4.2.2

Rotation Angle Estimation

  

This experiment compares the AM-estimator with the M- estimator in computing relative rotation of motion sequences. Consider a sequence of images in Fig.4.6. Corners can be detected from these images as in Fig.4.7 using the Wang-Brady detector [Wang and Brady 1991]. The data

where and , is a set of matched point pairs between two images. A previous work [Wang and Li 1994] shows that when the rotation axis is known, a unique solution can be computed using only one pair of corresponding points and the LS   solution can be obtained using m pairs by minimizing  

where

A unique solution exists for the LS problem [Wang and Li 1994]. It is determined by the equation

where , that is,

The above formulation is based on an assumption that all the pairs are correct correspondences. This may not be true in practice. For example, due to acceleration and deceleration, turning and occlusion, the measurements can change drastically and false matches, i.e. outliers, can occur. The LS estimate can get arbitrarily wrong when outliers are present in the data d. When outliers are present, the M-estimator can produce the more reliable estimate than the LS   estimator. The AM-estimator further improve the M-estimator to a significant extent.

The AM-estimator minimizes, instead of (4.17), the following  

where is an adaptive potential function. By setting and using , one obtains

Rearranging the above gives the following fixed-point equation  

where . It is solved iteratively with decreasing value.

  
Figure 4.7: Corners detected. Courtesy of H. Wang.

  
Figure 4.8: Rotation angles computed from the correspondence data containing 20% of outliers using the the LS   estimator (left ) and the M-estimator (right).

Figs.4.8 and 4.9 show the estimated rotation angles (in the vertical direction) between consecutive frames (the label on the horizontal axis is the frame number) and the corresponding standard deviations (in vertical bars) computed using the LS-, M- and AM-estimators. From Fig.4.8, we see that with 20% outliers, the M-estimator still works quite well while the LS   estimator has broken down. In fact, the breakdown point of the LS   estimator is less than 5%.

From Fig.4.9, we see that M- and AM-estimators are comparable when the data contains less than 20% of outliers. Above this percentage, the AM-estimator demonstrates its enhanced stability. The AM-estimator continues to work well when the M-estimate is broken down by outliers. The AM-estimator has the breakdown point of 60%. This illustrates that the AM-estimator has a considerably higher actual breakpoint than the M-estimator.