Perfect Reconstruction Two-Channel Wavelet Filter-Banks for Graph Structured Data

In this work we propose the construction of two-channel wavelet filterbanks for analyzing functions defined on the vertices of any arbitrary finite weighted undirected graph. These graph based functions are referred to as graph-signals as we build a framework in which many concepts from the classical signal processing domain, such as Fourier decomposition, signal filtering and downsampling can be extended to graph domain. Especially, we observe a spectral folding phenomenon in bipartite graphs which occurs during downsampling of these graphs and produces aliasing in graph signals. This property of bipartite graphs, allows us to design critically sampled two-channel filterbanks, and we propose quadrature mirror filters (referred to as graph-QMF) for bipartite graph which cancel aliasing and lead to perfect reconstruction. For arbitrary graphs we present a bipartite subgraph decomposition which produces an edge-disjoint collection of bipartite subgraphs. Graph-QMFs are then constructed on each bipartite subgraph leading to "multi-dimensional" separable wavelet filterbanks on graphs. Our proposed filterbanks are critically sampled and we state necessary and sufficient conditions for orthogonality, aliasing cancellation and perfect reconstruction. The filterbanks are realized by Chebychev polynomial approximations.Comment: 32 pages double spaced 12 Figures, to appear in IEEE Transactions of Signal Processin

Fixed-analysis adaptive-synthesis filter banks

Subband/Wavelet filter analysis-synthesis filters are a major component in many compression algorithms. Such compression algorithms have been applied to images, voice, and video. These algorithms have achieved high performance. Typically, the configuration for such compression algorithms involves a bank of analysis filters whose coefficients have been designed in advance to enable high quality reconstruction. The analysis system is then followed by subband quantization and decoding on the synthesis side. Decoding is performed using a corresponding set of synthesis filters and the subbands are merged together. For many years, there has been interest in improving the analysis-synthesis filters in order to achieve better coding quality. Adaptive filter banks have been explored by a number of authors where by the analysis filters and synthesis filters coefficients are changed dynamically in response to the input. A degree of performance improvement has been reported but this approach does require that the analysis system dynamically maintain synchronization with the synthesis system in order to perform reconstruction. In this thesis, we explore a variant of the adaptive filter bank idea. We will refer to this approach as fixed-analysis adaptive-synthesis filter banks. Unlike the adaptive filter banks proposed previously, there is no analysis synthesis synchronization issue involved. This implies less coder complexity and more coder flexibility. Such an approach can be compatible with existing subband wavelet encoders. The design methodology and a performance analysis are presented.Ph.D.Committee Chair: Smith, Mark J. T.; Committee Co-Chair: Mersereau, Russell M.; Committee Member: Anderson, David; Committee Member: Lanterman, Aaron; Committee Member: Rosen, Gail; Committee Member: Wardi, Yora

A Discrete Fourier Transform Based Subband Decomposition Approach For The Segmentation Of Remotely Sensed Images

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Bilişim Enstitüsü, 2006Thesis (M.Sc.) -- İstanbul Technical University, Institute of Informatics, 2006Yüksek LisansM.Sc

Dual-Tree Complex Wavelet Transform in the Frequency Domain and an Application to Signal Classification

We examine Kingsbury's dual-tree complex wavelet transform in the frequency domain, where it can be formulated for standard wavelet filters without special filter design and apply the method to the classification of signals. The obtained transforms achieve low shift sensitivity and better directionality compared to the real discrete wavelet transform while retaining the perfect reconstruction property

A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity

The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping "pictures". We hope that this paper will contribute to the appreciation and apprehension of a stream of current research directions in image understanding.Comment: 65 pages, 33 figures, 303 reference