The Number of Forms for Two-Dimensional Images

In this paper, forms for gray-level images are investigated. These forms are defined by the decomposition of an $(N\times M)$-dimensional rectangle belonging to a two-dimensional integer grid into non-overlapping connected domains with boundaries passing along the grid edges. An expression for the number $L(M;N)$ of possible forms is derived as the sum of elements of a matrix raised to a power, for which a recursive relation is written. Then an estimate for the rate of exponential increase of the quantity $L(M;N)$ for $M\to\infty$, $N\to\infty$ is obtained.