Multirezolucijski valići i diskretne transformacije

master's thesis

• Author: Toni Vidović
• Mentor: Damir Bakić (mentor)
• Committee member: Damir Bakić (predsjednik povjerenstva)
• Committee member: Ilja Gogić (član povjerenstva)
• Committee member: Juraj Šiftar (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-07-13, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom smo radu izložili osnove teorije ortonormiranih valića u jednoj dimenziji. Prikazali smo kako konstruirati valiće na temelju multirezolucijske analize, no važno je napomenuti kako to nisu ni blizu svi ortonormirani valići u \(L^2(\mathbb{R})\). Nakon same konstrukcije, važno je bilo uvesti jednostavnije karakterizacije kako ortonormiranih valića, tako i multirezolucijske analize i pripadne skalirajuće funkcije. U zadnjem smo poglavlju prikazali neke primjene ortonormiranih valica. Uveli smo algoritme za kompresiju i rekonstrukciju podataka bazirane na valićima koji se koriste, primjerice, u kompresiji slika. Kroz valićne pakete, koji se koriste u analizi signala, pokazali smo kako dobiti bolju lokalizaciju frekvencije, ako nam originalna valićna nije dostatna.

Abstract (english)

In this thesis we have introduced basic theory of orthonormal wavelets in one dimension. We have shown how to construct wavelets from multiresolutional analysis; however, it is important to emphasise that there are wavelets that do not arise from a multiresolutional analysis on \(L^2(\mathbb{R})\). After construction, it was important to introduce simpler characterizations of orthonormal wavelets, multiresolutional analysis and associated scaling function. In the last chapter we have shown some applications of orthonormal wavelets. We introduced algorithms for data compression and data reconstruction based on wavelets, which are used, for instance, in image compression. Using wavelet packets, which are one of the tools for signal processing, we have shown how to achieve better frequency localization, if the original wavelet frequency localization was not enough.

• Keywords: konstruiranje valića na temelju multirezolucijske analize; ortonormirani valići; skalirajuće funkcije; kompresija slika
• Keywords (english): wavelets construction from multiresolutional analysis; orthonormal wavelets; scaling function; image compression
• Language: croatian
• URN:NBN: urn:nbn:hr:217:198248
• Study programme: Title: Applied Mathematics; 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: 2019-02-19 14:18:23