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Post-processing module: the NNIPDA-UKF filter
This section presents the post-processing module, that is responsible with filtering multiple echoes (i.e targets) from the set of observations obtained by the pre-processor. We consider a number nt of independent targets with system states denoted as {xit}nti=1. Since the trajectories of individual targets are considered independent, echo tracking can be conducted separately on each individual target state. For the i−th echo, the state system is given by the state-transition equation (II.5a) and the observation equation (II.5b) xit= ft(xi t−1) + vit, (II.5a)zit = Htxit+ wit, i = 1, · · · , nt, (II.5b)where the state vector xit2 X takes values in the state space X. For the sake of generality, weconsider xit as a vector of dimension dx and X Rdx. However, in sections II.4 and II.5, we will consider more specific models (e.g. x = ). The state transition function ft(·) represents an added prior, i.e. a model on the DOA trajectory. The covariance Qi t of the model noise vit controls the confidence in the model. The notation xit is kept for the sake of coherence with the tracking and dynamical-systems literature. zi t is the system noisy observation supplied bythe pre-processing module, as discussed in the previous section. Ht is the observation matrix.Whenever, the observation zit only consists in maxima locations of the spectrogram (as depictedin Eq. II.4), the observation reduces to a scalar and Ht = 1. The observation noise witwithvariance Rit depends on the acquisition parameters and the pre-processing module. The noisesequences vitand wit are independent, white random processes characterized by their first two moments. Trajectory initialization is usually done by the “two out of three” heuristic rule, in other words, two observations falling inside a given gate within three consecutive range bins are assumed to represent a target. However, automatic trajectory initialization is beyond the scope of this chapter.
The pre-processing module detects the presence of a target (i.e. an echo) having state xit , with non-unitary probability of detection pD(xit), and estimates the DOA of the target. Conditionedon the target state, pD(xit) is by definitionpD(xit) , Pr{zit 2 Zt|xit}. (II.6) pD(x) conveys the fact that zt may not belong to the ˆmt maxima of S(, t). In other words, the backscattering amplitude of the trajectory can be less than the largest ˆmt maxima. Furthermore, Zt possibly contains clutter (false) DOA estimates since interfering echoes are present along the bathymetrically informative observation zt. Performance analysis of model-order estimators in array processing, involving AIC or MLD penalties, have been presented in [57, 58]. In [59], the dependence of pD on the state x was analyzed and addressed at the expense of increased algorithmic complexity. For the applications considered in this chapter, employing pD = pD(x) = 0.9 8 x 2 X provides a compromise between performance and computational complexity.
The objective of model-based filters is to obtain an estimator ˆxi t|t , E{xit |Z1:t} of the state{xit}nt i=1 given the sequence of current and past observations, denoted by Z1:t. In the tracking literature, the sequence of filtered state vectors ˆxi t|t is usually called trajectory or track. All recursive filtering algorithms consist in two phases: prediction and update. Prediction aims to obtain an estimate ˆxi t|t−1 by using the target model. The update step acts like a correction to the predicted state value ˆxt|t−1, by incorporating the current observation Zt to obtain ˆxt|t. The magnitude of the correction takes into account the noise covariances Qt and Rt. In the case of Gaussian processes wt and vt, and linear ft(·), the Kalman [60] filter offers the Minimum Mean Squared Error (MMSE) estimator as closed-form equations for both the prediction and the update step. Whenever the noises are only characterized by their first two moments, the Kalman filter supplies the optimal linear MMSE from the family of linear estimators. Furthermore, for non-linear systems, a sub-optimal but computationally attractive solution is given by the Unscented Kalman Filter (UKF) [61, 62]. The UKF only requires the first two moments of the noise terms vt and wt. Indeed, the UKF only supposes uni-modal distributions, the Gaussian hypothesis not being strictly required. The mean-squared estimates supplied by the UKF amount to an approximate posterior distribution. To validate the model of Eqs. (II.5a) and (II.5b) and the aforementioned hypotheses, we employ Goodness-of-Fit (GoF) tests on simulated and real data.
Trajectory quality measure
As stated in the introduction, an object on the sea bottom creates an acoustic shadow, where the corresponding trajectory estimates must not be considered for bathymetry reconstruction. In order to decide whether a trajectory is valid, we introduce an existence process. For trajectory i at time t, the trajectory existence process i t is supposed to evolve according to a Hidden Markov Chain (HMC) with two internal states: trajectory exists, denoted i t = 1 and trajectory does not exist, denoted i t = 0. The estimated trajectory existence probability is defined as i t|t = Pr{i = 1|Z1:t} and represents a measure of quality associated with trajectory i. By thresholding i t|t we are able to extinguish trajectory estimates over shadow areas and terminate trajectories. The transitions between the Markov states are given by the transition matrix with elements []kl = kl , Pr{i t = l|i t−1 = k} with k, l 2 {0, 1}. Note that, for simplicity, we suppose a homogeneous HMC. However, IPDA filters can also cope with time-dependent transition probabilities. Typical values for these transition probabilities, also used throughout this chapter, are 00 = 11 = 0.98 [73]. In other words, transitions occur infrequently. is identical for all trajectories i. A detailed derivation of thepredicted and updated trajectory existence probability is given in Appendix A, while here we only recall the main results. The updated track existence probability i t|t is given by:
Multi-path echoes separation
In Fig. II.7 a) we observe NN-IMM-IPDA results for bottom and sea-surface echoes overlaid onto the Capon spectrogram of the corresponding ping line. Pre-processing was done with the Capon beamformer and the AIC [52] method for number of sources estimation. Observe the echo separation achieved by the multiple-trajectory filter with the nearest neighbor association scheme. The separation of the bottom and the sea-surface echoes ensures spurious-free bathymetric reconstruction. Bottom-echo tracking is conducted using the set of models: FHB (II.23) with model noise variance QFHB = 0.01 deg2, and RW model (II.24) with QRW = 0.1 deg2. The IMM [74, Ch.11.6.6] framework that merges the two model estimates, employs sequentiallyestimated model probabilities μt|t showcased in Fig. II.7 b). Model probabilities μFHB t|t and μRW t|t indicate the model likelihood given current and past observations, thus providing a geometrical segmentation of the underlying bottom profile. Since the bottom echo has an arch-shaped curve typical of flat and horizontal bottoms, the FHB model is, for most of the times, more likely than the RW model. The bottom trajectory existence probability t|t, given by Eq. ((A.17)) and presented in Fig. II.7 c) is also computed sequentially and indicates trajectory quality. Whenever t|t decreases below the threshold value, set to 0.5, the trajectory estimates are extinguished (for slant range values between 35−40 m). The filtering residuals given by Eq. (II.17b) are employed for GoF tests yielding the acceptance of the whiteness hypothesis with p−values of 0.318 and 0.283 for the Box-Pierce and Ljung-Box tests, respectively. Since the sea-surface echo exhibits high variations, with a less evident trend, a single RW model was employed with QRW = 1 deg2 for DOA trajectory estimation. Sea-surface echo trajectory existence probability is given in Fig. II.7 d). GoF tests for the surface echo yield the acceptance of the whiteness hypothesis with p−values of 0.439 and 0.346 for the Bow-Pierce and Ljung-Box tests, respectively. In order to better visualize the trajectory existence probabilities t|t we present the results for 200 adjacent ping lines forming Fig. II.8. In Fig. II.8 a) we observe the lateral (or side-scan) amplitude image, while in Fig. II.8 b) we observe the corresponding instantaneous trajectory quality, t|t. Notice the correspondence between the two figures, notably concerning the two acoustic shadows where the trajectory existence probability is low. The ping line processed in Fig. II.7 is obtained from the set of ping lines presented in Fig. II.8 at the along track marker of 10m. Notice the echo interruption from Fig. II.7 a) caused by an acoustic-shadow patch. Thresholding t|t ensures acoustic-shadow detection, where tracking is stopped and bathymetric points are not reconstructed (lack of bottom-backscattered signal).
Table of contents :
Abbreviations and nomenclature
Acknowledgements
R´esum´e
Contexte
Les principaux d´efis
Objectifs et r´ealisations
Plan d´etaill´e de la th`ese
Conclusions
Perspectives
I Introduction
II Sonar echo tracking: a data-association approach
II.1 Introduction: Objectives and challenges
II.2 Underwater environment and pre-processing module
II.3 Post-processing module: the NNIPDA-UKF filter
II.3.1 Prediction
II.3.2 Update
II.3.3 Trajectory quality measure
II.4 Geometrical models
II.5 Results of NNIPDA-UKF
II.5.1 Simulated data results
II.5.2 Real data results
II.6 Chapter conclusions
III Sonar echo tracking: an intensity function approach
III.1 Point-process formalism for target tracking
III.2 Point-process theory
III.2.1 Preliminary definitions
III.2.2 A probability measure for PP
III.2.3 RFS and set integral
III.2.4 Moment measures of point processes
III.3 Examples of finite point processes
III.3.1 Independent and identically distributed cluster (iidc)
III.3.2 Poisson process
III.4 Error metrics for multi-target tracking
III.5 The PHD filter for TWS systems
III.5.1 TWS-PHD prediction
III.5.2 TWS-PHD update
III.6 TBD-PHD filter for phased arrays: a Gaussian point-target case
III.6.1 Introduction
III.6.2 Array signal and marked multi-target process
III.6.3 Approximate PHD filter for marked multi-target process
III.6.4 Results on simulated phased-array data
III.7 Chapter conclusions
IV Sonar echo tracking: the case of extended and impulsive targets
IV.1 Introduction
IV.2 Impulsive Sonar-signal distributions
IV.2.1 The SIRV model
IV.2.2 The joint-SIRV model
IV.3 Phased-array signal and multi-target formalism
IV.3.1 Extended and impulsive target model
IV.3.2 Marked process for extended and impulsive target signals
IV.4 Phased-array TBD-CPHD filter
IV.4.1 Phased-array TBD-CPHD prediction
IV.4.2 Phased-array TBD-CPHD intensity update
IV.4.3 Phased-array TBD-CPHD cardinality update
IV.4.4 TBD-CPHD Monte Carlo Implementation
IV.5 Simulation Results
IV.5.1 Tracking results
IV.5.2 Angular resolution analysis
IV.6 Real sonar data results
IV.7 Chapter conclusions
V Conclusions and perspectives
A Integrated PDAF (IPDAF)
B Observation noise covariance estimation
C Model validation
D Proof of p-value |H0 U(0, 1)
E Quadratic and Gaussian equalities
G Array narrow-band condition
H Statistical analysis of sonar signals
