27225

قضيه هاي نقطه ثابت براي تصاوير ديجيتال در توپولوژي ديجيتال

كارشناسي ارشد

رياضي- رياضي محض ـ آناليز

1399

1401/6/21

محمد باقر قائمي

رياضي

در اين پايان‌نامه در ابتدا به بيان مقدمات و پيش‌نيازهايي از توپولوژي و جبر خطي مي‌پردازيم. سپس تعاريف و قضيه‌هاي نقطه ثابت را بيان مي كنيم. در ادامه به بيان اصل انقباض باناخ براي تصاوير ديجيتال مي‌پردازيم و كاربرد قضيه نقطه ثابت باناخ را براي تصاوير ديجيتال بيان مي‌كنيم. در پايان به بيان و بررسي كاربردهاي نقطه ثابت در فشرده سازي تصاوير مي‌پردازيم.

1401/08/09

FIXED POINT THEOREMS IN DIGITAL IMAGES AND APPLICATIONS TO FRACTAL IMAGE COMPRESSION

9/12/2023 12:00:00 AM

In this Thesis 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 nondecreasing 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.

فشرده سازي تصوير , قضيه نقطه ثابت باناخ , متر اقليدسي

Image compression , Banach Fixed Point Theorem , Euckidean metric

Gholamreza Niyazi

Dr. Ghaemi