Hierarchical flow element with 1 child each level for clarity. (L) Identity mapping. (C) Actual mapping. (R) Printout of shifts for each level of the element. All shifts for identity element should be 0. Note that shifts for actual element vary as the size of the domain decreases. This phenomenon will be the basis for the remaining work on this page.
Same as above, but each level has its range preshifted by the sum of the shifts for all levels above. Thus, the shifts for lower levels should approach 0. Although the identity flow behaves as expected, the actual flow does not converge with decreasing domain size, as was also seen for the previous non-shifted hierarchical flow. The partial sum of incremental shifts for each level gives the same result as in the previous non-shifted flow, indicating that the preshifting does not add anything to the calculation.
(T) Unshifted and (B) preshifted concentric hierarchical flow elements. In this case, the unshifted elements maintain good estimation of the flow at reduced size, and the preshifted elements also tend to converge. However, the utility of this approach, and the underlying variability of the flow estimation with respect to domain size, are still issues to be resolved.
This sequence of pictures shows the candidate (peak) vectors at a single point for flow domains decreasing in size from 128x128 pixels down to 4x4 pixels. At large domain sizes, there is only one peak in the parametric correlation surface. At 32x32 pixels, several peaks appear. By 4x4 pixels, there are 39 peaks in an apparent random distribution.
A close-up of the same region as above using FITSVector to manually estimate the flow in a small domain. Note the variation of dx values (6th column in table) for nearby vectors. Although all vectors point in the correct direction, this example makes clear the difficulties encountered by an automatic flow estimation program, especially at points in the map for which there are no clear discernable features or signal.
In this picture, an aggregate flow consisting of 25 individual flow elements in a 5x5 grid centered about the indicated domain is used to estimate the flow. The green arrows show the maximum peak values for each of the 25 individual flows, translated to the center of the domain. The yellow arrow is the average of all 25 flows. All vectors are scaled by 5x for clarity. 6 different domain sizes, from 128x128 down to 4x4 pixels are shown. In addition, the ratio of the variance magnitude to the mean magnitude of the population is shown at the upper right of each plot. Initially the variance is 0 in both axes, indicating that all 25 flow vectors are identical. As the domain size decreases, the variance to mean ratio increases substantially. Note that this function is not monotonic.
Text file showing values of all individual flow vectors, and associated population statistics, for the aggregate flows shown above.
Locations of 5 positions chosen for testing the adaptive flow element shown in the pictures below.
An adaptive flow element was created based on the aggregate flow element shown above. This element adjusts its domain size larger and smaller until the ratio of the variance magnitude to the mean magnitude is closest to a specified target value. For the positions shown above, the target value was 0.125. These positions where chosen so that some were close to discernable features, while others were not. In order to reach the target ratio, flow elements which were not located on discernable features required a larger set of domains. Ideally, the adaptive flow element will use the smallest domain size which does not exceed the specified variance / mean ratio, so as to better localize the flow.
Variance / mean ratios for the positions indicated above for all domain sizes from 128x128 down to 1x1 pixels. Note that these plots are not monotonic, although they generally start with a variance of 0 for 128x128 domain sizes and have the largest ratios for smaller domain sizes. Also note that plots for positions near to discernable features (e.g. position 4) show lower ratios for smaller domains, and a more constant average flow magnitude, than do plots for positions away from discernable features (e.g. position 5).
Values of domain size, mean x, y, and magnitude, variance x, y, and magnitude, and variance / mean ratio for all positions. Based on this table, one criterion for adaptive flow elements might be to use the mean value for the smallest domain size (other than 1x1) for which the variance / mean ratio is < 0.01 (appears as 0.00 in table).
Positions and estimated flow vectors for 50 points manually entered with FITSVector. Note that it is especially difficult to cover the entire map, due to the paucity of stable discernable features contained in both map images. Based on this exercise (which took about an hour to complete), the automated tracking software actually appears to be doing a good job of estimating the flow from the available data, especially in relatively featureless regions.
The next step is to apply the adaptive flow element demonstrated above to estimating the flow at every pixel in the maps. An iterated procedure such as relaxation labelling or simulated annealing may then be applied to the resulting field.