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, L"akV,w4p 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, MW^,l=kqW) 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? ^nYS@ 那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? !?o661+b v^ a.
b 8()L }@y *.UM[Wo \)=X=yn2 function z = zernfun(n,m,r,theta,nflag) yE(> R(^ %ZERNFUN Zernike functions of order N and frequency M on the unit circle. s 9,?"\0Zm % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N rTiW % and angular frequency M, evaluated at positions (R,THETA) on the %8 )GuxG* % unit circle. N is a vector of positive integers (including 0), and "0F =txduS % M is a vector with the same number of elements as N. Each element U}55;4^LX % k of M must be a positive integer, with possible values M(k) = -N(k) aD aQ7i % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, <Q06<{]R8 % and THETA is a vector of angles. R and THETA must have the same +)#d+@- % length. The output Z is a matrix with one column for every (N,M) MVW2%6 % pair, and one row for every (R,THETA) pair. !4 4 )=xW % =gCv`SFW % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike J
00%,Ju_ % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), =rV*iLy % with delta(m,0) the Kronecker delta, is chosen so that the integral =y; tOdj % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, QfuKpcT& % and theta=0 to theta=2*pi) is unity. For the non-normalized NJG-~w % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. AR i_m % P#/k5]g % The Zernike functions are an orthogonal basis on the unit circle. #<X+)B6t % They are used in disciplines such as astronomy, optics, and k#8,:B2 % optometry to describe functions on a circular domain. FnN@W^/z % qm-G=EX % The following table lists the first 15 Zernike functions. aHosu=NK % Iz/o|o]# % n m Zernike function Normalization #{)=%5=c % -------------------------------------------------- j$ h.V#1z % 0 0 1 1 3%?01$k % 1 1 r * cos(theta) 2 JG xuB*} % 1 -1 r * sin(theta) 2 *]Nd
I % 2 -2 r^2 * cos(2*theta) sqrt(6) U[/k=}76 % 2 0 (2*r^2 - 1) sqrt(3) sgdxr!1?y % 2 2 r^2 * sin(2*theta) sqrt(6) -hav/7g % 3 -3 r^3 * cos(3*theta) sqrt(8) \$Xo5f< % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) cD&53FPXC % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 'u }|~u?m % 3 3 r^3 * sin(3*theta) sqrt(8) F6*n,[5( % 4 -4 r^4 * cos(4*theta) sqrt(10) b
!FX]d1~k % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) c <8s\2 % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) @EZ@X/8{& % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^EGe%Fq*x] % 4 4 r^4 * sin(4*theta) sqrt(10) D2 o,K&V % -------------------------------------------------- YGP.LR7 % 9Xb,Swo~ % Example 1: isaDIl;L/ % ) -+u8# % % Display the Zernike function Z(n=5,m=1) z; 6Tp % x = -1:0.01:1; jM8e2z3 % [X,Y] = meshgrid(x,x); 8A{n9>jrb % [theta,r] = cart2pol(X,Y); hqW4.|&\c % idx = r<=1; 5mwtlC':l? % z = nan(size(X)); p\]Mf#B % z(idx) = zernfun(5,1,r(idx),theta(idx)); R}MdBE % figure fZK&h. % pcolor(x,x,z), shading interp 4,CQJ % axis square, colorbar hj@< wU % title('Zernike function Z_5^1(r,\theta)') P?GHcq$\ % 1Wd?AyTY, % Example 2: QiB^U^f % <aJdm!6 % % Display the first 10 Zernike functions {-*+G] % x = -1:0.01:1; E/mp.f2! % [X,Y] = meshgrid(x,x); D_oGhQYY4 % [theta,r] = cart2pol(X,Y); cn&\q.!fh % idx = r<=1; _-aQ.p ?T % z = nan(size(X)); lub(chCE[ % n = [0 1 1 2 2 2 3 3 3 3]; QXZjsa_| % m = [0 -1 1 -2 0 2 -3 -1 1 3]; gBQK % Nplot = [4 10 12 16 18 20 22 24 26 28]; HvSKR1wL\ % y = zernfun(n,m,r(idx),theta(idx)); u_[^gS7 % figure('Units','normalized') FB{4& ; % for k = 1:10 yyke"D % z(idx) = y(:,k); f/t1@d! % subplot(4,7,Nplot(k)) <11pk % pcolor(x,x,z), shading interp va \5
% set(gca,'XTick',[],'YTick',[]) iM;7V*u % axis square O,(p><k$/ % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) o<@b]ukl& % end
PZZTRgVc % o@TxDG % See also ZERNPOL, ZERNFUN2. 1"J\iwN3 N1rBpt Fy!uxT-\ % Paul Fricker 11/13/2006 R/8>^6 t]?u<KD< 'f0*~Wq| B0Ql1x#x V?U->0>Z4 % Check and prepare the inputs: gJn|G#! % ----------------------------- `$j"nP F_ if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) CAg\-*P| error('zernfun:NMvectors','N and M must be vectors.') MQc|j'vEY end .]+Z<5Fo :A%|'HxH3 Vy-N3L if length(n)~=length(m) d<mj=V@bd error('zernfun:NMlength','N and M must be the same length.') !~5;Jb>s[/ end L'k) (=:9pbP X;JptF^ n = n(:); b X.S` m = m(:); 9Z}Y2:l' if any(mod(n-m,2)) 4qq+7B error('zernfun:NMmultiplesof2', ... kL;sA'I:S 'All N and M must differ by multiples of 2 (including 0).') &oJ= end \Z0-o&;w tRU+6D
<w P_11N9C if any(m>n) 7FL!([S5i error('zernfun:MlessthanN', ... s5? 1w 'Each M must be less than or equal to its corresponding N.') PLDg'4DMg end rUjK1A{V SP][xdN7 f\CJ |tKX if any( r>1 | r<0 ) hES_JbX}] error('zernfun:Rlessthan1','All R must be between 0 and 1.') J7:VRf|,?( end Vae}:8'} H8d%_jCr 3|4jS"t{f if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) PveY8[i error('zernfun:RTHvector','R and THETA must be vectors.') qW8sJ= end f0rM 4"1 rEwEdyK 61e)SIRz9I r = r(:); -*8 |J; theta = theta(:); ?+-uF} length_r = length(r); @~pIyy\_ if length_r~=length(theta) MY>mP error('zernfun:RTHlength', ... HGqT"NJr 'The number of R- and THETA-values must be equal.') 2pR+2p` end y8"8QH ut8v&i1? &NbhQY`k % Check normalization:
Q)eYJP=W % -------------------- -xg$qvK if nargin==5 && ischar(nflag) Db"jzMW. isnorm = strcmpi(nflag,'norm'); ,l-tLc if ~isnorm x6Q,$B error('zernfun:normalization','Unrecognized normalization flag.') (>O'^W\3p end yhzC 9nTH else ziUEA>m*/ isnorm = false; sPMCN's end gA0:qEL\ )C^ZzmB ]PWK^-4P %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Wk1o H % Compute the Zernike Polynomials ]%AmX-U %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %JUD54bBt W&E?#=*X m-V_J`9" % Determine the required powers of r: N l~'W % ----------------------------------- J~.8.]gXW m_abs = abs(m); Ph@hk0dgr/ rpowers = []; {4B{~Qe; for j = 1:length(n) TmI~P+5w rpowers = [rpowers m_abs(j):2:n(j)]; $tKz|H) end (jj=CLe rpowers = unique(rpowers); "u#,#z_ WdQR^'b$ 7p"4rL % Pre-compute the values of r raised to the required powers, y5>X0tT % and compile them in a matrix: 0hJ,l. % ----------------------------- .gZ1}2GF= if rpowers(1)==0 ^FO&GM2a rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); dMn0nc+ rpowern = cat(2,rpowern{:}); A7U]wW9 rpowern = [ones(length_r,1) rpowern]; =3K}]3f else :SBB3G)| rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 6ZvGD}/ rpowern = cat(2,rpowern{:}); FU]jI[ end C/34K( jU~q~e7Te HLYog+? % Compute the values of the polynomials: ~`Uil= % -------------------------------------- <L!9as]w y = zeros(length_r,length(n)); 94uAt&&b( for j = 1:length(n) }
O:Y?Wq^ s = 0:(n(j)-m_abs(j))/2; &k\`!T1 pows = n(j):-2:m_abs(j); 5?] Dn k.o for k = length(s):-1:1 a!?JVhD& p = (1-2*mod(s(k),2))* ... =}F}XSvXH prod(2:(n(j)-s(k)))/ ... c&>S prod(2:s(k))/ ... %s&"gWi prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... wqx9 prod(2:((n(j)+m_abs(j))/2-s(k))); (FVHtZi7 idx = (pows(k)==rpowers); ya`Z eQ-p y(:,j) = y(:,j) + p*rpowern(:,idx); zE,1zBS< end 3UR'*5|' CdZS"I if isnorm gUa-6@ y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Z'!Ii+'6 end J?R\qEq% end a_z1S Z2[ % END: Compute the Zernike Polynomials \+iZdZD %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z;'5A2 !-tP\%' Zb&5)&'X % Compute the Zernike functions: #[odjSb % ------------------------------ xj<
K6 idx_pos = m>0; w ]%EJ|' idx_neg = m<0; BV"l;&F[ f%[0}.wp \281X z = y; y8.3tp if any(idx_pos) RKb{QAK!v z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); "=Xky,k end JHJIjYG>P if any(idx_neg) Y@ l>4q") z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 16-1&WuY@ end r?%,#1|$$ ZCAg)/ O-uf^S4 % EOF zernfun E51'TT9
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