| 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, #Vnkvvv 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, pI1-cV,` 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? S4Pxc
]! 那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? XPavReGf xzdf^Ce HCIU!4rH _:ReN_0 |T<_ 5Ik function z = zernfun(n,m,r,theta,nflag) B?OFe'* %ZERNFUN Zernike functions of order N and frequency M on the unit circle. /74QMx? % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 8^kGS-+^ % and angular frequency M, evaluated at positions (R,THETA) on the /,BD#| % unit circle. N is a vector of positive integers (including 0), and ]P9l jwR % M is a vector with the same number of elements as N. Each element AgWa{.`f: % k of M must be a positive integer, with possible values M(k) = -N(k) 1NbG>E#Ol % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, a1g,@0s % and THETA is a vector of angles. R and THETA must have the same v3*_9e % length. The output Z is a matrix with one column for every (N,M) d8DV[{^ % pair, and one row for every (R,THETA) pair. yjO1 Ol % ^\hG"5# % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike m~w[~flgZ % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi),
YC*"Thuu % with delta(m,0) the Kronecker delta, is chosen so that the integral NyaQI<5D % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, aEBu *`-j % and theta=0 to theta=2*pi) is unity. For the non-normalized UBv,=v % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. nyX2|m& % ,[N%Q# % The Zernike functions are an orthogonal basis on the unit circle. i"1Mfz~e % They are used in disciplines such as astronomy, optics, and T tfo^ksw % optometry to describe functions on a circular domain. J9..P&c\ % :W"~
{~#? % The following table lists the first 15 Zernike functions. aacpM[{f % *Hg>[@dP0 % n m Zernike function Normalization l?\jB\, % -------------------------------------------------- >d(~#Z` % 0 0 1 1 2pZXZ % 1 1 r * cos(theta) 2 cA&9e< % 1 -1 r * sin(theta) 2 gK+4C % 2 -2 r^2 * cos(2*theta) sqrt(6) d}OTO10 % 2 0 (2*r^2 - 1) sqrt(3) Lt2u,9 % 2 2 r^2 * sin(2*theta) sqrt(6) *o]L|Vu % 3 -3 r^3 * cos(3*theta) sqrt(8) @tF\p
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 9-sw!tKx % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Av$]|b % 3 3 r^3 * sin(3*theta) sqrt(8) _Mi5g_ % 4 -4 r^4 * cos(4*theta) sqrt(10) %9
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E % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) oF vfCrd % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) :vYYfs& % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?#Ge.D~u % 4 4 r^4 * sin(4*theta) sqrt(10) w3N[9w?1 % -------------------------------------------------- W= ig.- % y3vdUauOn % Example 1: K>
%Tq % adlV!k7RG % % Display the Zernike function Z(n=5,m=1) <3L5"77G6 % x = -1:0.01:1; 'Oxy$U
% [X,Y] = meshgrid(x,x); "H2EL}3/] % [theta,r] = cart2pol(X,Y); &`h{iK7 % idx = r<=1; '"`IC\N^ % z = nan(size(X)); HsxVZ.dS % z(idx) = zernfun(5,1,r(idx),theta(idx)); ;[(oaK@+n % figure O],T,Z?z % pcolor(x,x,z), shading interp 1kz\IQ{ % axis square, colorbar 3v(* 5 % title('Zernike function Z_5^1(r,\theta)') SP@ >vl+; % V#v`(j% % Example 2: bkRLC_/d % 8bxfj<O, % % Display the first 10 Zernike functions
#+JG(^%B % x = -1:0.01:1; %Celc#v % [X,Y] = meshgrid(x,x); CZ8KEBl % [theta,r] = cart2pol(X,Y); 65L6:}# % idx = r<=1; "<6G6?sz % z = nan(size(X)); ag;Q F % n = [0 1 1 2 2 2 3 3 3 3]; !H#bJTXB % m = [0 -1 1 -2 0 2 -3 -1 1 3]; yZAS# ko}} % Nplot = [4 10 12 16 18 20 22 24 26 28]; PYQ;``~x % y = zernfun(n,m,r(idx),theta(idx)); [m+2(I1 % figure('Units','normalized') \1d( 9jR % for k = 1:10 "_P;2N6 % z(idx) = y(:,k); AJt+p&I[J % subplot(4,7,Nplot(k)) 1f%1*L0>@ % pcolor(x,x,z), shading interp [2>yYr s_= % set(gca,'XTick',[],'YTick',[]) zy?.u.4L % axis square -N6f1>}pE % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) eP.wOl % end #||}R[~P" % EJj.1/]|r % See also ZERNPOL, ZERNFUN2. Uq[>_"} ^/uA?h:]\ czA5n % Paul Fricker 11/13/2006 :8I9\eet3 Q}`0W[a
~ 9Q.rMs>qj 09|K>UC)v <qtr % Check and prepare the inputs: ^pxX]G] % ----------------------------- z-BXd if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }\hVy(\c error('zernfun:NMvectors','N and M must be vectors.') *RDn0d[ end 6uv#de .>q8W u$h
4lIl if length(n)~=length(m) .RE:;<|w error('zernfun:NMlength','N and M must be the same length.') dFRsm0T end ?e`^P VX].3=T8 :=}BN n = n(:); &@G:G( m = m(:); Ua<5U5 if any(mod(n-m,2)) Ld\R:{M" error('zernfun:NMmultiplesof2', ... d<% z
1Dj2 'All N and M must differ by multiples of 2 (including 0).') I+BHstF5um end ) dn(G@5 O80<Z#%j` 3Ko/{f if any(m>n) H0:E(}@ error('zernfun:MlessthanN', ... wZG\>9~ 'Each M must be less than or equal to its corresponding N.') X]'{(?Ch end lun#^ J _?<|{O .nB0 h if any( r>1 | r<0 ) <
nXL error('zernfun:Rlessthan1','All R must be between 0 and 1.') cU
R kP` end bmJ5MF]_fG %WSo b@f8 ;ZH3{ if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 6{x(.= error('zernfun:RTHvector','R and THETA must be vectors.') nePfuG]Q end fg
s!v7 #eQJEajv5 zepm!JR1 r = r(:); *Y,x|F theta = theta(:); #J@[Wd length_r = length(r); RzxNbeki[W if length_r~=length(theta) yQU_>_!n error('zernfun:RTHlength', ... (XeE2l2M 'The number of R- and THETA-values must be equal.') ks"|}9\%< end 34z"Pm YHkn2]^#A $RYa6"` % Check normalization: V\{clJ\U % -------------------- e7@ojOQ% if nargin==5 && ischar(nflag) H+1-] 'g` isnorm = strcmpi(nflag,'norm'); OSlvwH%(EE if ~isnorm <L`KzaA error('zernfun:normalization','Unrecognized normalization flag.') `q?8A3A end wr5AG<%( else E7NV ^4h isnorm = false; @AGn{q end r) HHwh{9 i8`Vv7LF z,Medw6[ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% qo p^;~ % Compute the Zernike Polynomials _hN\10ydY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %oq{L]C(rf RLw;(*(g " |Xk2U % Determine the required powers of r: [f)cL6AeF % ----------------------------------- 8s"%u ) m_abs = abs(m); jNTjSX rpowers = []; (mgS"zPS for j = 1:length(n) *vflscgt rpowers = [rpowers m_abs(j):2:n(j)]; wN`jE0
{ end e91aK rpowers = unique(rpowers); {/"2Vk<H8 (0j}-iaQEZ hakKs.U|[ % Pre-compute the values of r raised to the required powers, 9)}[7Mg:C % and compile them in a matrix: HIQ_%L4] % ----------------------------- " 7!;KHc if rpowers(1)==0 qm./|#m> rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); RMK"o? rpowern = cat(2,rpowern{:}); "^4_@ oo rpowern = [ones(length_r,1) rpowern]; G;&-\0>W else t0o`-d( rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 21
O'M rpowern = cat(2,rpowern{:}); K&nE_.kbl end '>&^zgr %`OJ.:k sp#p8@Cj % Compute the values of the polynomials: >xF/Pl % -------------------------------------- [pl'| B y = zeros(length_r,length(n)); PUF/#ck for j = 1:length(n) (&}i`}v_ s = 0:(n(j)-m_abs(j))/2; |K6REkzr pows = n(j):-2:m_abs(j); 0ZBJ~W for k = length(s):-1:1 8)O[Aq:: p = (1-2*mod(s(k),2))* ... TT'[qfAI prod(2:(n(j)-s(k)))/ ... vI2^tX9 prod(2:s(k))/ ... (^@ra$. prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... bLe<G prod(2:((n(j)+m_abs(j))/2-s(k))); :z4)5=
6M idx = (pows(k)==rpowers); &{>cZh}\ y(:,j) = y(:,j) + p*rpowern(:,idx); 2@9Tfm(= end iMI lZ |UK} if isnorm [ JpKSTg[ y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); LJ*q 1
;<E end X}tVmO? end vWRju*Z& % END: Compute the Zernike Polynomials IIg^FZ*]_ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~V&aUDO>/ zN!ZyI$nqP j:k[90 % Compute the Zernike functions: 9A}# 6 % ------------------------------ F">Qpgt idx_pos = m>0; 4G$|Rx[{, idx_neg = m<0; *$p2*%7Ne q8^^H$<Db MP_'D+LS z = y; Gs9:6 if any(idx_pos) BDq%'~/^ z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); o>/YAX:.!T end WpF2)R}G= if any(idx_neg) <4_X P.N z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Rn{iaM2Y< end nX(+s*Y+w *8#i$w11M oN{Z+T : % EOF zernfun 3T"j)R_=l
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