Comptacion Visual 1997
Genetic Algorithms for Fractal Image and Image Sequence Compression Luc´ıa Vences Isaac Rudomin Instituto Tecnol´ ogico de Estudios Superiores de Monterrey, Campus Estado de M´ exico. Km 4 Carretera Lago de Guadalupe, Atizapan de Zaragoza. phone: 011 (525) 326-5614 email:
[email protected]
Abstract In this paper we present a method that uses Genetic Algorithms (GAs) to find a Local Iterated Function System (LIFS) that encodes a single image. By doing this, the time needed to achieve this LIFS is reduced by about 30% compared with Barnsley’s method if similar image quality is desired. If less quality is acceptable, using a GA we can vary the time the encoding will take by changing parameters such as population size and number of generations allowed. The algorithm was extended to deal with image sequences by using the population that has evolved for a given image as the initial population for the next image in the sequence.
Key Words: Fractal Image Compression, Genetic Algorithms 1. Introduction The development of a wide range of 1 applications has led to increased research attention to data compression and particularly to image compression. Among the techniques for image compression, the method based on the theory of Local Iterated Function Systems (LIFS) has received of late a great deal of attention. The basis for this technique, known as the fractal inverse problem, is to find a LIFS whose attractor is close to a given image. One limitation of this method is that it takes a long time to compress an image. In this paper, we present a technique to compress an image using GAs. We also present an extension of this method to deal with image sequences. GAs are used to obtain, from a randomly generated population c Isaac Rudomin 1997.
of LIFS, the one whose attractor is the first frame in the sequence. Once a code for the first frame has been obtained, the population that is generated in the process is used as the initial population for the next frame. In the next section, we present the theoretical basis for fractal image compression, the collage theorem which is the key for this technique, some of the algorithms that have been used for fractal image compression (FIC) and general background for GAs. The third section is dedicated to the details of our algorithm: the representation of the compressed image as a chromosome, the fitness measure used, the genetic operators applied and some other improvements to the simple genetic algorithm. The application to the fractal compression of image sequences is also discussed here. In the results section, we examine the performance of our algorithm for both a single image and image sequences. Lastly, we discuss the results, analyze some
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L. Vences and I. Rudomin / Genetic Algorithms for Fractal Image and Image Sequence Compression
of the limitations of the method, and review possible directions for further research.
If E is a compact non empty subset such that ωi (E) ⊂ E and m [
W (E) =
2. Theoretical Foundations
ωi (E),
(5)
i=1
In this section we present the basic theory that is involved in fractal image compression and GAs.
we define the k-th iteration of W, W k (E), to be W 0 (E) = E, W k (E) = W (W k−1 (E)),
2.1. Self-affine and self-similar transformations
for k ≥ 1, then we have that
The fractal image compression algorithm is based on the fractal theory of self-similar and self-affine transformations. Some basic definitions: 1. A self-affine transformation W :
