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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.
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:
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