Deconvolution Between Matrix and Sub Matrix

Question

I have a matrix a=[[1 2 3]; [4 5 6]; [7 8 9]] and a submatrix b=[[5 6];[8 9]].

Is there a method in matlab for deconvolving (a,b) ?

I am looking for a method fo recognize the presence of a submatrix in a possible giant matrix. By a sort of deconvolution I expect to obtain something like a matrix with zeros all around and 1 in the place where the submatrix is present.

In the above example, a 1 in the right-down corner.

Answer 1

There is a better explanation here.

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Let's talk about 1D deconvolution for simplicity sake.

Your signal can be represented as a vector, and convolution is multiplication with a tridiagonal matrix.

For example:

Your vector/signal is:

    V1
    V2
    ...
    Vn

Your filter (convolving element) is:

  [b1 b2 b3];

So the matrix is nxn: (Let it be called A):

[b2 b3 0  0  0  0.... 0]
[b1 b2 b3 0  0  0.... 0]
[0  b1 b2 b3 0  0.... 0]
.....
[0  0  0  0  0  0...b2 b3]

Convolution is:

  A*v;

And de-convolution is

  A^(-1) * ( A) * v;

Obviously, in some cases de-convolution is not possible. Then you will have singular A. But if A^-1 exists, you need to compute it, and apply it on the result.

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For 2D case, it is a bit more complex, but the idea is the same.