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, <_o).hE{ 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, mE|?0mRA % 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? "s$$M\)T 那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? {WYJQKs8 cdBD.sg GkOZ=ej ]~YY#I": l2Gtw*i_I function z = zernfun(n,m,r,theta,nflag) yYdow.b! %ZERNFUN Zernike functions of order N and frequency M on the unit circle. S2;u!f % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N aEL^N0\d % and angular frequency M, evaluated at positions (R,THETA) on the #N?VbDK9_ % unit circle. N is a vector of positive integers (including 0), and xF/u('A % M is a vector with the same number of elements as N. Each element ?M<q95pL % k of M must be a positive integer, with possible values M(k) = -N(k) >}"9heF % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, W(gOidKKz % and THETA is a vector of angles. R and THETA must have the same yi29+T7j4S % length. The output Z is a matrix with one column for every (N,M) -Lo3@:2i % pair, and one row for every (R,THETA) pair. !_yWe % b.N$eJlQ& % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike (dH "b
* % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), lG1\41ZxB % with delta(m,0) the Kronecker delta, is chosen so that the integral (aeS+d x % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, r5>1n/+6 % and theta=0 to theta=2*pi) is unity. For the non-normalized k1.h |&JJN % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. n|p(Cb#G % Wb1?>q % The Zernike functions are an orthogonal basis on the unit circle. ~; V5*t % They are used in disciplines such as astronomy, optics, and SsY:gp_ % optometry to describe functions on a circular domain. prk@uYCa = % ^t2b`n60 % The following table lists the first 15 Zernike functions. :{g;J % 99KW("C1F % n m Zernike function Normalization +u[^@>_I0 % -------------------------------------------------- ]jB`"to*} % 0 0 1 1 (:9=M5d % 1 1 r * cos(theta) 2 2FE13{+f % 1 -1 r * sin(theta) 2 +jPJv[W % 2 -2 r^2 * cos(2*theta) sqrt(6) (zmLMG(R % 2 0 (2*r^2 - 1) sqrt(3) ~WW!P_wI, % 2 2 r^2 * sin(2*theta) sqrt(6)
A)5;ae % 3 -3 r^3 * cos(3*theta) sqrt(8) p0|PVn.^h % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ,6EFJVu
\ % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) x@p1(V. % 3 3 r^3 * sin(3*theta) sqrt(8) 6)h~9iK % 4 -4 r^4 * cos(4*theta) sqrt(10) qlNB\~HCe % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) YXlaE=9bn % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) L!c.1Rf_ % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) VE $Kdo^ % 4 4 r^4 * sin(4*theta) sqrt(10) H"; !A=0 % -------------------------------------------------- .',d*H))E7 % GzN /0:b % Example 1: .gJv})Vi % <9/?+) % % Display the Zernike function Z(n=5,m=1) >4^,[IO/ % x = -1:0.01:1; _qf$dGqc
% [X,Y] = meshgrid(x,x); M/abd 7q % [theta,r] = cart2pol(X,Y); KKRj#m(:! % idx = r<=1; s}93nv*ez % z = nan(size(X)); {5NE jUu{j % z(idx) = zernfun(5,1,r(idx),theta(idx)); Q>yO,H| % figure ?X'l&k> % pcolor(x,x,z), shading interp L2Z-seE % axis square, colorbar ?Z2_y- % title('Zernike function Z_5^1(r,\theta)') ZWb\^N % GTocN1,Z~a % Example 2: qCI0[U@ % E5X#9;U8E" % % Display the first 10 Zernike functions ($X2SIZh % x = -1:0.01:1; *G"}m/j- % [X,Y] = meshgrid(x,x); ?58*#'r % [theta,r] = cart2pol(X,Y); 89YG
` % idx = r<=1; zLSha\X % z = nan(size(X)); ]^6r7nfR6| % n = [0 1 1 2 2 2 3 3 3 3]; =KW~k7TaN % m = [0 -1 1 -2 0 2 -3 -1 1 3]; !F08F>@D % Nplot = [4 10 12 16 18 20 22 24 26 28]; h @2.D|c)g % y = zernfun(n,m,r(idx),theta(idx)); 87-z=>IU % figure('Units','normalized') Q-} cB % for k = 1:10 P_F0lO % z(idx) = y(:,k); cq4sgQ?sW % subplot(4,7,Nplot(k)) p1']+4r% % pcolor(x,x,z), shading interp Rebo.6rG % set(gca,'XTick',[],'YTick',[]) vm.%)F#@ % axis square ?2<V./2F % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) q!as~{! % end j-k]|0ea} % `G<|5pe % See also ZERNPOL, ZERNFUN2. T( CTU/a-, A,;[9J2\& @ [<B:Tqo % Paul Fricker 11/13/2006 5gZ* ja%IGaH;s 3Lm7{s?=Z- |o#pd\ =GL^tAUJ % Check and prepare the inputs: +<^c2diX % ----------------------------- S.*.nv if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) %TDY &@i= error('zernfun:NMvectors','N and M must be vectors.') =PmIrvr'[5 end UJ^-T+fut yUX<W'-Hev P] Xl if length(n)~=length(m) '=(@3ggA: error('zernfun:NMlength','N and M must be the same length.') L[. )!c8k end 0F%V+Y\R yC9~X='D v4W<_
7L_ n = n(:); 13MB1n m = m(:); ;*>':-4 if any(mod(n-m,2)) l*|m(7s error('zernfun:NMmultiplesof2', ... [w}KjV/yi 'All N and M must differ by multiples of 2 (including 0).') xX\A&9m end hEfFMi=a` 3
Bn9Ce= QV_Ep8 if any(m>n) )'e9(4[V1 error('zernfun:MlessthanN', ... 7KZ>x*o 'Each M must be less than or equal to its corresponding N.') AxiCpAS;J end X~rHNRIU PaBqv] F=V_ACU if any( r>1 | r<0 ) ke5_lr( error('zernfun:Rlessthan1','All R must be between 0 and 1.') C''[[sw'K end zF_aJ+i:~ r=ht:+m !345 if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) K~jN"ev error('zernfun:RTHvector','R and THETA must be vectors.') csms8J end QUi=ZD1 3.D|xE]g +KHk`2{y~ r = r(:); G-G\l?R( theta = theta(:); H
>1mi_1 length_r = length(r); .ot[_*A.FD if length_r~=length(theta) 6a*OQ{8 error('zernfun:RTHlength', ... rtk1 8U- 'The number of R- and THETA-values must be equal.') o;J_"'kP end C:P.+AU"` ~n9- <dX7{="& % Check normalization: nCSXvd/ % -------------------- Yc~c(1VRz if nargin==5 && ischar(nflag) U66 zm9
3& isnorm = strcmpi(nflag,'norm'); 0?\d%J!"S if ~isnorm ]?j[P=\ error('zernfun:normalization','Unrecognized normalization flag.') Avo"jN*<d end vV /fTO else a3(q;^v isnorm = false; = ms
o1 end S0-/9h UZ3oc[#D=] '/K-i.8F %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% GeCyq%dN % Compute the Zernike Polynomials A]mXV4RmI %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2a[_^v $v t 4tXLI;' PU{7s % Determine the required powers of r: +}@6V4BRn % ----------------------------------- {;Ispx0m m_abs = abs(m); T0Zv. rpowers = []; 'UL"yM for j = 1:length(n) $XO#qOW rpowers = [rpowers m_abs(j):2:n(j)]; Tq=OYJq5U end B;mt11M rpowers = unique(rpowers); uZ7~E._ $ h<l Y]!{
nW % Pre-compute the values of r raised to the required powers, a]u1_ $) % and compile them in a matrix: %$.]g % ----------------------------- @Zd/>' if rpowers(1)==0 U,)@+?U+h rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); < &~KYu\r rpowern = cat(2,rpowern{:}); jM DG rpowern = [ones(length_r,1) rpowern]; ;\N${YIn else 8I*WVa$l rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); rezH5d6z62 rpowern = cat(2,rpowern{:}); C!r9+z)< end sVJwe\! 'dTg\
Qv <!M ab} % Compute the values of the polynomials: !O~5<tA[#1 % -------------------------------------- N#? Ohz y = zeros(length_r,length(n)); 4)=\5wJDg1 for j = 1:length(n) :6Oh ?y@ s = 0:(n(j)-m_abs(j))/2; 7ZVW7%,zF pows = n(j):-2:m_abs(j); =7WE for k = length(s):-1:1 56R)631]p p = (1-2*mod(s(k),2))* ... ]rP'\a prod(2:(n(j)-s(k)))/ ... MGzuQrl{H prod(2:s(k))/ ... y $K#M prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... \.7O0Q{ prod(2:((n(j)+m_abs(j))/2-s(k))); E6NrBPm idx = (pows(k)==rpowers); R^=)Ucj y(:,j) = y(:,j) + p*rpowern(:,idx); "Lp"o end G~\ SI. xRx8E;Q@h? if isnorm H _%yh,L y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); eVYUJ, end z
a^s%^:yK end (YJ]}J^ % END: Compute the Zernike Polynomials uBe1{Z %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i+z;tF` ? <.U, TdAHw
@( % Compute the Zernike functions: "?~u*5 % ------------------------------ FGP~^Dr/ idx_pos = m>0; ]EzX$T idx_neg = m<0; 3)J0f+M>dv )@]Y1r4U '0!IF&p' z = y; 'h6Vj6 if any(idx_pos) *qLOr6 z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); %7$oig\wE end \5wC&|WEB if any(idx_neg) !PfI e94{` z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); !%x=o& end ,DT=( &x(^=sTHI Z-!W#
% EOF zernfun nFn@Z'T$N
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