Sharif Digital Repository

Graph signals are sets of values residing on sets of nodes that are connected via edges. Graph Neural Networks (GNNs) are a type of machine learning model for working with graph-structured data, such as graph signals. GNNs have applications in graph classification, node classification, and link prediction. They can be thought of as learnable filters. In this thesis, our focus is on graph filters and enhancing the performance of GNNs. In the first part, we aim to reduce computational costs in graph signal processing, particularly in graph filters. We explore methods to transform signals to the frequency domain with lower computational cost. In the latter part, we examine regulations in... 

Over-complete transforms due to their maneuverability in signal representation have been under focus during the last decade. Different properties for the representation can be useful in different applications. These properties includes minimum ℓ2 representation, minimum ℓ1 representation, minimum ℓ0 representation and so on. Among these properties, minimum ℓ0 representation (also known as sparse representation) has been shown to be efficient in many applications including image denoising, data compression, blind source separation and so on, and create a new approach in signal processing area named sparse signal processing. Sparse signal processing is based on two principles, the first one is... 

Image compression is a noteworthy problem in image processing field. Transform coding provides a scheme to confront the image compression problem. The discrete cosine transform (DCT) is used in the majority of image compression standards by transform coding. The DCT can efficiently represent smooth signals, but it becomes inefficient when the image contains arbitrary-shaped discontinuities. As an example, piecewise-smooth images (i.e., an image that contains multiple smooth areas separated with arbitrary-shaped boundaries), which are widely used in 3-dimensional image representation, cannot well represent by the DCT. Therefore, replacing the DCT with an adaptive transform can improve image... 

The problem of inferring a graph from a set of graph signals over it plays a crucial role in the field of Graph Signal Processing (GSP). When provided with a graph that best models the structure of data, the GSP algorithms can offer high data processing capability. However, a meaningful graph of data is not always available, hence in some applications, the graph needs to be learned from the data itself. When the data is corrupted and missing, this task becomes even more challenging. In this paper, we present a graph learning algorithm that is capable of learning the underlying structure of data from an incomplete and noisy dataset of graph signals. We propose an algorithm that jointly... 

Sparse representation and consequently, dictionary learning have been two of the great importance topics in signal processing problems for the last two decades. In sparse representation, each signal has to be represented as a linear combination of some basic signals, which are called atoms, and their collection is called a dictionary. To put it in other words, if complete dictionaries such as Fourier or Wavelet dictionaries are used for the representation of signals, the representation will be unique, but not sparse. On the other hand, if overcomplete dictionaries are used, we will confront with too many representations, and the goal of sparse representation is to find the sparsest one. ... 

Tensors or multi-way arrays are useful tools to identify unknown quantities thanks to the uniqueness of their decomposition. Tensor decompositions have been widely applied to obtain unknown components with physical meanings in many applications such as medical image and signal processing, hyperspectral images analysis, chemometrics, etc.In this thesis, we investigate the application of tensor decomposition for probability estimations, which are required for some targeted data/text mining tasks such as unsupervised clustering of data/documents. Besides criticizing the existing tensor decomposition algorithms for probability estimations, we propose to apply some proper constrained tensor... 

The aim of tracking is localization and positioning of position-variant object through consecutive times. The essence of this object determines the application of tracking. For example this object can be the satellite, mobile, certain object in sequential movie or etc. The particle filter as an estimation filter is a method that provides us the solution of tracking Problem. Therefore this thesis is devoted to particle filter and its application in tracking. But tracking problem needs some prior information; one of them is access to measurements relating to object position. In situations that the measurement equation which is related to object position has ambiguity we need another mechanism... 

Brain-Computer Interface (BCI) presents a way for brain’s direct connection with external world. BCI system is composed of three parts: 1) Signal acquisition, 2) Signal processing and 3) External device control. The main part of this system is signal processing which includes three subparts: 1) Feature extraction, 2) Dimension reduction and 3) Signal separation and classification. In this thesis, we focus on the signal processing section in BCI systems. One of the most successful works done in signal processing is the use of covariance matrices in feature extraction from brain signals. Since covariance matrices are positive semi-definite and symmetric, they belong to certain manifolds called... 

Nowadays, using sparsity of signals has been utilized in diverse applications in signal processing community. Two important applications of signal sparsity are sparse source separation and sparse signal representation. These two problems are joined with a Sparse Component Analysis (SCA) framework. In SCA, the problem is divided into two subproblems which are matrix estimation and sparse vector estimation. In this thesis, a MAP-based algorithm is suggested for sparse vector estimation with a Bernoulli-Gaussian distribution for sparse vector elements. To reduce the complexity, an iterative Bayesian algoritm is used in which an steepest-ascent is utilized for maximization. A complete... 

Deep neural networks have been shown to perform very well in many classical machine learning tasks, including classification. However, it has been shown that these models are vulnerable to very small, and often imperceptible, adversarial perturbations of their input data, which makes it difficult to apply neural networks in security-critical areas. Finding the sparse solution of an underdetermined system of linear equations, which is the basis of sparse representation theory, is of significant importance in signal processing. Since finding such a solution requires minimizing the ℓ0 norm of a vector, which in turn requires using a combinatorial search, several methods for ℓ0 norm... 

Compressed Sensing (CS) technique refers to the digitalization process that efficiently reduces the number of measurements below the Nyquist rate while preserving signal structure. This technique was originally developed for the analysis of vector datasets. An x ∈R^n vector is transformed into an y ∈R^m vector so that n≪m. For a sufficient number of measurements, this transformation has been shown to preserve the signal structure. Therefore, the technique has been applied to machine learning applications.2D-CS was further developed for matrices (image datasets) so that they could be directly applied to matrices without flattening. X ∈R^(n×n) is transformed into Y ∈R^(m×m) via 2D-CD such... 

The main motivation behind compressive sensing is to reduce the sampling rate at the input of a discrete-time signal processing system. However, if for processing the sensed signal one requires to reconstruct the corresponding Nyquist samples, then the data rate will be again high in the processing stages of the overall system. Therefore, it is preferred that the desired processing task is done directly on the compressive measurements, without the need for the reconstruction of the Nyquist samples. This thesis addresses the cases in which the processing task is “detection and/or estimation”. Firstly, a detector/estimator is proposed for compressed sensing radars, which does not need to... 

Image inpainting is the process of reconstructing lost parts of damaged images based on collected local information, or even general information, as prior knowledge. The image inpainting’s objective is to improve the damaged images, for example restoring missing pixels caused by folding, erasing background’s text in an image, removing watermarks from an image, or editing an image such as eliminating an object or a person from the image. The majority of the image inpainting algorithms approaches the problem by signal restoration from remaining samples or iterative methods to complete the damaged images. Algorithms based on samples, algorithms based on partial differential equations, and... 

The problem of linear prediction is one of the traditional issues in signal processing. With the genesis of graph signal processing, the prediction of signals defined on graphs has been recently addressed. Some existing methods provide approaches for predicting samples of a graph signal based on a known adjacency matrix among nodes. On the other hand, some studies have used graph smoothing technique, which ensures that the estimated signals remain smooth on the graph. Furthermore, graph neural networks have been proposed recently, and some research has considered methods for predicting graph signals using a graph neural network, which is possible by using extracted features from the training... 

Blind Source Separation (BSS) aims to recover multiple source signals from mixtures without prior knowledge of the sources or the mixing process. This manuscript focuses on speech separation in convolutive mixtures, aiming to recover speech signals from mixtures that are linear combinations of the filtered versions of original signals. Most recent methods for blind source separation in both instantaneous and convolutive mixtures typically involve transforming time-domain mixed signals into the time-frequency domain using Short-Time Fourier Transform (STFT) and conducting the separation in this domain. Broadly, the separation process is followed by two main approaches. The first involves... 

Sparse decomposition has recently attracted the attention of many researchers in dierent areas of signal processing. In mathematical viewpoint, sparse decomposition is nding a sparse solution of an Underdetermined System of Linear Equations (USLE). This concept has many applications in dierent signal processing elds including blind sourse separation, optical character recognition and image processing. In this thesis, we investigate the application of sparse decomposition in image denoising. One of the image denoising methods which is related to sparse decomposition concepts is the \transfrom domain method". In this method, the noisy image is rst transformed to another domain, and then noise... 

In recent years, Machine Learning models have been developed in Signal Processing, Computer Vision and Neuroscience areas. There are two categories of Machine Learning models which are supervised and unsupervised learning models. Regression and classification problems are two popular problems examples of supervised learning models. From unsupervised learning problems, we can mention the clustering problem. Support Vector Regression (SVR), Decision Tree Regression and Bagging Ensemble Regression models are some important models of the regression problem. For classification problems, we can also mention to Support Vector Classification, Decision Tree Classification, and Bagging Ensemble... 

One of the problems in signal processing is supervised classification. In supervised classification, the goal is to learn the structures and patterns of a dataset using a set of labeled data called the training dataset to correctly classify data samples that are not used in the training data but follow the same pattern and structure. One approach to this problem that has recently received attention is neural networks. Although this approach has good performance in applications, in order to perform well, they require a large amount of data and many trainable parameters, which result in high computational complexity. Another approach to this problem is dictionary learning-based classification.... 

Blind separation of mixed graph signals is one of the new topics in the field of graph signal processing. However, similar to the most proposed methods for separating traditional signals, it is assumed that the number of observed signals is equal to or greater than the number of sources. In this thesis, we show that a signal can be uniquely decomposed into the summation of a set of smooth graph signals, up to the indeterminacy of their DC values. From the blind source separation point of view, this is like the separation of a set of graph signals from a single mixture, contrary to traditional blind source separation in which at least two observed mixtures are required. Moreover, we... 

Graph signal separation is a new topic in the field of graph signal processing that aims to recover graph signals from their linear combinations, taking into account the relationship between the signals and their corresponding graphs. Among the existing methods for separating graph signals from observing only one mixture, a recently published approach assumes the smoothness of the signals and minimizes the smoothness criterion of the signals on their related graphs. In this thesis, the closed-form solution of this method is obtained and the reconstruction error of the graph signals is calculated from it and the performance of this method is evaluated. It is also shown by numerical...