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, ?.~E:8 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, \L}aTCvG 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? E9TWLB5A)( 那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? hZf0q 2 wR
+C> 7.-Q9xv O`1_eK~1< 37Ux2t function z = zernfun(n,m,r,theta,nflag) pYIm43r H %ZERNFUN Zernike functions of order N and frequency M on the unit circle. #8iRWm0*6 % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N "_t2R &A % and angular frequency M, evaluated at positions (R,THETA) on the u^T)4~( % unit circle. N is a vector of positive integers (including 0), and @T[}]e % M is a vector with the same number of elements as N. Each element xU+c?OLi % k of M must be a positive integer, with possible values M(k) = -N(k) DjUif "v % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, eFS;+?bu % and THETA is a vector of angles. R and THETA must have the same Y5e6|b| % length. The output Z is a matrix with one column for every (N,M) k2DT+}u7G % pair, and one row for every (R,THETA) pair. [F{q.mZj % gBb+Q, % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 5:v"^"S z % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), NF+^ % with delta(m,0) the Kronecker delta, is chosen so that the integral %_C!3kKv~ % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, W,dqk=n % and theta=0 to theta=2*pi) is unity. For the non-normalized ,?g}->ZB % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. {#"[h1 % >KXSb@ % The Zernike functions are an orthogonal basis on the unit circle. W@U<GF1 % They are used in disciplines such as astronomy, optics, and I?c "\Fe % optometry to describe functions on a circular domain. mTXeIng? % |^p7:)cy % The following table lists the first 15 Zernike functions. 6S7 =+> % @H[)U/. % n m Zernike function Normalization +|(-7" % -------------------------------------------------- t;X
!+ % 0 0 1 1 =yo?] ZS % 1 1 r * cos(theta) 2 2VObj7F % 1 -1 r * sin(theta) 2 r5yp
jT^ % 2 -2 r^2 * cos(2*theta) sqrt(6) 9>,$q"M}? % 2 0 (2*r^2 - 1) sqrt(3) Xm,w.|dx % 2 2 r^2 * sin(2*theta) sqrt(6) 6t@kft>Nv % 3 -3 r^3 * cos(3*theta) sqrt(8) ajB4Lj,:r % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) _0^f % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) eT8(O36% % 3 3 r^3 * sin(3*theta) sqrt(8) NvCq5B$C % 4 -4 r^4 * cos(4*theta) sqrt(10) *b#00)d
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 2_i/ F)W % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) g=W1y % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) vzDoF0Ts*p % 4 4 r^4 * sin(4*theta) sqrt(10) PNVYW?l % -------------------------------------------------- qQ\&] % x[XN;W& % Example 1: O*%
1 % qy@v,a % % Display the Zernike function Z(n=5,m=1) R%l6+Okr % x = -1:0.01:1; "Z xM,kI % [X,Y] = meshgrid(x,x); 8K(3{\J[V % [theta,r] = cart2pol(X,Y); cTlitf9 % idx = r<=1; xZ2^lsY % z = nan(size(X)); ,au-g)IFZ % z(idx) = zernfun(5,1,r(idx),theta(idx)); ]M2<b:yo % figure YT:])[gVV % pcolor(x,x,z), shading interp xF|P6GXg % axis square, colorbar G.Z4h/1< % title('Zernike function Z_5^1(r,\theta)') ^\|Hz\"* % [fVtQ@-S! % Example 2: & !0 [T
% X{2))t%
% % Display the first 10 Zernike functions _g{*;?mS % x = -1:0.01:1; lJZ-*"9V % [X,Y] = meshgrid(x,x); W>jgsR79M % [theta,r] = cart2pol(X,Y); { zGM[A % idx = r<=1; 4n1-@qTPF~ % z = nan(size(X)); gN"Abc % n = [0 1 1 2 2 2 3 3 3 3]; P|M#S9^] % m = [0 -1 1 -2 0 2 -3 -1 1 3]; :.xdG>\n3 % Nplot = [4 10 12 16 18 20 22 24 26 28]; x.gRTR`7( % y = zernfun(n,m,r(idx),theta(idx)); 8Ter]0M& % figure('Units','normalized') /eFudMl % for k = 1:10 +[W_Jz % z(idx) = y(:,k); Fh)`A5# % subplot(4,7,Nplot(k)) 5Z
(1& % pcolor(x,x,z), shading interp 42 6l:>D( % set(gca,'XTick',[],'YTick',[]) JjO="Cmk/ % axis square |n9q4*dN % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) PH,MZ"Z% % end R
2.y=P8N % E]Wnl\Be % See also ZERNPOL, ZERNFUN2. <<Zt.!hS $inpiO|s 1rhEk|pGZ % Paul Fricker 11/13/2006 ZAKNyA2 /K+GM8rtE ZH
o#2{F >J!J: 3WH"NC-O< % Check and prepare the inputs: Z{'.fq2A % ----------------------------- FPg5!O% if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) )
N\Nw mx error('zernfun:NMvectors','N and M must be vectors.') ]J`yh$a end 52RFB!Z[ =aL=SC+ hu=b, if length(n)~=length(m) h ~\bJ*Zp error('zernfun:NMlength','N and M must be the same length.') 49/j9#hr end :)cn&'l(S K/^70;/!. D7'P^*4_B n = n(:); FNQR sNi m = m(:); K9-?7X if any(mod(n-m,2)) dV~yIxD}C* error('zernfun:NMmultiplesof2', ... BK+(Uf;g 'All N and M must differ by multiples of 2 (including 0).') !21#NCw end ,F4_ps?( OfSy _#aEK x+mfQcSD& if any(m>n) R78=im7 error('zernfun:MlessthanN', ... oM ')NIW@ 'Each M must be less than or equal to its corresponding N.') O&ur|&v end rSGt`#E-s. M=HP!hn 4 nIs+ if any( r>1 | r<0 ) k@,&'imx error('zernfun:Rlessthan1','All R must be between 0 and 1.') wK0= I\WN9 end E`^?2dv+/ R^nkcLFb/q 8ec6J*b if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) #fF~6wopV error('zernfun:RTHvector','R and THETA must be vectors.') nWrknm end k!%[W,* .H.#W1` "q-,140_ r = r(:); yUZ;keQ_Tw theta = theta(:); &7gL&AY8 length_r = length(r); !W^b:qjJ if length_r~=length(theta) ?2;gmZd7 error('zernfun:RTHlength', ... %Q)3*L 'The number of R- and THETA-values must be equal.') - %ul9} . end }w,^]fC: Z(' iZ'55F 3I rmDT % Check normalization: zsQhydTR % -------------------- |'C{nTX if nargin==5 && ischar(nflag) Ym)8L. isnorm = strcmpi(nflag,'norm'); x{$~u2| if ~isnorm W?*]'0 error('zernfun:normalization','Unrecognized normalization flag.') ]A;{D~X^w end I 0/enL else %*>ee[^L , isnorm = false; `ViFY
end 9c/&+j #i#4h<R ,mu=#}a@} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~|LlT^C % Compute the Zernike Polynomials H;&^A5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ciq'fy ?1r>t"e5
>&1MD} % Determine the required powers of r: hXvg<Rf % ----------------------------------- $@[`/Uh m_abs = abs(m); tkN5|95 rpowers = []; B/*`u for j = 1:length(n) kJ;fA|(I rpowers = [rpowers m_abs(j):2:n(j)]; 1T{A(<:o$ end n1X.]|6' rpowers = unique(rpowers); rv(Qz|K@ 7~t,Pt) mP1EWh| % Pre-compute the values of r raised to the required powers, t+R8{9L- % and compile them in a matrix: S{v [65 % ----------------------------- -SZW[T<N" if rpowers(1)==0 \2F$FRWo rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); '>GZB rpowern = cat(2,rpowern{:}); 9~6FWBt rpowern = [ones(length_r,1) rpowern]; !y8/El else wKjL}1.k rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); (6xrs_ea rpowern = cat(2,rpowern{:}); kWv)+ end tMWDKatb g3p*OYf ~*Fbs! ;, % Compute the values of the polynomials: LuM[*_8 % -------------------------------------- qusX]Tstz y = zeros(length_r,length(n)); {b|:q>Be8 for j = 1:length(n) vgfLI}|5 s = 0:(n(j)-m_abs(j))/2; $'SWH+G pows = n(j):-2:m_abs(j); wnf'-dw] for k = length(s):-1:1 ryd*Ha">I p = (1-2*mod(s(k),2))* ... [LwmzmV+F prod(2:(n(j)-s(k)))/ ... *c\:ogd prod(2:s(k))/ ... 9-<EeV_/ prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... mk)F3[ke prod(2:((n(j)+m_abs(j))/2-s(k)));
vOb=> idx = (pows(k)==rpowers); Iz'*^{Ssm y(:,j) = y(:,j) + p*rpowern(:,idx); O-rHfIxY end & | |