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The complexity of compressing subsegments of images described by finite automata. (English) Zbl 1010.68075

Summary: We investigate how the compression size of the compressed version of a two-dimensional image changes when we cut off a part of it, e.g. extract a photo of one person from a photo of a group of people, when compression is considered in terms of finite automata. Denote by \(c(T)\) the compression size of a square image \(T\) in terms of deterministic automata, it is the smallest size of a deterministic acyclic automaton \(A\) describing \(T\). The corresponding alphabet of \(A\) has only four letters, corresponding to four quadrants. We consider an independent useful combinatorial interpretation of \(c(T)\) in terms of regular subsquares of \(T\). Denote by \(\varPsi (n)\) the largest compression size \(c(R)\) of a square subsegment \(R\) of the image \(T\) such that \(c(T)=n\). We show that there is a constant \(c>0\) such that: \[ cn^{2.5}\leqslant \varPsi (n)\leqslant n^{2.5} \] For weighted automata we show that the compression size grows only linearly, if \(T\) is described by a weighted automaton with \(n\) states and \(m\) edges then a subimage \(R\) can be described by a similar automaton having \(O(n)\) states and \(O(m)\) edges.
We also show how to construct efficiently (in linear time w.r.t. the total size of the input and the produced output) the compressed representation of subsegments given the compressed representation of the whole image.

MSC:

68Q45 Formal languages and automata
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