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An interest is often present in knowing evolving variables that are not directly observable; this is the case in aerospace, engineering control, medical imaging, or data assimilation. What is at hand, though, are time-varying measured data, a model connecting them to variables of interest, and a model of how to evolve the variables over time. However, both models are only approximation and the observed data are tainted with noise. This is an ill-posed inverse problem. Methods, such as Kalman filter (KF), have been devised to extract the time-varying quantities of interest. These methods applied to this inverse problem, nonetheless, are slow, computation wise, since they require large matrices multiplications and even matrix inversion. Furthermore, these methods are not usually suitable to impose some constraints. This article introduces a new iterative filtering algorithm based on alternating projections. Experiments were run with simulated moving projectiles and were compared with results using KF. The new optimization algorithm proves to be slightly more accurate than KF, but, more to the point, it is much faster in terms of CPU time.

Tracking moving objects or a substance is a common optimization and control problem [

The novel filtering algorithm presented in this article aims, as does KF, to find an estimate

Tracking a moving object (or moving objects) with large unknown variables and few given data, as in aerospace, medical imaging, or data assimilation, is an ill-posed inverse problem. This ill-posedness is further inflated by physical degradation of the acquired data causing noise. Filtering algorithms, as the one presented in this article, are therefore very suitable tools.

The remainder of the paper is organized as follows. Section

A new filtering algorithm is introduced. Without loss of generality, it is illustrated, with an example of tracking moving objects. It could be applied to any instance where KF was and is used when modeled by the two linear state-space equations (

To illustrate the new algorithm, tracking moving objects is studied here where detectors, such as radars or cameras, measure their positions over time. The goal is to estimate their positions, velocities, and/or acceleration, all at once, given a few noisy measurements of their positions. Kalman filter can average out the noise of the measured data and gives somehow smooth tracks of these over time moving objects in space [

Velocity and acceleration can be easily derived from the position. Someone could be interested in finding only the positions and can easily derive the velocities and acceleration therefrom; however, having velocities and acceleration helps to build the state evolution of (

Detectors are exploited to register the positions of the moving objects. The problem is then modeled as follows. Let

To make sense of this linear evolution model, consider the following basic 2D example with a constant time increment

Let

For instance, consider example

Note that this is just an illustrating example with very small sizes;

The optimal

The covariance matrices

The index

The KF update [

The approach here to solving for both linear state-space equations (

The authors in [

First, two convex sets are defined of

Minimize

Minimize

The sequence

The goal here is to solve for both linear state-space equations (

Recall that the aim is to find an estimate

The functional to be minimized at each recursion step

It suffices now to use the change of variables, given in step 2 of the Alternating Projections Filter (APF) algorithm

To solve the linear state-space problem given by (

Assume the recursive steps up to time

To get

Make

Do

Compute

The update formula for the next estimate is

Observe that the two steps

The temporal regularization parameter

The experiment is done with different large values of

The

Experiment parameters.

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Tracking of moving objects: simulated positions in blue, noisy measurements of positions in black, Kalman filter (KF) recovering of positions in red, and Alternating Projections Filter (APF) recovering of positions in green. Running CPU times of both algorithms depend on the size

The three curves in Figure

The relative error between the simulated and both reconstructed curves has a median around 30%. Running CPU times of both algorithms depend on the size

A novel filtering algorithm APF, to be applied to the linear case, was presented; it could be implemented in aerospace and other fields. It could also be applied, for instance, anytime tracking of moving objects is desired, such as in medical imaging and data assimilation where large size systems are involved. Experiments were run comparing APF and KF in terms of accuracy and computer speed. Quality of reconstructed curves is about the same in both algorithms, although APF performs slightly better than KF. More importantly, APF is up to 25 times faster than KF, seconds instead of minutes. As it is the case with KF, APF algorithm filters out errors from modeling the dynamical system and the noise from the data. Both algorithms insure temporal regularization and output an optimal recursive estimate. However, APF does not use any matrix-matrix multiplication and does not necessitate any matrix inversion. Furthermore, APF does not need to calculate, update, or store any covariance matrix. This is not the case for KF regarding these last three properties. Indeed, these three properties are at the heart of making APF take much less CPU time compared to KF, so that APF is very suitable for large scale systems such as the ones in aerospace. APF could be used in any discipline which has used, for instance, KF or in any field that is interested in time-varying variables such as financial risk assessment/evaluation and forecasting or control. The results substantiate the efficiency of this novel APF algorithm.

Time parameter

Maximum value of

Position vector of a moving object at time

Evolution matrix from position

Measured data vector at time

System matrix relating

Expected value of a vector

Covariance matrix of a vector

Error vector in modeling the transition from

Covariance of

Noise vector of measured

Covariance of

The data used to support the findings of this study are included within the article.

The author declares no conflicts of interest.