| 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, =nw,*q + 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, 8x`Kl( 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? )|MIWgfWN 那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? t4C<#nfo P~n8EO1r 6j?FRs oPp!*$V !7ph,/P$7 function z = zernfun(n,m,r,theta,nflag) 6zELe.tq %ZERNFUN Zernike functions of order N and frequency M on the unit circle. xNocGtS % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 7=; D0SS % and angular frequency M, evaluated at positions (R,THETA) on the 5/zf
x % unit circle. N is a vector of positive integers (including 0), and QZ6[*_Z6 % M is a vector with the same number of elements as N. Each element 6sO % k of M must be a positive integer, with possible values M(k) = -N(k)
$@5%5 % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 'nC3:U % and THETA is a vector of angles. R and THETA must have the same
#_?426Wfs % length. The output Z is a matrix with one column for every (N,M) dxk;@Tz % pair, and one row for every (R,THETA) pair. >Iu]T{QNO % s@.`"TF.7 % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike )w^GPlh % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), A%.J%[MVz % with delta(m,0) the Kronecker delta, is chosen so that the integral />2A<{6\=P % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 0\8*S3,q % and theta=0 to theta=2*pi) is unity. For the non-normalized uEc0/a :. % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. /8 e2dw:
\ % 6~:W(E} % The Zernike functions are an orthogonal basis on the unit circle. /5L' 9e % They are used in disciplines such as astronomy, optics, and tjBh$) % optometry to describe functions on a circular domain. n1fEdaa7g % *%P>x}6w3 % The following table lists the first 15 Zernike functions. !V$6+?2 % UrD=|-r` % n m Zernike function Normalization vLi/ '|7 % -------------------------------------------------- &/J.0d-*`` % 0 0 1 1 7.w*+Z>z % 1 1 r * cos(theta) 2 ]
P:NnKgK % 1 -1 r * sin(theta) 2 ~0'_K1(H % 2 -2 r^2 * cos(2*theta) sqrt(6) ^&f{beU9 % 2 0 (2*r^2 - 1) sqrt(3) P5yJO97 % 2 2 r^2 * sin(2*theta) sqrt(6) {[YqGv=fF % 3 -3 r^3 * cos(3*theta) sqrt(8) BLl%D % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) tdMP,0u % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) s0~05{ % 3 3 r^3 * sin(3*theta) sqrt(8) I?^Q084 % 4 -4 r^4 * cos(4*theta) sqrt(10) q\\8b{~ % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1]D/3! % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) kxr6sO~ % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) XwHu:v'= % 4 4 r^4 * sin(4*theta) sqrt(10) !0>!tW % -------------------------------------------------- }QX2:a % [[/ }1% % Example 1: Mhu53DT % J]kP` % % Display the Zernike function Z(n=5,m=1) !0!P.Q8>& % x = -1:0.01:1; )V9Mcr*Ce6 % [X,Y] = meshgrid(x,x); 1)P<cNj % [theta,r] = cart2pol(X,Y); []6ShcqJ[v % idx = r<=1; FcA)RsMI* % z = nan(size(X)); Allt]P> % z(idx) = zernfun(5,1,r(idx),theta(idx)); }(f.uN_v % figure V8KTNt% % pcolor(x,x,z), shading interp iECC@g@a % axis square, colorbar zezofW]a % title('Zernike function Z_5^1(r,\theta)') !R] CmK % Tv*1q.MB % Example 2: 0,"n-5Im % mCC:}n"# % % Display the first 10 Zernike functions G>_42Rp % x = -1:0.01:1; "FLD%3l % [X,Y] = meshgrid(x,x); yPzULO4 % [theta,r] = cart2pol(X,Y); Xd19GP! % idx = r<=1; [+:mt</HN % z = nan(size(X)); 8zWBXV % n = [0 1 1 2 2 2 3 3 3 3]; OxmlzQ"vM % m = [0 -1 1 -2 0 2 -3 -1 1 3]; %(dV|,|v % Nplot = [4 10 12 16 18 20 22 24 26 28]; m"?'hR2 % y = zernfun(n,m,r(idx),theta(idx)); Hd=D#u=A4{ % figure('Units','normalized') 3u"J4%zg|L % for k = 1:10 fRv
S@ % z(idx) = y(:,k); NejsI un% % subplot(4,7,Nplot(k)) DS[l,x % pcolor(x,x,z), shading interp YfrTvKX % set(gca,'XTick',[],'YTick',[]) SS45<!iy % axis square $&n240( % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) mNBpb} % end r=P$iG'& % _I
-0, % See also ZERNPOL, ZERNFUN2. @tjZvRtZ %DND&0` B4*X0x % Paul Fricker 11/13/2006 )l[7;ZIw$ oRvm*"8B "z(fBnv i}$N& <'4!G"_EP % Check and prepare the inputs: eqUn8<<s % ----------------------------- D\_*,Fc if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) O+8ApicjTc error('zernfun:NMvectors','N and M must be vectors.') EDa08+Y end K9z_=c+ Ie`SWg*WL Vp{RX8?. if length(n)~=length(m) Lct+cKKU error('zernfun:NMlength','N and M must be the same length.') > {LJ#Dc6 end 8HH.P`Vk# GD6'R"tJ /kviO@jm4( n = n(:); tWX+\ | m = m(:); M;Mdz[Q if any(mod(n-m,2)) wHN`-
5% error('zernfun:NMmultiplesof2', ... UNZVu~WnF 'All N and M must differ by multiples of 2 (including 0).') h?pGw1Q end sVm'9k I!0 $%
]F r^o}Y if any(m>n) H@&"M% error('zernfun:MlessthanN', ... k}Clq;G 'Each M must be less than or equal to its corresponding N.') +IOKE\,Y end qU
x7S(a Lw2YP[CR "*>QxA%c4 if any( r>1 | r<0 ) dE9aE# o error('zernfun:Rlessthan1','All R must be between 0 and 1.') uwS'*5tU end fY+ .#V J<P/w%i2 :#!F 7u if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) MgK(gL/&[ error('zernfun:RTHvector','R and THETA must be vectors.') 8KdcLN@ end =B{$U~} 7[?}kG Z`1o#yZ r = r(:); CPCB!8-5 theta = theta(:); 4#Nd;gM2 length_r = length(r); qI%9MI;BV if length_r~=length(theta) Y8CYkJTAD- error('zernfun:RTHlength', ... }y=n#%|i. 'The number of R- and THETA-values must be equal.') []fj~hj end XuAc3~HAd W,oV$ s^ 1p5q}">z % Check normalization: ah@GSu;7 % -------------------- >8HRnCyp/ if nargin==5 && ischar(nflag) FUv)<rK isnorm = strcmpi(nflag,'norm'); w7ABnX if ~isnorm _q!ck0_ error('zernfun:normalization','Unrecognized normalization flag.') ojs/yjvx end H-y-7PW*~ else F9G$$%Q-Z isnorm = false; YwTtI ID% end _@3O` "kuBjj2 ewlc ^` %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BOcEL%+ % Compute the Zernike Polynomials 2!& ;ZcT, %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `fE:5y M(f*hOG{Y ;0}"2aGY % Determine the required powers of r: .;sPG % ----------------------------------- a{YVz\?d} m_abs = abs(m); B!C32~[ rpowers = []; Qz90 mb for j = 1:length(n) oM7-1O rpowers = [rpowers m_abs(j):2:n(j)]; YO4ppL~xe end KE1@z] rpowers = unique(rpowers); [~H`9Ab= ;iI2K/ 3 FQ&VM6_ % Pre-compute the values of r raised to the required powers, ^\t">NJ^ % and compile them in a matrix: GnHf9
JrR % ----------------------------- ll^O+>1dO if rpowers(1)==0 4>eg@s N rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); JZ0+VB-3U rpowern = cat(2,rpowern{:}); `)_FO]m}jS rpowern = [ones(length_r,1) rpowern]; :<G+)hIK else T{2//$T? rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ,%^0 4sl rpowern = cat(2,rpowern{:}); ^~od*: end }#M|3h;q9+ UYUdIIoL }Q_i#e(S % Compute the values of the polynomials: cPSpPx % -------------------------------------- G_@H:4$3 y = zeros(length_r,length(n)); N3)EG6vE* for j = 1:length(n) Os;\\~e5 s = 0:(n(j)-m_abs(j))/2; ,`b9c=6; pows = n(j):-2:m_abs(j); D1a4+AyI for k = length(s):-1:1 #e[5O|V~ p = (1-2*mod(s(k),2))* ... V7<}
;Lzm prod(2:(n(j)-s(k)))/ ... Nt?B(.G prod(2:s(k))/ ... gB#t"s) prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ty
?y&~axk prod(2:((n(j)+m_abs(j))/2-s(k))); jC=_>\<|X* idx = (pows(k)==rpowers); LvaF4Y2v y(:,j) = y(:,j) + p*rpowern(:,idx); Qpc>5p![3 end $I%]jAh6 xe'*%3-v) if isnorm %!RQ:?= y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); (nm&\b~j end ;pJ7k23( end tFCeE=4% % END: Compute the Zernike Polynomials S_zE+f+
2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3$TpI5A $=
gv IN"qJ3<k % Compute the Zernike functions: "cZ.86gG`: % ------------------------------ y!j1xnzki idx_pos = m>0; *hba>LZ idx_neg = m<0; ]4PG[9J@ '" 6VfF)* rBaK$Ut z = y; :hr%iu if any(idx_pos) vhKD_}}aP z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 3JwmLGj} end aE+E'iL if any(idx_neg) >i'3\ z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)');
}0I ! n@ end z
'j%.Dd8 sy~mcH:%+ ry:tL0;;e# % EOF zernfun <5
okwcJ^
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