| 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, |)72E[lL 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, DS,FVh".| 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? 8#d1}Y 那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? C^\*|=*\ mC[U)` ey !7"K>m< HC;I0&v> ZR01<V function z = zernfun(n,m,r,theta,nflag) mB|mt+ %ZERNFUN Zernike functions of order N and frequency M on the unit circle. L@k;L % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N eV{FcJha % and angular frequency M, evaluated at positions (R,THETA) on the ~&j`9jdOj % unit circle. N is a vector of positive integers (including 0), and ;KZtW % M is a vector with the same number of elements as N. Each element pJ*#aH[ySP % k of M must be a positive integer, with possible values M(k) = -N(k) S'-`\%@7 % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 1J{z}yPHc % and THETA is a vector of angles. R and THETA must have the same a+>W % length. The output Z is a matrix with one column for every (N,M) j~L1~@ % pair, and one row for every (R,THETA) pair. Ignv|TYG % $qUta<o2@ % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike D`~{[cv)\ % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), |n6Q % with delta(m,0) the Kronecker delta, is chosen so that the integral (P>eWw\0 % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 3!oQmG_T % and theta=0 to theta=2*pi) is unity. For the non-normalized :rs\ydDUF % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. <%3SI. % nwZr3r % The Zernike functions are an orthogonal basis on the unit circle. | 8L`osg % They are used in disciplines such as astronomy, optics, and sc $QbO c % optometry to describe functions on a circular domain. +S5_J&~ % #L IsL % The following table lists the first 15 Zernike functions. @<TfA>*VJ % Z/05 wB % n m Zernike function Normalization 2eR+dT % -------------------------------------------------- yDkDtO`K % 0 0 1 1 ^B!?;\4IM % 1 1 r * cos(theta) 2 lKhh=Pc2 % 1 -1 r * sin(theta) 2 > v!c\ % 2 -2 r^2 * cos(2*theta) sqrt(6) !E:Vn *k; % 2 0 (2*r^2 - 1) sqrt(3) {|J2clL % 2 2 r^2 * sin(2*theta) sqrt(6) .iN*V|n % 3 -3 r^3 * cos(3*theta) sqrt(8) JTh=JHJ % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Nj-rZ%& % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) b;{"lJ:+Z % 3 3 r^3 * sin(3*theta) sqrt(8) eZod}~J8 % 4 -4 r^4 * cos(4*theta) sqrt(10) *JC{G^|Y % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) v4>"p!_C % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) \;:@=9` % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) HFx"fT % 4 4 r^4 * sin(4*theta) sqrt(10) //u76nQ % -------------------------------------------------- ^RyTK|SQ % rD
U6 5j % Example 1: (F
' % w$;*~Qc % % Display the Zernike function Z(n=5,m=1) Y7V&zF{ % x = -1:0.01:1; +ZA\M:^b % [X,Y] = meshgrid(x,x); BvW gH.OX % [theta,r] = cart2pol(X,Y); l
75{JxZX % idx = r<=1; y3[)zv % z = nan(size(X)); 7C?mD75j % z(idx) = zernfun(5,1,r(idx),theta(idx)); :+^$?[6] % figure Cbg#Yz~/ % pcolor(x,x,z), shading interp 5m7Ax]\ % axis square, colorbar {i}Q}OgYq % title('Zernike function Z_5^1(r,\theta)') G1^!e j % L8tLW09 % Example 2:
<d&)|W % |21hY % % Display the first 10 Zernike functions UG'U
D" % x = -1:0.01:1; H'\ EA(v+ % [X,Y] = meshgrid(x,x); LP-Q'vb<= % [theta,r] = cart2pol(X,Y); yW(+?7U % idx = r<=1; zomNjy* % z = nan(size(X)); X|1YGZJ % n = [0 1 1 2 2 2 3 3 3 3]; @d^h/w % m = [0 -1 1 -2 0 2 -3 -1 1 3]; )9 jQ_ % Nplot = [4 10 12 16 18 20 22 24 26 28]; /P"\+Qp % y = zernfun(n,m,r(idx),theta(idx));
<m:wuNEM % figure('Units','normalized') h}&IlDG % for k = 1:10 FYS83uq0 % z(idx) = y(:,k); O Lup`~ % subplot(4,7,Nplot(k)) 6:tr8 X_ % pcolor(x,x,z), shading interp vl~ % set(gca,'XTick',[],'YTick',[]) ;L%~c4`l~m % axis square
F-,{+B66 % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) dTQvz9 C % end b e%*0lr % *`.{K12T % See also ZERNPOL, ZERNFUN2. B]F7t4Y! l4reG:uYG R./ 6Q1 % Paul Fricker 11/13/2006 h:sG23@= kD7(}N8YR iQ"F`C `#8R+c=$ k+M-D~@5H % Check and prepare the inputs: m e{SVG{ % ----------------------------- O9)}:++T if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) '\\Cpc_g error('zernfun:NMvectors','N and M must be vectors.') BQ0\+ end ah9',( (! %\&dFwb xumv I{ if length(n)~=length(m) *v/*_6f* error('zernfun:NMlength','N and M must be the same length.') VVl-cU end q#3X*!) 1^^D :tt S]=Vr%irX n = n(:); M)^9e? m = m(:); 1u+(rVQN if any(mod(n-m,2)) H5 hUY'O error('zernfun:NMmultiplesof2', ... Yb{t!KL 'All N and M must differ by multiples of 2 (including 0).') Hvo27THLo end z5vI0 N$ _ u2 $Xc<K_Z if any(m>n) ye2Oh7 error('zernfun:MlessthanN', ... y<d#sv(s 'Each M must be less than or equal to its corresponding N.') w/6@R 4)p end 'FFc"lqj (SA*9% htym4\Z= if any( r>1 | r<0 ) * =@pdQkR error('zernfun:Rlessthan1','All R must be between 0 and 1.') lXKZNCL end _/ZY&5N U~t(YT kq0m^` if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) :%MWbnVSC, error('zernfun:RTHvector','R and THETA must be vectors.') b.;}Hq> end qG]PUc>j _I4sy=tYXK B{ "<\g r = r(:); Ngnjr7Q={T theta = theta(:); =LnAMl#9 length_r = length(r); )aSkUytg"
if length_r~=length(theta) ayp}TYh* error('zernfun:RTHlength', ... \]%U?`A 'The number of R- and THETA-values must be equal.') 3/FB>w gt end e*H$c?7NL 0{F.DDiNT e r$ 'c % Check normalization: */E{s? % -------------------- L ai"D[N if nargin==5 && ischar(nflag) --kK<9J7 isnorm = strcmpi(nflag,'norm'); ^&HYnwk if ~isnorm I r~X#$Upc error('zernfun:normalization','Unrecognized normalization flag.') KL4/"$l] end sXu+F2O else W$S.?[X isnorm = false; X2v'9 x end 8F1!9W7 mM.&c5U =w-H ) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F}>`3//u % Compute the Zernike Polynomials (xL=X%6a %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |=s3a5sl $ cSZX#\ J~.kb k % Determine the required powers of r: Jiq[VeLe % ----------------------------------- %R"Fx$tQ m_abs = abs(m); ez{&Y>n rpowers = []; Lt_]3go for j = 1:length(n) y e'5A rpowers = [rpowers m_abs(j):2:n(j)]; }R$%MU5:: end ;rgsPVbVf rpowers = unique(rpowers); YPl{5= gp=0;#4
4 o@. !Z8 % Pre-compute the values of r raised to the required powers, X;h~s:LM
% and compile them in a matrix: '! (`? % ----------------------------- 1~Nz6 if rpowers(1)==0 "Q1hP9xV rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); z/b*]"g, rpowern = cat(2,rpowern{:}); =xoTH3/,> rpowern = [ones(length_r,1) rpowern]; 14RL++ else -eTGRr rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); rtm28|0H' rpowern = cat(2,rpowern{:}); 16vfIUtb end GcuZPIN%D Lrq&k40y AI2CfH#:C % Compute the values of the polynomials: 71_N9ub@z % -------------------------------------- 0W> ",2|z y = zeros(length_r,length(n)); X}$S|1CjO for j = 1:length(n) F <(Y s = 0:(n(j)-m_abs(j))/2; 6F2}|c pows = n(j):-2:m_abs(j); (C&f~U for k = length(s):-1:1 ,P^"X5$ p = (1-2*mod(s(k),2))* ... !?{5ET,gtN prod(2:(n(j)-s(k)))/ ... _EP}el prod(2:s(k))/ ... zw?6E8$h prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... +Ji dP prod(2:((n(j)+m_abs(j))/2-s(k))); bGZy0. idx = (pows(k)==rpowers); NQmDm!-4 y(:,j) = y(:,j) + p*rpowern(:,idx); /1*\*<cs end F~EriO dSbV{*B;> if isnorm Mtu8zm y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi);
qcqf9g end A.n1|Q# end ;I>`!|mT % END: Compute the Zernike Polynomials f4q-wX_1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 945psG@| b> |oU 9 wc=B(a| % Compute the Zernike functions: "LYob}_z % ------------------------------ XZhhr1-<a idx_pos = m>0; ; ?!sU idx_neg = m<0; \2Yh I0skW <T['J]k% ;Bm{_$hf= z = y; Rzp-Q5@MY if any(idx_pos) m9/a!|fBE z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); q_!3<.sf end FXbNmBXF if any(idx_neg) sB $!X@ z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); fI6F};I5}T end @I%m}>4Jm DGcd|>q }+z}vb % EOF zernfun q;rU}hAzG0
|
|