A Spectral Method for Generating Surrogate Graph Signals

The increasing availability of network data is leading to a growing interest in processing of signals on graphs. One notable tool for extending conventional signal-processing operations to networks is the graph Fourier transform that can be obtained as the eigendecomposition of the graph Laplacian. In this letter, we used the graph Fourier transform to define a new method for generating surrogate graph signals. The approach is based on sign-randomization of the graph Fourier coefficients and, therefore, the correlation structure of the surrogate graph signals (i.e., smoothness on the graph topology) is imposed by the measured data. The proposed method of surrogate data generation can be widely applied for nonparametric statistical hypothesis testing. Here, we showed a proof-of-concept with a high-density electroencephalography dataset.

Graph Signal Processing

Ntorina Thanou

Over the past few decades we have been experiencing an explosion of information generated by large networks of sensors and other data sources. Much of this data is intrinsically structured, such as traffic evolution in a transportation network, temperature values in different geographical locations, information diffusion in social networks, functional activities in the brain, or 3D meshes in computer graphics. The representation, analysis, and compression of such data is a challenging task and requires the development of new tools that can identify and properly exploit the data structure. In this thesis, we formulate the processing and analysis of structured data using the emerging framework of graph signal processing. Graphs are generic data representation forms, suitable for modeling the geometric structure of signals that live on topologically complicated domains. The vertices of the graph represent the discrete data domain, and the edge weights capture the pairwise relationships between the vertices. A graph signal is then defined as a function that assigns a real value to each vertex. Graph signal processing is a useful framework for handling efficiently such data as it takes into consideration both the signal and the graph structure. In this work, we develop new methods and study several important problems related to the representation and structure-aware processing of graph signals in both centralized and distributed settings. We focus in particular in the theory of sparse graph signal representation and its applications and we bring some insights towards better understanding the interplay between graphs and signals on graphs. First, we study a novel yet natural application of the graph signal processing framework for the representation of 3D point cloud sequences. We exploit graph-based transform signal representations for addressing the challenging problem of compression of data that is characterized by dynamic 3D positions and color attributes. Next, we depart from graph-based transform signal representations to design new overcomplete representations, or dictionaries, which are adapted to specific classes of graph signals. In particular, we address the problem of sparse representation of graph signals residing on weighted graphs by learning graph structured dictionaries that incorporate the intrinsic geometric structure of the irregular data domain and are adapted to the characteristics of the signals. Then, we move to the efficient processing of graph signals in distributed scenarios, such as sensor or camera networks, which brings important constraints in terms of communication and computation in realistic settings. In particular, we study the effect of quantization in the distributed processing of graph signals that are represented by graph spectral dictionaries and we show that the impact of the quantization depends on the graph geometry and on the structure of the spectral dictionaries. Finally, we focus on a widely used graph process, the problem of distributed average consensus in a sensor network where sensors exchange quantized information with their neighbors. We propose a novel quantization scheme that depends on the graph topology and exploits the increasing correlation between the values exchanged by the sensors throughout the iterations of the consensus algorithm.

EPFL2016

Architecture of a Real-Time Platform Independant GPS L1 Software Receiver

Personal digital assistants or mobile phones applications are not anymore restricted to multimedia or wireless communications, but have been extended to handle Global Positioning System (GPS) functionalities. Consequently, the growing market of GPS capable mobile devices is driving the interest of software receiver solutions as they provide several advantages with respect to traditional hardware implementations. First, they share the same system resources such as the processor, embedded memory and power with other system units, reducing both the size and the costs of their integration. Second, they can be easily reprogrammed – via a firmware update – for incorporating the latest developments, such as the exploitation of the future satellites signals or some improved multipath mitigation techniques. Finally, they offer a more flexible solution for rapid research and development as compared to conventional hardware receivers where the chip design is fixed and obtained after a long integration process. With the increasing performance of modern processors, it becomes now feasible to implement in software a multi-channel GPS receiver operating in real-time. However, a major problem with the software architecture is the large computing resources required for the digital signal processing. Former studies have demonstrated that a straightforward transposition of traditional hardware based architectures into software would lead to an amount of integer operations which is not suitable for today's fastest computers. From this observation, several strategies have been proposed in the literature in order to reduce the complexity of the receiver operations. The first one relies on the utilization of advanced microprocessor instructions set which provides the capability of processing vectors of data by operating on multiple integer values at the same time. This results in significant gains in execution speed, but also severely limits the portability of the code, since the operations are tied to specific processors architectures. Another alternative consists in exploiting the native bitwise representation of the signal. The data bits are stored in separate vectors on which logical parallel operations can be performed. The objective is to take advantage of the universality, high parallelism, and speed of the bitwise operations for which a single integer operation translates into a few simple parallel logical relations. However, the inherent drawback of the bitwise processing is the lack of flexibility as the complexity becomes bit-depth dependent. This thesis has been carried out in the framework of a two-year industrial project (2007-2009) in collaboration with U-blox AG in Thalwil. It aimed to the realization of a multi-channel, platform-independent real-time GPS L1 software receiver. The main challenge of this project consisted in providing real-time performances while keeping the portability of the code to make the receiver suitable for any type of software implementation. In that sense, new techniques and algorithms have been developed for optimizing the processing chain in order to lower the processor load. The main idea consists in regrouping data which share the same characteristics, and process them in batches instead of sequentially. This way, it becomes possible to progressively reduce the data throughput and consequently the amount of operations to perform. A completely new receiver architecture has been proposed and validated through the realization of a functional prototype, thus demonstrating the feasibility of the concept.

EPFL2010

Signal Processing in Space and Time

Francisco Pereira Correia Pinto

Sound waves propagate through space and time by transference of energy between the particles in the medium, which vibrate according to the oscillation patterns of the waves. These vibrations can be captured by a microphone and translated into a digital signal, representing the amplitude of the sound pressure as a function of time. The signal obtained by the microphone characterizes the time-domain behavior of the acoustic wave field, but has no information related to the spatial domain. The spatial information can be obtained by measuring the vibrations with an array of microphones distributed at multiple locations in space. This allows the amplitude of the sound pressure to be represented not only as a function of time but also as a function of space. The use of microphone arrays creates a new class of signals that is somewhat unfamiliar to Fourier analysis. Current paradigms try to circumvent the problem by treating the microphone signals as multiple "cooperating" signals, and applying the Fourier analysis to each signal individually. Conceptually, however, this is not faithful to the mathematics of the wave equation, which expresses the acoustic wave field as a single function of space and time, and not as multiple functions of time. The goal of this thesis is to provide a formulation of Fourier theory that treats the wave field as a single function of space and time, and allows it to be processed as a multidimensional signal using the theory of digital signal processing (DSP). We base this on a physical principle known as the Huygens principle, which essentially says that the wave field can be sampled at the surface of a given region in space and subsequently reconstructed in the same region, using only the samples obtained at the surface. To translate this into DSP language, we show that the Huygens principle can be expressed as a linear system that is both space- and time-invariant, and can be formulated as a convolution operation. If the input signal is transformed into the spatio-temporal Fourier domain, the system can also be analyzed according to its frequency response. In the first half of the thesis, we derive theoretical results that express the 4-D Fourier transform of the wave field as a function of the parameters of the scene, such as the number of sources and their locations, the source signals, and the geometry of the microphone array. We also show that the wave field can be effectively analyzed on a small scale using what we call the space/time-frequency representation space, consisting of a Gabor representation across the spatio-temporal manifold defined by the microphone array. These results are obtained by treating the signals as continuous functions of space and time. The second half of the thesis is dedicated to processing the wave field in discrete space and time, using Nyquist sampling theory and multidimensional filter banks theory. In particular, we show examples of orthogonal filter banks that effectively represent the wave field in terms of its elementary components while satisfying the requirements of critical sampling and perfect reconstruction of the input. We discuss the architecture of such filter banks, and demonstrate their applicability in the context of real applications, such as spatial filtering and wave field coding.

EPFL2010

Architecture of a Real-Time Platform Independant GPS L1 Software Receiver

EPFL2010

Signal Processing in Space and Time

Francisco Pereira Correia Pinto

EPFL2010

Graph Signal Processing

Ntorina Thanou

EPFL2016