on the use of IFS for track fusion

J. Li, P. Yip H. Leung E. Bossè

Communications Research Lab., Surface Radar Division, Command and Control Div.,McMaster University, Defence Research Estab., Defence Research Estabi.,Hamilton, Ontario, Ottawa, Ontario, Valcartier, Quebec,Canada, L8S 4K1 . Canada, K 1A 0K2. Canada, GOA 1RO.

ABSTRACT

In a multiple radar tracking environment, measurements form different sensors observing thesame target or track are required to be combined optimally in order to provide accurate information.One problem that has to be overcome before a weighted combination of the measurements can be madeis the possible difference in scanning periods used by the sensors. The different scanning periodsproduce different resolutions that must be reconciled before the data are fused.

Iterated function systems (IFS) have been used successfully for interpolation and datacompression. When a measured track is to be fused with another of different resolution, the underlyingproblem is one of accurate interpolation. Tracks, be they linear or curvilinear, have certain amount ofself-similarity as a geometrical object, just as natural coastlines are found to be fractal. Linear andpiece-wise linear IFS have been shown to provide excellent interpolation and compression even for non-fractal objects. In this work, we report two interpolation schemes based on linear IFS for tracksmeasured at different resolutions. Simulations using linear and curvilinear tracks are performed andthe results are compared to those using linear interpolations.

Keywords: Track Fusion, Iterated Function Systems, Fractal Interpolations.

1. INTRODUCTION

In a multiple radar tracking environment, the combination of information from differentsensors defines a data or track fusion problem. The purpose of data or track fusion is to integrateinformation from different measurement devices in order to improve the knowledge regarding thelocation of the target being tracked. This integration process raises several questions. One suchquestion relates to the problem of systematic errors involved in transforming the measurements fromphysically separated sensors to a common reference frame. These are referred to as registration errors,and must be removed before the data sets can be effectively fused. Another question refers to theproblem that the sensors may be scanning the target at different rates or scanning periods, thusproducing tracks of different resolutions. Fusion of tracks can only proceed when all measured trackshave been obtained at the same level of resolution. In this work, we concentrate on this secondquestion and make the assumption that the measured tracks to be fused have already been corrected forregistration errors.

Consider the measured track to be a discretely sampled representation of a true continuoustrack produced by a target. When two sensors are sampling the true track at different rates, themeasurements are obtained at different time instants. The underlying problem of producingmeasurements for the missing or mis-matched time instants for one or the other sensor can be viewed as

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one of effective and accurate interpolation. A direct and intuitive approach is to use linear orpolynomial interpolation to produce the required missing measurements. Such a non-parametricapproach fails to recognize the dynamical nature of the physical system producing the tracks. Tracks,be they linear or curvilinear, are produced by dynamical systems. While the dynamics are complex, thetracks often display hints of self-similarity, much as natural coastlines formed by complex naturalprocesses are found to be self-similar fractal objects [1]. An interpolation scheme which, in some way,takes this self-similarity into account should prove to be more appropriate. The Iterated FunctionSystem (IFS) [2] approach is just such a scheme. This approach is parametric in nature, since a modelbased on IFS is built to represent the mechanism of producing the measurements. Unlike the traditionalapproach of studying time series using an auto-regressive (AR) or auto-regressive moving average(ARMA) model, the IFS approach takes into account the self-similar nature of the object beingmodelled. To this extent, as will be shown in this work, the IFS would prove to be superior in ourproblem of interpolation. Two simple schemes are proposed in this report, using the IFS to model themeasured tracks. Parameters for the IFS are determined by the track measurements and then used tointerpolate for the missing or mis-matched points.

In Section 2, a brief overview is given on the theoiy of self-affme IFS fractal model and themethod of determining the relevant parameters. The contraction factors, which are the most importantparameter of the IFS fractal model, are used in Section 3, where two schemes are described forinterpolating the missing points. The data models generating artificial tracks are presented in Section 4and the paper concludes with discussions on the computer simulation results in Section 5.

2. OVERVIEW OF THE IFS FRACTAL INTERPOLATIN MODEL

Self-affme IFS fractal model assumes strict self-affinity for a continuous graph, so that a sub-interval of the graph or sequence can be obtained from the entire graph or sequence by a sheartransformation. More specifically, suppose y(t) is a continuous function of time t, for t0 < t < tL sothat y= y(t0) and yL=y(tL). If the graph {y(t), t0 < t < tL} is self-a.ffme, then for any pair of points

{t.j, y..j } and {t, yj} with t..1 <t1 , forming the boundary of an interpolation sub-interval, thereexists a shear (or self-affme) transformation 0j:

(;)=(: )÷() (2.1)

which maps the entire sequence onto the sub-interval bounded by the points {t..i, Yj-i } and {t,yj}. In(2.1), a, c, d, e and f are parameters of the shear transformation. The end points of the graph{t0, yo} and {tL, YL} are constrained to map onto the points {t..i, Yi-i } and {t,Yj} respectively sothat,

.(0 = and (t . (2.2)\YO) \Yi-i) \YL) \Yi)

When d, the contraction factor for this map, is given, all the other parameters are determined becauseof the constraint (2.2) and are given by,

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where n = 0, 1 , ..., L are the time indices for the entire sequence of measurements. If this is the sheartransformation used to map onto a sub-interval bounded by (p, Yq) fld (q, Yq) q >p' the values of mand om are given by

m=int(an +e),

and m average of all °k 'with k [m - (1/2) , m + (1/2)] (2.9)

The contraction factor d is derived by minimizing the distance between the mapped points and theactual points in the interpolation interval from (p. Yp) to (q, Yq) This is referred to as the analyticalapproach and provides a contraction factor given by,

(L

n=Od= L (2.10):A2n=O

where A= Y [&'o (1 n)YLI and B Ym [nYp (1 n)Yq] with (L n)/L.

In [3], an iterative procedure was developed to determine the proper interpolation sub-intervals. In thecase of using IFS for track interpolation, it is not possible to use this iterative approach. Theinterpolation sub-intervals are determined by the common lime instants among the two tracks to befused. In this way, our application of the IFS approach for interpolation is sub-optimal.

3. IFS FRACTAL INTERPOLATION OF TRACK MEASUREMENTS

Let zj(k1) = [x1(k1), y1(kj)} and z2(k2) = [x2(k2), y2(k2)], be the position measurements bysensor 1 and sensor 2 respectively, of a planar track covering the same time duration T. We assumedifferent scan periods &, & for these sensors, so that Kj = T/&j and K2= T/&2 are themaximum index values for k1 and k2. Without loss of generality, we assume that K1> K2. Now wedescribe two schemes for the interpolation of the track measurements z2(k2) to generate z2(kj) at thefmer time resolution At, in order that z1(k1) and z2(kj) can be fused.

In order to combine measurements, the interpolation sub-intervals for z1(k1) and those forz2(k2) must coincide in time. This requirement is illustrated in Figure 1.

I I I

t0 OAtj tj=k1pt1 t2k12&1 tL= tKj&j

I I

t0= Oztt2 tjk21At2 t2—k22Lt2 tL= t1K2Et2

Figure 1 TheM Common Interpolation Sub-intervals

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These M sub-intervals will be the interpolation intervals for both tracks. It also means that the scanperiods must have a ratio that is rational, i.e. &j/&2 must be rational. We now describe the twoschemes for interpolation.

3.1 Self-interpolation using IFS

(a) Determine the M interpolation intervals [0, k211, [k21, k22], ..., [k2,M..J, k2M] for the

measurements z2(k2), noting that k2M= K2.

(b) Using (2.10), determine the two sets of contraction factors, one for the x-components andone for the y-components of z2(k2).

(c) Compute the two sets of parameters based on these contraction factors to defme the affmemaps, one set for x-components and the other for the y-components.

(d) Using the sets of affme maps (the approximate attractors) interpolate at step size of Atj togenerate z2(kj)

Figure 2 depicts this approach.

Sensor 1 _____________________________________

Fuse

Sensor 2 _____ IFS Analysis Interpolate

Figure 2 Self-Interpolation

3.2 Cross-interpolation using IFS

(a) Determine the M interpolation intervals [0, k1j], [k11, k12], ... kiM] for themeasurements zj(kj), noting that klM= Kj.

(b) Using (2.10), determine the two sets of contraction factors, one for the x-components andone for the y-components.

(c) Based on these contraction factors from z1(k1), and the boundary measurements for theintervals [0, k21], [k21, k22], ..., [k2,JW1, k2M] in z2(k2) compute the parameters thatdefme the two sets of affine maps, one for the x-components and one for the y-components.

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(d) Using the sets of affme maps (the approximate attractors) interpolate at step size of At1 to.generate z2(kj).

Figure 3 depicts this approach.

Sensor 1 ____IFS Analysis*

4-Contraction I Fuse

Factors :#

Sensor 2_ Affme _________ InterpolateMaps

Figure3 Cross-interpolation

We note that the main difference between these two schemes is the origin of the contractionfactors. In the self-interpolation approach, the contraction factors come from the measurements z2(k2)and interpolation is applied on z2(k2). In the cross-interpolation approach, the contraction factors areobtained from the measurements zj(k1) and interpolation is applied on z2(k2). One may surmise that inthe second scheme, some fusing of information has already occurred before an actual fusing algorithmis applied to zj(k1) and z2(kj).

4. DATA MODELS

4.1 For self-interpolation using IFS

(A) Linear motion: In this case, the target is assumed to be moving at a uniform speed alonga linear path, with random perturbation. For sensor i, the x-component data are generatedby:

x(k+1) = x(k) + &j ('k)

= j(kj) + u(k)

u(k +1) = p u(k) + i = 1, 2; k = 0, 1 , K (4.1)

where {x(k), 1(k)} represents the x-components of the position and velocity at the timeinstant k. u1 and p are perturbations and v is white noise. The data model is completed byadding a white noise i and a possible bias to x1. Thus, for sensor i the measurementsof the x-components are:

x(k) + + (4.2)

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A similar set of equations generates the y-component measurements.

(B) Curvilinear motion: An elliptic path is generated by the equations:

x(k+1 ) = x(k) cos (& ) - a y(k) sin (& )

y(k+1 ) = (1/a) x(k) sin (Ate ) + y(k) cos (& 9)

i = 1,2 ; k= 0,1,..., K (4.3)

where is the constant angular speed and a is the ratio between the semi-major and the semi-minor axes. The data models are then completed by adding measurement noise and possiblebias as in (4.2).

4.2 For cross-interpolation using IFS

(A) Linear motion: A constant velocity linear motion is used to generate the x- andy-components:

xj(k) = xO + S k&y(k) = Y0 + Sy k i= 1,2 ;k= 0,1, .... K (4.4)

where s and Sy are the constant speeds, and (xo, yo) is the initial point of the track. Again,measurement noise and possible bias are added to complete the model.

(B) Curvilinear motion: A circular motion with constant angular speed is used to simulate thepath,

x(k) = r sin ( k& 9 +

y(k) = rcos (k& 9 + 0) i = 1, 2; k =0,1,..., K (4.5)

where r is the radius of the circular path, 9 is the constant angular speed and is the initialphase. Adding measurement noise and possible bias completes the data model.

5. Simulation results and discussions

5.1 Self-interpolation using IFS

The data model here is based on (4.1) and (4.3) for the linear and curvilinear motionsrespectivley. In this case, the measurement noise for the two sensors is independent white noise. Sensor1 samples at 5 seconds intervals and sensor 2 samples at 12 seconds intervals. This results in aminimum common interpolation interval of 60 seconds duration and the total time duration of thesequences is 300 seconds. Figures 4 and 5 show the linear and curvilinear tracks from the two sensors,

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with random perturbations indicated by the mean squared errors shown, and no bias. Figures 6 and 7show simulation results of fusing two track measurements with opposite but equal biases, one using theIFS self-affme interpolation and the other using linear interpolation. The fusion after interpolation iseffected by simple averaging of the two track values at the same time instants. It is consideredadequate for our purposes of comparing the performance of IFS interpolation with linear interpolation.Other fusing algorithms could certainly be applied. Figures 8 and 9 show the corresponding resultswith the elliptical tracks. We summarize these results in Table 1, which shows the normalized meansquared errors of the fused track from the true track.

Track IFS interpolation Linear Interpolation1

Linear 0.0072 0.0069Elliptic

1

0.0066 0.0070

Table 1 Normalized MSE of fused tracks

It should be noted that linear interpolation is performed within each time interval of 12 seconds for thesecond sensor, while the interpolation interval for the IFS approach is 60 seconds. The resultsconfinns the efficiency of the IFS for compression.

5.2 Cross-interpolation using iFS

The data models are based on (4.4) and (4.5). &1 and &2 are set at 1 unit and 1.5 unitsrespectively. Number of measurements used are Kj=150 and K2= 100. No biases are introduced andthe perturbation c is set at different levels for comparison. The interpolation interval is set at 6 units.

Fusion after interpolation is by averaging. Again, linear interpolation is performed for each of the 1.5unit interval for the second sensor to produce the missing measurements in the second track. Table 2and Table 3 summarize the performance comparisons in terms of the mse of the fused track from thetrue track.

IFS Interpolation Linear Interpolation

c1=0.8Track slope a1= 0.2 a1=0.4

-

a1=0.6 a1=0.8 a1.= 0.2 a1=0.4 a1=0.6

-1 0.0482 0.1540 0.3627 0.7529 0.2179 0.3273 0.4782 0.76510 0.0347 0.1199 0.233 1 0.4942 0.0350 0.1742 0.3 196 0.54001 0.0352 0.1518 0.3589 0.6466 0.2215 0.3237 0.5522 0.8692

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Table 2 Cross-interpolation on linear tracks

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; IFS Interpolation Linear Interpolation-r 0.2 a1=O.4 a1=0.6 cy=O.8 a1= 0.2 cYi=0.4 oi=0.6 a1=O.8

.

50 0.0706 0.2226 0.4319 0.7728 0.4396 0.5578 0.6875 1.0292100 0.1418 0.3116 0.4570 0.7867 1.6805 1.7226 1.9287 2.0513150 0.2632 0.4260 0.8208 1.2221 3.6936 3.8763 4.0812 4.5397

Table 3 Cross-interpolation on circular tracks

From these tables, we observe that cross-interpolation performs better on average than self-interpolation. It is as expected since the shorter scan period of sensor 1 should provide a betterapproximate attractor to be used as an interpolation function. While the improvement over linearinterpolation is not substantial for linear tracks, it seems more significant for the circular track. Thedeterioration of performance as the noise level is increased indicates the sensitivity of the IFS shearmaps to random noise.

6. ACKNOWLEDGEMENT

The authors wish to thank Dr. Y. Thou for providing assistance in data generation andsimulations.

7. REFERENCES

1 . M. Bamsley, Fractals Everywhere, Academic Press, New York, 1988.

2. S. Demko, L. Hodges, and B. Naylor, "Construction of fractal objects with iterated functionsystems," Comput. Graph., Vol. 19, #3, pp. 271-278, 1985.

3. D. Maze! and M. Hayes, "Using iterated function systems to model discrete sequences," IEEETrans. Sig. Proc. Vol.40, #7, pp.1724-1734, 1992.

4. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, New York,1990.

5. M. Bamsley, J. Elton, and D. Hardin,"Recurrent iterated function systems," Constr. Approx. Vol.5,pp. 3-31, 1989.

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18x_ii

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—60 —40 —20 0 20 40 60 80 100x

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Linear Fused Track using Linear Interpolation

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