Transitivity Error

The transitivity error of the unidirectional and inverse consistent image registration algorithms was evaluated by measuring the difference between the identity mapping and the composition the transformations from image

-to-

, and

-to-

. The average and maximum transitivity errors were calculated using Eqs. 3 and 4. The transitivity error was measured by concatenating the transformations

-to-

, and

-to-

for the phantom and CT data sets. Twenty data sets were used to evaluate the transitivity error in the case of MRI brain mapping. For the MRI data, the transitivity error was computed for twenty groups of three data sets i.e., 1-to-2-to-3-to-1, 2-to-3-to-4-to-2, etc.

Figures 6, 9 and 10 show one example of the spatial location of the transitivity error for each of the phantom, CT, and MRI experiments, respectively. These figures show that there is some correlation of the transitivity error to the object structure since some of the object structure is seen in these images. This is less pronounced in the transitivity error image for the phantom.

We looked at the question of whether the transitivity error of

-to-

differed spatially and/or quantitatively from the transitivity error from

-to-

, and

-to-

for the phantom data. All of the transitivity error images for these experiments were nearly identical to those shown in Fig. 6. In general, the transitivity error images for these cases will be different from each other since they are computed in different coordinate systems and are correlated with the size of the object in the image. Therefore, the transitivity error will be distributed over a larger region in a coordinate system containing a large object as compared to a coordinate system containing a smaller object.

Tables 1 and 3 tabulate the average and maximum transitivity errors for the phantom, CT, and MRI experiments. These tables show that the inverse consistent registration algorithm reduced the maximum transitivity error by 60 % for the phantom data, 30% for the CT data, and 37 % on average for the MRI data compared to the unidirectional algorithm. Likewise, the average transitivity error was reduced by 50% for the phantom data, 70 % for the CT data, and 50 % on average for the MRI data. These results clearly demonstrate that the inverse consistency algorithm improves the performance of the registration compared to the unidirectional algorithm with respect to reducing the transitivity error.

**Table 3:** Transitivity Errors for 3D MRI Experiment over the brain region of interest for algorithms using the inverse consistency constraint (w/ ICC) and without (unidirectional).
Exp.	Avg Error			Max Error
	unidir.	w/ ICC	Ratio	unidir.	w/ ICC	Ratio
	(voxels)	(voxels)		(voxels)	(voxels)
01-02-03	2.23	1.65	1.35	28.0	24.8	1.13
02-03-04	1.77	1.26	1.41	16.3	12.9	1.26
03-04-05	2.13	1.54	1.38	13.8	11.9	1.15
04-05-06	2.12	1.47	1.44	11.7	9.4	1.24
05-06-07	2.24	1.60	1.40	14.2	11.8	1.20
06-07-08	2.15	1.59	1.35	16.0	14.9	1.07
07-08-09	1.96	1.38	1.42	15.5	12.4	1.25
08-09-10	2.20	1.44	1.53	17.1	11.8	1.45
09-10-11	1.96	1.27	1.54	14.5	10.1	1.44
10-11-12	2.01	1.23	1.63	16.0	7.1	2.26
11-12-13	1.54	0.95	1.61	10.6	7.8	1.36
13-14-15	1.59	1.05	1.52	10.6	10.9	0.97
14-15-16	1.70	0.97	1.75	14.3	6.3	2.28
15-16-17	1.67	0.94	1.78	10.0	6.1	1.64
16-17-18	1.86	1.18	1.58	14.1	12.5	1.13
17-18-19	2.05	1.42	1.44	12.7	10.4	1.21
18-19-20	1.87	1.21	1.54	15.2	12.1	1.25
19-20-01	1.87	1.21	1.55	12.4	8.3	1.50
20-01-02	2.15	1.55	1.39	14.7	12.1	1.21
Average	1.95	1.31	1.50	14.6	11.2	1.37
Sd. Dev	0.22	0.23	0.13	3.8	4.1	0.35