Gaussian Approximation to the Flow Distribution
We implement this approach by approximating the distributions by two-dimensional Gaussians, characterised by their means and standard deviations. The following section is an account, largely technical, of how the distributions are represented and how the flow relaxation algorithm, equation 52 translates into an updating algorithm for the means and standard deviations of the Gaussians. The results are given as equations 94 and 95 for the means and standard deviations respectively. The rationale underlying this approximation is that the mean and standard deviation of a Gaussian distribution represent the most likely value of the flow at a point together with the reliability with which it is known.It is convenient to utilise the statistical nomenclature for Gaussian distributions
 |
[66] |
generalised to two dimensions as
 |
[67] |
If pxu, σ) denotes a 2D Gaussian probability distribution having mean u(x) and standard deviation σ(x), with x=(x, t), then the Gaussian approximation to the flow distribution at x can then be expressed as
 |
[68] |
As the system of flow relaxation achieves consistency we may expect that σ(x) → 0 and, in the limit of zero uncertainty,
 |
[69] |
The great benefit of the approximation is that, instead of an arbitrary function at each pixel, the flow distribution can now be represented by two vectors, the mean flow u = ui +vj together with its standard deviation σ = σxi + σyj
The relaxation algorithm for the flow field distribution becomes
 |
[70] |
However, since the updated distribution is also approximated by a Gaussian, characterised by a mean vector and its standard deviation, it will be sufficient to formulate and implement the relaxation algorithms for these quantities. Now the updated mean at x is given by
 |
[71] |
while the updated variance is
 |
[72] |
Thus we may calculate the updated values directly. Approximations to these quantities are provided by the three lemmas proved in the appendix.
Relaxed Flow Distributions
Using Lemma 1 of the appendix it follows that the smoothing of the flow at a pixel, px(u,σ), by the velocity correlation function, equation 47, is |
[73] |
where
 |
[74] |
As a result equation 44 may be evaluated using equation 46 to yield
 |
[75] |
The velocity integration of equation 44 has been performed. The estimated velocity distribution, qx(u), is a sum of Gaussians; it too will be approximated by a Gaussian,
 |
[76] |
where, by applying Lemma 2 of the appendix to equation 75,
 |
[77] |
with
 |
[78] |
Correspondingly for the variance, again from Lemma 2,
 |
[79] |
Equations 77 and 79 are the mean and variance of the estimated flow at x from its neighbouring flows. The multiplicative factor which enters the relaxation of the flow, equation 52, takes the form
 |
[80] |
where
|
[78] |
For x=x'
 |
[81] |
with
 |
[82] |
while, for x ≠ x',
 |
[83] |
Since the factors which appear in the integrand have all been approximated by Gaussian distributions, we may use Lemma 3 of the appendix to evaluate the integral, obtaining
 |
[84] |
It follows from the above that, for x ≠ x',
 |
[85] |
The complete updating factor for the flow relaxation is given by equation 80, we must therefore evaluate the effect of the exponential factor on qx(u).
Now, for x'=x, rxx(v) ≅ 0 whenever x is an interior point of a region of smooth flow. This is, in general, not so for x≠x', however, since the flow distributions are normalised, ∫dvx'(v) = 1, while Qx(u) is arbitrary up to a constant multiplicative factor, we may replace rx'x(v) by r'x'x(v), where
 |
[86] |
from whence, analogous to the above derivation of equation 85, we obtain
 |
[87] |
Close to convergence at an interior point of a region of smooth flow,
 |
[88] |
therefore r'x'x(v) will be small, and consequently, as a first approximation, we may expand the exponential factor of equation 80, it following that
 |
[89] |
By making use of Lemma 2 we approximate rx'x(v)qx(v) by a Gaussian distribution of mean u(x',x) and standard deviation σ1(x',x) where
 |
[90] |
with the distribution weighted by the factor
 |
[91] |
The approximation to the relaxation factor becomes
 |
[92] |
This enters the flow relaxation, equation 52. Again using Lemma 2, we may derive the approximation
 |
[93] |
where the mean is
 |
[94] |
and the variance is given by
 |
[95] |
The flow relaxation, equation 52,
 |
[96] |
becomes
 |
[97] |
The updated mean flow vector, ü(x), is given by equation 94 and the updated standard deviation σ(x) follows from equation 95.
Updating the Flow Field
Let the estimate of the flow at x=(x,t) which is obtained by relaxation be |
[98] |
with standard deviations of the components of σx and σy respectively. Let the estimate of the flow at x' = (x, t+Δt) that is obtained from the space and time derivatives via equation be
 |
[99] |
with standard deviations of the components of õx and õy. The updated, i.e. before relaxation, estimates of the flow components are taken as
 |
[100] |
where τx, τy is zero if a flow boundary has crossed (x,y) in the x (respectively y) direction during the time interval from t to t+Δt, and is otherwise unity. These factors serve to decouple the flow fields on either side of a flow edge. The corresponding estimates of the standard deviations are obtained from
 |
[101] |
The above estimates are then input to the relaxation labelling algorithm for flow consistency.
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