Sažimajuće očitavanje signala

master's thesis

• Title (english): Summarizing signal reading
• Author: Karlo Grozdanić
• Mentor: Azra Tafro (mentor)
• Committee member: Azra Tafro (predsjednik povjerenstva)
• Committee member: Zvonko Iljazović (član povjerenstva)
• Committee member: Ljiljana Arambašić (član povjerenstva)
• Committee member: Dijana Ilišević (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-03-02, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovome radu prezentiramo metodu sažimajućeg očitavanja koja nam omogućuje rekonstrukciju signala iz vrlo malog broja mjerenja, uz uvjet rijetkosti signala u nekoj bazi. Rekonstrukciju dobivamo preko problema minimizacije \(l_1\) norme, koja pod određenim uvjetima daje ista rješenja kao problem minimizacije \(l_0\) norme. U prvom poglavlju formuliran je problem sažimajućeg očitavanja i dana je usporedba s klasičnim pristupima sažimanju podataka. Uvedene su definicije i pojmovi koji se kasnije koriste. U drugom poglavlju detaljno je predstavljena teorija sažimajućeg očitavanja: pokazani su uvjeti pod kojima će postupak biti uspješan te su dane granice točnosti rekonstrukcije ovisno o parametrima mjerenja. U trećem poglavlju pokazali smo kako sažimajuće očitavanje radi u problemima rekonstrukcije slika, te smo uveli poboljšanja na sam postupak koja dovode do boljih rezultata rekonstrukcije.

Abstract (english)

In this paper, we investigate compressive sensing, a technique that allows for the reconstruction of a signal from a limited number of measurements, provided that the signal exibits sparsity in some orthonormal basis. The signal is reconstructed by solving a \(l_1\) minimization problem is solved, which in the right conditions, yields a solution that is equivalent to a \(l_0\) minimization problem. In the first chapter, we formulate the compressive sensing problem and give compare it to the standard approach for signal compression. Key definitions and concepts are introduced. In the second chapter, we present a comprehensive theory of compressive sensing, demonstrating the conditions under which the method is guaranteed to be successful, and we providing the bounds on the reconstruction error based on the measurement process. In the third chapter, we demonstrate the application of compressive sensing to image reconstruction and modifications that significantly improve the reconstruction accuracy.

• Keywords: problem minimizacije \(l_1\) norme; problem minimizacije \(l_0\) norme; teorija sažimajućeg očitavanja; problemi rekonstrukcije slika; parametri mjerenja
• Keywords (english): \(l_1\) minimization problem; (l_0\) minimization problem; theory of summarizing reading; image reconstruction problems; measurement parameters
• Language: croatian
• URN:NBN: urn:nbn:hr:217:303523
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2023-03-27 08:17:12