FIXED POINT THEOREMS IN DIGITAL IMAGES AND APPLICATIONS TO FRACTAL IMAGE COMPRESSION
U. P. DOLHARE *
D. S. M. College, Jintur, Dist. Parbhani, Maharashtra,431509, India.
V. V. NALAWADE
S. G. R. G. Shinde Mahavidyalaya, Paranda, Dist. Osmanabad, Maharashtra, 413502, India.
*Author to whom correspondence should be addressed.
In this research paper we prove some fixed point theorems for digital images. Ege and Karaca stated and proved Banach contraction principle for digital images. Main objective of the research article is to present another generalization of the well known Banach contraction mapping principle for digital images. We generalize the principle by replacing the contraction condition of Banach by a condition that involves monotone non-decreasing function. In the second result, we use a weakly uniformly strict digital contraction to prove the existence of unique fixed point for digital images. The basic concepts about the digital images are mentioned. We give an important application of our fixed point theorem to compression of digital images. Fractal image compression is one of the popular technique for compressing a digital image. It is based on the self similarity search of the image. But it has a major drawback of computational intensity in encoding a digital image. Computational intensity increases the time of data transmission. In this paper a technique is proposed to bring down the time of data transmission. In an image compression, it is a challenge to either maximize the image quality for a stipulated data transmission time or to minimize the data transmission time for a given quality of an image to be transmitted. To achieve this goal, a constant contractive factor in conventional fractal image compression is replaced by the non-linear contractive mapping. This leads to significantly better reconstruction of image in lesser time. Finally we mention some conclusions about our research article.
Keywords: Digital image, digital metric space, fixed points, fractals, fractal image compression