COLLAGE-BASED INVERSE PROBLEMS FOR IFSM WITH ENTROPY MAXIMIZATION AND SPARSITY CONSTRAINTS
DOI:
https://doi.org/10.5566/ias.v32.p183-188Keywords:
collage theorem, entropy, fractal transforms, iterated function systems with mappings, sparsityAbstract
We consider the inverse problem associated with IFSM: Given a target function f, find an IFSM, such that its invariant fixed point f is sufficiently close to f in the Lp distance. In this paper, we extend the collage-based method developed by Forte and Vrscay (1995) along two different directions. We first search for a set of mappings that not only minimizes the collage error but also maximizes the entropy of the dynamical system. We then include an extra term in the minimization process which takes into account the sparsity of the set of mappings. In this new formulation, the minimization of collage error is treated as multi-criteria problem: we consider three different and conflicting criteria i.e., collage error, entropy and sparsity. To solve this multi-criteria program we proceed by scalarization and we reduce the model to a single-criterion program by combining all objective functions with different trade-off weights. The results of some numerical computations are presented. Numerical studies indicate that a maximum entropy principle exists for this approximation problem, i.e., that the suboptimal solutions produced by collage coding can be improved at least slightly by adding a maximum entropy criterion.References
Barnsley MF, Ervin V, Hardin D, Lancaster J (1985). Solution of an inverse problem for fractals and other sets. Proc Nat Acad Sci USA 83:1975--77.
Barnsley MF (1989). Fractals everywhere. New York: Academic Press.
Barnsley MF, Demko S (1985). Iterated function systems and the global construction of fractals. Proc Roy Soc London Ser A 399:243--75.
Barnsley MF, Hurd L (1993). Fractal image compression. Massachussetts: A.K. Peters.
Centore P, Vrscay ER (1994). Continuity of attractors and invariant measures for Iterated Function Systems. Canad Math Bull 37(3):315--29.
Demers M, Kunze H, La Torre D (2012). On random iterated function systems with greyscale maps. Image Anal Stereol 31(2):109--120.
Fisher Y (1995). Fractal image compression, theory and application. New York: Springer-Verlag.
Forte B, Vrscay ER (1995). Solving the inverse problem for function and image approximation using iterated function systems. In: Dynamics of Continuous, Discrete and Impulsive Systems 1(2).
Forte B, Vrscay ER (1999). Theory of generalized fractal transforms. In: Fisher Y, ed. Fractal Image Encoding and Analysis. NATO ASI Series F. Vol. 159. New York: Springer Verlag.
Ghazel M, Freeman GH and Vrscay ER (2003). Fractal image denoising. IEEE Trans Image Proc 12(12):1560--78.
Hutchinson J (1981). Fractals and self-similarity. Indiana Univ J Math 30:713--47.
A comparative simulation study on the IFS distribution function estimator. Nonlinear Anal Real World Appl 6(5):858--73.
Approximating distribution functions by iterated function systems. J Appl Math Dec Sci 1:33--46.
Kunze H, La Torre D, Vrscay ER (2008). From iterated function systems to iterated multifunction systems. Commun Appl Nonlinear Anal 15(4):1--15.
Kunze H, La Torre D, Vrscay ER (2012). Solving inverse problems for DEs using the collage theorem and entropy maximization. Appl Math Letters 25
(12): 2306--2311.
La Torre D, Vrscay ER, Ebrahimi M, Barnsley M (2009). Measure-valued images, associated fractal transforms and the affine self-similarity of images. SIAM J Imaging Sci 2(2):470--507.
La Torre D, Vrscay ER (2009). A generalized fractal transform for measure-valued images. Nonlinear Anal 71 (12): e1598-e1607.
La Torre D, Vrscay ER (2011). Generalized fractal transforms and self-similarity: recent results and applications. Image Anal Stereol 30(2):63--76.
La Torre D, Vrscay ER (2012). Fractal-based measure approximation with entropy maximization and sparsity constraints. AIP Conf. Proc. 1443:63--71.
Lu N (2003). Fractal imaging. New York: Academic Press.
Vrscay ER, Saupe D (1999). Can one break the 'collage barrier' in fractal image coding? In: Dekking M, LevyVehel J, Lutton E, Tricot C eds. Fractals: Theory and Applications in Engineering, London: Springer-Verlag. 307--23.
