Curvelet Based Multiresolution Analysis of Graph Neural Networks

Bharat Bhosale
2014 International Journal of Applied Physics and Mathematics  
Multiresolution techniques are deeply related to image/signal processing, biological and computer vision, scientific computing, optical data analysis. Improving quality of noisy signals/images has been an active area of research in many years. Although wavelets have been widely used in signal processing, they have limitations with orientation selectivity and hence, they fail to represent changing geometric features along edges effectively. Curvelet transform on the contrary exhibits good
more » ... ruction of the edge data by incorporating a directional component to the conventional wavelet transform and can be robustly used in the analysis of complex neural networks; which in turn are represented by graphs, called Graph Neural Networks. This paper explores the application of curvelet transform in the analysis of such complex networks. Especially, a technique of Fast Discrete Curvelet Transform de-noising with the Independent Component Analysis (ICA) for the separation of noisy signals is discussed. Two different approaches viz. separating noisy mixed signals using fast ICA algorithm and then applying Curvelet thresholding to de-noise the resulting signal, and the other one that uses Curvelet thresholding to de-noise the mixed signals and then the fast ICA algorithm to separate the de-noised signals are presented for the purpose. The Signal-to-Noise Ratio and Root Mean Square Error are used as metrics to evaluate the quality of the separated signals. the field of neuroscience such as study of models of neural networks, anatomical connectivity, and functional connectivity based upon functional magnetic resonance imaging (fMRI), electroencephalography (EEG) and magnetoencephalography (MEG). The recent applications of network theory to neuroscience such as modeling of neural dynamics on complex networks; graph theoretical analysis of neuroanatomical networks; and applications of graph analysis to study functional connectivity with fMRI, EEG and MEG are discussed at length [1] . A graph is an abstract representation of complex network. Many types of relations and process dynamics in physical, biological, social and information systems can be modeled with graphs. Graph analysis has been used in the study of models of neural networks, anatomical connectivity, and functional connectivity. These developments in the theory of complex networks have inspired new applications in the upcoming field of neuroscience including neural networks. Many practical problems can be represented by graphs that can be used to model different types of relations and process dynamics in physical, social and information systems. In mathematics, graphs are useful in geometry and certain parts of topology, e.g. Knot Theory. Algebraic graph theory has close links with group theory. In computer science, graphs are used to represent networks of communication, data organization, computational devices, the flow of computation, etc. Graph theory is also used to study molecules in chemistry and physics. In condensed matter physics, the three dimensional structure of complicated simulated atomic structures can be studied quantitatively by gathering statistics on graph-theoretic properties related to the topology of the atoms. For example, Franzblau's shortest-path (SP) rings. In chemistry, a graph makes a natural model for a molecule, where vertices represent atoms and edges bonds. This approach is especially used in computer processing of molecular structures, ranging from chemical editors to database searching. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems. Likewise, graph theory is useful in biology and conservation efforts where a vertex can represent regions where certain species exist (or habitats) and the edges represent migration paths, or movement between the regions. This information is important when looking at breeding patterns or tracking the spread of disease, parasites or how changes to the movement can affect other species. In sum, many underlying relationships among data in several areas of science and engineering, e.g., computer vision, molecular chemistry, molecular biology, pattern recognition, and data mining, can be represented in terms of graphs and the graph theoretic approach can be employed to analyse such complex data. In recent years many important properties of complex networks have been delineated and studied the relationship between the structural properties, nature of dynamics taking place on these networks. For instance, the 'synchronizability' of complex networks of coupled oscillators can be determined by graph spectral analysis. These developments in the theory of complex networks have inspired new applications in the field of neuroscience such as study of models of neural networks, anatomical connectivity, and functional connectivity based upon functional magnetic resonance imaging (fMRI), electroencephalography (EEG) and magnetoencephalography (MEG). The recent applications of network theory to neuroscience such as modelling of neural dynamics on complex networks; graph theoretical analysis of neuroanatomical networks; and applications of graph analysis to studies of functional connectivity with fMRI, EEG and MEG are discussed at length. The complex networks like neural networks are also represented with the help of graphs by showing the computational elements, neurons of the network. Each node corresponds to one neuron and the arrows usually denote weighted sums of the values from other neurons. Recently various approaches have been unified in neural network model called graph neural networks (GNN), which is used for processing the data represented in graph domains. The GNN are of two kinds viz. Biological neural networks (BNN) and Artificial neural networks (ANN) [2] . BNNs are of objective existence, in which the neurons are linked as a network in a certain order, e.g. human neural network is the most intelligent network system. The ANN are aimed at modeling the organization principles of central neural
doi:10.7763/ijapm.2014.v4.304 fatcat:74m5ue53ebahdi57izusgmffbq