al data that can be used to search for potentially hazardous asteroids. Further, these surveys have the potential to allow us to detect and track fainter objects. However, these improvements greatly increase the combinatorics of the problem reinforcing the need for tractable algorithms.we introduce a new methodology for track initiation. Instead of treating track initiation as a sequential decision problem, we exhaustively consider all possible linkages. Thus we provide an exact algorithm for linkages. We then introduce a multiple tree algorithm for tractably finding the linkages. We compare this approach to an adapted version of multiple hypothesis tracking using spatial data structures and show how the use of multiple trees can provide a significant benefit.Definitiontrack initiation problem consists of taking sets of observations from different time steps and linking together those observations that fit a desired model without initial estimates of the track parameters. Figure 2 shows a simple one dimensional example with five time steps and a linear model. The sets of linked observations are shown as open circles with their linear models as dashed lines.
Figure 2: A set of one dimensional observations linked together by linear tracks. The white circles are the observations that correspond to the linear tracks (dashed lines).
the linkage problem can be phrased as a filtering problem. At each time step k we observe Nk points from both the underlying set of tracks and noise. Given a set of observations at K distinct time steps, we want to return all tuples of observations such that:
1. the tuple contains exactly one observation per time step, and
2. it is possible for a single track to exist that passes within given thresholds of each observation.we wish to filter the? Kk=1 Nk possible tuples down to just those tuples that could be feasible tracks.observations consist of real-valued coordinates in D dimensional space, with xi indicating the ith observation. These coordinates are the dependent variables of the track. We use ti to indicate the independent variable of the ith observation. Although in many of the applications below ti will correspond to the time of the observation, it can be used to represent any independent variable.second condition specifies a constraint on the observations fit to the underlying model. A tuple of observations (xI1, ···, xIK) is valid only if there exists a track g such that: ? L [d]? xIi [d]? g (tIi) [d]? ? H [d]? d, i (1)
Equation 1 states that a track g is feasible for a tuple of observations if it falls within some bounds [g (tIi) [d] +? L [d], g (tIi) [d] +? H [d]] of each observation xIi in each dimension d. The thresholds? L and? H provide upper lower bounds on the fit. Figure 3 shows an example of a feasible triplet using linear tracks and one feasible track for these points. The track is allowed to pass anywhere within the error bars around each point.
Figure 3: Three points that are compatible for linear tracks.
above definition of feasibility is compatible with a range of statistical noise models. For example, we can define an arbitrary observation noise model for the points on a track and set the thresholds in each dimension to be the 95% confidence interval for the noise in this dimension. Figure 4 shows an example of this. Further, we can vary? L and? H to account for systematic or time varying errors.
Figure 4: An arbitrary probability distribution and the resulting bounds. The circle denotes the observed location and the upper and lower bars indicate the acceptable locations for the track.
contrast to the flexibility for noise models, it should be noted that the above criteria does not allow for a concept known as process noise. This means that we assume the track always follows the model. For example, a linear track model can not account for changes in velocity. This is briefly discussed in Section 8.discussion below focuses on two major types of tracks: linear and quadratic. The quadratic track is simply a quadric function of time:
(t)=a · t2 + b · t + c (2)
can be used to describe physical motions of objects undergoing constant acceleration. The linear track is a linear function of time:
(t)=b · t + c (3)
can be used to describe the physical motion of objects traveling at a constant velocity. In addition, the linear model can be used for such queries as finding lines or edges described by the observations. While much of our discussion and techniques presented below will also apply to other track models, we restrict the discussion to the linear and quadratic models to keep the discussio...
