218
shown in Fig. 3 left and middle, respectively. In Fig. 3 (right) we show the flow field computed with the LAC method
in sec. 4. The error measures used are the relative error and the angle between (u,v,1) and (à, ,1) where (u, v) and
(, à) are the ground-truth and estimated flow, respectively. Furthermore, we compare not only to the transformed
ground-truth flow but also to the transformation (3) of a flow estimated as usual in the cartesian plane. The error with
respect to the latter should be considered as a lower bound for the error expected. The density is the fraction of the
estimates with smallest singular value above a threshold which varies for the four estimation techniques.
We first present (Tab.1) the angle- and relative errors for the polar transform of the sequence obtained with angular
resolution of 512 samples per 360 degrees and radial resolution equal to the original (256). As expected the error is lower
when we compare the polar estimates to the transformed cartesian estimates. Regarding the local constancy assumption
applied on the polar (LCT) and the cartesian (LCC) plane the errors are about the same for the same density. However,
this density is achieved for appropriately chosen high threshold for the LCT method. The superiority of the LCC method
is shown if we compare it to the performance of the LCT method with the same threshold (LCT-thr).
Transformed ground truth Transform of the cartesian estimate
Technique | av. ang. err. | av. rel. err. | density | av. ang. err. | av. rel. er density
LCT 5.64729 18.74557 0.54977 3.68909 12.36976 0.54977
LCT-th 7.26917 23.65154 0.77566 3.78853 13.56477 0.77566
LCC 6.02655 19.11052 0.48914 3.63529 12.04593 0.48914
Table 1: Error statistics for the polar transform of the “Marbled Block” sequence (see text for explanation).
The log-polar transform of the “Marbled Block” sequence is obtained with angular resolution of 128 samples/360 de-
grees and radial resolution of 45 samples for the radial range [32..356]. It should be noted that the angular resolution
is one-fourth of the angular resolution of the polar transform, therefore the error is due to both the logarithmic and the
polar subsampling. The errors with respect to the transformed ground-truth values are not significantly higher than to
the transformed estimates what means that the effects of noise and poor gray-value structure in the original are inferior
to the subsampling effects. The errors are shown for about the same density in order to compare the performance at
points where the coefficient matrices are regular in all methods. We are thus strict to the methods using assumptions in
the cartesian plane. Even with such a high relative error we can use the log-polar transform for 3D-analysis. We show
in [Daniilidis, 1995] that the 3D-translation direction can be computed with only 5 degrees error.
Transformed ground truth Transform of the cartesian estimate
Technique | av. ang. err. | av. rel. er | density | av. ang. err. | av. rel. er density
LCT 5.34357 34.53929 0.71908 4.26153 32.02543 0.71908
LCC 5.79178 38.03186 0.56144 4.80384 34.88554 0.56144
LAT 5.30501 34.45435 0.75163 4.29278 32.22566 0.75163
LAC 5.10992 34.02044 0.67358 4.28657 32.28787 0.67358
Table 2: Error statistics for the log-polar transform of the “Marbled Block” sequence (see text for explanation).
References
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IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop "From Pixels to Sequences", Zurich, March 22-24 1995
