Alireza Ghodrati

An analytical comparison of three spatio-temporal regularization methods for dynamic linear inverse problems in a common statistical framework

Dynamic inverse problems, which occur in medical imaging and other fields, are inverse problems in which the quantities to be reconstructed vary in time, although they are related to the measurements through spatial operators only. Traditional methods solve these problems by frame-by-frame reconstruction, then extract temporal behaviour of the objects or regions of interest through curve fitting and other image-based processing. These approaches solve the inverse problem while exploiting only the spatial relationship between the object and the measurement data at each time instant, without using any temporal dynamics of the underlying process, and thus are not optimal unless the solution is temporally uncorrelated. If the spatial operators are linear, and if one, by contrast, solves the whole spatio-temporal process jointly, it falls into the category of general linear least-squares problems. Such approaches are generally difficult, both due to the challenge of modelling the temporal dynamics appropriately as well as to the high dimensionality of the associated large linear system. Several recent reports have approached this problem in different ways, making different prior assumptions on the spatial and temporal behaviour. In this paper we discuss three such approaches, which have been introduced from different points of view, in a common statistical regularization framework, and illuminate their relationships. The three methods are a state-space model, the separability condition and a multiple constraints model. The key result is that there is a clear relationship among the three methods; specifically, the inverse of the spatio-temporal autocovariance matrix has a block tri-diagonal form, a Kronecker product form or a Kronecker sum form, respectively. Some simple simulation examples are presented to illustrate the theoretical analysis.

Wavefront-based models for inverse electrocardiography

We introduce two wavefront-based methods for the inverse problem of electrocardiography, which we term wavefront-based curve reconstruction (WBCR) and wavefront-based potential reconstruction (WBPR). In the WBCR approach, the epicardial activation wavefront is modeled as a curve evolving on the heart surface, with the evolution governed by factors derived phenomenologically from prior measured data. The body surface potential/wavefront relationship is modeled via an intermediate mapping of wavefront to epicardial potentials, again derived phenomenologically. In the WBPR approach, we iteratively construct an estimate of epicardial potentials from an estimated wavefront curve according to a simplified model and use it as an initial solution in a Tikhonov regularization scheme. Initial simulation results using measured canine epicardial data show considerable improvement in reconstructing activation wavefronts and epicardial potentials with respect to standard Tikhonov solutions. In particular the WBCR method accurately finds the anisotropic propagation early after epicardial pacing, and the WBPR method finds the wavefront (regions of sharp gradient of the potential) both accurately and with minimal smoothing

RR interval analysis for detection of Atrial Fibrillation in ECG monitors

We investigate two RR irregularity measures suitable for Atrial Fibrillation (AFIB) detection in ECG monitors, one based on the absolute deviation and the other based on the difference between successive RR intervals. A sequence of RR intervals is fed to the irregularity measures after applying certain constraints on length and beat classifications to provide criteria for detection of AFIB. Receiver Operating Curves (ROC) are used to analyze and compare the performance of the two methods against MIT-BIH Arrhythmia Database, MIT-BIH AFIB Database and a proprietary AFIB Database.

Methods of Solving Reduced Lead Systems for Inverse Electrocardiography

In the context of inverse electrocardiography, we examine the problem of using measurements from sets of electrocardiographic leads that are smaller than the number of nodes in the associated geometric models of the torso. We compared several methods to estimate the solution from such reduced-lead measurements sets both with and without knowledge of prior statistics of the measurements. We present here simulation results that indicate that deleting rows of the forward matrix corresponding to the unmeasured leads performs best in the absence of prior statistics, and that Bayesian (or least-squares) estimation performs best in the presence of prior statistics

A method to reconstruct activation wavefronts without isotropy assumptions using a level sets approach

We report on an investigation into using a Level Sets based method to reconstruct activation wavefronts at each time instant from measured potentials on the body surface. The potential map on the epicardium is approximated by a two level image and the inverse problem is solved by evolving a boundary, starting from an initial region, such that a filtered residual error is minimized. The advantage of this method over standard activation-based solutions is that no isotropy assumptions are required. We discuss modifications of the Level Sets method used to improve accuracy, and show the promise of this method via simulation results using recorded canine epicardial data.

Analysis of spatial-temporal regularization methods for linear inverse problems from a common statistical framework

In some medical imaging problems, the quantity to image is time-varying but related to the measurements by spatial dynamics only. Traditional methods solve the associated inverse problem separately at each time instant. Several recent reports take advantage of prior knowledge and/or measurement temporal behavior to solve jointly in space and time. In this paper we discuss three such approaches, which have been introduced in distinct mathematical contexts, from a common statistical regularization framework, and illuminate their relationships, advantages and disadvantages.

A level sets algorithm for the inverse problem of electrocardiography

One of the key features of the hearts electrical activity is the activation wavefront. The purpose of this work is to reconstruct the activation wavefront at each time instant from the measured potentials on the body surface by using a level sets based algorithm. We approximated the potential map of the heart surface as a two level image and we solve the inverse problem by evolving a boundary starting from an initial region, such that the residual error is minimized. The advantage of this method over standard activation-based solutions is that we do not require any isotropy assumption about cardiac activation.

Some Spatio-Temporal Approaches to Inverse Electrocardiography

In inverse electrocardiography one tries to accurately and meaningfully characterize cardiac electrical activity from electrical potential measurements on the body surface and a volume model of the torso. This is a typical ill-posed bioelectric field problem requiring constraints (regularization). One source of constraints is the strongly spatio-temporal nature of cardiac electrical activity. However formulating such constraints in a tractable fashion can be challenging. We review the major approaches used for this problem, and present work on a middle ground between simple non-electrophysiological constraints and strong electrophysiological constraints that eliminate useful complexity in solutions.

Inverse electrocardiography in the framework of dynamic imaging problems

We describe several current approaches which include temporal information into the inverse problem of electrocardiography. Some of these approaches operate directly on potential-based source models, and we show how three recent methods, introduced with rather distinct assumptions, can be placed in a common framework and compared. Others operate on parameterized models of the cardiac sources, and we discuss briefly how recent developments in curve evolution methods for inverse problems may allow more physiologically complex parametric models to be employed.

A wavefront-based constraint for potential surface solutions in inverse electrocardiography

Inverse electrocardiography in recent years has generally been approached using one of two quite distinct source models, either a potential-based approach or an activation-based approach. Each approach has advantages and disadvantages relative to the other, which are inherited by all specific methods based on a given approach. Recently our group has been working to develop models which can bridge between these two approaches, hoping to capture some of the most important advantages of both. In this work we present one such effort, which we term wavefront-based potential reconstruction (WBPR). It is a modification of standard regularization methods for potential-based inverse electrocardiography, into which we incorporate a constraint based on a wavefront-like approximation to the potential-based solution. Initial results indicate significant improvement with respect to localization and characterization of the wavefront in simulations using both epicardially and supra-ventricularly paced heartbeats