浅谈压缩感知（十七）：测量矩阵之有限等距常数RIC的计算

有限等距常数（RestrictedIsometry Constant, RIC）是与有限等距性质（Restricted IsometryProperty, RIP）紧密结合在一起的一个参数。

一、RIC定义：

在前面的一篇RIP文章中其实已经提到了，这里在贴出该定义：



CS满足条件：S阶RIP性质只要要求0<δS<1就可以了，而RIC是指满足RIP的最小δS。

二、RIC计算：

RIC与特征值的关系：

三、MATLAB代码

function y=RIPText(x,t)
% calculete the kth Restricted Isometry Constant of measurement matrix x.
% Modified on March 13th, 2012.
% Reference: Wei Dai, Olgica Milenkovic. Subspace Pursuit for Compressive Sensing
% Signal Reconstruction. IEEE Trans. ona Information Theory. vol.55, no.5,
% 2230-2249. May, 2009
%calculate the Restricted Isomentry Constant of matrix constructed by
%random n columns of the original matrix x.
%created by Li Bo in March 16th, 2011
%Harbin Institute of Technolgy
n=size(x,2);%the numbers of column of original matrix x
Num=1:n;%create a row of numbers from one to n with iterval 1
Com=combntns(Num,t);% all the combination of the selected t columns from total n columns
ma=size(Com,1);% the number of combinations
delta=zeros(1,ma);% a vector used to store the restricted isometry constant candidate
for i=1:ma
b=x(:,Com(i,:));% new matrix constructed by the t selected columns
e=eig(b'*b-eye(t));%calculate all the eigen values of matrix b'*b-eye(t)
delta(i)=(max(abs(e)));%select the largest absolute eigen value as restricted isomentry candidate
end
y=max(delta);% select the largest one as restricted isometry constant

用此代码来验证矩阵其实是不现实的，原因解释如下，在程序运行过程中容易导致内存不足等问题。

四、参考文章

http://blog.csdn.net/jbb0523/article/details/44622819