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, ~qe%Yq 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, H~ZV*[A` 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? RrUBpqA 那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? qTZFPfyU !Z
VU,b> xGTP;NT_H kmzH'wktt Bqma\1cgb function z = zernfun(n,m,r,theta,nflag) -*5Rnx|Y{ %ZERNFUN Zernike functions of order N and frequency M on the unit circle. F}Vr:~ % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N *5 w{8 % and angular frequency M, evaluated at positions (R,THETA) on the Z{&cuo.@<] % unit circle. N is a vector of positive integers (including 0), and D}8EER b % M is a vector with the same number of elements as N. Each element Eu"_MgD % k of M must be a positive integer, with possible values M(k) = -N(k) N?Lb % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, rZ8`sIWQt % and THETA is a vector of angles. R and THETA must have the same Y0eE-5F, % length. The output Z is a matrix with one column for every (N,M) PkI:*\R % pair, and one row for every (R,THETA) pair. dy_:-2S % MSf;ZB % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 8@so"d2e % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), {s. = )0V % with delta(m,0) the Kronecker delta, is chosen so that the integral w$JvB5O % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, N('&jHF % and theta=0 to theta=2*pi) is unity. For the non-normalized >EY3/Go> % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. TB0
5?F % jy-{~xdg[ % The Zernike functions are an orthogonal basis on the unit circle. I? ,>DHUX % They are used in disciplines such as astronomy, optics, and Lemui) % optometry to describe functions on a circular domain. M4as % w@,zFV % The following table lists the first 15 Zernike functions. E>l~-PaZY % 98^V4maR: % n m Zernike function Normalization 7uzkp&+: % -------------------------------------------------- #%DE; % 0 0 1 1 / m=HG^! % 1 1 r * cos(theta) 2 2}8v(%s p % 1 -1 r * sin(theta) 2 |1j["u1 % 2 -2 r^2 * cos(2*theta) sqrt(6) dAuJXGo % 2 0 (2*r^2 - 1) sqrt(3) j]`PSl+w % 2 2 r^2 * sin(2*theta) sqrt(6) HTG%t/S % 3 -3 r^3 * cos(3*theta) sqrt(8) 41&\mx
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) KCs[/] % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) B_.%i+ZZ % 3 3 r^3 * sin(3*theta) sqrt(8) ~@}Bi@* % 4 -4 r^4 * cos(4*theta) sqrt(10) a\r\PBi % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) M3.do^ss % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) s0vDHkf8 % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) E>K!Vrh-L % 4 4 r^4 * sin(4*theta) sqrt(10) ov, hI>0!D % -------------------------------------------------- q<M2,YrbAI % hIT+gnhh % Example 1: 79;<_(Y % $&=S#_HQS % % Display the Zernike function Z(n=5,m=1) Hm*/C4B` % x = -1:0.01:1; u A<n % [X,Y] = meshgrid(x,x); Hl,W=2N % [theta,r] = cart2pol(X,Y); m;,N)<~ % idx = r<=1; ?32&]iM
oW % z = nan(size(X)); FYpzQ6s~ % z(idx) = zernfun(5,1,r(idx),theta(idx)); [@.!~E)P % figure ~A\GT$ % pcolor(x,x,z), shading interp +L;e^#>d % axis square, colorbar |!4K!_y % title('Zernike function Z_5^1(r,\theta)') +{oG|r3L % z:wutqru % Example 2: wfH^<jY)E % a^I\ /&aw' % % Display the first 10 Zernike functions XuFYYx~ ^3 % x = -1:0.01:1; BI%$c~wS % [X,Y] = meshgrid(x,x); {N+$Q' % [theta,r] = cart2pol(X,Y); @u6B;)'l % idx = r<=1; p;>ec:z3M % z = nan(size(X));
XwJ7|cB % n = [0 1 1 2 2 2 3 3 3 3]; EFM5,gB.m % m = [0 -1 1 -2 0 2 -3 -1 1 3]; Y^wW2-,m % Nplot = [4 10 12 16 18 20 22 24 26 28]; %WjXg:R % y = zernfun(n,m,r(idx),theta(idx)); A
PEE~ % figure('Units','normalized') JkbQyn % for k = 1:10 =%TWX[w % z(idx) = y(:,k); nWw":K<@Q_ % subplot(4,7,Nplot(k)) + R~'7*EI % pcolor(x,x,z), shading interp ZbdZrE$ % set(gca,'XTick',[],'YTick',[]) I=`U7Bis" % axis square pOIJH =# % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) g,!L$,/F % end #V~me % vg32y /l]S % See also ZERNPOL, ZERNFUN2. X}Ai-D [M=7M}f; !$gR{XH$] % Paul Fricker 11/13/2006 Yi.N& | |