Alternative Models

The model, as outlined in this chapter, is deemed an appropriate and sufficiently flexible model for the type of data we are considering. Two of the possible variations on the model which were considered are outlined in this section.

3.3.1 A Fibre-Process Generated Cox Process

In Section 3.2.1 it was suggested that points could be modelled as a Cox process generated by a fibre process. In a Cox process, points are Poisson distributed with in- tensity function given by a random field overW which, in turn, depends on the fibre process. Such a model gives rise to an independent point process unlike our model where points are perturbed from anchor points, which are Dirichlet-distributed along a fibre.

If, in the construction of our model, we choose a Dirichlet parameter αDir = 1,

so that anchor points are uniformly distributed along the fibres, then the resulting point process is a Cox process. Variations on this Cox process are found by choosing an alternative likelihood functionL(pi|yi, Zi= 1), or indeed, dropping the auxiliary

variables and anchor points, and determining a likelihoodL(F|yi).

For example, the likelihood could be a function of the distance fromyi to thenearest

fibre. The intensity function of the resulting Cox process is not quite the same as that of the Cox process derived from our construction, where the intensity is determined

by integrating a kernel along each fibre. However for a field of orientations with low curvature, these intensity measures can be nearly equal. The advantage of using anchor points in the Cox process construction is that the model can be easily extended to control the regularity of points along fibres.

3.3.2 Towards an Unbiased Fibre Process

As described in Section 3.2.1, fibres are identified by partial integral curves of a field of orientations. We have chosen a prior distribution for the fibre process based on the sampling mechanism of drawing a reference point uniformly at random from the window W and integrating the field of orientations to a random length in each direction. However, this prior fibre distribution is biased, in that some regions of the window are more likely to contain a random fibre than others due to the inherent curvature of the field of orientations. The bias is depicted in Figure 3.4, where a large number of fibres have been drawn from the prior distribution. There is clearly a long region from (140,143) to (170,129) that is integrated by a greater density of fibres than other areas of the window.

Figure 3.4: A cropped window showing a large sample of fibres drawn from the prior fibre distribution with a diverging field of orientations. There is clearly a bias on the number of fibres we would expect the curve S, orthogonal to the field of orientations, to intersect.

that intersect the curve segmentS which runs perpendicular to the field of orienta- tions. A natural condition to impose, in order to reduce the bias on the density of fibres, is to require thatm(S) is proportional to the length ofS. However, we need to construct a fibre process model satisfying this condition.

One approach is to model the number of fibres intersectingSas a time-homogeneous birth-death process by settingStto be the curve perpendicular to the field of orienta-

tions, conditioned on two streamlines on which its end points are located, and meet- ing one of these streamlines at pointt, measured in arc length along the streamline (see Figure 3.5). By taking the two streamlines to be very close, we can arbitrarily choose either streamline to measure the arc length along.

Figure 3.5: A section of Figure 3.4 motivating the construction of a birth-death pro- cess. The number of fibres integrating curveStwill vary ast increases or decreases

fromt0. The two thick streamlines that St connects are assumed fixed.

The birth-death process describes the number of fibresXtintersecting Stastvaries

(from−∞ to∞ in principle). This means thatXt → Xt+ 1 at rate bt, the birth

rate, and Xt → Xt−1 at rate dtXt, where dt is the death rate, and both bt and

dt are functions of time. It is possible to estimate birth and death rates which

ensure that the mean number of fibres intersectingSt is proportional to the length

of St, for a chosen pair of streamlines. By considering the birth and death rates

for an arbitrarily close pair of streamlines we can draw samples from an unbiased distribution of fibres - where ‘unbiased’ is defined as ‘the mean number of fibres

intersecting St is proportional to the length ofSt.

In practice, the prior fibre distribution described in Section 3.2.1 leads to simple calculations, and generally there is sufficient data largely to eliminate this bias from the posterior. However, finding an unbiased fibre-process prior is both of theoretical interest and informative as to how unbiased and biased priors differ.

3.4

Conclusions

In this chapter we presented a Bayesian hierarchical model for a general point pro- cess exhibiting clustering around an unknown number of curvilinear features. The Bayesian approach is motivated by the inherent complexity of the data clustering and the prior belief that there exists a random fibre process with random points clustered about each fibre. In Chapter 5 we will show how Markov chain Monte Carlo (MCMC) methods can be used to draw samples from the posterior distribu- tion of fibres and other parameters (e.g. signal/noise allocations and fibre lengths), given an instance of the random point process. Examples and statistics of samples drawn from the MCMC methods are given in Chapter 6.

Fibres are identified as partial integral curves of a field of orientations. This is a relatively novel characterisation of fibres in the study of fibre-generated point processes, yet integral curves have been used in other areas such as image analysis (Kass and Witkin, 1987), diffusion tensor imaging (Mori et al., 2001) and fingerprint topology (Sherlock and Monro, 1993). An important consideration is how the field of orientations should be estimated so that integral curves produce high likelihoods, this is the focus of the following chapter.