Obrada slika pomoću varijacijskog računa i diferencijalnih jednadžbi

master's thesis

• Title (english): Image processing using calculus of variations and differential equations
• Author: Lucija Tolja
• Mentor: Marko Erceg (mentor)
• Committee member: Marko Erceg (predsjednik povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Committee member: Eduard Marušić-Paloka (član povjerenstva)
• Committee member: Ilja Gogić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-09-29, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Obrada slika matematički se može opisati s različitih aspekata, no u ovom radu odabran je pristup preko varijacijskog računa i parcijalnih diferencijalnih jednadžbi. Osnovni zadaci pri obradi slika su raščišćavanje, izoštravanje, nadopunjavanje i segmentacija. Kada jednom digitalnu sliku definiramo kao matematički element, možemo analizirati modele koji se bave jednim od navedena četiri zahtjeva pri obradi slika. U prvom poglavlju motiviran je spomenuti pristup te su dani kratki opisi osnovnih alata koji se koriste. Drugo poglavlje opisuje različite načine reprezentacije slika kao matematičkih elemenata te sugerira kako se prostor BV slika pokazuje najpogodnijim za opis modela obrade. U trećem poglavlju opisuje se Rudin, Osher, Fatemi TV model rasčišćavanja te su pokazani rezultati implementacije metode u Phyton-u. Četvrto poglavlje opisuje izoštravanje, koje je inverzni problem, opet preko Rudin, Osher, Fatemi TV modela. U petom poglavlju opisano je nadopunjavanje, a u šestom segmentacija, oboje preko Mumford - Shahovih modela.

Abstract (english)

Image processing can be mathematically described from different aspects, but in this thesis variational calculus and partial differential equations were chosen. Main tasks in image processing are denoising, deblurring, inpainting and segmentation. Once we represent an image as a mathematical object, we can analyse models which deal with one of four tasks in image processing. In the first chapter mentioned approach is motivated and brief description of tools used is given. Second chapter describes different ways to represent an image as a mathematical object and suggest the space of BV images as the most suitable. Third chapter describes Rudin, Osher, Fatemi TV denoising model and gives the results of the implementation of the method in Python. Fourth chapter describes deblurring, which is an inverse problem, again via Rudin, Osher, Fatemi TV model. Fifth chapter describes inpainting and sixth chapter describes segmentation, both via Mumford - Shah models.

• Keywords: raščišćavanje; izoštravanje; nadopunjavanje; segmentacija; digitalna slika; Rudin; Osher; Fatemi TV model; Mumford-Shahovi modeli; Python
• Keywords (english): clearing; sharpening; filling; segmentation; digital image; Rudin; Osher; Fatemi TV model; Mumford-Shah models; Python
• Language: croatian
• URN:NBN: urn:nbn:hr:217:827697
• 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: 2022-11-09 12:41:50