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, C&K%Q3V 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, qdvGBdF 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? H\8.T:> 那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? U/wY;7{)# )6^b\` r]"
> |4x&f!%m VqbMFr<k function z = zernfun(n,m,r,theta,nflag) qS+'#Sn %ZERNFUN Zernike functions of order N and frequency M on the unit circle. FxD\F % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ?^5W.`Y2i % and angular frequency M, evaluated at positions (R,THETA) on the Y-7x**I % unit circle. N is a vector of positive integers (including 0), and N{L ]H_= % M is a vector with the same number of elements as N. Each element %[&cy' % k of M must be a positive integer, with possible values M(k) = -N(k) ^ D?;K8a-l % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Uw<Lt"ls. % and THETA is a vector of angles. R and THETA must have the same 82efqzT % length. The output Z is a matrix with one column for every (N,M) 2gb49y~ % pair, and one row for every (R,THETA) pair. "JbFbcj % 6D/5vM1 % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike IeZ}`$[H % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ;{j:5+' % with delta(m,0) the Kronecker delta, is chosen so that the integral K;gm^ % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, c|hKo[r) % and theta=0 to theta=2*pi) is unity. For the non-normalized laKMQLtv % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. wC..LdSR % ,K8PumM_ % The Zernike functions are an orthogonal basis on the unit circle. VCkhK9(N % They are used in disciplines such as astronomy, optics, and (gs"2 % optometry to describe functions on a circular domain. z2wR]G5! % nYTI\f/8v % The following table lists the first 15 Zernike functions. |o|0qG@g % +SZ#s:#SE % n m Zernike function Normalization /M\S^!g@ % -------------------------------------------------- 3,S5>~R= % 0 0 1 1 v=iz*2+X % 1 1 r * cos(theta) 2 M@
! {m % 1 -1 r * sin(theta) 2 akrEZ7A % 2 -2 r^2 * cos(2*theta) sqrt(6) e8--qV#< % 2 0 (2*r^2 - 1) sqrt(3) ?v8B;="#w % 2 2 r^2 * sin(2*theta) sqrt(6) YmNBtGhT % 3 -3 r^3 * cos(3*theta) sqrt(8)
=y[eQS$ % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) j;J4]]R;o % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) `b# w3 2 % 3 3 r^3 * sin(3*theta) sqrt(8) Gk'J'9* % 4 -4 r^4 * cos(4*theta) sqrt(10) H:a(&Zb % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) P;mmK&& % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) +w%MwPC7` % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) htkn#s~= % 4 4 r^4 * sin(4*theta) sqrt(10) V~NS<!+q % -------------------------------------------------- *~:4&$ % |!flR? OU % Example 1: rR^VW^|f % "a>%tsl$K % % Display the Zernike function Z(n=5,m=1) Cf@WjgR
% x = -1:0.01:1; oT_k"]~Q~2 % [X,Y] = meshgrid(x,x); enDjP % [theta,r] = cart2pol(X,Y); 57%:0loW % idx = r<=1; rF>:pS,`& % z = nan(size(X)); $l 0^2o= % z(idx) = zernfun(5,1,r(idx),theta(idx)); h8$lDFo % figure ){5$8 % pcolor(x,x,z), shading interp . m@Sk`s % axis square, colorbar kYmkKl_ % title('Zernike function Z_5^1(r,\theta)') mWfzL'* % .y#@~H($ % Example 2: '!b1~+PV % 7z5AI!s_ % % Display the first 10 Zernike functions Ym2![FC1 % x = -1:0.01:1; yLipuMNV % [X,Y] = meshgrid(x,x); ]* ': % [theta,r] = cart2pol(X,Y); AL3zE=BL % idx = r<=1; &5~bJ]P % z = nan(size(X)); +p>tO\mo % n = [0 1 1 2 2 2 3 3 3 3]; G %Wjtrpj % m = [0 -1 1 -2 0 2 -3 -1 1 3]; Z_ak4C % Nplot = [4 10 12 16 18 20 22 24 26 28]; z;2kKQZm % y = zernfun(n,m,r(idx),theta(idx)); GbBcC#0 % figure('Units','normalized') q: ?6 % for k = 1:10 nH/V2>Lm % z(idx) = y(:,k); cQA;Y!Q# % subplot(4,7,Nplot(k)) Ro$l/lXl8t % pcolor(x,x,z), shading interp u01x}Ff~6 % set(gca,'XTick',[],'YTick',[]) `G@]\)-! % axis square ?2aglj*"v, % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) at/bes W % end rB<
UOe % @m<xpel % See also ZERNPOL, ZERNFUN2. CRw.UC\ diaLw QZYD;&iY& % Paul Fricker 11/13/2006 wS hsu_(i |36d<b Io i%:oO
KI 6Y`eYp5A Zb^0EbV % Check and prepare the inputs: Kj;Q;Ii % ----------------------------- DrC4oxS 1 if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) MOJKz!% error('zernfun:NMvectors','N and M must be vectors.') =NQDxt} end S)>L 0^M1 ?|w>."F es6!p 7p? if length(n)~=length(m) Z[[qW
f error('zernfun:NMlength','N and M must be the same length.') x32hO; end 5.q2<a : 4703\
HK ;1'X_tp n = n(:); 9!NL<}]{ m = m(:); h'|{@X if any(mod(n-m,2)) p$`71w)'[ error('zernfun:NMmultiplesof2', ... (1^AzE%U+Z 'All N and M must differ by multiples of 2 (including 0).') wzwEYZN(q end m6a`Ok P '-N `u$3Y zn@<>o8hU if any(m>n) }~DlOvsq error('zernfun:MlessthanN', ... Pv|g.hH9m 'Each M must be less than or equal to its corresponding N.') mCb(B48]%X end
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s2 z | (JxtQqQg G3
rTzMO if any( r>1 | r<0 ) !ow:P8K? error('zernfun:Rlessthan1','All R must be between 0 and 1.') >B!E 6ah end z"Miy HIsIW%B jhgS@g=@ZC if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) >E;kM
B error('zernfun:RTHvector','R and THETA must be vectors.') 'y[74?1 end #>iBu:\J |r>+\" X _~/F- r = r(:); aMe]6cWHV> theta = theta(:); r'/&{?Je/ length_r = length(r); Kkcb'aDR if length_r~=length(theta) ?<)4_ error('zernfun:RTHlength', ... Tsch:r S 'The number of R- and THETA-values must be equal.') ' ^E7T'v% end b"Hc==` &&T\PspM Z}#'.y\ f % Check normalization: CT1@J-np % -------------------- 1y'8bt~7Pf if nargin==5 && ischar(nflag) VwvL isnorm = strcmpi(nflag,'norm'); o&0fvCpW if ~isnorm o@:${>jw error('zernfun:normalization','Unrecognized normalization flag.') _N)/X|=~s end 5.! OC5tO else gR1vUad7 isnorm = false; |>|f?^ end ?1w{lz(P {*mf Is (KxI* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G5*"P!@6 % Compute the Zernike Polynomials ]8@s+N %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jV{?.0/h| D+#OB|&Dn I]Ev6>=; % Determine the required powers of r: xB-\yWDZe
% ----------------------------------- :j^IXZW m_abs = abs(m); M^IEu} rpowers = []; K|L&mL&8 for j = 1:length(n) PWci D '! rpowers = [rpowers m_abs(j):2:n(j)]; qlSI| @CO end c"KN;9c, rpowers = unique(rpowers); |BGB60}]f 7[=\bL lCafsIB % Pre-compute the values of r raised to the required powers, +pUG6.j% % and compile them in a matrix: ]31>0yj[Q % ----------------------------- Z9wKjxu+ if rpowers(1)==0 9K!kU6Gh rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); )7]la/0 rpowern = cat(2,rpowern{:}); (A(j.[4a rpowern = [ones(length_r,1) rpowern]; s>J\h else D/[;Y<X#V rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); \-Vja{J] rpowern = cat(2,rpowern{:}); CP0;<}k end :j2?v(jT_l M$u.lI H 48YX(HI % Compute the values of the polynomials: ^4/
% -------------------------------------- 0<i8
;2KD y = zeros(length_r,length(n)); |j}D2q= for j = 1:length(n) '4KN s = 0:(n(j)-m_abs(j))/2; /a,"b8 pows = n(j):-2:m_abs(j); lA{JpH_Y8s for k = length(s):-1:1 jOUM+QO p = (1-2*mod(s(k),2))* ... dNu?O>= prod(2:(n(j)-s(k)))/ ... X9
N4 prod(2:s(k))/ ... o$QC:%[# prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ^[x6p}$ prod(2:((n(j)+m_abs(j))/2-s(k))); *@I/TX'\rY idx = (pows(k)==rpowers); 7Pe<0K)s( y(:,j) = y(:,j) + p*rpowern(:,idx); 5GK> ~2c( end ^(kmF UV,Z @.&KRAZ if isnorm ?B+]Ex(\B, y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ^HhV?Iqg end o 9rZ&Q< end oRo[WQla % END: Compute the Zernike Polynomials bvW3[ V %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LpK? C<?x BOflhoUX >,x&L[3 % Compute the Zernike functions: l{I.l % ------------------------------ Sx:JuK@ idx_pos = m>0; P5KpFL`B idx_neg = m<0; 6t\0Ui r>#4Sr A^c
( z = y; 9!_JV;2 if any(idx_pos) +iqzj-e&e[ z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); HV&i! M@T end B/*\Ih9y if any(idx_neg) ^
Paf -/ z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 00B,1Q HP end b*(,W x)jc >*/:"!u % EOF zernfun {[4.<|26
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