| 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, fWc|gq 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, TL$EV>Nr 这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? 6VP`evan 那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? :<Y}l-x 1n5(S<T >T2LEW VV4Gjc >(w2GD? function z = zernfun(n,m,r,theta,nflag) 4/ kv3rv %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ?bZovRx % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N s>k Uh % and angular frequency M, evaluated at positions (R,THETA) on the &6 s) X % unit circle. N is a vector of positive integers (including 0), and 3f" %G\ % M is a vector with the same number of elements as N. Each element n79QJl/ % k of M must be a positive integer, with possible values M(k) = -N(k) VErv;GyV % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, n\Fp[9+Z\ % and THETA is a vector of angles. R and THETA must have the same Jj~EiA % length. The output Z is a matrix with one column for every (N,M) tWTKgbj( % pair, and one row for every (R,THETA) pair. EN{]Qb06A % 8dD2 % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike -K,-h[o % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ^,l_{ % with delta(m,0) the Kronecker delta, is chosen so that the integral |Fm6#1A@ % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, LMi:%i%\ % and theta=0 to theta=2*pi) is unity. For the non-normalized Uoya3#4 G % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 5uq3\a % 6u`F
d# % The Zernike functions are an orthogonal basis on the unit circle. F' U 50usV % They are used in disciplines such as astronomy, optics, and y@2epY?{ % optometry to describe functions on a circular domain. UYk>'\%H0 % p4IZ
% The following table lists the first 15 Zernike functions. 7 n]65].t % C%*k.$#r! % n m Zernike function Normalization O#wpbrJ % -------------------------------------------------- O }9KJU % 0 0 1 1 -jgysBw+Xb % 1 1 r * cos(theta) 2 L
%ip> % 1 -1 r * sin(theta) 2 8+]hpa,q % 2 -2 r^2 * cos(2*theta) sqrt(6) 3lV^B[$ % 2 0 (2*r^2 - 1) sqrt(3) f\'{3I29 % 2 2 r^2 * sin(2*theta) sqrt(6) EbeI{-'aF % 3 -3 r^3 * cos(3*theta) sqrt(8) K{n{KB&_& % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) q-nSLE+_; % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) @
'@:sM_ % 3 3 r^3 * sin(3*theta) sqrt(8) ~~/xRs % 4 -4 r^4 * cos(4*theta) sqrt(10) KH\b_>wU2 % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1@u2im-O % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ~GE$myUT\p % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) qE'9QQ>:b % 4 4 r^4 * sin(4*theta) sqrt(10) V8eB$in % -------------------------------------------------- ]9}HEu;1M % =rdY
@ % Example 1: tXJUvish % QwhRNnE= % % Display the Zernike function Z(n=5,m=1) l5l>d62 % x = -1:0.01:1; ikE<=:pe % [X,Y] = meshgrid(x,x); Fnk_\d6Ma % [theta,r] = cart2pol(X,Y); n| GaV % idx = r<=1; hOhS) % z = nan(size(X)); .0R v(Y % z(idx) = zernfun(5,1,r(idx),theta(idx)); _gKe%J& % figure 2pdvWWh3l % pcolor(x,x,z), shading interp u?sVcD[ % axis square, colorbar r`c_e)STO % title('Zernike function Z_5^1(r,\theta)') R/"x}B1d % -
0?^#G}3} % Example 2: 5*[2yKsTi % (98Nzgxgx} % % Display the first 10 Zernike functions eY{+~|KZ % x = -1:0.01:1; qj cp65^ % [X,Y] = meshgrid(x,x); '!f5?O+E % [theta,r] = cart2pol(X,Y); bc
, p} % idx = r<=1; 2lL,zFAq % z = nan(size(X)); ?FfC % n = [0 1 1 2 2 2 3 3 3 3]; EGl^!.' % m = [0 -1 1 -2 0 2 -3 -1 1 3]; VLBE'3Qg1 % Nplot = [4 10 12 16 18 20 22 24 26 28]; r>GZ58i % y = zernfun(n,m,r(idx),theta(idx)); sB69R:U; % figure('Units','normalized') OFje+S % for k = 1:10 }w4QP+ x % z(idx) = y(:,k); V.wqZ {G % subplot(4,7,Nplot(k)) dMR3)CO % pcolor(x,x,z), shading interp W2uOR{
'? % set(gca,'XTick',[],'YTick',[]) U-n;xX0= % axis square I(=V}s2 % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Ts~L:3oaQ % end l }XU59 % |lv|!]qAma % See also ZERNPOL, ZERNFUN2. Zw
wqSyuGf !%dN<%Ah .f+TZDUO % Paul Fricker 11/13/2006
]({~,8s Q]p(u\* 4Vq%N vFEQ7qI \c^45<G2qA % Check and prepare the inputs: eW|^tH % ----------------------------- %kgkXc~6|x if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 3]?#he error('zernfun:NMvectors','N and M must be vectors.') 1
hg}(Hix end aZbw]0q@o _#vrb;.+ 7t.!lh5G% if length(n)~=length(m) /PsnD_s]5 error('zernfun:NMlength','N and M must be the same length.') epgPT'^ end 3j3N!T9 |F#1C9]P ~)(\6^&=| n = n(:); ]vuwkn+) m = m(:); GKcv<G208 if any(mod(n-m,2)) h,"4SSL error('zernfun:NMmultiplesof2', ... A/`%/0e 'All N and M must differ by multiples of 2 (including 0).') q{+_
<2U| end &IlU|4`R% qTQBt} *{+G=d if any(m>n) 2h%z ("3/ error('zernfun:MlessthanN', ... CW<N: F.9 'Each M must be less than or equal to its corresponding N.') 2U-3Q]/I} end ]5%/3P,/ 8mQmi` bu51$s?B if any( r>1 | r<0 ) jbR0%X2 error('zernfun:Rlessthan1','All R must be between 0 and 1.') yV^s,P1 end n9s iX VsA'de!V4[ >|;aIa@9 if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ^%`wJ.c error('zernfun:RTHvector','R and THETA must be vectors.') hdVdcnM end Q?X>E3=U MMj9{ou H8"@iE, r = r(:); }K3x theta = theta(:); D1&A,2wO length_r = length(r); Bm]8m=p if length_r~=length(theta) 'R_g">B. error('zernfun:RTHlength', ... XQS9,Hl 'The number of R- and THETA-values must be equal.') p ]d]QMu end *E{2J:` ciMzf$+G$ E4hLtc^
+ % Check normalization: x.q+uU$^ % -------------------- :^(>YAyHj^ if nargin==5 && ischar(nflag) -{ZWo:,r~q isnorm = strcmpi(nflag,'norm'); w}=5ElB if ~isnorm \<g*8?yFs error('zernfun:normalization','Unrecognized normalization flag.') ~s5SZK* end 2p"WTd else :>=\. \ isnorm = false; YY!Rz[/ end ,TFIG^Dvq O?|gp<=d KGg3 !jY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z4\=*ic@ % Compute the Zernike Polynomials QqU!Najf %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !50[z: LGtIm7 Y0X-Zqk' % Determine the required powers of r: r9dyA5oD % ----------------------------------- bcYF\@}; m_abs = abs(m); hvaSH69*m rpowers = []; ukUGvK for j = 1:length(n) q|),`.eh\ rpowers = [rpowers m_abs(j):2:n(j)]; )+6MK(<" end F|!){=
rpowers = unique(rpowers); LEtG|3Dx ctGL-kp ?F3h)(} % Pre-compute the values of r raised to the required powers, r
>nG@A % and compile them in a matrix: m|G'K[8 % ----------------------------- $b mLu=9 if rpowers(1)==0 yYfsy?3 rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); g
pOC`=
rpowern = cat(2,rpowern{:}); 1aTB%F rpowern = [ones(length_r,1) rpowern]; OtNd,U.dE else _D+J!f^ rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ;&)-;l7M rpowern = cat(2,rpowern{:}); FIsyiSY<j end \7'+h5a aYSCw3C< ?pd8w#O % Compute the values of the polynomials: :n-]>Q>5=k % -------------------------------------- Uw7h=UQh y = zeros(length_r,length(n)); mVpMh#zw for j = 1:length(n) y9Us n8 s = 0:(n(j)-m_abs(j))/2; b"{'T]"*j pows = n(j):-2:m_abs(j); 2_Z ? #Y for k = length(s):-1:1 <Pi|J-Y p = (1-2*mod(s(k),2))* ... :w^Ed%>y7 prod(2:(n(j)-s(k)))/ ... K>@+m prod(2:s(k))/ ... Bn &Ws prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... &eX!#nQ_. prod(2:((n(j)+m_abs(j))/2-s(k))); W*I(f]8:y` idx = (pows(k)==rpowers); Iepsz y(:,j) = y(:,j) + p*rpowern(:,idx); ZNvEW end ys kO OD!& .% if isnorm |3KLk ?2 y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); TtTj28k7 end "[ZB+-|[0 end '?p<lu^^B % END: Compute the Zernike Polynomials ~XmLX)vO/ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dx@-/^. 9j6QX~, t,+nQ9 % Compute the Zernike functions: |$
lM#Ua % ------------------------------ F}/S:(6LF2 idx_pos = m>0; Su/6Q$0 t idx_neg = m<0; \6 Zr 7E79-r&n A"dR{8&0 z = y; |#cm`v if any(idx_pos) DrY:9[LP z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); eEv@}1~ end GQUe!G9 if any(idx_neg) U"^kH| z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 4i(JZN? end SPY|K 9\BT0kx D!Pq4'd( % EOF zernfun )9"_J9G
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