qL}:ha? end +/DT#}JE }<g-0&GLm wUcp_)aE| % Compute the values of the polynomials: ~=Q Tv8 % -------------------------------------- H:.l:PJ y = zeros(length_r,length(n)); %S]g8O[}nl for j = 1:length(n) Vl%jpjqP s = 0:(n(j)-m_abs(j))/2; CC.ri3+. pows = n(j):-2:m_abs(j); [?nM)4d for k = length(s):-1:1 `rZS\A p = (1-2*mod(s(k),2))* ... fQ_(2+FM prod(2:(n(j)-s(k)))/ ... ?$Ii_. prod(2:s(k))/ ... IhPX/P prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... )m.U"giG++ prod(2:((n(j)+m_abs(j))/2-s(k))); N_| '`]D idx = (pows(k)==rpowers); .fD%*- y(:,j) = y(:,j) + p*rpowern(:,idx); <K
GYwLk end KIYs[0*k I#9q^,,F if isnorm iBaz1pDc y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); |}naI_Qudv end 22r$Ri_> end tLo_lLn*~% % END: Compute the Zernike Polynomials J0,;F9<C#X %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .ly K
,p qotWWe# jC_m0Iwc % Compute the Zernike functions: klSA Y % ------------------------------ FgTWym_ idx_pos = m>0; 2^4OaHY88 idx_neg = m<0; 9jW"83*5 i4H,Ggb 0t(js_ z = y; [LDY;k~5+ if any(idx_pos) %)p?&_ z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); xDEjeM G end tl+ 9SBl if any(idx_neg) `$i/f(t6` z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); C9L_`[9DO end p^*A&7d:P ,#Mt10e{ OS sYmF % EOF zernfun oJe`]_XZ
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