| 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, e5s=@-[ 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, QY^v*+lr\ 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? 1ti9FQ 那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ;8~tt I ;Y^.SR" ]c+HD* 'm4v)w<y# apkmb< function z = zernfun(n,m,r,theta,nflag) oEuV&m|yX %ZERNFUN Zernike functions of order N and frequency M on the unit circle. F?!X<N{ % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Dcep^8' % and angular frequency M, evaluated at positions (R,THETA) on the @V7HxW7RX % unit circle. N is a vector of positive integers (including 0), and S^,q{x*T % M is a vector with the same number of elements as N. Each element a(s%3"*Q % k of M must be a positive integer, with possible values M(k) = -N(k) Ec/-f`8 % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, q83!PI % and THETA is a vector of angles. R and THETA must have the same O$K?2- % length. The output Z is a matrix with one column for every (N,M) >!gW]{ % pair, and one row for every (R,THETA) pair. -Wt(t2 % s?,\aSsU@ % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike x\R%hGt % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), }#Q?\ % with delta(m,0) the Kronecker delta, is chosen so that the integral 4#fgUlV % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 9R1S20O % and theta=0 to theta=2*pi) is unity. For the non-normalized UWPzRk#s" % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. FRR`<do5$, % )Bb:?!EuEH % The Zernike functions are an orthogonal basis on the unit circle. DOa%|H'P % They are used in disciplines such as astronomy, optics, and %
k}+t3aF % optometry to describe functions on a circular domain. 'Cp]Q@]\ % v6#i>n~x, % The following table lists the first 15 Zernike functions. s~)I1G % \Q~HL_fy|Y % n m Zernike function Normalization z7PmyU
> % -------------------------------------------------- px~ :'U % 0 0 1 1 #$?!P1 % 1 1 r * cos(theta) 2 dJf#j?\[ % 1 -1 r * sin(theta) 2 ;&~9k?v7L % 2 -2 r^2 * cos(2*theta) sqrt(6) O~bJ<O=? % 2 0 (2*r^2 - 1) sqrt(3) +W}dO# % 2 2 r^2 * sin(2*theta) sqrt(6) C
U 8s* % 3 -3 r^3 * cos(3*theta) sqrt(8) 0%b!ARix % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) iYR`|PJi % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) f.{/PL % 3 3 r^3 * sin(3*theta) sqrt(8) [izP1A$r#Q % 4 -4 r^4 * cos(4*theta) sqrt(10) '#ow9w+^ % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) '0tNo.8K % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 1(4}rB3 % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ae3=o8p % 4 4 r^4 * sin(4*theta) sqrt(10) DFvj % -------------------------------------------------- EA=EcUf' % rWS],q=c % Example 1: 8oxYgj&~X % {@*l ,[,5- % % Display the Zernike function Z(n=5,m=1) ZBxV& | |