Graph Slepians To Strike A Balance Between Local And Global Network Interactions: Application To Functional Brain Imaging

Brain function exhibits coordinated activity patterns that are also reflected in anatomy, a finding that can be harnessed to constrain the dynamics of functional time series to the underlying structure while performing various signal processing operations. Graph signal processing (GSP) is such a framework, which we here equip with a new tool to uncover localised functional brain interactions. The functional magnetic resonance imaging (fMRI) signal is projected onto a collection of Slepian vectors defined on a graph extracted from structural and diffusion MRI data. This decomposition allows a multi-bandwidth description of signals that are maximally concentrated within a subset of nodes, as is often the case for neural activity. On simulated data, we compare this technique to classical Laplacian and localised Laplacian filtering. We then present, on real fMRI data, an illustration of the Slepians potential to retrieve localised interaction patterns in the context of a visual stimulation task.

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

Electrical neuroimaging based on biophysical constraints

Roberto Martuzzi

This paper proposes and implements biophysical constraints to select a unique solution to the bioelectromagnetic inverse problem. It first shows that the brain's electric fields and potentials are predominantly due to ohmic currents. This serves to reformulate the inverse problem in terms of a restricted source model permitting noninvasive estimations of Local Field Potentials (LFPs) in depth from scalp-recorded data. Uniqueness in the solution is achieved by a physically derived regularization strategy that imposes a spatial structure on the solution based upon the physical laws that describe electromagnetic fields in biological media. The regularization strategy and the source model emulate the properties of brain activity's actual generators. This added information is independent of both the recorded data and head model and suffices for obtaining a unique solution compatible with and aimed at analyzing experimental data. The inverse solution's features are evaluated with event-related potentials (ERPs) from a healthy subject performing a visuo-motor task. Two aspects are addressed: the concordance between available neurophysiological evidence and inverse solution results, and the functional localization provided by fMRI data from the same subject under identical experimental conditions. The localization results are spatially and temporally concordant with experimental evidence, and the areas detected as functionally activated in both imaging modalities are similar, providing indices of localization accuracy. We conclude that biophysically driven inverse solutions offer a novel and reliable possibility for studying brain function with the temporal resolution required to advance our understanding of the brain's functional networks.

Elsevier2004

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

EPFL2010

Signal Processing in Space and Time

Francisco Pereira Correia Pinto

EPFL2010