41 research outputs found
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
Sampling and Reconstruction of Sparse Signals on Circulant Graphs - An Introduction to Graph-FRI
With the objective of employing graphs toward a more generalized theory of
signal processing, we present a novel sampling framework for (wavelet-)sparse
signals defined on circulant graphs which extends basic properties of Finite
Rate of Innovation (FRI) theory to the graph domain, and can be applied to
arbitrary graphs via suitable approximation schemes. At its core, the
introduced Graph-FRI-framework states that any K-sparse signal on the vertices
of a circulant graph can be perfectly reconstructed from its
dimensionality-reduced representation in the graph spectral domain, the Graph
Fourier Transform (GFT), of minimum size 2K. By leveraging the recently
developed theory of e-splines and e-spline wavelets on graphs, one can
decompose this graph spectral transformation into the multiresolution low-pass
filtering operation with a graph e-spline filter, and subsequent transformation
to the spectral graph domain; this allows to infer a distinct sampling pattern,
and, ultimately, the structure of an associated coarsened graph, which
preserves essential properties of the original, including circularity and,
where applicable, the graph generating set.Comment: To appear in Appl. Comput. Harmon. Anal. (2017
From spline wavelet to sampling theory on circulant graphs and beyond– conceiving sparsity in graph signal processing
Graph Signal Processing (GSP), as the field concerned with the extension of classical signal processing concepts to the graph domain, is still at the beginning on the path toward providing a generalized theory of signal processing. As such, this thesis aspires to conceive the theory of sparse representations on graphs by traversing the cornerstones of wavelet and sampling theory on graphs.
Beginning with the novel topic of graph spline wavelet theory, we introduce families of spline and e-spline wavelets, and associated filterbanks on circulant graphs, which lever- age an inherent vanishing moment property of circulant graph Laplacian matrices (and their parameterized generalizations), for the reproduction and annihilation of (exponen- tial) polynomial signals. Further, these families are shown to provide a stepping stone to generalized graph wavelet designs with adaptive (annihilation) properties. Circulant graphs, which serve as building blocks, facilitate intuitively equivalent signal processing concepts and operations, such that insights can be leveraged for and extended to more complex scenarios, including arbitrary undirected graphs, time-varying graphs, as well as associated signals with space- and time-variant properties, all the while retaining the focus on inducing sparse representations.
Further, we shift from sparsity-inducing to sparsity-leveraging theory and present a novel sampling and graph coarsening framework for (wavelet-)sparse graph signals, inspired by Finite Rate of Innovation (FRI) theory and directly building upon (graph) spline wavelet theory. At its core, the introduced Graph-FRI-framework states that any K-sparse signal residing on the vertices of a circulant graph can be sampled and perfectly reconstructed from its dimensionality-reduced graph spectral representation of minimum size 2K, while the structure of an associated coarsened graph is simultaneously inferred. Extensions to arbitrary graphs can be enforced via suitable approximation schemes.
Eventually, gained insights are unified in a graph-based image approximation framework which further leverages graph partitioning and re-labelling techniques for a maximally sparse graph wavelet representation.Open Acces
Filter Bank Fusion Frames
In this paper we characterize and construct novel oversampled filter banks
implementing fusion frames. A fusion frame is a sequence of orthogonal
projection operators whose sum can be inverted in a numerically stable way.
When properly designed, fusion frames can provide redundant encodings of
signals which are optimally robust against certain types of noise and erasures.
However, up to this point, few implementable constructions of such frames were
known; we show how to construct them using oversampled filter banks. In this
work, we first provide polyphase domain characterizations of filter bank fusion
frames. We then use these characterizations to construct filter bank fusion
frame versions of discrete wavelet and Gabor transforms, emphasizing those
specific finite impulse response filters whose frequency responses are
well-behaved.Comment: keywords: filter banks, frames, tight, fusion, erasures, polyphas
Compression of an ECG Signal Using Mixed Transforms
Electrocardiogram (ECG) is an important physiological signal for cardiac disease diagnosis. With the increasing use of modern electrocardiogram monitoring devices that generate vast amount of data requiring huge storage capacity. In order to decrease storage costs or make ECG signals suitable and ready for transmission through common communication channels, the ECG data
volume must be reduced. So an effective data compression method is required. This paper presents an efficient technique for the compression of ECG signals. In this technique, different transforms have been used to compress the ECG signals. At first, a 1-D ECG data was segmented and aligned to a 2-D data array, then 2-D mixed transform was implemented to compress the ECG data in the 2-
D form. The compression algorithms were implemented and tested using multiwavelet, wavelet and slantlet transforms to form the proposed method based on mixed transforms. Then vector quantization technique was employed to extract the mixed transform coefficients. Some selected records from MIT/BIH arrhythmia database were tested contrastively and the performance of the
proposed methods was analyzed and evaluated using MATLAB package. Simulation results showed that the proposed methods gave a high compression ratio (CR) for the ECG signals comparing with other available methods. For example, the compression of one record (record 100) yielded CR of 24.4 associated with percent root mean square difference (PRD) of 2.56% was achieved