| 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, FuZ7xM, 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, ldJ:A*/M6 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? K#=)]qIk 那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? k-LB %\p :eK;:pN KfXE=v{t <uugT9By #-9;Hn4x function z = zernfun(n,m,r,theta,nflag) 2EubMG %ZERNFUN Zernike functions of order N and frequency M on the unit circle. "RG.27 % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N hi>sDU<x % and angular frequency M, evaluated at positions (R,THETA) on the W*q[f!@ % unit circle. N is a vector of positive integers (including 0), and zS*X9|p % M is a vector with the same number of elements as N. Each element \ORNOX: % k of M must be a positive integer, with possible values M(k) = -N(k) nj*B-M\p % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, MorR&K % and THETA is a vector of angles. R and THETA must have the same >Xq:?}-m2 % length. The output Z is a matrix with one column for every (N,M) )fz)Rrr % pair, and one row for every (R,THETA) pair. 6#+&_#9 % &)Fp % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike V~+{douq % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), a.a5qwG % with delta(m,0) the Kronecker delta, is chosen so that the integral llbj-9OZL % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, C:n55BE9 % and theta=0 to theta=2*pi) is unity. For the non-normalized a*d>WN.;U % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. UW+|1Bj_: % eKlh }v % The Zernike functions are an orthogonal basis on the unit circle. #c5 NFU}9 % They are used in disciplines such as astronomy, optics, and TxYxB1C) % optometry to describe functions on a circular domain. 3S-n sMs. % nT0FonK> % The following table lists the first 15 Zernike functions. }LNpr % L Ty[) % n m Zernike function Normalization gqaENU> % -------------------------------------------------- OLc/Vij; % 0 0 1 1 n&=3Knbd@d % 1 1 r * cos(theta) 2 8CxC`*L( % 1 -1 r * sin(theta) 2 &eQF[8 , % 2 -2 r^2 * cos(2*theta) sqrt(6) ^tIi;7k % 2 0 (2*r^2 - 1) sqrt(3) 00'R1q4 % 2 2 r^2 * sin(2*theta) sqrt(6) e,qc7BJzK % 3 -3 r^3 * cos(3*theta) sqrt(8) 2G8f4vsC[ % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) %|[+\py$Q % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) B:=*lU.n % 3 3 r^3 * sin(3*theta) sqrt(8) u>j:8lhtV % 4 -4 r^4 * cos(4*theta) sqrt(10) D+/27# % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Pew-6u" % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) d-g&TSGd % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4r!8_$fN?G % 4 4 r^4 * sin(4*theta) sqrt(10) BlQu9{=n % --------------------------------------------------
N3Ub|$}q % ajuwP1I % Example 1: q9w6 6R % 9u/ "bj % % Display the Zernike function Z(n=5,m=1) )/h~csy:~ % x = -1:0.01:1; xtyzy@)QL % [X,Y] = meshgrid(x,x); K
oPTY^ % [theta,r] = cart2pol(X,Y); ]R/VE"- % idx = r<=1; 89:Y s= % z = nan(size(X)); m M!H}| % z(idx) = zernfun(5,1,r(idx),theta(idx)); 2E^zQ>;01 % figure "gXz{$q % pcolor(x,x,z), shading interp `#hdb=3 % axis square, colorbar |HXI4MU" % title('Zernike function Z_5^1(r,\theta)') \3(d$_:b % ;Y#~2eYCz % Example 2: T_O\L[]p* % 2~+_T % % Display the first 10 Zernike functions r#wMd9]) % x = -1:0.01:1; FA?xp1E % [X,Y] = meshgrid(x,x);
]Kb % [theta,r] = cart2pol(X,Y); }!b9L] % idx = r<=1; _B)LRD+Hj % z = nan(size(X)); 2xH9O{ % n = [0 1 1 2 2 2 3 3 3 3]; ZKyK#\v< % m = [0 -1 1 -2 0 2 -3 -1 1 3]; mmm025. % Nplot = [4 10 12 16 18 20 22 24 26 28]; dL'hC#!h % y = zernfun(n,m,r(idx),theta(idx)); l?v-9l M % figure('Units','normalized') QA\eXnR % for k = 1:10 iIu % z(idx) = y(:,k); CXGq>cQ=d % subplot(4,7,Nplot(k)) VZ{aET! % pcolor(x,x,z), shading interp 09`5<9/ % set(gca,'XTick',[],'YTick',[]) a9qB8/Gg[ % axis square t0p^0 % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) #q40 >)] % end S P)$K= % /|Za[ % See also ZERNPOL, ZERNFUN2. &yv%"BPV ,/{mRw% "|V{@)!t % Paul Fricker 11/13/2006 g4_DEBh N7k<q=r- w~QUG^0Fx O`U&0lKi' @47MJzC % Check and prepare the inputs: o0^'xVv % ----------------------------- \[oU7r}?/V if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) !EuU
@+ error('zernfun:NMvectors','N and M must be vectors.') 'WkDpa end EAp6IhW{ FqAW>< \2)a.2mAz if length(n)~=length(m) gUzCDB^.: error('zernfun:NMlength','N and M must be the same length.') iD#HBo end )h&s.k 1$ez}k, [TvH7ott'1 n = n(:); {; ]:}nA m = m(:); 8=OK8UaU if any(mod(n-m,2)) fw,ruROqD error('zernfun:NMmultiplesof2', ... 'm9f:iTr 'All N and M must differ by multiples of 2 (including 0).') 7
N+;K0 end n}PK0 )vO;=%GQ ~` v7 if any(m>n) V*xT5TljS- error('zernfun:MlessthanN', ... z|[#6X6tT 'Each M must be less than or equal to its corresponding N.') aW]!$ end ,A9pj k' qN}kDT 2
|w;4 if any( r>1 | r<0 ) I
<`9ANe error('zernfun:Rlessthan1','All R must be between 0 and 1.') G=a.Wff end Z{RRhJ $Z(fPKRN/ ]YYjXg}% if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) )[Bwr
bn error('zernfun:RTHvector','R and THETA must be vectors.') [RG&1~ end /-JBzU$ XbdoTriE Yf
>SV # r = r(:); t_ 5b theta = theta(:); q1a}o% length_r = length(r); YUd*\_ if length_r~=length(theta) Vc|r(lM error('zernfun:RTHlength', ... J;4x-R$W 'The number of R- and THETA-values must be equal.') "H\'4'hg end }yCJ#} 9.ZhkvR4A "f\2/4EIl % Check normalization: 'gd3 w~ % -------------------- [?$ZB),L8 if nargin==5 && ischar(nflag) 2)]C' isnorm = strcmpi(nflag,'norm'); 6r"uDV #0 if ~isnorm B
MU@J error('zernfun:normalization','Unrecognized normalization flag.') TtEc~m end YgiwtZ5FY else rBLkowDP* isnorm = false; =ZM #_uW end Y,K): ~T gv$6\1 o8\@R %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V !G&Aen % Compute the Zernike Polynomials <y1V2Np %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Q/r0p> X||Z>w}v 6J0HaL % Determine the required powers of r: -IhFPjQ % ----------------------------------- .QOQqU*2I m_abs = abs(m); .b>1u3 rpowers = []; %J4]T35^2 for j = 1:length(n) _KiaeVE rpowers = [rpowers m_abs(j):2:n(j)]; ,!_ end "8|y rpowers = unique(rpowers); 'SF+P)Kmz (.\GI D+i B;tU+36nM % Pre-compute the values of r raised to the required powers, |,M&ks % and compile them in a matrix: 3;=nQ{0b % ----------------------------- h+F@apUS if rpowers(1)==0 ?6.vd]oNO rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 8>a/x , rpowern = cat(2,rpowern{:}); n's3!HQY[ rpowern = [ones(length_r,1) rpowern]; W Da;wt else
3U=q3{%1 rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); cC
w,b] rpowern = cat(2,rpowern{:}); YAnt}]u!" end L(Q v78F Q(h,P+ EJY[M % Compute the values of the polynomials: p#~'xq % -------------------------------------- Im%|9g;P y = zeros(length_r,length(n)); [^t"Hf for j = 1:length(n) ,}F2l|x_ s = 0:(n(j)-m_abs(j))/2; j{N;2#.u pows = n(j):-2:m_abs(j); tVQfR*= for k = length(s):-1:1 t ]{qizfOB p = (1-2*mod(s(k),2))* ... \V`O-wcJ]S prod(2:(n(j)-s(k)))/ ... hKjvD.6]% prod(2:s(k))/ ... _`Ey),c _ prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... eU_|.2 prod(2:((n(j)+m_abs(j))/2-s(k))); Zy@35;r idx = (pows(k)==rpowers); 5}
|O y(:,j) = y(:,j) + p*rpowern(:,idx); {;^booq end k9UmTvX 2#&9qGR if isnorm D4'"GaCv y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); (WiA end GyJp!
xFB end ^T"9ZBkb % END: Compute the Zernike Polynomials V[,/Hw~d% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8yax.N
j =:`1!W0I pVn6>\xa % Compute the Zernike functions:
U,)Ngnd % ------------------------------ A@*P4E`xp idx_pos = m>0; VpMpZ9oM< idx_neg = m<0; mH*42XC* b_ Sh#d& >JS\H6 z = y; Pgf$GXE if any(idx_pos) *{tn/ro6a z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); FOpOS?Cr' end !Jb?rSJ.h if any(idx_neg) ?
Ldw\ z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); V S2p"0$3D end @]tFRV VA4vAF K @"m0 % EOF zernfun KrVF>bq+
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