Nour Zalmai

On sparsity by NUV-EM, Gaussian message passing, and Kalman smoothing

Normal priors with unknown variance (NUV) have long been known to promote sparsity and to blend well with parameter learning by expectation maximization (EM). In this paper, we advocate this approach for linear state space models for applications such as the estimation of impulsive signals, the detection of localized events, smoothing with occasional jumps in the state space, and the detection and removal of outliers. The actual computations boil down to multivariate-Gaussian message passing algorithms that are closely related to Kalman smoothing. We give improved tables of Gaussian-message computations from which such algorithms are easily synthesized, and we point out two preferred such algorithms.

Local statistical models from deterministic state space models, likelihood filtering, and local typicality.

Surprisingly many signal processing problems can be approached by locally fitting autonomous deterministic linear state space models to the data. In this paper, we introduce local statistical models for such cases and discuss the computation both of the corresponding estimates and of local likelihoods for different models.

Blind deconvolution of sparse but filtered pulses with linear state space models

The paper considers the problem of joint system identification and input signal estimation of an unknown linear system from noisy observations of the output signal. The input signal is assumed to be sparse, and each individual input pulse may affect the system in its own (and unknown) way. Based on ideas from sparse Bayesian learning, we derive an efficient expectation maximization (EM) algorithm for jointly estimating all unknown quantities. Unlike related prior work, the proposed algorithm does not alternate between estimating the input signal and estimating the system parameters; instead, all unknown quantities are jointly updated in each EM step. We give closed-form expressions for these EM updates, which can be efficiently computed by Gaussian message passing.

Gesture recognition from magnetic field measurements using a bank of linear state space models and local likelihood filtering

Detecting and inferring the trajectory of a moving magnet from magnetic field measurements is a challenge due to a wide range of time scales and amplitudes of the recorded signals and limited computational power of devices embedding a magnetometer. In this paper, we model the magnetic field measurements using a bank of autonomous linear state space models and provide an efficient algorithm based on local likelihood filtering for reliably detecting and inferring the gesture causing the magnetic field variations.

Signal detection and discrimination for medical devices using windowed state space filters

We introduce a model-based approach for computationally efficient signal detection and discrimination, which is relevant for biological signals. Due to its low computational complexity and low memory need, this approach is well-suited for low power designs, as required for medical devices and implants. We use linear state space models to gain recursive, efficient computation rules and obtain the model parameters by minimizing the squared error on discrete-time observations. Furthermore we combine multiple models of different time-scales to match superpositions of signals of variable length. To give immediate access to our method, we highlight the use in several practical examples on standard and on esophageal ECG signals. This method was adapted and improved as part of a research and development project for medical devices.

Unsupervised feature extraction, signal labeling, and blind signal separation in a state space world

The paper addresses the problem of joint signal separation and estimation in a single-channel discrete-time signal composed of a wandering baseline and overlapping repetitions of unknown (or known) signal shapes. All signals are represented by a linear state space model (LSSM). The baseline model is driven by white Gaussian noise, but the other signal models are triggered by sparse inputs. Sparsity is achieved by normal priors with unknown variance (NUV) from sparse Bayesian learning. All signals and system parameters are jointly estimated with an efficient expectation maximization (EM) algorithm based on Gaussian message passing, which works both for known and unknown signal shapes. The proposed method outputs a sparse multi-channel representation of the given signal, which can be interpreted as a signal labeling.

Autonomous state space models for recursive signal estimation beyond least squares

The paper addresses the problem of fitting, at any given time, a parameterized signal generated by an autonomous linear state space model (LSSM) to discrete-time observations. When the cost function is the squared error, the fitting can be accomplished based on efficient recursions. In this paper, the squared error cost is generalized to more advanced cost functions while preserving recursive computations: first, the standard sample-wise squared error is augmented with a sample-dependent polynomial error; second, the sample-wise errors are localized by a window function that is itself described by an autonomous LSSM. It is further demonstrated how such a signal estimation can be extended to handle unknown additive and/or multiplicative interference. All these results rely on two facts: first, the correlation function between a given discrete-time signal and a LSSM signal can be computed by efficient recursions; second, the set of LSSM signals is a ring.

Inferring depolarization of cells from 3D-electrode measurements using a bank of linear state space models

Cell depolarization runs essentially in a uniform motion along the muscular tissue, which creates transient electrical potential differences measurable by nearby electrodes. Inferring the depolarization speed and direction from measurements is of great interest for physicians. In cardiology, this is part of the inverse ECG problem which often requires a large number of electrodes and intense computational power even if the simple common model of the single equivalent moving dipole (SEMD) is applied. In this paper, we model a depolarization process as a straight-line movement of a SEMD. We provide an efficient algorithm based on linear state space models that infers the SEMD movement using only 3 measurement channels from a tetrahedral electrode and with the presence of interferences. Our algorithm is tested both on simulated and experimental data.

Tomographic reconstruction using a new voxel-domain prior and Gaussian message passing

The paper proposes a new prior model for gray-scale images in 2D and 3D, and a pertinent algorithm for tomographic image reconstruction. Using ideas from sparse Bayesian learning, the proposed prior is a Markov random field with individual unknown variances on each edge, which allows for sharp edges. Such a prior model remarkably captures and preserves both the edge structures and continuous regions of natural images while being computationally attractive. The proposed reconstruction algorithm is an efficient EM (expectation maximization) algorithm where the actual computations essentially reduce to scalar Gaussian message passing. Simulation results show that the proposed approach works well even with few projections, and it yields (slightly) better results than a state-of-the art method.