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Additional info for A Stochastic Approach to Content Adaptive Digital Image Watermarking

Example text

That is the point in the geometric center of the triangle, the vector [1/3 1/3 1/3]. Every Markov process must have at least one such point that remains stationary when the P transformation is applied. t It is the point or set of points satisfying the equation 1t = 1tP t This feature is a consequence of Brouwer's fixed point theorem. 26) 34 THE BASIC MARKOV PROCESS ........ " "- "'- Shrinkage factor = 1 "" " \.. \.. '\ \ \ \ \ \ 1 \ \ \ \ \ \ \ \ \ \ I I ......... ' ........ 9 (a) Transitio n diagram for the periodic process.

Thus we refer to one-, two-, three-, etc. chain processes as monodesmic, duodesmic, tridesmic, etc. A process with more than one chain but with the exact number of chains unspecified will be called a polydesmic process. We shall not think of transient states as being members of chains, but we shall associate them with chain. A transient state is associated with a chain if it is possible for the process to enter that chain from the transient state. Therefore, a transient state must be associated with at least one chain.

We shall soon understand not only its form, but how that form is achieved. However, let us anticipate one result. It is the rule rather than the exception that the rows of are identical. For reasons that will become clear presently, we call a Markov process that has a with equal rows a monodesmic process. We shall soon develop precise conditions for a process to be monodesmic. However, a sufficient condition for a process to be monodesmic is that the process be able to make a transition (have a non-zero transition probability) from any state to any other state.