circulant matrix

PRESENTED BY, LEKSHMY A SASI

ROLL NO:10 MTECH CSE

CIRCULANT AND BLOCK CIRCULANT MATRIX

Estimate the unknown object vector x from the noisy measurements y.

g=H f + vH is assumed to be knownf=H-1g – H-1v H=N 2 * N 2 matrix f, g, v is N 2 * 1 matrix for an N*N image

Two major problem It is extremely sensitive to noise and it has been

shown that one needs impossibly low levels of noise for the method to work

The solution of equation requires the inversion of an N2 * N2 square matrix, with N typically being 500,which is formidable task even for modern computers.

Is there any way by which matrix H can be inverted?

Yes, for the case of homogeneous linear degradation ,matrix H can easily be inverted because it is a block circulant matrix.

When is a matrix block circulant? A matrix H is block circulant if it has the following

structure.

Where H0, H1, Hm-1 are partitions of matrix H and they are themselves circulant matrices.

When is a matrix circulant? A matrix D is circulant if it has the following

structure:

N=6

In such a matrix ,each column can be obtained from the previous one by shifting all elements one place down and putting the last element at the top.

Why can a block circulant matrix can be inverted easily? Because we can easily find their eigen values and

eigen vectors.

K=0,1,2,….M-1It can be shown then by direct substitution

that.Dω(k)=λ(k)ωkHow does the knowledge of the eigen values

and eigen vectors of a matrix help in inverting the matrix? If we form matrix W which has the eigen vectors of

matrix D as its columns, we know that we can write D=w^w-1

THANK YOU