| 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, i];P!Gm 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, (<8}un 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? yMTO 5~U{ 那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? :7mHPe}( Am_>x8z u6Lx3 32j}ep.* 7 )rL<+ function z = zernfun(n,m,r,theta,nflag)
E)ZL+( %ZERNFUN Zernike functions of order N and frequency M on the unit circle. KIag(!& % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Lj9RF<39g % and angular frequency M, evaluated at positions (R,THETA) on the }m~MN4 l % unit circle. N is a vector of positive integers (including 0), and 7CvBE;i % M is a vector with the same number of elements as N. Each element "WUS?Q % k of M must be a positive integer, with possible values M(k) = -N(k) zsJermF,O % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, nw0#gDI| % and THETA is a vector of angles. R and THETA must have the same v8j3
K % length. The output Z is a matrix with one column for every (N,M) R&J?XQ % pair, and one row for every (R,THETA) pair. J9p4\=9 % "=T&SY % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 2\QsF,@`YU % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), m!ueqV" % with delta(m,0) the Kronecker delta, is chosen so that the integral -THMTRFz % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, lg~7[=%k# % and theta=0 to theta=2*pi) is unity. For the non-normalized _]pu"hZz4 % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ^W,5A;*3 % V3cKbk7~ % The Zernike functions are an orthogonal basis on the unit circle. aR/?YKA % They are used in disciplines such as astronomy, optics, and
mPk'a % optometry to describe functions on a circular domain. \m
GY'0 % 6/Xs}[iJ % The following table lists the first 15 Zernike functions. kuV7nsXiQ % ZcQu9XDIt % n m Zernike function Normalization <7`zc7c]# % -------------------------------------------------- nGkSS_X % 0 0 1 1 =4a:)g' % 1 1 r * cos(theta) 2 \'4~@ % 1 -1 r * sin(theta) 2 "cPg_-n % 2 -2 r^2 * cos(2*theta) sqrt(6) yi>AogQ, % 2 0 (2*r^2 - 1) sqrt(3) wz*iwd- % 2 2 r^2 * sin(2*theta) sqrt(6) &Xqxuy
]J % 3 -3 r^3 * cos(3*theta) sqrt(8) '.(Gg%*\. % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) -6HwGfU % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) qul#)HI % 3 3 r^3 * sin(3*theta) sqrt(8) I}3F'}JV< % 4 -4 r^4 * cos(4*theta) sqrt(10) v#d\YV{I % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) pUb1#= % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) =I@t%Y % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) oDz|%N2s| % 4 4 r^4 * sin(4*theta) sqrt(10) P<<+;'] % -------------------------------------------------- {YzCgf % "J 1A9| % Example 1: AcPLJ!y % 2!Dz9m3 % % Display the Zernike function Z(n=5,m=1) h@!p:] % x = -1:0.01:1; JhFbze> % [X,Y] = meshgrid(x,x);
j)?M % [theta,r] = cart2pol(X,Y); \7r0]& _ % idx = r<=1; O
{1" I % z = nan(size(X)); o8 JOpD % z(idx) = zernfun(5,1,r(idx),theta(idx)); Nc7"`!;-
% figure pg4W?N` % pcolor(x,x,z), shading interp &uK(. @ % axis square, colorbar , ~O>8VbF % title('Zernike function Z_5^1(r,\theta)') =cS&>MT % G`Nw]_
Z_ % Example 2: 0\P5=hD)K % Zj2 si % % Display the first 10 Zernike functions *9^8NY] % x = -1:0.01:1; P 0,]`w % [X,Y] = meshgrid(x,x); oS fr5
i % [theta,r] = cart2pol(X,Y); -WlYHW % idx = r<=1; f^uiZb % z = nan(size(X)); EfrQ~`\ % n = [0 1 1 2 2 2 3 3 3 3]; F@i>l{C % m = [0 -1 1 -2 0 2 -3 -1 1 3]; &q-&%~E@ % Nplot = [4 10 12 16 18 20 22 24 26 28]; i/x |c!E % y = zernfun(n,m,r(idx),theta(idx)); i6'=]f'{ % figure('Units','normalized') d:(Ex^^ % for k = 1:10 &zdS9e-fF % z(idx) = y(:,k); f+cb83}n] % subplot(4,7,Nplot(k)) S4x9k{Xn % pcolor(x,x,z), shading interp yYA*5
7^A % set(gca,'XTick',[],'YTick',[]) "GO!^ZG] % axis square G%
tlV&In % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) {EoYU\x % end /iU<\+ H % *#T:
_ % See also ZERNPOL, ZERNFUN2. DM^0[3XuV5 '~D4%WKT (p-q>@m % Paul Fricker 11/13/2006 $oBs%.Jp yE8D^M|g .<%tu 0 &n6{wtBP d`^3fr'.4A % Check and prepare the inputs: 1K#>^!?M
% ----------------------------- o$*(N if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) {N4 'g_ error('zernfun:NMvectors','N and M must be vectors.') 41X`. end 5n3yc7NPP ys9:";X;} r3'J{-kl if length(n)~=length(m)
XIInI error('zernfun:NMlength','N and M must be the same length.') 4$xVm,n|
end ,a #>e 0=$/ Lh[0B.g< n = n(:); w2
Y%yjCV m = m(:); D
S U`(` if any(mod(n-m,2)) ip-X r|Bq error('zernfun:NMmultiplesof2', ... CvU$Fsb 'All N and M must differ by multiples of 2 (including 0).') C+NN.5No end !mlfG"FE J@5iD ?Q"andf if any(m>n) <?.eU<+O`S error('zernfun:MlessthanN', ... d{S'6*`D 'Each M must be less than or equal to its corresponding N.') duG!QS: end (47?lw
& Z@zo~*o Cqr{Nssu if any( r>1 | r<0 ) D6bYg ` error('zernfun:Rlessthan1','All R must be between 0 and 1.') "\o#YC end t"VT['8 _k@cs^
S_P&Fv if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) I$;`^z error('zernfun:RTHvector','R and THETA must be vectors.') jFN0xGZ end a|t~&\@ Y_%:%J L`nW&;w' r = r(:); !n-Sh<8 theta = theta(:);
|vs5N2_ length_r = length(r); =yod if length_r~=length(theta) )L b` 4B error('zernfun:RTHlength', ... r2RJb6 'The number of R- and THETA-values must be equal.') VIAq$iu7 end kLgkUck8] #*iUZo r&LZH.$oh % Check normalization: lh;fqn` % -------------------- hz:7W8 if nargin==5 && ischar(nflag) 'zUV(K?2] isnorm = strcmpi(nflag,'norm'); %0Ur3 if ~isnorm Ch"wp/[ error('zernfun:normalization','Unrecognized normalization flag.') S`s]zdUTP end vkG#G]Qs"; else 0F)v9EK(W4 isnorm = false; mF
1f( end !ZTghX}D m,HE4`g q1rj!7 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pT,8E(*l2 % Compute the Zernike Polynomials QM(xMq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% T_*inPf D\Ez~.H xa)p, % Determine the required powers of r: _^_3>}y5op % ----------------------------------- /r7xA}se^ m_abs = abs(m); cSPQ
NYU: rpowers = []; 89M'klZ for j = 1:length(n) if&bp , rpowers = [rpowers m_abs(j):2:n(j)]; z6`0Uv~ end i]Mem M- rpowers = unique(rpowers); fqI67E$59 #da{3>z: _$UJ'W})/ % Pre-compute the values of r raised to the required powers, 7T/BzXr,B % and compile them in a matrix: $#rkvG_w % ----------------------------- q(n"r0)= if rpowers(1)==0 KS*,'hvY rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Z
)c\B rpowern = cat(2,rpowern{:}); uw3vYYFX rpowern = [ones(length_r,1) rpowern]; 1]/;qNEv else d[6 'w ? rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); WaB0?jI rpowern = cat(2,rpowern{:}); y[b8rv end 1'f_C<.0 z|Y54o3 ;a?<7LIx % Compute the values of the polynomials: CU|E-XPW % -------------------------------------- 6Amt75RY y = zeros(length_r,length(n)); CR$wzjP j for j = 1:length(n) (xo`*Q,+ s = 0:(n(j)-m_abs(j))/2; >5t!
Xt pows = n(j):-2:m_abs(j); I0x)d` for k = length(s):-1:1 :E-$:\V0}k p = (1-2*mod(s(k),2))* ... M0$MK> prod(2:(n(j)-s(k)))/ ... bll[E}E|3 prod(2:s(k))/ ... fnq 3ic"V prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 6,5h4[eF* prod(2:((n(j)+m_abs(j))/2-s(k))); B .y}S idx = (pows(k)==rpowers); ~HIj+kN y(:,j) = y(:,j) + p*rpowern(:,idx); aV$kxzEc end Al?%[-u U5C]zswL if isnorm G_1r&[N3 y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); A O5&Y.A# end vb[0H{TT2 end q0}u%Yz % END: Compute the Zernike Polynomials ;U3:1hn %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C<_\{de|9 2-@)'6"n '1D$ ; % Compute the Zernike functions: P%:?"t+J`; % ------------------------------ AO8 #l
YP? idx_pos = m>0; C`r:jA<LC, idx_neg = m<0; #HV5M1mb AZ(zM.y!#_ D _dv8 z = y; (l%?YME if any(idx_pos) rf=l1GW z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); E2M<I;:EA end E#_/#J]UQn if any(idx_neg) -F?97&G$ z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Stwg[K0< end I\TSVJk^Xi +IS6l*_y>6 cD]H~D}M % EOF zernfun f+9WGNpw
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