Formal Definitions Equivalent for Two Dimensional Hidden Markov Models

In order to build formal definitions for general Two-dimensional Hidden Markov Models(2-D HMM) and to facilitate their studies in theory and application,this paper summarizes five strict definitions of 2-D HMM with their equivalence proven by high-order cases on low-order ones.These equivalent results are helpful to understand the nature of 2-D HMM and to provide a strict basis of formalization for further research.