Karel Culik II and Simant Dube. Affine automata and related techniques for generation of complex images. Theoretical Computer Science, 116:373--398, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....1 Introduction Picture generating devices, such as iterated function systems, chain code picture grammars, turtle geometry picture grammars, cellular automata, random context picture grammars and collage grammars, specify in general infinite picture sequences or languages. See, for example, [PJS92,CD93,MRW82,PL90,EvdW99, DK99]. Although they are intended and used for modelling visible phenomena of various kinds, due to their infinity additional prerequisites are required to make the generated pictures visible. Depending on the resolution of the chosen output medium, be it a display or a printer, the generated pictures ....

Karel Culik II and Simant Dube. Affine automata and related techniques for generation of complex images. Theoretical Computer Science, 116:373--398, 1993.

....Dimensions of Some Fractals . 24 6 Conclusions 27 1 Introduction There are a series of approaches which use formal languages in order to describe fractals, or, more generally, pictures. Probably the most prominent examples are L systems [30, 31] and finite automata [4, 5, 6, 7]. Further links provide hypergraphbased ideas like collage grammars [19, 44] and cellular automata [43, 20] Another approach to fractals is via Iterated Function Systems (IFS) cf. 2, 9, 11] An IFS is composed of a metric space X and a set fF (1) F (n)g of mappings on X . If all F (i) ....

....: a x . 3. The above mentioned solution K equals ff F (x) x : IN f1; ngg. Mappings x : IN S where S is a finite set (alphabet) are known in the theory of formal languages as w words (cf. 40] It also makes sense to consider IFS where not all mappings F (i) are contractive (cf. [1, 6, 26]) Should this be the case, Item 1 from above is not fulfilled for all x. Then one has to single out those x for which the sequence (1) converges independently of the start a 2 X . In practice, this condition is very intriguing and depends heavily on the mappings F (i) In the above mentioned ....

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K. Culik II and S. Dube, Affine automata and related techniques for generation of complex images, Theor. Comp. Sci., 116:373--398, 1993.

.... of pictures addressed by sequences obeying a given regular expression is just a logarithm of the maximum modulus of the connection matrix of the underlying finite automaton (which is in fact the topological entropy of the set of sequences specified by the underlying automaton) Culik II and Dube [26], 27] investigated several methods for fractal image description (compression) based on iterative function systems driven by a prescribed set L of symbolic sequences. Typically, the set L was taken to be a regular language. Recently, there has been an extensive research activity in fractal and ....

K. Culik(II) and S. Dube, "Affine automata and related techniques for generation of complex images," Theoretical Computer Science, vol. 116, no. 2, pp. 373--398, 1993.

.... for sequencing transformations, thus LRIFS s incorporate the earlier generalizations committed to a particular mechanism, such as hierarchical IFS s [4] sofic systems [1] recurrent IFS s [3] Markov IFS s [21] mixed IFS s [5] controlled IFS s [17] and mutually recursive function systems [7, 8]. Several other authors considered similar generalizations without giving them a name. Visualization of the attractors of generalized IFS s has been addressed by Hart [9] referring to his earlier results with DeFanti [10] This paper is organized as follows. Sections 2 and 3 summarize the ....

K. Culik II and S. Dube. Affine automata and related techniques for generation of complex images. Manuscript, University of South Carolina in Columbia.

.... for sequencing transformations, thus LRIFS s incorporate the earlier generalizations committed to a particular mechanism, such as hierarchical IFS s [4] sofic systems [1] recurrent IFS s [3] Markov IFS s [21] mixed IFS s [5] controlled IFS s [17] and mutually recursive function systems [7, 8]. Several other authors considered similar generalizations without giving them a name. Visualization of the attractors of generalized IFS s has been addressed by Hart [9] referring to his earlier results with DeFanti [10] This paper is organized as follows. Sections 2 and 3 summarize the ....

K. Culik II and S. Dube. Affine automata and related techniques for generation of complex images. Manuscript, UniversityofSouth Carolina in Columbia.

..... 24 6 Conclusions 27 IFS and Control Languages 3 1 Introduction There are a series of approaches which use formal languages in order to describe fractals, or, more generally, pictures. Probably the most prominent examples are L systems [30, 31] and finite automata [4, 5, 6, 7]. Further links provide hypergraphbased ideas like collage grammars [19, 44] and cellular automata [43, 20] Another approach to fractals is via Iterated Function Systems (IFS) cf. 2, 9, 11] An IFS is composed of a metric space X and a set fF (1) F (n)g of mappings on X . If all F (i) ....

....: a x . 3. The above mentioned solution K equals ff F (x) x : IN f1; ngg. Mappings x : IN S where S is a finite set (alphabet) are known in the theory of formal languages as w words (cf. 40] It also makes sense to consider IFS where not all mappings F (i) are contractive (cf. [1, 6, 26]) Should this be the case, Item 1 from above is not fulfilled for all x. Then one has to single out those x for which the sequence (1) converges independently of the start a 2 X . In practice, this condition is very intriguing and depends heavily on the mappings F (i) In the above mentioned ....

[Article contains additional citation context not shown here]

K. Culik II and S. Dube, Affine automata and related techniques for generation of complex images, Theor. Comp. Sci., 116:373--398, 1993.

.... the main part of the paper four of the well known classes of picture generating devices found in the literature, namely collage grammars [22] mutually recursive function systems (a generalised type of iterated function systemsv [2] which is also called hierarchical iterated function system; cf. [4, 27]) context free chain code grammars [25] and 0L systems with turtle interpretation [28] are translated into tree based picture generators. In each case the equivalence of the traditional device and the tree based variant is shown. This establishes a sound formal basis for future work as well as ....

....2 R, 0 c 1, such that dist(a(x) a(y) c Delta dist(x; y) for all x; y 2 R d . Every sequence w = a 1 Delta Delta Delta a k 2 contr(d) yields an operation Hw on (R d ) the Hutchinson operator determined by w, which is defined by Hw (p) a 1 (p) Delta Delta Delta [ a k (p) see [2, 27, 4, 5] for this definition and the following ones) 6 P R d is bounded if it is contained in a ball of finite radius and closed if the limit of every converging sequence of points in P is in P , too. 15 There are mainly three types of IFSs one can encounter in the literature: the ordinary IFS, ....

[Article contains additional citation context not shown here]

Karel Culik II and Simant Dube. Affine automata and related techniques for generation of complex images. Theoretical Computer Science, 116:373--398, 1993.

....Sierpinski grammars seem to be related to results in the area of so called iterated function systems (see, e.g. Barnsley [Bar88] This connection should be investigated. 3) A comparison of collage grammars with other syntactic approaches to fractal geometry, like the work by Culik II and Dube [CD90, CD91], would be interesting. 4) Moreover, one should compare collage grammars with other syntactic devices for the generation of pictures, like chain code picture languages (see, e.g. Maurer, Rozen10 berg, and Welzl [MRW82] Dassow [Das89] Kim [Kim90] graphical interpretation of L systems (see, ....

K. Culik II, S. Dube. Affine automata and related techniques for generation of complex images. Lecture Notes in Computer Science 452, 224--231, 1990. To appear in Theoretical Computer Science.

....of the addresses of the nodes assigned value 1 (black) Regular sets of strings are specified by finite automata or regular expressions [13] Therefore, finite automata can be used to specify (regular) multiresolution images. This idea has been recognized independently by several authors [1, 2, 4, 14], but has not led to a successful image data compression method. The finite automata method of image specification has been extended to gray scale images, represented by Weighted Finite Automata (WFA) 6, 7] A theoretical WFA inference algorithm was proposed in [7] This algorithm finds a WFA ....

K. Culik II and S. Dube, Affine automata and related techniques for generation of complex images, Proceedings of MFCS 1990, Lecture Notes In Computer Science 452, 224-231, Springer, Berlin (1990).

....languages in order to describe infinite IFS (IIFS) This is the link between formal language theory and fractal geometry on which we elaborate in this paper. More results on this topic are contained in [30, 31, 73, 36, 33, 32, 35, 34] Another approach, called MRFS by Culik, can be found in [58, 4, 8, 72, 17, 18, 19, 53, 64, 22, 20] mostly under different names where basically images of closed regular languages are used 1 As regards our notations, we refer the reader to the table of symbols , Section 7. 2 Note that Kuich in [50] also considered only unambiguous contextfree grammars when treating structure ....

K. Culik, II and S. Dube. Affine automata and related techniques for generation of complex images. Technical report, Dep. of Computer Science, University of South Carolina, July 1991.

....how well the network captures the topological n block structure of the training sequence S. For example, starting from the Sierpinski triangle [3] n block structure (figure 11e) corresponding to a stochastically trivial initial one state solution with three (almost equally probable) symbol loops [13, 12], RNN eventually learns to generate n block structure (shown in figures 12a,d) closely approximating that of the training sequence S. Finally, in figure 11b, we show the final configuration of the Kohonen self organizing map (SOM) with St(10=3) topology used in construction of the machine MCBR . ....

K. Culik(II) and S. Dube. Affine automata and related techniques for generation of complex images. Theoretical Computer Science, 116(2):373--398, 1993.

.... of pictures addressed by sequences obeying a given regular expression is just a logarithm of the maximum modulus of the connection matrix of the underlying finite automaton (which is in fact the topological entropy of the set of sequences specified by the underlying automaton) Culik II and Dube [26], 27] investigated several methods for fractal image description (compression) based on iterative function systems driven by a prescribed set L of symbolic sequences. Typically, the set L was taken to be a regular language. Recently, there has been an extensive research activity in fractal and ....

K. Culik(II) and S. Dube, "Affine automata and related techniques for generation of complex images," Theoretical Computer Science, vol. 116, no. 2, pp. 373--398, 1993.

....the field of picture generation the idea seems to be new, however. In the main part of the paper four of the well known classes of picture generating devices found in the literature, namely collage grammars [HK91] mutually recursive function systems (or hierarchical iterated function systems; cf. [Bar88, CD93, PJS92]) contextfree chain code grammars [MRW82] and 0L systems with turtle interpretation [PL90] are translated into tree based picture generators. In each case the equivalence of the traditional device with its tree based counterpart is shown. This establishes a sound formal basis for future work as ....

....R, 0 c 1, such that dist(a(x) a(y) c Delta dist(x; y) for all x; y 2 R d . Every sequence w = a 1 Delta Delta Delta a k 2 contr(d) yields an operation Hw on (R d ) the Hutchinson operator determined by w, which is defined by Hw (p) a 1 (p) Delta Delta Delta [a k (p) see [Bar88, PJS92, CD93] for this definition and the following ones) There are mainly three types of IFSs one can encounter in the literature: the ordinary IFS, the IFS with condensation, and the mutually recursive function system (MRFS [CD93] called hierarchical IFS in [PJS92] In order to obtain a comprehensive ....

[Article contains additional citation context not shown here]

Karel Culik II and Simant Dube. Affine automata and related techniques for generation of complex images. Theoretical Computer Science, 116:373--398, 1993.

.... of pictures addressed by sequences obeying a given regular expression is just a logarithm of the maximum modulus of the connection matrix of the underlying finite automaton (which is in fact the topological entropy of the set of sequences specified by the underlying automaton) Culik II and Dube [6, 7] investigated several methods for fractal image description (compression) based on iterative function systems driven by a prescribed set L of symbolic sequences. Typically, the set L was taken to be a regular language. Recently, there has been an extensive research activity in fractal and ....

K. Culik(II) and S. Dube. Affine automata and related techniques for generation of complex images. Theoretical Computer Science, 116(2):373--398, 1993.

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