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Decentralized Kalman filtering theory & its applications  

(one technologically logical approach to sensor fusion; another is to use merely one Kalman filter, depending on the situation)

Key Benefits:

Tap into our 40+ year reservoir of hands-on experience in estimation and tracking applications (for navigation & for Early Warning Strategic surveillance radar)
Saves customers’ time and money
Reduces customers’ project risks

Capabilities:

Consulting for engineering design, analysis, and performance evaluation of alternative estimation algorithms (including decentralized versions for sensor fusion, if decentralized KF are really needed). Sometimes use of a centralized KF suffices.

Click here to download a 214KByte pdf file which conveys our view on the problems with Covariance Intersection.

Click here to see insightful high level essay below on decentralized Kalman filtering status.

An Historical Account of our experience therein (more detail below):

Kerr, T. H., Stability Conditions for the RelNav Community as a Decentralized Estimator-Final Report, Intermetrics, Inc. Report No. IR-480, Cambridge, MA, 10 August 1980, for NADC (Warminster, PA) for Leonard Chin.
Kerr, T. H., and Chin, L., “A Stable Decentralized Filtering Implementation for JTIDS RelNav Proceedings of IEEE Position, Location, and Navigation Symposium (PLANS), Atlantic City, NJ, 8-11 December 1980.
Kerr, T. H., and Chin, L., “The Theory and Techniques of Discrete-Time Decentralized Filters,” in Advances in the Techniques and Technology in the Application of Nonlinear Filters and Kalman Filters, edited by C. T. Leondes, NATO Advisory Group for Aerospace Research and Development, AGARDograph No. 256, Noordhoff International Publishing, Lieden, 1981.
Carlson, N. A., Kerr, T. H., Sacks, J. E., Integrated Navigation Concept Study, Intermetrics Report No. IR-MA-321, 15 June 1984, for ITT (Nutley, NJ).
Kerr, T. H., “Decentralized Filtering and Redundancy Management Failure Detection for Multi-Sensor Integrated Navigation Systems,” Proceedings of the National Technical Meeting of the Institute of Navigation (ION), San Diego, CA, 15-17 January 1985.
Kerr, T. H., “Decentralized Filtering and Redundancy Management for Multisensor Navigation,” IEEE Trans. on Aerospace and Electronic Systems, Vol. AES-23, No. 1, pp. 83-119, Jan. 1987 (minor corrections appear on p. 412 of May and on p. 599 of July 1987 issues of same journal).
Kerr, T. H., “Comments on ‘Estimation Using a Multirate Filter’,” IEEE Transactions on Automatic Control, Vol. AC-34, No. 3, p. 384, March 1989.
Kerr, T. H., “Comments on ‘Federated Square Root Filter for Decentralized Parallel Processes’,” IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-27, No. 6, November 1991.
Kerr, T. H., “Streamlining Measurement Iteration for EKF Target Tracking,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 27, No. 2, March 1991 (minor correction appears in Nov. 1991 issue).
Kerr, T. H., “Extending Decentralized Kalman Filtering (KF) to 2-D for Real-Time Multisensor Image Fusion and\or Restoration,” Signal Processing, Sensor Fusion, and Target Recognition V, Proceedings of SPIE Conference, Vol. 2755, Orlando, FL, pp. 548-564, 8-10 April 1996.
Kerr, T. H., “Extending Decentralized Kalman Filtering (KF) to 2D for Real-Time Multisensor Image Fusion and\or Restoration: Optimality of Some Decentralized KF Architectures,” Proceedings of the International Conference on Signal Processing Applications & Technology (ICSPAT96), 7-10 October 1996.

Sharing some of our Insights:

 DECENTRALIZED OR DISTRIBUTED EXTENDED KALMAN-LIKE FILTERING AS A BASIS FOR SENSOR FUSION IN NEXT GENERATION EARLY WARNING RADAR

Historical Precedents and Benefits to be Accrued from using Decentralized Kalman Filters

The perspective conveyed here on the utility of Decentralized Kalman filtering is the direct result of our (TeK Associates) research and navigation applications experience in this area for over three decades [1]-[8], [10], [11], as cited below. We later became familiar with the models and constraints of radar target tracking for strategic Reentry Vehicle (RV) targets [21], [23]-[26]. We cite the important building block contributions of others (as done in each of our extensive bibliographies appearing in our references, especially in [6], [9], [11], [24], [25]). We do not overlook the use of Kalman filters in image restoration [9], [50], [51] nor the use of more advanced random process statistical techniques in the image processing arena (e.g., [52]).

For over 40+ years, Inertial Navigation Systems (INS), consisting of a constellation of gyros and accelerometers for owncraft position determination (for sea-, land-, airborne, missile-borne, space-borne platforms), have been successfully handled using a single unified Kalman filter to compensate for the deleterious gyro drift rate error of the INS (that increases with time). This compensation or INS correction is accomplished using measurement updates from multiple independent measurement sensors constituting navigation aides such as Omega, Loran-C (hyperbolic or phase-shift), VOR/DME, Navsat, GPS, bathymetric (sonar) map matching (for ships and submarines), visual sightings of geographic sites of known location when flown over (denoted as “Mark-on-Top”), optical star sensors for star fixes, horizon sensors, etc. Each is utilized at different fix rates (many simultaneously) in an algorithmic scheme known as “adaptive” or “aided” INS navigation [27]-[29].

A variation on the above approach uses decentralized Kalman filters [4]-[6], one dedicated to its own particular navaid sensor, then the results are incorporated into a final single result. The motivation for this particular architecture is three-fold: 

 

  1. Participating sub-systems can be tied together for a unified global solution; 
  2. In case of system failures (or battle damage/data dropouts), adverse situation can be accommodated by “doing the best one can with remaining participating subsystems still available” for fail-safe, limp home operations or Fault Detection, Identification (or Isolation) and Reconfiguration (FDIR) as a redundancy management scheme to improve overall system availability; 
  3. Divide and conquer for the computer burden as the sum of the cube of each individual subsystem state sizes for CPU time expended (instead of the cube of the sum of the individual participating state sizes for a unified Kalman filter) and, likewise, the computer memory required goes as the square of the state size for similar savings in a decentralized mechanization.  

Decentralized EKF or Kalman Filters

To avoid single point vulnerabilities, it is prudently recommended (in [4]-[6]) that more than just one collating filter be used (each having the capability of putting together the results of all of the participating sub-system filters). This is not an Interactive Multiple Model (IMM) formulation. However, IMM could constitute any of the subsystem filters.

As a precedent, decentralized Kalman filters were used for C-4 Trident and C-4 backfit Poseidon submarine navigation in the 1970’s and ’80’s within Ships Inertial Navigation System (SINS), Electrostatically Supported Gyro Monitor (ESGM), and joint SINS/ESGM operation (each system having its own filter running simultaneously) to ultimately provide only one output to the Navigator (and to Fire control). Decentralized Kalman filters also naturally arise in networked radio systems that attempt to provide navigation connectivity, such as done in the Navy version of the Joint Tactical Information Distribution System (JTIDS) [1], [2], by Air Force Integrated Communications Navigation and Identification for Avionics (ICNIA), and possibly useful in the current Joint Tactical Radio Systems (JTRS), pronounced Jitters. [Motivation for JTRS is that a smaller, lighter software radio will enable future combat troops in the field to carry more water instead of heavy communications equipment, as the standard pragmatic trade-off. A drawback to the use of JTRS beyond this field situation is that current platforms already have adequate radio communications and JTRS will have to match current form, fit, and function, F3, so that any changes in power consumption, or cooling, or volume expected for JTRS (even if all are less) within a line replaceable unit (LRU) will still cause a huge expense in backfitting platforms to accommodate JTRS or anything new. Another problem is that in order to be flexible enough to accommodate a change in radio protocol (i.e., JTRS changing mode from one particular radio system to another for its advertised interoperability or to change frequency, or to change waveform (of the current 30 waveforms in the JTel collection) or to change communications protocol, there is a need to reboot the processor. The time constrains for doing so are tight. A maximum of 4 seconds to reboot the operating system may suffice (an objective being sought, as pushing the envelop pretty hard) but an implementation taking or needing as much as 40 seconds to reboot is much too long to be practically accommodated by JTRS as it needs to be agile in switching modes.] As a pleasant surprise, Microsoft’s Windows XPe using Ardence’s (previously Venturcom’s) ReadyOn appears to be able to reboot within 7 seconds, as a feat demonstrated for all in the audience to see at the Real-Time Embedded Computer Conference (RTECC) in Framingham, MA on 24 May 2005. Use of such an operating system (a reduced footprint duplicate for embedded processors of a Desktop Windows XP) in conjunction with Ardence’s excellent third party tool, ReadyOn within an embedded architecture leverages all experience with the existing wide set of readily available and familiar favorite software development tools and Integrated Development Environments (IDE’s) currently on Desktop or Laptop Personal Computers under Windows XP that can now be used for developing the software eventually intended for the embedded target machines. Moreover, in the embedded environment, there are tools that suppress any pop-up messages that usually need a mouse click or keyboard key press that may be routinely encountered in a desktop or laptop software implementation that would otherwise plague an embedded implementation (that is likely without any monitor screen, or mouse, or keyboard in the target application). Similarly, recall that in DOS on a desktop machine running Microsoft Windows, programmers can routinely suppress DOS messages to the user or error reports that would normally appear within a DOS Window on the monitor screen merely by redirecting the message to a file and then killing the file (by deleting it or its contents). After software development is complete for the target machine, Microsoft has automated the task of selecting the subset of software that is to be ported over to the target machine to support successful running of what the user specifies and to automatically include any dependencies that the casual or less experienced user may be unaware of. In this way, the final target software footprint may be kept small without exceeding hardware resources. To that end, alternate reboot strategies (e.g., use of flash) are also supported by Windows XPe for target processor systems with hard disk absent. Such is the situation now for embedded devices. A wireless development path also exists within the Microsoft Windows XPe framework which adheres to existing standards and doesn’t force a proprietary turnkey approach. All these new and recent Microsoft software products and high quality, compatible, third party tools makes it easier to pursue Commercial Off-The-Shelf (COTS) implementations using readily available hardware processors with known capabilities and characteristics and low expense because they are already manufactured in high volumes and already have a path in place to avoid future obsolescence (so as to not disappoint its existing, substantial customer base). This is ideal risk mitigation that also leverages the experience of the army of PC programmers already available as a payoff for sticking with the Microsoft approach. The conventional wisdom of the 1980’s and 1990’s is still just as true: “Never bet against Bill Gates!” A word of caution: the cognizant older programmers are likely to also be familiar with “objects”, “classes”, and “collections”, but may tend to avoid using these constructs and methods in favor of using earlier alternate constructs and methods that are standard in Structured Programming because use of “objects”, “classes”, and “collections” is more time consuming in implementation, which is a real-time constraint consideration in embedded applications. Use of Assembly Language may still be useful for achieving the necessary speed in computationally intensive situations of repetitive and/or intensive Signal Processing. Prevalence of “Banker’s Rounding” or “Gaussian Rounding” instead of, or in place of, standard “scientific rounding” in many existing standard PC products should be viewed as a boon and not a bust, as recently revealed by many numerical analysts. We at TeK Associates agree with this favorable assessment of the effect of “Banker’s Rounding”, as encountered and directly tested within our own considerable computational numerical DSP experience.

Benefits of Decentralized Filters in RV Tracking

The motivation for decentralized Kalman filtering in radar or optical target tracking is different from that arising in Navigation. Two (or more) synchronized decentralized estimates of the same target state vector, as viewed from different sensors with, perhaps, different perspective views, different segments of the electromagnetic spectrum utilized (e.g., as with conventional radar and laser radar) to exploit inherent target characteristics, different independent noise contamination intensities (that can be averaged out) is the primary motivation for such a pursuit (along with the fact that the result of several separate Multi-Target Tracking (MTT) decisions no longer need to be reconciled later if the MTT algorithm only resides with the higher level collating filter). However, the initially unaltered covariances of estimation error associated with each separate radar or optical sensor’s view would not necessarily exhibit a partial-ordering in a completely general sensor fusion application but, instead, likely be skewed off from each other in tilt and overall size. However, use of collocated conventional radar along with laser radar may yield one target ellipsoid contained entirely within another due to the greater resolution and smaller azimuth error incurred for laser optics. Sometimes the term “sensor fusion” is used in the Target Tracking context. 

It is suspected that certain U.S. strategic radars currently operate autonomously in a stand-alone manner but take direction and report to a Master Center. At some time in the future, it may be desirable to provide a joint solution to target tracking and at that time decentralized filtering will likely provide the theoretical basis for accomplishing this. Such an endeavor should be especially helpful for situations involving multi-target tracking by helping to eliminate some of the ambiguities associated with crossing targets (as seen from just one radar sensor) by simultaneously providing more than one geographic perspective and a natural solution since targets would then be seen from more than one aspect angle (without needing to incur the delay of waiting for more time to elapse to see the same targets later in their trajectory, as was previously the case before the individual targets could be resolved).

An example of a radar for tracking Strategic Targets (e.g., Reentry Vehicles)

    

The use of decentralized filters would be a rational basis for combining both optical and radar target tracking (or for combining any other sensors). [Missile depicted is a U.S. Peacekeeper.]

Ref. [10] is an excellent overview, in depth numerical ranking, and clear interpretation of all of the various approaches to sensor fusion that have occurred in the target tracking literature over the last two decades and culminates in an algorithmic improvement to Covariance Intersection (CI) that [10] attributes to the pioneering work of the late Fred C. Schweppe’s unknown but bounded approach [11], where, instead of embracing the assumption of Gaussian noises being present, Schweppe uses an entirely deterministic approach of circumscribing the set of potential outcomes arising at each discrete-time step of a linear system’s output within an ellipsoid. Schweppe’s approach, although creative, was notoriously conservative and was never used in applications nor was it admired as frequently as what is reported on in [12], as more representative of Schweppe’s genius. (Although overlooked in [10], parallel developments were occurring in navigation, as similar techniques were being investigated, e.g., [1]-[9]; some “Federated Kalman Filter” approaches were criticized by Larry Levy (JHU/APL) [13]; but he was refuted in [9] regarding Jason Speyer’s exact 1979 version of decentralized estimation [18], where just the estimation portion was extricated by Chang [19] (also see [14]). In [9], it was also pointed out that the crux of several more recent approaches to decentralized filtering harkens back to the structural equations presented in [20] (although they neglected to reference [20], as a published precedent). Ref. [10] demonstrates that the approximate algorithms of the CI approach should only be used with extreme caution since uncertainty increases as more measurements are used (and fused) as a counterintuitive, extremely unsettling result stated and proved simply and convincingly in [10, following Eq. 22]. In a standard Kalman filter, the covariance of estimation error, P(k), usually decreases (or at least should not increase) as measurements from more independent sensors are used at that discrete time step “k” (where multiplication by the step size delta is implied by convention). The same is also the case with an Extended Kalman Filter (EKF) or even with an Iterated Extended Kalman Filter (IEKF), as used in radar target tracking for nonlinear system models and/or measurement models, as arise in reentry vehicle (RV) target tracking [21]

Certain attempts to handle non-zero process noise in decentralized filtering have their own drawbacks [8]. The four different decentralized filtering architectures for possible sensor fusion developed by the SDI Panel 10+ years ago (by Dr. Oliver Drummond and also presented at his 1997 short course at SPIE conference in Orlando, FL) were all predicated (as he rigorously acknowledged) on there being zero process noise present, otherwise these structures are useless or detrimental in situations where process noise is present (so they are inappropriate to use for indoatmospheric tracking wherethe process noise intensity matrix Q is definitely not zero). Ref. [8] discusses a decentralized version of filter fusion where the presence of non-zero process noise covariance was not well accommodated because only a small fraction of Q was apportioned to each participating filter (thus causing each participating filter to perceive the system as being more benign than actual and as a consequence was not tuned to the true underlying situation) and therefore likely to fail in its objective of adequately tracking for each, which then adversely affects the collated whole (since each of N subset filters fails to track a single target system that is much noisier than it expects) the aggregate cannot be much better and the computer burden is N times larger than would be the case if just one correctly modeled single filter were used [thus this particular approach actually defies the fundamental reason for seeking a decentralized solution].

Dr. Dana Sinno (Lincoln Laboratory of MIT) has investigated self-organizing networks of Kalman filter-based sensors (not unlike the electronic acoustic sensors dispensed into the jungles of Viet Nam in the late 1960’s and 1970’s in a failed attempt to monitor Viet Cong activity in the vicinity). Now the sensors have a degree of intelligence and an ability to hierarchically self-organize (like the 1970’s vintage JTIDS RelNav multilateration did for the U.S. Navy in the 1970’s and subsequent decades) and automatically turn-off to conserve power when noisy activity is absent. These smart sensors can be interrogated and the master sensor reports back results to a command center (which can be one of many to avoid a single point vulnerability). [Only ideal exact initial conditions were presumed for each target model for each local Kalman-filter-model-based sensor making each correspond exactly to the simulated mathematical truth model, so results may be less encouraging when practical initialization is eventually invoked for the local sensor model of the system’s target dynamics. Recall that the local sensor models are differential equation-based and apparently assume only constant velocity. Once ideal exact initial conditions are inserted in such a differential equations-based or difference equation-based model, the targets’ trajectories are completely determined precisely even without any measurements being available if Q=0 and no process noise is present. Subsequently providing sensor measurements to the Kalman tracking filter is now just “icing on the cake” by improving the on-line Kalman filter computed covariance but the target locations are already known precisely without deviation (such would not be the case if the targets were allowed to accelerate later after tracking had been initiated-as long as it was not the same constant acceleration as provided in the initial conditions). Even the use of available (but noisy) Doppler measurements from the radar sensors degrade the accuracy available from using mere position measurements alone in this overly idealized experimental situation (as a first step). A real difficulty at present is how to handle multi-target tracking at the sensor level. It is likely that this aspect will only be handled at the level of the supervising Command Center(s) as master station (s).] This was an IR&D project reported on at the 2004 MIT Lincoln Laboratory ASAP Workshop. Dr. Sinno is now performing follow-on activities under a DARPA contract at Lincoln Laboratory (~2004) that is, hopefully, more realistic. (The late Prof. George N. Saridis [RPI] had published a book on self-organizing systems [17].) Other precedents are reported, both recently and historically, as [33] and [34], respectively. Also see [36]- [49].

A Recent Approach that Almost Scores a Goal

Reference [22] appears to be much further along in usefulness and creativity in this area but evidently still misses the mark regarding practical realism, as explained next. Interactive Multiple Model (IMM) Filtering approach is implemented in [22] (and elsewhere) only for targets described by linear systems, even under maneuvers. Reentry Vehicles are known to be nonlinear in both the dynamics and in the algebraic measurement model. Ref. [22] also considers Particle Filtering (PF) and Probabilistic Data Association (PDA) approaches as well. As with almost all PF examples to date, the state space target dynamics model used within the calculations is planar and consists of only 4 states. The complexity of PF implementation as a CPU computational burden increases drastically as the state dimension increases. Practical Early Warning Radar target models must consist of at least six states (3 position, 3 velocity), and must be nonlinear to realistically reflect the action of an inverse square gravity and the oblateness of the earth (effects that cannot be ignored in a realistic EWR application). Ref. [22] does not reflect Bit Error Rates (BER) nor possible requests for retransmission within the communications channels as a consequence of BER (an activity that can cause measurements to the filter to get out of sequence but can be routinely handled appropriately as long as each measurement is properly time-tagged or time-stamped). Ref. [22] does not consider having adequate processor capability at each sensor site, as would be the case for EWR with two way communication links present to Colorado Springs’ Cheyenne Mountain, and which is more compatible with methodology of decentralized filtering, reported in References [9], [14], [18]-[20].

Dirty Tricks that Result in Greater Estimation Accuracy in Simulations but fail in Practice

Several open literature simulation evaluations of alternative estimation approaches for strategic target tracking have in fact been previously reported (e.g., [30]) but all with accuracy results obtained under a somewhat artificial scenario of use (viz., invoking only 4 tracking filter states and assuming only planar motion despite target object not being under the influence of central forces exclusively but projectile treated as if it were and not only that but the observing radar is only at the launch point and in the same plane as the target trajectory) thus leaving prior accuracy quantifications somewhat questionable at best as they relate to actual missile defense since this prior overly benign scenario lacks realism. We appreciate novelty and encourage creativity but we also desire that all major test conditions for algorithm evaluation be realistic and representative of the actual application scenario, as enforced and explained further within our study.

Further elucidating our perception regarding lack of realism by several recent evaluations (e.g., [30]) using reentry model being completely planar, central forces in 3-D give rise to trajectories that are confined entirely to be within a specific plane (known, historically, as the osculating plane). From a celestial mechanics course, one learns about central force motions and associated properties. Such studies reveal that for a central force field (like inverse squared gravity) the following cross-product , where r is the position vector from the origin of a coordinate system erected at the center of gravity of the earth as focus to the location of the projectile, defines the normal to the plane in which the motion of the projectile is confined. In actual radar applications, the ideal behavior is not precisely obtainable because of the range-Doppler ambiguity encountered (as associated with use of practical radar measurements), so there are slight errors present between measured range and its associated range-rate to some degree which, further, corrupts the accuracy of the effective estimates  and that degrades the estimate of the normal to the osculating plane to which the projectile is confined, to the degree of departure from the ideal as indicated from the calculation of , where the individual contributing errors are and . The effect of regression of nodes and rotation of apsides (defined above) aggravates the problem of instantaneously estimating the correct plane of projectile confinement even more since, instead of being merely constant and fixed, the osculating plane now moves due to the oblateness of the earth. While several recent open literature simulations of RV target tracking treat it as a problem that is entirely planar, the real world problem in Missile Defense is to actually figure out what plane the trajectory is situated in and how it is posed. By ignoring these real world effects, such simulations, as a consequence reap greater accuracy than is likely in practice, where four other out-of-plane errors arise (that are ignored). Results were better in these simulations than can be obtained in practice since they assumed the plane of motion was already known precisely (and treat the component of out-of-plane errors as being nonexistent and zero in the tally of total error incurred). Part of the real problem should be to deduce in what plane the target is traveling. In some cases, such as in benign pre-planned test shots from Vandenberg AFB in CA toward the Kwajalein Atoll [where the Tradex radar used for tracking these RV test shots is located there at KREMS (along with Altair, Alcor, and MMW). The L-band Tradex radar has MTT for up to 63 simultaneous targets appearing within the same mechanically scanned 0.61o, 6 dB beamwidth pencil-beam of the 25.6 meter diameter antenna, but has a 600 meter blind zone behind the primary target cluster grouping] in the Marshall Islands using our own cooperative target Reentry Vehicles, one may know the launch point precisely as well as the aim point and our own missiles may have the reentry angle completely known to us for tracking purposes (but for actual missile defense, the defender’s radar usually only has a view of the target during a portion of midcourse through reentry and we know that anticipated launch points can be varied via use of wheeled or rail vehicles or via submerged submarines and that a sophisticated enemy can also suddenly vary several parameters relating to reentry drag during end-game). Some recent simulations assume that the observing stationary ground-based radar is at the launch point (and in the plane of the target trajectory, as a considerable advantage of having almost perfect initial conditions for the target tracking algorithms almost from the start, which is an unrealistic assumption for Missile Defense [where, typically, initial conditions for each target must be deduced from detection and confirmation waveforms, available pulse patterns, or from other supplementary information and EWR uses phased arrays and electronic scanning]). More realistic approaches usually have more of a handicap in this aspect and must approximate the initial conditions used in the Extended Kalman filter for each candidate target.

Moreover, when atmospheric drag enters into the picture, the problem is no longer an exclusively central force problem (otherwise guaranteed to be planar). Any tilt of the reentry body forces the trajectory out of the expected osculating plane, which recent simulation studies treat as being perfectly known but did not list this among the assumptions, as were missed in [30].

When awarded a contract, TeK Associates can Illustrate the Efficacy of Decentralized EKF’s for UEWR using MatLAb/TK-MIP

We will implement the existing RVCC EKF [31] (with parameters as in [32]) and view target RV’s simultaneously from two UEWR sites and combine the EKF estimates from these two sites to demonstrate the benefit of this approach for resolving two crossing targets early on in their trajectory.

 

Viewing RV Target Simultaneously from Two Radar and Combining the Tracking Results

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General Principles Associated with Sensor Fusion and Kalman Filters for Networked Sensor Arrays and Virtual Sensors: straight talk and common sense

 Despite the evolution and abstraction terminology, nomenclature, or current parlance in vogue in sensor fusion and net-centric considerations, as identified in [53], the situations that must be properly handled and the underlying principles involved are fairly simple and straight forward, as discussed below.

Since the 1970’s, several alternative navaids from which to obtain “fixes” (of sufficient accuracy and frequency of occurrence) have been used in conjunction with a single Kalman filter to update owncraft INS position and velocity assessment by using these fixes as corrections or “resets” to prevent the deleterious effect of accumulating gyro drift that would otherwise continue to increase and further degrade the accuracy of the onboard INS with the passing of time.

Terrain map-matching (TERCOM) historically involved using 3 or 4 “Janus configured” simultaneous active radar views aboard a single platform in order to correlate actual current terrain measurements with an onboard master map (quantized to a certain practical resolution) to obtain a navaid position fix in order to update and correct the primary onboard Inertial Navigation System present within many terrestrial airborne applications (e.g., in cruise missiles, Unmanned Air Vehicles, aircraft). 

In the 1980’s, and 1990’s for navigation applications, the use of decentralized Kalman filters were proposed (and used) to handle situations where some navaid sensors were failed or unavailable because of environmental concerns (sunspot activity, enemy jamming), onboard electronic failures, or during owncraft maneuvers that incur wing shading of antennas, or from battle damage. Such approaches were pursued to obtain margins of safety via redundancy. There was only a single state variable model of primary interest throughout: position and velocity of owncraft, presented as “aided” INS readouts. All that needs to be conveyed to a central collator (Master Filter onboard the parent platform) from each participating sensor sub-filter is the following 2n+1 entity: (state estimate, computed inverse covariance times the state estimate, time of fix), where n is the state size of the filter model in use (usually 6 states). Since the databuses stretch for relatively short distances aboard aircraft from where the measurement data was received, the delay incurred is small and for a single master filter, the out of sequence participant reports can still be handled to give the best current state estimate without any reservations (other than those caused by certain approximations perhaps being invoked merely to lessen the processing burden). The point is that no uncalibrated simplifying approximations need be invoked and the result is that the current best estimate of owncraft position and velocity can be calculated in a straight forward way even if navaid sensor measurements are all taken at different times (as long as these times are known and conveyed). In ship-borne applications, the adverse effect of sensor measurement data senescence is worse because data buses are usually much longer and incur greater delay.

Sensor fusion for Strategic Defense against reentry vehicles in still interested in position and velocity of the objects but now there may be several identified targets (some being real treats and others being decoys, chaff, jammers, or other obstacles designed to obscure the situation). The multi- sensors (in this case strategic radars at different known locations) can pass their target state estimates: to a central (and back-up) location where all the data can be collated and multi-target deciphering algorithms can now be applied only once to the end result of collated sensor views. However, this ideal situation can be aggravated by the communications time delay incurred from packet switching Bit Error Rate (BER) between sensor and collating stations (where a Master Kalman filter resides).

Sensor fusion for sonobuoy arrays have the same characteristics as described above since now GPS derived sonobuoy position and its associated Doppler velocity component contribution due to the water currents can be obtained. Similarly for dispersed acoustic sensors designed to be self organizing since GPS derived position location and time stamps can be used. Max Nikias and Jerry Mendel (USC) have good ideas for how to handle sensor arrays of different configurations and phase centers, where sensors can fail and on how to effect a healing of sorts despite such failures of individual participating arrays.

The situation for fusing alternate imaging sensor results is much more complicated. For image registration, we need to worry about having a common scale and common orientation for all participating images before information can be fused. Ideally, the views should be simultaneous with the same color scale and aspect angle of the sensors. (Some surveillance aircraft have quickly rotating mirrors so that all imaging sensors see the same view through the same aperture almost simultaneously.) Only stereo views would seek having different aspect angles from the viewing sensors of comparable resolution. Virtual sensors like Synthetic Aperture Radar (SAR) have a larger effective aperture due to the known motion of the sensor platform. These days, the moving platform upon which the sensor is attached can be an aircraft or a satellite. However, to fuse data from two different comparable sensors, if ferreting out moving targets is the goal, all views should be at the same time otherwise the moving targets of interest will be dimmed rather than reinforced from differing images of the same scene at different times. If only fixed landmarks are of interest, the two views of the same scene can be at different times and the data of interest will still reinforce to improve the assessment from combined data. Drastically different imaging sensors need to be reduced to a common resolution and color scale (especially if synthetic color is invoked within any sensor processing). Two Dimensional Kalman filtering may be used in sensor fusion of two (or more) comparable sensor views of the same scene. Issues for InfraRed (IR) are: (1) is the image light on a dark background or dark on a light background; (2) are they using a common gray scale?. Consistency is needed throughout in order to fuse information conveyed in the underlying images including use of a common imaging picture format, usually .bmp or its equivalent. IR imaging also needs to be almost simultaneous or else drastic changes can occur due to effects of the interaction of cloud cover and bright sunlight as it heats or cools a scene of interest. Obviously, this is not a problem for radar imaging. However, radar imaging is not usually real-time so it has its own issues being that by the time a radar image is ready for use, the time stamp from other imaging approaches may be very different for the same scene. The known INS/GPS velocity of the platform can be used to help reduce effects of motion blur associated with the moving platform upon which the sensors are mounted and other gyros can be used as well to stabilize the onboard cameras during use.

For networked radars on U.S. Navy surface ships in the 1980’s, GRIDLOCK was the name of an approach that constituted a preferred method of round-robin radar calibration of several participating ships by comparing the known location of friendly target blips (as reported via radio as, say, using Joint Tactical Information Distribution System [JTIDS]/Relative Navigation [RelNav] or using the Naval Tactical Data System [NTDS]) while keeping track of the radar location of unknown or unidentified more potentially threatening target blips. Using this approach, and through repetitive use of it, eventually the radar biases of the various networked participants could be reduced or removed as being negligible. 

We have an abiding interest in Bayesian Nets and appreciate their potential both in defense applications (click for PowerPoint presentation or click here for a pdf version) and in commercial applications.

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REFERENCES:

[1] T. H. Kerr, Stability Conditions for the RelNav Community as a Decentralized Estimator-Final Report, Intermetrics, Inc. Report No. IR-480, Cambridge, MA, 10 August 1980 for NADC (Warminster, PA).

[2] T. H. Kerr, and L. Chin, “A Stable Decentralized Filtering Implementation for JTIDS RelNav Proceedings of IEEE Position, Location, and Navigation Symposium (PLANS), Atlantic City, NJ, 8-11 December 1980.

[3] T. H. Kerr, and L. Chin, “The Theory and Techniques of Discrete-Time Decentralized Filters,” in Advances in the Techniques and Technology in the Application of Nonlinear Filters and Kalman Filters, edited by C. T. Leondes, NATO Advisory Group for Aerospace Research and Development, AGARDograph No. 256, Noordhoff International Publishing, Lieden, 1981.

[4] N. A. Carlson, T. H. Kerr, J. E. Sacks, Integrated Navigation Concept Study, Intermetrics Report No. IR-MA-321, 15 Jun. 1984, for ITT (Nutley, NJ).

[5] T. H. Kerr, “Decentralized Filtering and Redundancy Management Failure Detection for Multi-Sensor Integrated Navigation Systems,” Proceedings of the National Technical Meeting of the Institute of Navigation (ION), San Diego, CA, 15-17 Jan. 1985.

[6] T. H. Kerr, “Decentralized Filtering and Redundancy Management for Multisensor Navigation,” IEEE Trans. on Aerospace and Electronic Systems, Vol. 23, No. 1, Jan. 1987 (corrections: May and Jul. 1987).

[7] T. H. Kerr, “Comments on ‘Estimation Using a Multirate Filter’,” IEEE Transactions on Automatic Control, Vol. 34, No. 3, Mar. 1989.

[8] T. H. Kerr, “Comments on ‘Federated Square Root Filter for Decentralized Parallel Processes’,” IEEE Transactions on Aerospace and Electronic Systems, Vol.27, No. 6, Nov. 1991.

[9] T. H. Kerr, “Extending Decentralized Kalman Filtering (KF) to 2D for Real-Time Multisensor Image Fusion and\or Restoration: Optimality of Some Decentralized KF Architectures,” Proceedings of the International Conference on Signal Processing Applications & Technology (ICSPAT96), Boston, MA, 7-10 Oct. 1996.

[10] C. Y. Chong, S. Mori, “Convex Combination and Covariance Intersection Algorithms in Distributed Fusion,” Proc. of 4th International Conf. on Information Fusion, Montreal, CA, Aug. 2001.

[11] F. C. Schweppe, Uncertain Dynamic Systems, Prentice-Hall, Englewood Cliffs, NJ, 1973.

[12] T. H. Kerr, “An Analytic Example of a Schweppe Likelihood Ratio Detector,” IEEE Trans. on Aerospace & Electronic Systems, Vol. 25, No. 4, Jul. 1989.

[13] L. J. Levy, “Sub-optimality of Cascaded and Federated Filters,” Proc. of 53rd Annual ION Meeting: Navigation Technology in the 3rd Millennium, Cambridge, MA, Jun. 1996.

[14] A. G. O. Mutambra, Decentralized Estimation and Control Systems, CRC Press, NY, 1998.

[15] S. Alfano, M. L. Greer, “Determining if Two Solid Ellipsoids Intersect,” AIAA Journal of Guidance, Control, and Dynamics, Vol. 26, No. 1, Jan.-Feb. 2003.

[16] T. H. Kerr, “Comments on ‘Determining if Two Solid Ellipsoids Intersect’,” AIAA Journal of Guidance, Control, and Dynamics, Vol. 28, No. 1, Jan.-Feb. 2005.

[17] G. N. Saridis, Self-Organizing Control of Stochastic Systems, Control and Systems Theory Series, Vol. 4, Marcel Dekker, NY, 1977.

[18] J. L. Speyer, “Computation and Transmission Requirements for a Decentralized Linear-Quadratic-Gaussian Control Problem,” IEEE Trans. on Automatic Control, Vol. 24, No. 2, Apr. 1979.

[19] T.-S. Chang, “Comments on ‘Computation and Transmission Requirements for a Decentralized Linear-Quadratic-Gaussian Control Problem’,” IEEE Trans. on Automatic Control, Vol. 25, No. 3, Jun. 1980.

[20] A. Klinger, “Information and Bias in Sequential Estimation,” IEEE Trans. on Automatic Control, Vol. 13, No.1, Feb. 1968. 

[21] T. H. Kerr, “Streamlining Measurement Iteration for EKF Target Tracking IEEE Transactions on Aerospace and Electronic Systems, Vol. 27, No. 2, Mar. 1991 (correction: Nov. 1991).

[22] R. Evans, V. Krishnamurthy, G. Nair, L. Sciacca, “Networked Sensor Management and Data Rate Control for Tracking Maneuvering Targets,” IEEE Transactions on Signal Processing, Vol. 53, No. 6, Jun. 2005. [Interactive Multiple Model (IMM) Filtering approach is again implemented only for targets described by linear systems, even under maneuvers. Strategic Reentry Vehicles are targets that are well known to be adequately modeled only as being nonlinear in both the dynamics and in the algebraic measurement model. This article also considers Particle Filtering (PF) and Probabilistic Data Association (PDA) approaches as well. As with almost all PF examples to date, the state space target dynamics model used within the calculations is planar and consists of only 4 states. The complexity of PF implementation as a CPU computational burden increases drastically as the state dimension increases. Practical Early Warning Radar target models must be at least six states (3 position and 3 velocity states), and must be nonlinear to realistically reflect the action of an inverse square gravity and the oblateness of the earth (effects that cannot be ignored in a realistic EWR application). Article doesn’t reflect Bit Error Rates (BER) or possible requests for retransmission within the communications channels as a consequence of BER. Article does not consider having adequate processor capability at each sensor site, as would be the case for EWR with two way communication links present to Colorado Spring’s Cheyenne Mountain, and which is more compatible with methodology of decentralized filtering, reported in references [9], [14], [18]-[20].]

[23] T. H. Kerr, “Developing Cramer-Rao Lower Bounds to Gauge the Effectiveness of UEWR Target Tracking Filters,” Proceedings of AIAA/BMDO Technology Readiness Conference and Exhibit, Colorado Springs, CO, 3-7 Aug. 1998.

[24] T. H. Kerr, UEWR Design Notebook-Section 2.3: Track Analysis, TeK Associates, Lexington, MA, XonTech Report No. D744-10300, 29 Mar. 1999.

[25] T. H. Kerr, “Considerations in whether to use Marquardt Nonlinear Least Squares vs. Lambert Algorithm for NMD Cue Track Initiation (TI) Calculations,” TeK Associates Technical Report No. 2000-101, Lexington, MA, (for Raytheon, Sudbury, MA), 27 Sep. 2000.

[26] H. S. Satz, T. H. Kerr, “Comparison of Batch and Kalman Filtering for Radar  Tracking,” Proceedings of 10th Annual AIAA/BMDO Conference, Williamsburg, VA, 25 Jul. 2001 (Unclassified).

[27] T. H. Kerr, “Improving C-3 SSBN Navaid Utilization (U),” TASC Technical Information Memorandum TIM-1390-3-1, Reading, MA, Aug. 1979 (Secret) for the Navy: SP-2413.

[28] T. H. Kerr, “Modeling and Evaluating an Empirical INS Difference Monitoring Procedure Used to Sequence SSBN Navaid Fixes,” Proceedings of the Annual Meeting of the Institute of Navigation, U.S. Naval Academy, Annapolis, Md., 9-11 June 1981. (Selected for reprinting in Navigation: Journal of the Institute of Navigation, Vol. 28, No. 4, Winter 1981-82).

[29] T. H. Kerr, “Sensor Scheduling in Kalman Filters: Evaluating a Procedure for Varying Submarine Navaids,” Proceedings of 57th Annual Meeting of the Institute of Navigation, Albuquerque, NM, 9-13 Jun. 2001.

[30] A. Farina, B. Ristic, D. Benvenuti, “Tracking a Ballistic Target: Comparison of Several Nonlinear Filters,” IEEE Trans. on Aerospace and Electronic Systems, Vol. 38, No. 3, Jul. 2002.

[31] F. E. Daum, and R. J. Fitzgerald, “Decoupled Kalman Filters for Phased Array Radar Tracking,” IEEE Trans. on Automatic Control, Vol. 28, No. 3, Mar. 1983.

[32] P. E. Zadunaisky, “Small Perturbations on Artificial Satellites,” IEEE Trans. on Aerospace and Electronic Systems, Vol. 39, No. 4, Oct. 2003.

[33] T. Eren, W. Whiteley, A. S. Morse, P. N. Belhumer, B. D. O. Anderson, “Sensor and Network Topologies of Formations with Direction, Bearing, and Angle Information,” 42nd IEEE Conf. on Decision and Control, Vol. 3, pp. 3064-3069, 9-12 Dec. 2003.  

[34] R. J. Kenefic, P. L. Goulette, “Sensor Netting Via the Discrete Time Extended Kalman Filter,” IEEE Trans. on Aerospace and Electronic Systems, Vol. 17, No. 4, pp. 3064-3069, Jul. 1981.

[35] Budianu, C., Ben-David, S., Tong, L., “Estimation of the Number of Operating Sensors in Large-Scale Sensor Networks with Mobile Access,” IEEE Trans. on Signal Processing, Vol. 54, No. 5, pp. 1703-1715, May 2006. 

[36] Rabbat, M. G., Nowak, R. D., “Decentralized Source Localization and Tracking,” Proceedings of International Conference on Acustics, Speech, and Signal Processing, Vol. 3, Montreal, QC, Canada, pp. 921-924, May 2004.

[37] Ribeiro, A., Giannakis, G. B., “Bandwidth-Constrained Distributed Estimation Using Wireless Sensor Networks-Part I: Gaussian Case,” IEEE Trans. on Signal Processing, Vol. 54, No. 3, pp. 1131-1143, Mar. 2005.  

[38] He, T., Ben-David, S., Tong, L., “Nonparametric Change Detection and Estimation in Large-Scale Sensor Networks,” IEEE Trans. on Signal Processing, Vol. 54, No. 4, pp. 1204-1217, April 2006.

[39] Marano, S., Matta, V., Willett, P., Tong, L., “Support-Based and ML Approaches to DOA Estimation in a Dumb Sensor Network,” IEEE Trans. on Signal Processing, Vol. 54, No. 4, pp. 1563-1567, April 2006. 

[40] Zhang, X., Willet, P., Bar-Shalom, Y., “Uniform Versus Nonuniform Sampling when Tracking in Clutter,” IEEE Trans. on Aerospace and Electronic Systems, Vol. 42, No. 2, pp. 388-400, Apr. 2006.

[41] Yaesh, I., Shaked, U., “Discrete-Time Min-Max Tracking,” IEEE Trans. on Aerospace and Electronic Systems, Vol. 42, No. 2, pp. 540-547, Apr. 2006.

[42] Dogandzic, A., Zhang, B., “Distributed Estimation and Detection for Sensor Networks Using Hidden Markov Random Field Models,” IEEE Trans. on Signal Processing, Vol. 54, No. 8, pp. 3200-3215, Aug. 2006.

[43] Delouille, V., Neelamani, R. N., Baraniuk, R. G., “Robust Distributed Estimation Using the Embedded Subgraphs Algorithm,” IEEE Trans. on Signal Processing, Vol. 54, No. 8, pp. 2998-3010, Aug. 2006.

[44] Ait-El-Fquih, B., Desbouvries, F., “Kalman Filtering in Triplet Markov Chains,” IEEE Trans. on Signal Processing, Vol. 54, No. 8, pp. 2957-2963, Aug. 2006.

[45] Huang, M., Dey, S., “Dynamic Quantizer Design for Hidden Markov State Estimation Via Multiple Sensors With Fusion Center Feedback,” IEEE Trans. on Signal Processing, Vol. 54, No. 8, pp. 2887-2896, Aug. 2006. 

[46] Ma, W.-K., Vo, B.-N., Singh, S. S., Baddeley, A., “Tracking an Unknown Time-Barying Number of Speakers Using TDOA Measurements: A Random Finite Set Approach,” IEEE Trans. on Signal Processing, Vol. 54, No. 9, pp. 3291-3304, Sep. 2006. 

[47] Xiao, J.-J., Luo, Z.-Q., Giannakis, G. B., “Performance Bounds for the Rate-Constrained Universal Decentralized Estimators,” IEEE  Signal Processing Letters, Vol. 14, No. 1, pp. 47-50, Jan. 2007. 

[48] Goodman, N. A., Stiles, J.-C., “On Clutter Rank Observed by Arbitrary Arrays,” IEEE Trans. on Signal Processing, Vol. 55, No. 1, pp. 178-186, Jan. 2007.

[49] Aldosari, S. A., Moura, J. M. F., “Detection in Sensor Networks: The Saddlepoint Approximation,” IEEE Trans. on Signal Processing, Vol. 55, No. 1, pp. 327-340, Jan. 2007.

[50] Galatsanos, N. P., Chin, R. T., “Restoration of Color Images by Multichannel Kalman filtering,” IEEE Trans. on Signal Processing, Vol. 39, No. 10, pp. 2237-2252, Oct. 1991.

[51] Galatsanos, N. P., Katsaggelos, A. K., Chin, R. T., Hillery, A. D., “Least Squares Restoration of Multichannel Images,” IEEE Trans. on Signal Processing, Vol. 39, No. 10, pp. 2222-2236, Oct. 1991.

[52] Chang, M. M., Tekalp, A. M., Erdem, A. T., “Blur Identification Using the Bispectrum,” IEEE Trans. on Signal Processing, Vol. 39, No. 10, pp. 2323-2325, Oct. 1991.

[53] Blasch, E., Salerna, J., Kadar, I., Bierman, J., Chong, C., Das, S., “Resource Management Coordination with Level 2/3 Fusion Issues and challenges,” IEEE AES Systems Magazine, Vol. 23, No. 3, pp. 32-46, Mar. 2008.

[54] Wettergren, T. A.,  “Performance of Search via Track-Before-Detect for Distributed Sensor Networks,” IEEE Trans. on Aerospace and Electronic Systems, Vol. 44, No. 1, pp. 314-325, Jan. 2008.

[55] Liang, Q., Cheng, X., “KUPS: Knowledge-Based Ubiquitous and Persistent Sensor Networks for Threat Assessment IEEE Trans. on Aerospace and Electronic Systems, Vol. 44, No. 3, pp. 1060-1069, July 2008.

[56]  Iyengar, S. S., Presard, L., Min, H. (eds.), Advances in Distributed Sensor Integration: Application and Theory, Prentice-Hall PTR, Upper Saddle River, NJ, 1995.

[57] Pao, L. Y., “Measurement Reconstruction Approach for Distributed Multisensor Fusion,” AIAA Journal of Guidance, Control, and Dynamics, Vol. 19, No. 4, pp. 842-847, Jul.-Aug. 1996.

[58] Grigoryan, A. M., Grigoryan, M. M., Brief Notes in Advanced DSP: Fourier Analysis in MatLab, CRC Press, Taylor & Francis, Boca Roton, FL, 2009 (especially see Chapter 6, pp. 285-330, “Fourier Transforms and Multiresolution”).

[59] Henry Stark (ed.), Application of Optical Fourier Transforms, Academic Press, NY, 1982.

[60] Roberto Brunelli, Template Matching Techniques in Computer Vision: Theory and Practice, John Wiley & Sons, Ltd., West Sussex, UK, 2009.

[61] William K. Pratt, Digital Image Processing, Wiley-Interscience Publications, John Wiley & Sons, Inc., NY, 1991.

[62] Anil K. Jain, Fundamentals of Digital Imaging Processing, Thomas Kailath (series editor), Prentice-Hall, Englewood Cliffs, NJ, 1989.

[63] Arregui, F. J. (ed.), Sensors Based on Nanostructural Materials, Springer Scientific & Business Media, LLC, NY, 2009.

[64] Righini, G. C., Tajani, A., Cutolo, A., An Introduction to Optoelectronic Sensors, World Scientific, NY, 2009.

[65] Junichiro Toriwaki, Hiroyuki Yoshida, Fundamentals of Three-Dimensional Digital Processing, Springer-Verlag, London, 2009.

[66] Julien Bourgeois, Wolfgang Minker, Time-Domain Beamforming and Blind Source Separation: Speech Input in the Car Environment, Springer Science + Business Media, NY, 2009.

[67] Japkowicz, N., Shah, M., Evaluating Learning Algorithms: a classification perspective, Cambridge University Press, UK, 2011.

[68] Soysal, G., Efe, M., “Data fusion in a multistatic radar network using covariance intersection and particle filtering,” Proceedings of the 14th International Conference on Information Fusion (FUSION), Chicago, Il, 5-8 July 2011.

[69] Hall, D. L., McMullen, S. A. H., Mathematical Techniques in Multisensor Data Fusion, Artech House, Inc., Norwood, MA, 2004.
https://books.google.com/books?id=HxjDMcJWLYwC&pg=PA167&lpg=PA167&dq=Particle+Filter+and+Covariance+Intersection&source=bl&ots=kcfwV1r9pp&sig=jbdRc5mR16-0EJ2UbOgvpsA6ZfY&hl=en&sa=X&ved=0CDIQ6AEwAmoVChMI0MTT4dn0xwIVwz4-Ch2VgQOS#v=onepage&q=Particle%20Filter%20and%20Covariance%20Intersection&f=false 

[70] ftp://labattmot.ele.ita.br/ele/ivo/Leitura/SLAM_P2_D_Whyte.pdf 

[71] Bahador Khaleghi, Alaa Khamis, Fakhreddine O. Karray, Saiedeh N. Razavi, “Multisensor data fusion: A review of the state-of-the-art,” Information Fusion, Vol. 14, No. 1, pp. 28–44, Jan.2013.

[72] Bahador Khaleghi, Alaa Khamis, Fakhreddine O. Karray, Saiedeh N. Razavi, “Corrigendum to `Multisensor data fusion: A review of the state-of-the-art’ Information Fusion, Vol. 14, No. 4, page 562, Oct. 2013.

[73] Son, J. S., Thomas, G., Flores, B. C., Range-Doppler Radar Imaging and Motion Compensation, Artech House, Boston, 2001.

[74] Soysal, G., Efe, M., “Data fusion in a multistatic radar network using covariance intersection and particle filtering Proceedings of the 14th International Conference on Information Fusion (FUSION), Chicago, IL, 5-8 Jul. 2011.

[75] Hall, D. L., McMullen, S. A. H., Mathematical Techniques in Multisensor Data Fusion, Artech House, Inc., Norwood, MA, 2004.
https://books.google.com/books?id=HxjDMcJWLYwC&pg=PA167&lpg=PA167&dq=Particle+Filter+and+Covariance+Intersection&source=bl&ots=kcfwV1r9pp&sig=jbdRc5mR16-0EJ2UbOgvpsA6ZfY&hl=en&sa=X&ved=0CDIQ6AEwAmoVChMI0MTT4dn0xwIVwz4-Ch2VgQOS#v=onepage&q=Particle%20Filter%20and%20Covariance%20Intersection&f=false

[76] ftp://labattmot.ele.ita.br/ele/ivo/Leitura/SLAM_P2_D_Whyte.pdf

[77] Dougherty, E. R., Laplante, P. A., Introduction to REAL-TIME IMAGING, Tutorial Texts in Optical Engineering, Vol. TT19, Donald C. O'Shea (Series Editor at GIT), Copublished by SPIE Optical Engineering Press, Bellingham, Washington, USA, IEEE Press, New York, 1995.

[78] M. L. Hernandez, A. D. Marrs, N. J. Gordon, S. R. Maskell, and C. M. Reed, “Cram้r-Rao bounds for non-linear filtering with measurement origin uncertainty,” in Proceedings of the 5th International Conference on Information Fusion, Vol. 1, pp. 18–25, July 2002.
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