The interpolation to a certain extent smooths the initial estimates from one pixel to the next by applying weighting factors based on surrounding pixels. This is not generally enough to create a smooth flowfield, since our estimates are based on the partial derivatives of intensity in three dimensions, which can vary rapidly from one pixel to the next.
This introduces the need for a process which removes inconsistency between neighbouring pixels within the regions determined by the spatial segmentation process, known as relaxation. We do not relax across boundaries, because these are taken to determine the areas of the image which are distinct, and distinct areas of the image may represent distinct objects with individual velocities. The relaxation is an iterative process, which proceeds by slowly changing the value at a given point based on its neighbours values.
Cross-entropy of Flow Distributions
Let pxΔ(u, ε) be the probability that a pixel at x has optic flow velocity u and is an interior or boundary pixel of the flow field relative to direction Δ according as ε = 0 or 1 respectively. We can regard pxΔ(u, ε) as representing a combination of a field of optic flow, defined on the pixel locations, together with a field of boundary segments, defined on the interpixel locations. As probability is conserved |
[29] |
for all x and Δ.
We assume that the flow and boundary fields are independent and hence that the probability factorises as
 |
[30] |
with
 |
[31] |
Let qxΔ(u, ε) be an estimate of pxΔ(u, ε) obtained from the (six) pixels which form a neighbourhood δx of x.
 |
[32] |
The cross-entropy between the current probability distribution pxΔ(u, ε) and that estimated from its neighbours qxΔ(u, ε) is
 |
[33] |
Due to the assumption of flow and boundary field independence, equation , the entropy splits into separate region and boundary components
 |
[34] |
with
 |
[35] |
where the factor of six comes from the four spatial and two temporal neighbouring pixels, and
 |
[36] |
Relaxation algorithms will be derived for the separate terms and then iterated alternately until convergence.
Flow estimation
Let p(x,u; x', v) be the joint probability for adjacent pixels at x and x' to have flow velocities u and v respectively. We assume that, if there is a flow boundary between x and x', then the flow velocities are uncorrelated |
[37] |
while, in the absence of a boundary,
 |
[38] |
Now, if there is a flow boundary between x and x', then the
probabilities, pxΔ(0) and px'Δ(0), of the pixels being interior to a flow region relative to direction Δ satisfy the conditions
 |
[39] |
while, if both are interior to the flow field
 |
[40] |
It follows that an estimate of the probability of flow u at x in terms of that at x' is given by
 |
[41] |
with Δ = x' - x, since, if there is a boundary between x and x', then
 |
[42] |
whereas
 |
[43] |
... if both x and x' are interior to the flow field.
We wish to take account of all of the neighbours of x in forming the estimate. Distributions which are highly localised are more reliable, therefore we wish to give them greater weight when forming the estimate. An expression which does so is given by
 |
[44] |
where δ'x is an inclusive neighbourhood of x, viz x and its neighbours,
 |
[45] |
and the compatibility coefficient between flows at x and x' is given by
 |
[46] |
with σ(x') being the standard deviation of the flow at x'=(x', t'). We approximate the velocity correlation function by an isotropic Gaussian, viz.
 |
[47] |
σ0-2 is a measure of the average correlation between the flow fields at adjacent pixels.
The stationary points of the flow distortion (entropy of local flow information) are solutions of [Bfh]
 |
[48] |
where
 |
[49] |
Now, from equation 44,
 |
[50] |
Therefore
 |
[51] |
with Cx' x(v,u) defined in equation 46. The relaxation labelling algorithm [Bfh] for the entropy of local flow information (the cross-entropy) is given by
 |
[52] |
This algorithm will be applied iteratively in conjunction with the corresponding algorithm for the entropy of boundary information, which will now be derived
Cross-Entropy of Boundary Distributions
The cross-entropy of boundary distortion (entropy of local boundary information) is |
[53] |
where the local estimate of pixel x being a boundary relative to direction Δ is given as in Haddon and Boyce [HB90] by
 |
[54] |
with
 |
[55] |
i.e:
 |
[56] |
with ⊥Δ a translation generator normal to Δ. The estimate of interior probability then follows from
 |
[57] |
Stationary points of the entropy of boundary information are solutions of
 |
[58] |
where
 |
[59] |
and Φ(x) is an estimate of ΣΔpxΔ(1) in terms of pxu.
Now, from equation 53
 |
[60] |
with, from equation 54
|
[60] |
hence
 |
[61] |
In order to obtain Φ(x), an estimate of ΣΔpxΔ(1) in terms of pxu, we consider
 |
[62] |
Since, by Minkowski's inequality,
 |
[63] |
hence...
 |
[64] |
... is a suitable estimate of ΣΔpxΔ(1). The boundary relaxation thereby becomes
 |
[65] |
The two relaxations, equation 52 and 65, are applied alternately until convergence.
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