| 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, -V;Y4,:c 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, tnUfi8\ob 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? [err$ 那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? *#>(P \5&Mg81 4+>yL+sC%v @!oN]0`F; +`wr{kB$~ function z = zernfun(n,m,r,theta,nflag) m%u`#67oK %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 0qNmao4E_ % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N =(hBgNH % and angular frequency M, evaluated at positions (R,THETA) on the I2HV{1(i % unit circle. N is a vector of positive integers (including 0), and iCpm^ XT % M is a vector with the same number of elements as N. Each element sE&nEc % k of M must be a positive integer, with possible values M(k) = -N(k) p@~Y[a = % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, kSEA % and THETA is a vector of angles. R and THETA must have the same *LhwIY % length. The output Z is a matrix with one column for every (N,M) 3<<wHK;) % pair, and one row for every (R,THETA) pair. 5@1h^wv % ,=C ipL9] % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike PTe$dPB % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 'fK=;mM % with delta(m,0) the Kronecker delta, is chosen so that the integral gYN;Fu-9Z % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, zUJXA:L9 % and theta=0 to theta=2*pi) is unity. For the non-normalized i7T#WfF % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. G3|23G.~)( % !{V`N|0
% The Zernike functions are an orthogonal basis on the unit circle. u,iiS4'Ze % They are used in disciplines such as astronomy, optics, and 32+N?[9
* % optometry to describe functions on a circular domain. L|APX y]> % XOqHzft h6 % The following table lists the first 15 Zernike functions. =.S2gO > % W:n\,P % n m Zernike function Normalization m`zd0IRTP % -------------------------------------------------- 74
ptd, % 0 0 1 1 =aj|auu % 1 1 r * cos(theta) 2 =3hJti9[ % 1 -1 r * sin(theta) 2 n5xG4.#G % 2 -2 r^2 * cos(2*theta) sqrt(6) `::j\3B&Y- % 2 0 (2*r^2 - 1) sqrt(3) o(v` % 2 2 r^2 * sin(2*theta) sqrt(6) IA1O]i
S % 3 -3 r^3 * cos(3*theta) sqrt(8) n3J,`1*ct % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) u]B
b ^[ % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) o>]w76A^( % 3 3 r^3 * sin(3*theta) sqrt(8) sdXchVC % 4 -4 r^4 * cos(4*theta) sqrt(10) uq:'`o-1 % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )I<VH+6 % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) S WsD]rn % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ')F@em % 4 4 r^4 * sin(4*theta) sqrt(10) b N>Ar % -------------------------------------------------- %E,-dw % ?b^<Tny % Example 1: w\t % 3 5-FD{ % % Display the Zernike function Z(n=5,m=1) ]6=opvm % x = -1:0.01:1; <9=RLENmY" % [X,Y] = meshgrid(x,x); $\4O r % [theta,r] = cart2pol(X,Y); ~c1~)QzZ % idx = r<=1; _;(QMeR % z = nan(size(X)); ,aWfGh#$ % z(idx) = zernfun(5,1,r(idx),theta(idx)); \<%FZT_4~ % figure qj*BV % pcolor(x,x,z), shading interp I GcR5/3 % axis square, colorbar wL0"1Ya % title('Zernike function Z_5^1(r,\theta)') Q"xDRQA % *OE>gg&?Nh % Example 2: Q%GLT,f1. % SR)@'-Wd % % Display the first 10 Zernike functions BYS>" % x = -1:0.01:1; Z[j-.,Qu % [X,Y] = meshgrid(x,x); [iSLn3XXRX % [theta,r] = cart2pol(X,Y); e8]mdU{) % idx = r<=1; 10/3 -)+ % z = nan(size(X)); ^T@-yys % n = [0 1 1 2 2 2 3 3 3 3]; T?5F0WKi % m = [0 -1 1 -2 0 2 -3 -1 1 3]; YX2j;Y? % Nplot = [4 10 12 16 18 20 22 24 26 28]; Q;1$gImFz % y = zernfun(n,m,r(idx),theta(idx)); yFP#z5G % figure('Units','normalized') 3^&pb % for k = 1:10 b;|^62 % z(idx) = y(:,k); tQ?}x#J % subplot(4,7,Nplot(k)) `Sj8<O} % pcolor(x,x,z), shading interp GHWpL\A{8` % set(gca,'XTick',[],'YTick',[]) zjJyc? % axis square }KkH7XksF % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) lu.2ZQE % end {la^useg[ % t!Av[K % See also ZERNPOL, ZERNFUN2. 3?/} &l|B>{4v q`;URkjk % Paul Fricker 11/13/2006 |b7>kM}" *XzUqK 1r w>gR 9p$q@Bc [K9q+ % Check and prepare the inputs:
Q{Bj(f % ----------------------------- ,9M \`6 if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) pK1(AV'L error('zernfun:NMvectors','N and M must be vectors.') o_$r*Z|HG end p-/x Md 86} rz \S2'3SDd/ if length(n)~=length(m) d ly 0874 error('zernfun:NMlength','N and M must be the same length.') C"mb-n7s end n:!J3pR 42Ffx?Qmv 9hp0wi@W} n = n(:); >zhbipA m = m(:); wYawG$@_ if any(mod(n-m,2)) `") I[h error('zernfun:NMmultiplesof2', ... S I(8.$1 'All N and M must differ by multiples of 2 (including 0).') SO&;]YO end bo"I:)n; ]ogy`O > !c`1~a! if any(m>n) fu&]t8MJC error('zernfun:MlessthanN', ... a:]yFi:Su 'Each M must be less than or equal to its corresponding N.') +1623E end hP#&]W3: JuI,wA UX9o if any( r>1 | r<0 ) 6%xl}z]o error('zernfun:Rlessthan1','All R must be between 0 and 1.') RGKJO_*J2 end |3cR'|<Ual 6u7HO-aa tw')2UGg if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Yb\36| error('zernfun:RTHvector','R and THETA must be vectors.') 3G&1. 8 end 1|89-Ii] d$Xvax,C cK } Qu r = r(:); u@3w$"Pv1 theta = theta(:); =w5w=qB length_r = length(r); #,;k>2j0 if length_r~=length(theta) X[[=YCi0 error('zernfun:RTHlength', ... Dx%fW` 'The number of R- and THETA-values must be equal.') w{qYP end !14z4]b 5-QXvw(TH iB`m!g6$ % Check normalization: 'mM5l*{ % -------------------- q.t5L=l^
r if nargin==5 && ischar(nflag) `F@yZ4L3S isnorm = strcmpi(nflag,'norm'); lb('r"*. if ~isnorm {Q"<q`c error('zernfun:normalization','Unrecognized normalization flag.') o_5@R+& end U|QDV16f else BkF[nL*| isnorm = false; a`uT'g[* end P;/T`R=Vr" A!~o?ej xl^'U/ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% J @fE") % Compute the Zernike Polynomials o5R\7}]GE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a*8}~p, %Z?
o] L5W>in5( % Determine the required powers of r: N@$%0! % ----------------------------------- Id0F2 [ m_abs = abs(m); GSH{1VS_b rpowers = []; .%A2 for j = 1:length(n) @6SSk=9_S rpowers = [rpowers m_abs(j):2:n(j)];
"gz;Q end loJ0PY'}= rpowers = unique(rpowers); 5dk,!Cjg UK,P?_e '3Ie0QO]"% % Pre-compute the values of r raised to the required powers, v#FUD-Z % and compile them in a matrix: /WfxI>v % ----------------------------- _;{-w%Vf if rpowers(1)==0 86g+c rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); "q.uiz+1: rpowern = cat(2,rpowern{:}); !)=o,sVA rpowern = [ones(length_r,1) rpowern]; @gc"-V*-/ else De_</1Au!2 rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); [)A#9L~s= rpowern = cat(2,rpowern{:}); ~aG-^BAS end *@nUas2" ?
h%+2 $5r,Q{;$ % Compute the values of the polynomials: )QD}R36Ic % -------------------------------------- [Bo$? y = zeros(length_r,length(n)); >``GDjcJ for j = 1:length(n) @UJmbD{ s = 0:(n(j)-m_abs(j))/2; (T01hR& pows = n(j):-2:m_abs(j); p/~kw:I for k = length(s):-1:1 rYQ@"o0/Y p = (1-2*mod(s(k),2))* ... v_0!uT5~NE prod(2:(n(j)-s(k)))/ ... ~@a
R5Q>us prod(2:s(k))/ ... |5Pbc&mH8A prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... =jt_1L4 prod(2:((n(j)+m_abs(j))/2-s(k))); rUjr'O0 idx = (pows(k)==rpowers); r.;iO0[/ y(:,j) = y(:,j) + p*rpowern(:,idx); df& |Lc1J end C5UDez IcQ!A=lB if isnorm $lA,{Q y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); us%RQ8=k end ;7k7/f: end 4
G[hU4L % END: Compute the Zernike Polynomials rbbuSI %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Zd ,= 1 ,'^BgI, qhtAtP>i" % Compute the Zernike functions: <v k$eB8EC % ------------------------------ nn4Sy,cz idx_pos = m>0; XH$|DeAFM idx_neg = m<0; YCO:bBmp: [uQZD1<q t}*!UixE z = y; wtLMc if any(idx_pos) 0f/!|c z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); |9]K:A end MPN=K|* if any(idx_neg) ]_8I_VcQ z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); x)ZH;) end Gw^=kzh @(fY4]K $4JX#lkt % EOF zernfun #`0z=w/)
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