Sensor Data Fusion Using a Probability Density Grid

Transcription

1 Sensor Data Fusion Using a Probability Density Grid Derek Elsaesser Communication and avigation Electronic Warfare Section DRDC Ottawa Defence R&D Canada Derek.Elsaesser@drdc-rddc.gc.ca Abstract - A novel technique has been developed at DRDC Ottawa for fusing Electronic Warfare (EW) sensor data by numerically combining the probability density function representing the measured value and error estimate provided by each sensor. Multiple measurements are sampled at common discrete intervals to form a probability density grid and combined to produce the fused estimate of the measured parameter. This technique, called the Discrete Probability Density (DPD) method, is used to combine sensor measurements taken from different locations for the EW function of emitter geolocation. Results are presented using simulated line of bearing measurements and are shown to approach the theoretical location accuracy limit predicted by the Cramer-Rao Lower Bound. The DPD method is proposed for fusing other geolocation sensor data including time of arrival, time difference of arrival, and a priori information. Keywords: Sensor data fusion, discrete probability density, emitter location estimation. Introduction The Discrete Probability Density (DPD) method is a novel technique developed at Defence R&D Canada for numerically combining data containing uncertainty, such as Electronic Warfare (EW) sensor measurements [,, 3, 4, 5]. The basic assumption is that the sensor measurement of a given parameter can be modeled by some probability density function (PDF) over the range of possible values. The PDF may be modeled by a function, such as a Gaussian distribution, or by a set of empirical values that reflect the probability of the true value occurring within some range of the measured value. The basic theory of the DPD method is presented in Section. It describes the process for sampling independent PDFs at common discrete intervals to produce a set of discrete probability density vectors which are combined to form a joint DPD vector for estimating the mean and variance of the fused estimate of the measured parameter. Section 3 describes how this method is used to combine data from multiple sensor locations by projecting measurements of a common parameter onto a two-dimensional probability density grid. This is used to determine the most likely coordinates of the object being measured and an error estimate. In section 4 the DPD method is applied to emitter geolocation using line of bearing (LOB) measurements from multiple sensor locations [3]. The performance of the DPD method for emitter location accuracy is compared to the theoretical Cramer-Rao Lower Bound () [6] as a function of the number of measurements included. The use of the DPD to fuse other sensor data, such as time of arrival (TOA), time difference of arrival (TDOA), and a priori information, is presented in section 5. Discrete Probability Density Theory Data with uncertainty, such as sensor measurements, can be modeled by some PDF over the range of possible values. These PDFs may be represented by various distributions. Unlike techniques that use systems of equations to combine multiple sensor measurements and minimize the resulting error estimate, the DPD method combines probability density distributions of measurements directly by sampling each PDF at common intervals and calculating the joint product over the sample space. The resulting discrete probability density distribution is then used to calculate the estimated value and variance of the measured parameter. This concept is introduced using a one-dimensional DPD vector.. Discrete Sampling of Probability Density Functions All inputs are assumed to be estimates of some parameter,, epressed as a probability density function, f (), representing the uncertainty or error in the estimated value of [7]. The discrete probability density vector is formed by evaluating f () at discrete points over the range of possible values of. Let (n) represent the discrete values of over the range [a, b] sampled at intervals of with a delta function δ(n): b ( n) = δ ( n) n = () = a The DPD vector is formed by evaluating the PDF at each value of (n), with the integer indices of the DPD vector, n, then representing discrete values of : F ( n) = f ( ( n)) n = ()

2 This requires the total area under f () equals unity and the value of f () > over the range of possible values of. If the value of the PDF is zero over some range of, it implies that it is impossible for the measured value of to occur in this range. As the DPD vector is produced by discrete sampling of a PDF, the DPD method is not reliant on a specific probability distribution and can be used with a variety of probability density distributions, including non-linear distributions. As an eample of a DPD vector, consider the Gaussian PDF which is commonly used to represent sensor measurement errors that are normally distributed. It can be described by a mean, representing the estimated value, and variance, representing the uncertainty [4]. It can also be used in a system of equations representing the combination of multiple measurements for applications such as geolocation. A sensor measurement produces an estimate of = with a standard deviation of σ = 9.. The Gaussian PDF is sampled over the range [-, 5] at intervals of = 3.. The continuous Gaussian PDF and the resulting DPD vector are shown in Figure. They are normalized over the range of and the inde, n, of F (n) is translated back into for comparison with f (). f(), F() Continuous and Discrete Gaussian pdf; mean =., std = 9., d = 3..5 continuous pdf.45 discrete pdf Figure. Discrete sampling of a Gaussian PDF. Selecting a Sampling Interval Selection of an appropriate sampling interval,, is important as it directly impacts the computational cost of combining multiple measurements. In order to minimize the number of points,, sampled over the range of, the sampling interval must be selected so that the nature of the distribution of f () is maintained by F (n). As the aim is to determine the estimated value of,, and an error estimate in the form of the variance, σ, about, the sampling interval can be relatively large with respect to the variation in f (). For combining a set of Gaussian PDFs, a sampling interval of approimately σ i / (using the smallest value of σ i from the PDFs) has been found to be sufficient. Sampling at smaller intervals is generally found to have little effect on the resulting value of and σ, but increases computational cost proportionately with number of samples. Another factor to consider is the desired resolution of the resulting estimate of. Because the inde, n, of F (n) is used to determine, the sampling interval should be smaller than the desired resolution in..3 Joint DPD Vectors A joint DPD vector is formed by taking the product of each input PDF at common sample points, (n). For S sensor measurements this is epressed as: F ( s, n) = f ( s, ( n)) n = (3) where s = S independent measurements of the same target; f (s, ) is the array of PDFs representing these measurements over the same range [a, b] of ; and F (s, n) is the resulting array of DPD vectors of length. The joint discrete probability density vector, P (n), is determined by taking the product of all DPD vectors at each integer value of n: S ' P ( n) = F ( s, n) n = (4) s= then calculating the normalization constant C: C = ' P ( n) (5) resulting in the joint DPD vector: ' P ( n) = P ( n) C n = (6) It is assumed that the chosen sampling interval is small enough to realize the significant variations in P (n). The estimated value of the fused result is determined from the indices n weighted by P (n): n = np n ( ) and the estimated variance of n by: (7) σ = ( n n) P ( n) (8) n This is translated back into the domain of within the range [a, b] giving the estimated value of as: = n + a The standard deviation representing the combined error estimate is given by: σ = () σ n (9)

3 As the estimated value and error of the combined measurements, andσ, are produced by evaluating each PDF at discrete points, the DPD method can be used to combine measurements with different types of probability density distributions..4 Combining Sensor Measurements Benefits of the DPD method can be demonstrated by the following eample, where the target value, T =3, and each sensor measurement is modeled as a Gaussian PDF, PDF i (µ,σ), i = 3, with the mean being the measured value and the error estimate being the standard deviation. ote that each f () must always be greater than. It is assumed that PDF and PDF 3 represent accurate, unbiased, measurements of T, but PDF is a wild or biased estimate. These three PDFs are shown in Figure, sampled at interval =.5 in Figure 3. The values of and σ are calculated from the joint DPD vector shown in Figure 4. It is seen that the inclusion of the biased measurement has only a modest effect on the resulting estimated value, = 9.7, and error estimate, σ = 3.7. Gaussian(mean, std) for PDF(3., 3.), PDF(.5, 4.), PDF3(3.5, 5.).4 PDF PDF. PDF3 f() F() Figure. Three sensor measurements of T = Discrete Sampling of Multiple PDFs, d = PDF PDF PDF3 Figure 3. Discrete sampling of measurement PDFs P() Discrete Joint DPD, E[] = 9.7, STD[] = 3.7, d = Figure 4. Joint DPD vector estimate of T umerous Monte Carlo simulations were conducted using combinations of various standard deviations, bias errors, and large numbers of measurements []. It was found that the joint DPD estimate generally converged on the true value as the number of measurements increased. It was also seen that the estimate was resilient to inclusion of measurements with large bias errors, with the main effect being an increase in the magnitude of the error estimate. 3 Two Dimensional DPD Grids A common data fusion function is to combine sensor measurements from multiple reference points to determine an estimate of the target in two or more dimensions. An eample is emitter geolocation using LOB or TDOA measurements from multiple sensor positions [3]. The DPD method is applied by projecting the measurement PDF from each sensor onto a common grid of sample points. This requires that some transform function eists that will map the measured parameter into -dimenional space. This section describes using bearing measurements to produce a -dimensional location estimate. 3. Projecting a LOB into Two Dimensions The Area-Of-Interest (AOI) that includes the target object is defined over a region in and Y using a -dimenisonal grid of points at sample intervals and y. It is implied that the target object is located within these bounds. The AOI must be large enough to ensure that truncation of the probability distribution at the boundaries does not significantly affect the result and that the sample interval is small enough to realize each PDF. Let there be S independent LOB measurements from sensor positions ( i, y i ), i = S, each with a measured bearing µ i and error estimate σ i. As bearing is an angular measurement, a LOB measurement can be represented by a von Mises PDF [7]: f ( θ ) = ep( κ cos( θ µ )) / πi ( κ) θ < π ()

4 where θ is the variable in radian, µ is the mean, κ is the concentration (which is analogous to /σ ), and I (κ ) is a Bessel function of the first kind and order zero. Eamples of the von Mises PDF for LOBs with different values of κ are shown in Figure 5. ote that other probability density distributions can be used to model a LOB..6.4 Von Mises Probability Density Function LOB RMS error = 3 degrees, k = LOB RMS error = 6 degrees, k = 9.89 LOB RMS error = degrees, k =.7973 An eample of F Y (n, for a sensor at position (, ) with bearing estimate µ i = 45 and error estimate σ i = 3 is shown in Figure 7 as a color surface plot. It is seen that the value of F(, y) is constant along any bearing line from the sensor, regardless of distance, since the value of the LOB PDF is constant for a given angle... p(angle) angle (degrees) Figure 5. The von Mises PDF used to model a LOB A transform is required to project the -dimensional LOB measurement, represented as an angular PDF, onto the - dimensional grid in, Y. The value of the LOB PDF at each node in the grid is calculated using its angle relative to the sensor position. The angular transform function between the sensor location and a grid point is simply [5]: θ i (, y) = arctan(( y y i ) /( i )) π θi < π () The value of a LOB PDF is calculated using the von Mises PDF with κ = /σ for θ i (, y) µ i at each discrete point (n) and Y(, representing each (, y) value in the grid, as shown in Figure 6. i,y i,y Figure 6. Transforming a LOB PDF to a -D DPD grid This produces a -dimensional LOB DPD array: F Y = f ( θ ( ( n), Y ( )) n =..., m =... M (3) i where F Y (n, is the LOB DPD array of size M points. The value of F Y (n, at a given inde is the LOB PDF, f(θ i ), taken at discrete values of, y. θ i µ i Figure 7. Eample of a -D LOB DPD for µ=45, σ=3 3. Joint DPD Location Estimate For multiple LOB measurements the joint DPD array is calculated over a common M grid by: S P ' Y = FY ( s, n, n =..., m =... M s= (4) where F Y (s, n, is the set of S independent LOB DPD arrays using a common grid. This is normalized by: M C = ' P (5) Y m= to produce the joint DPD array representing the target object s location estimate: ' PY = PY (6) C For a grid of by M points, the resulting computational compleity of the joint DPD array is of O(S M). The -dimensional location estimate of the target, T, yt, is determined by first taking the Probability Mass Functions (PMF) of P Y (n, : M Y m= PMF ( n) = P n =... (7) Y PMF ( = P m =... M (8) Y The target location estimate can be determined by treating PMF (n) and PMF Y ( as -dimensional DPD vectors and

5 calculating the inde estimates, n ˆ, mˆ T T, as in equation 7. For cases where a large number of measurements correlate, the distribution of P Y (n, becomes eponential about the estimated location. The estimated location can then be found from the indices that have the largest values of PMF (n) and PMF Y (, respectively. As the indices are translated back into and y coordinates to provide the location estimate of the target, this approach has the drawback that the resolution of the target location is limited to the sampling interval. The location error estimate is determined by the variances and covariance of the joint DPD array about the indices of estimated target location: COV Y σ = PMF ( n) ( n nˆ ) (9) M m= σ = PMF ( ( m mˆ ) () Y M = m= P Y Y ( n nˆ ) ( m mˆ ) T T T T () These terms are scaled by the sampling interval and form a covariance matri for calculating an Elliptical Error Probable (EEP), which is commonly used to represent the error estimate [3]. Although the EEP assumes that the error distribution is Gaussian, which may not be the case for a DPD distribution, it is useful for comparing the DPD results to other geolocation techniques and the. In cases where a sufficiently large number of measurements correlate near a point, the joint DPD distribution is seen to approimate a Gaussian distribution. 4 for LOB Geolocation LOB data is used for location estimation using the technique commonly referred to as triangulation. The DPD method is applied to LOB geolocation and compared to the, which represents the performance bound of an unbiased location estimator for the relative sensor positions and error estimates, ecluding measurement biases. This is used to assess the accuracy of the DPD method and its resilience to bias errors. 4. Increasing the umber of LOBs The comparison of the DPD method to the is conducted using Monte Carlo simulation and averaged over, iterations. All LOBs are normally distributed, with σ = 3, from 4 to sensor sites arranged as a linear baseline across the bottom of an AOI. The details of the testing are provided in []. The AOI is defined as a 3 point grid with a sampling interval of meters. The effect on the DPD location estimate is compared to the for an increasing number of unbiased LOBs with LOB having a bias error. This is repeated for bias errors on LOB of -4, -,,, and 4 counterclockwise. An eample for LOBs and a 4 bias on LOB is shown in Figure 8. The location RMSE for increasing number of LOBs is shown in Figure 9. Location RMSE ( Location RMSE ( Location RMSE ( Location RMSE ( Location RMSE ( Y m Figure 8. Eample scenario with LOB bias = 4 4 LOB STD = 3. deg, LOB Bias = -4. deg umber of LOBs included in fi LOB STD = 3. deg, LOB Bias = -. deg umber of LOBs included in fi LOB STD = 3. deg, LOB Bias =. deg umber of LOBs included in fi LOB STD = 3. deg, LOB Bias =. deg umber of LOBs included in fi LOB STD = 3. deg, LOB Bias = 4. deg 4 DPD Location Estimate and LOB DPD m umber of LOBs included in fi Figure 9. Effect if increasing the number of unbiased LOBs with LOB bias error = -4, -,,, 4.

6 It is seen that the inclusion of the biased LOB from site has limited effect on the DPD location estimate. Even for the worst-case, a LOB bias error of, the location RMSE for the DPD method approaches the as the number of unbiased LOBs is increased. Even though the location RMSE for the DPD method is still larger than the, it must be remembered that the represents the ideal estimate with no bias errors. The DPD location error estimate (the EEP) also decreased as the number of LOBs increased []. This suggests that the DPD method is an unbiased location estimator, even in the presence of unknown sensor bias errors. Other tests have shown that the DPD method is resilient to large numbers of bias errors because the joint DPD distribution is determined mainly by measurements that correlate near the most common location []. This makes the DPD method well suited to real-world applications where sensor measurements are often affected by environmental conditions or systemic errors. 5 Other Geolocation Techniques The DPD method may be applied to other geolocation techniques based on the measurement of parameters such as time, frequency, or power. Time is used in TOA techniques such as radar and range estimation for signals with known timing characteristics. TDOA between multiple sites is a common technique for geolocation of non-cooperative signals. The measured receive power of a signal can be used to estimate range if sufficient detail is known of the transmitted power and propagation path. Any of these techniques can be used in a hybrid method with LOBs. Because each technique relies on a measurement of a parameter with various degrees of error, it can be modeled with some PDF; hence, the DPD method could be used to produce a location estimate. This section shows how the DPD method is applied to timebased measurements and hybrid techniques using a priori information. 5. Time Of Arrival Location Estimation In TOA, the measured parameter is the time between when the signal was emitted and when it was received at the sensor (or half the transit time in the case of radar signals). If the speed of signal propagation, υ, is known, the TOA from any point to sensor i can be calculated by: toa d v v i (, y) = = ( y yi ) + ( i ) () If the sampling interval is constant in, y, a TOA measurement for sensor s can be transformed into a - dimensional TOA DPD array by evaluating the PDF at each discrete node: F = f ( toas ( ( n), Y( )) n =..., m =... M (4) s s The joint DPD array is calculated by taking the product of all S sensor measurement DPD arrays at each common node, n, m: S P( n, = F( s, n, n =..., m =... M s= (5) Consider the following eample for TOA geolocation. There are four sensors deployed in a km km AOI and the target emitter is located in the center of the AOI as shown in Figure. Y (meters) TOA Scenario (meters) Figure. Scenario for TOA location estimation Each sensor has the ability to estimate the TOA with a standard deviation of ns. The PDF is assumed to be Gaussian with the mean being the actual transit time of the signal. The desired location resolution is meters (sample interval = meters). A TOA DPD array from sensor, with no bias errors or multi-path effects, is shown in Figure. Similar DPD arrays are produced for the TOA measurements from the other sensors and a Joint DPD array is produced. The location estimate is calculated as in section 3 and shown in Figure. 4 3 The measurement of the TOA can be modeled by a PDF, such as a Gaussian distribution with the estimated TOA being the mean, µ, and estimated error as the standard deviation, σ: ep( ( toa µ ) ) /(σ )) f ( toa) = (3) πσ

7 in a non-gaussian distribution of the joint DPD peak and a smaller EEP ellipse. The truncation of the DPD peak is equivalent to using the bounds of the AOI as a priori information, as discussed net. Figure. TOA DPD grid for sensor at (5,) TOA DPD 5% EEP:.,.m 9. deg Y (meters) Target EEP (meters) Figure. TOA scenario with target location estimate 5. Time Difference Of Arrival The DPD method can also be used to provide location estimates from TDOA data, where the emission time of the signal is unknown but the relative time difference of arrival of the signal between pairs of sensors can be measured. Like TOA, the TDOA PDF can be represented by a Gaussian function, as shown in equation 3, with the TDOA value being the difference of the TOA value from each node in the grid to a given pair of sensors. Consider an eample using the same sensor deployment scenario as shown in Figure with the target emitter located at grid (6,6), and a TDOA measurement error, σ TDOA = 8.8 ns. The four sensors provide si TDOA measurement pairs that are shown in Figure 3 to visualize the joint probability density grid, which is seen to ehibit some truncation at the boundary. This results in the major ais of the resulting DPD EEP being slightly smaller than the, as shown in Figure 4. The estimated location of the target is unaffected by the truncation of the DPD distribution; however, the variance calculations are limited to the points in the grid, resulting 4 3 Y (meters) Figure 3. TDOA DPD grid for target at (6,6) TDOA DPD 5% EEP: 6.,6.m 45. deg DPD (red) (meters) Figure 4. Effect of truncation on DPD location estimate 5.3 Fusion of a Priori Information (black) It is implicit in the definition of the boundaries of the AOI that the probability the target resides outside the AOI is negligible. Hence, it can be argued that this truncation represents the product of the sensor measurements with a priori knowledge of the target s possible location. This concept could be etended to include other a priori information, such as terrain data, that could be represented as a DPD distribution. For eample, if the target emitter is known to be mounted on a vehicle, then a street map could be used as a priori information and modeled by a DPD distribution, as shown in Figure 5, with a relatively high probability the target in on a street and relatively low probability it is located elsewhere.

8 Figure 5. Using a street map DPD as a priori information When the TDOA DPD grid shown in Figure 3 is combined with the street map DPD grid, the result is the improved location estimate shown in Figure 6. Y (meters) TDOA-MAP DPD 5% EEP: 6.,6.m 45. deg (meters) Figure 6. Improved location estimate using map This illustrates that DPD grid can be used to represent a variety of types of data and information that have some degree of uncertainty. The nature of the probability density distribution can be almost any form as long as it is non-zero at all points and normalized over the AOI. Multiple probability density distributions can be combined directly, regardless of the source, as long as they are projected onto a common grid. 6 Conclusion TDOA-MAP FI The DPD method is useful for a variety of applications as it does not require sensor measurement errors to be normally distributed. Research suggests that the DPD method is an unbiased estimator that provides a maimum-likelihood solution for geolocation. This is achieved at the cost of computational compleity, which is of O(S M) for S sensor measurements over an M grid. The DPD method has been demonstrated for geolocation in two dimensions using line of bearing, time of arrival, time difference of arrival, and hybrid methods []. It can easily incorporate a priori information in the form of nonlinear data including geo-spatial data such as street maps. It is epected that the DPD method can be etended to provide 3-dimensional location estimates and used with other geolocation techniques and data. The DPD method is well suited for geolocation applications involving large numbers of measurements from different sensors that eperience significant errors. This includes urban environments with large multi-path effects, or sensors mounted on mobile platforms having significant positional and orientation errors, such as land combat vehicles or a small aircraft. Future research includes characterization of the DPD method and its application to other geolocation and data fusion problems, including the use of geographic information. References [] Derek Elsaesser, The Discrete Probability Density Method For Electronic Warfare Sensor Data Fusion, DRDC Ottawa TR 6-4, Defence R&D Canada Ottawa, ovember 6. [] Derek Elsaesser, The Discrete Probability Density Method For Emitter Geolocation, Canadian Conference on Electrical and Computer Engineering 6, Conference proceedings (ISB: ), Ottawa, Ontario, 7- May 6. [3] Richard A. Poisel, Electronic Warfare Target Location Methods. The Discrete Probability Density Method, Artech House, Boston, MA, 5, pp [4] Derek Elsaesser and Richard Brown, The Discrete Probability Density Method for Emitter Geo-Location, DRDC Ottawa TM 3-68, Defence R&D Canada Ottawa, June 3. [5] Richard Brown and Derek Elsaesser, Probability Grid and Contours for Estimating Radar Locations, DREO TM -95, Defence R&D Canada Ottawa, ovember. [6] Don Torrieri, Statistical Theory of Passive Location Systems, IEEE Transactions on Aerospace and Electronic Systems, VOL AES-, o., pp , March, 984. [7] Edward Emond, A ew Mathematical Approach to Direction Finding, Project Report o. PR55, Operational Research and Analysis Establishment, Directorate of Mathematics and Statistics, Department of ational Defence, Canada, 989.