# How to Decompose a Separable Filter?

I have done some research on the Internet and I have found that a given 2D mask is separable if it exists only a singular value of that matrix. For example, given the following matrix:

$$A = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 1 \end{pmatrix}$$

If I do for instance in matlab svd(A), it gives me the vector $$[0\; 0\; 6]$$, thus, the filter is separable. Nevertheless, I would like to know if it is possible to calculate the vectors that multiplied (or convolved) give raise to $$A$$. As a matter of fact, I know that:

$$\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} \begin{pmatrix} 1 & 2 & 1 \end{pmatrix}= \begin{pmatrix} 1 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 1 \end{pmatrix}$$

However, I would like to calculate those vector for any separable matrix.

Thank you so much for your responses.

Indeed you can do that. You may look on my answer to How to Prove a 2D Filter Is Separable?

By the SVD for any filter $$A$$:

$$A = \sum_{i = 1}^{n} {\sigma}_{i} {u}_{i} {v}_{i}^{T}$$

Since we're talking about separable filter then:

$$A = {\sigma}_{1} {u}_{1} {v}_{1}^{T}$$

So the columns filter is $$\sqrt{\sigma}_{1} {u}_{1}$$ and the rows filter is $$\sqrt{\sigma}_{1} {v}_{1}^{T}$$.

Since what's important is that $$A = {\sigma}_{1} {u}_{1} {v}_{1}^{T}$$ you can actually choose any 2 vectors which their outer product is $${\sigma}_{1} {u}_{1} {v}_{1}^{T}$$. The choice isn't unique. For example you could take for columns $${\sigma}_{1}^{2} {u}_{1}$$ and $$\frac{1}{{\sigma}_{1}} {v}_{1}^{T}$$ for the rows.