The complexity of compressing subsegments of images described by finite automata

We investigate how the compression size of the compressed version of a two-dimensional image changes when we cut o3 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 "nite 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 (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 6 (n)6 n: 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. A preliminary version of the paper was presented at conference CPM’99 [10]. ∗Corresponding author. Department of Computer Science, Liverpool University, Chadwick Building Peach Street, L69 72F Liverpool, UK. E-mail addresses: [email protected]." (J. Karhum% aki), [email protected] (W. Plandowski), [email protected] (W. Rytter). 1Supported by Academy of Finland under grant 44087. 2Supported in part by KBN grant 8T11C03915 and in part by Academy of Finland under grant 44087. 0166-218X/02/$ see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S0166 -218X(02)00196 -8 236 J. Karhum2 aki et al. / Discrete Applied Mathematics 125 (2003) 235–254 We also show how to construct eLciently (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. ? 2002 Elsevier Science B.V. All rights reserved.