This question is motivated by my wrong observation 2. of this post. For reasons of clarity, I am going to put the same set up here too:
Definition Let $A = (a_{ij}) \in M_{m,n}(\{0,1\})$ and $\mathcal{M}_A = (m_{ij}) \in M_{m,n}(\{0,\dots,9\})$ be the matrix defined by $$m_{ij} = \Big| \big\{ a_{kl} \, \big| \, |k-i|\leq 1, |l-j|\leq 1, a_{kl} = 1 \big\} \Big|.$$ Let us call $A$ the picture matrix and $\mathcal{M}_A$ the corresponding data matrix.
Example
If $$A = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix},$$ then $$ \mathcal{M}_A = \begin{pmatrix} 2 & 3 & 2 \\ 4 & 5 & 3 \\ 3 & 4 & 2 \end{pmatrix}. $$
Observations and Thoughts
- Not every matrix can be a data matrix, for instance there is no picture matrix $A \in M_{1,2}(\{0,1\})$ such that $ \mathcal{M}_A = \begin{pmatrix} 0 & 2 \end{pmatrix} $.
- It is also not necessary that a data matrix $M$ admits a unique corresponding picture matrix. For instance (cf. Jaap Scherphuis's comment in the mentioned link), if $M = \begin{pmatrix} 1 & 1 \end{pmatrix}$, then both $A = \begin{pmatrix} 0 & 1 \end{pmatrix}$ and $A = \begin{pmatrix} 1 & 0 \end{pmatrix}$ are possible solutions for $\mathcal{M}_A = M$.
- User antkam noted that if a data has $3k+2$ rows (or columns), it has more than one corresponding picture matrix, see here.
Question Under which circumstance does a data matrix $M$ admit a unique corresponding picture matrix?
Further thoughts, references and remarks are highly appreciated!