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In this thesis, we propose the novel adaptive wavelet filter bank structures that are used to obtain efficient representations of the analyzed images. We present the lifting scheme structures for building adaptive wavelet decompositions based on the nonseparable quincunx sampling scheme. The resulting wavelet decompositions are adaptive to the local properties of the analyzed image. Despite the introduced adaptation, a desired number of vanishing moments is still retained. The proposed adaptation is performed in order to minimize the energy of detail coefficients on a neighborhood of each pixel of the analyzed image. The appropriate neighborhood is determined for each pixel separately by using the intersection of confidence intervals (ICI) rule. The application of the ICI rule improves the estimation of the filter bank parameters and makes it more robust to noise. The image denoising results are presented for both synthetic and real-world images. It is shown that the adaptive wavelet decompositions outperform the existing fixed decompositions in terms of denoising quality of images that contain periodic components, and in general they give more compact image representations. U ovoj disertaciji predlažemo nove adaptivne valićne filtarske strukture koje se koriste za dobivanje efikasnih reprezentacija analiziranih slika. U radu prikazujemo filtarske slogove temeljene na shemi podizanja za konstrukciju adaptivnih valićnih razlaganja zasnovanih na neseparabilnom quincunx uzorkovanju. Rezultirajuća valićna razlaganja su prilagođena lokalnim svojstvima analizirane slike. Unatoč uvedenoj adaptaciji, zadržan je željeni broj nul-momenata pripadajućihih vali ćnih funkcija. Adaptacija se vrši sa ciljem minimizacije koeficijenata detalja na okolini svakog slikovnog elementa. Odabir odgovarajuće okoline za svaki slikovni element vrši se korištenjem metode presjecišta invervala pouzdanosti. Primjena dotične metode poboljšava adaptaciju parametara filtarskog sloga i čini je robusnijom na šum. Rezultati uklanjanja šuma iz slike prikazani su za primjere sintetičkih kao i realnih slika. Pokazano je da dobivena adaptivna valićna razlaganja nadmašuju postojeća fiksna razlaganja s obzirom na kvalitetu uklanjanja šuma iz slika koje sadrže periodičke komponente, te općenito daju kompaktniji zapis slike.

In this thesis, we propose the novel adaptive wavelet filter bank structures that are used to obtain efficient representations of the analyzed images. We present the lifting scheme structures for building adaptive wavelet decompositions based on the nonseparable quincunx sampling scheme. The resulting wavelet decompositions are adaptive to the local properties of the analyzed image. Despite the introduced adaptation, a desired number of vanishing moments is still retained. The proposed adaptation is performed in order to minimize the energy of detail coefficients on a neighborhood of each pixel of the analyzed image. The appropriate neighborhood is determined for each pixel separately by using the intersection of confidence intervals (ICI) rule. The application of the ICI rule improves the estimation of the filter bank parameters and makes it more robust to noise. The image denoising results are presented for both synthetic and real-world images. It is shown that the adaptive wavelet decompositions outperform the existing fixed decompositions in terms of denoising quality of images that contain periodic components, and in general they give more compact image representations.

More efficient coding, modelling and image analysis generate the need for search for better methods of the multiresolution image analysis, i.e. for more efficient wavelet filter bank structures. This thesis is based on a previous research of one-dimensional filter bank structures that had the possibility to adapt filter parameters to the properties of the analyzed signal. In this thesis we present the construction of a two-dimensional adaptive wavelet filter bank that is based on a lifting scheme. The filter bank is nonseparable, based on a quincunx polyphase decomposition and nonseparable filters. The lifting scheme has been chosen since it allows for an easy construction of space varying filter banks with a perfect reconstruction property. The proposed filter bank adapts to the analyzed image for every pixel in all decomposition levels while still preserving the good properties of the wavelet decomposition. A number of vanishing moments is guaranteed by the fixed part of the filter bank. Without degrading the overall filter bank properties, the variable part can be changed in order to adapt to the analyzed image. The paper explores various one-dimensional and two-dimensional adaptation methods based on the least squares criterion. Adaptation results have been shown for a number of synthetic and real-world images. Effects of lossy image reconstruction and impact of filter coefcients' quantization to the efficiency of the image decomposition have been presented. Učinkovitije kodiranje, modeliranje i analiza slika stvaraju potrebu za traženjem boljih metoda višerezolucijskog razlaganja slika odnosno potrebu za učinkovitijim strukturama dvodimenzionalnih wavelet filtarskih slogova. Rad se temelji na prethodnom istraživanju struktura jednodimenzionalnih filtarskih slogova koji su imali mogućnost adaptacije filtarskih parametara svojstvima signala. U ovom radu prikazujemo izvedbu dvodimenzionalnog adaptivnog wavelet filtarskog sloga koji se temelji na shemi podizanja. Filtarski slog je neseparabilan, temelji se na quincunx polifaznom razlaganju i neseparabilnim filtrima. Shema podizanja je odabrana jer omogućava jednostavnu izvedbu prostorno promjenjivih filtarskih slogova sa svojstvima savršene rekonstrukcije. Predloženi filtarski slog se prilagođuje analiziranoj slici u svakom slikovnom elementu u svim razinama razlaganja u isto vrijeme zadržavajući dobra svojstva wavelet razlaganja. Dovoljan broj nul-momenata pridruženih wavelet funkcija zagarantiran je nepromjenjivim dijelom filtarskog sloga. Promjenjivi dio filtarskog sloga može se prilagođivati svojstvima slike bez narušavanja cjelokupnih svojstava filtarskog sloga. Istražene su razne jednodimenzionalne i dvodimenzionalne metode adaptacije zasnovane na kriteriju najmanjih kvadrata. Rezultati adaptacije prikazani su za razne sintetske i realne slike. Prezentirati su rezultati rekonstrukcije slike s gubicima i utjecaj kvantizacije filtarskih koeficijenata na učinkovitost adaptivnog razlaganja slike.