Inverse Consistency Error

The inverse consistency error measures the degree to which the algorithms produced a consistent pointwise correspondence mapping between a pair of images. The average and maximum inverse consistency errors are computed using Eqs. 1 and 2. These error measures compare the composition of the forward and reverse transformations to the identity mapping and are minimized when the composed transformation equals the identity mapping.

Figure 6 shows typical 2D phantom registration results using the unidirectional and inverse consistent registration algorithms. The result of transforming image A to B (see Fig. 3) using both registration algorithms is shown in column (a) and the absolute intensity difference between the deformed image A and the target image B is shown in column (b). These transformed images and intensity differences were typical of all 12 registrations³. Notice that both registration algorithms did a good job in matching image A to B. Figure 7 shows typical registration results using both algorithms for CT data. This figure shows that both algorithms did a good job matching the large infant head (CT A) to the small infant head (CT B) (see Fig. 4). Figure 8 demonstrates typical registration results when matching MRI brain data. Again the unidirectional and inverse consistent registration algorithms performed about the same with respect to absolute intensity differences.

The absolute intensity difference images in Figures 6, 7, and 8 illustrate that one component of the registration error (as measured by intensity difference) occurs at the object edges. These figures also demonstrate that there is very little difference between the two registration algorithms with respect to the intensity mismatch. Therefore, it is not sufficient to measure the absolute intensity difference alone to evaluate the performance of the unidirectional and inverse consistent registration algorithms.

One reason for this registration error at the object edges is due to the fact that the linear-elastic model can only accommodate global nonrigid shape differences. The misregistration at the edges is also due to the balancing of the similarity and the regularization terms in the minimization problems defined by Eqs. 5 and 7. This occurs in both the unidirectional and consistent linear-elastic algorithms because the regularization term prevents the template image from fully deforming into the target data.

The performance of the unidirectional and inverse consistent registration algorithms can be distinguished by comparing the inverse consistency error of each algorithm. Figures 6, 9, and 10 show the spatial location of the inverse consistency error for the phantom, CT, and MRI experiments. Each figure shows the result of concatenating the A-to-B transformation with the B-to-A transformation. The spatial inverse consistency error shown in these figures are typical of the spatial inverse consistency error images for all the other registrations. The color bar associated with each set of results shows the range of the inverse consistency constraint error in pixels/voxels over the whole 2D or 3D image.

Figure 9: Columns (a) and (b) show the absolute magnitude ICC error for the A-to-B-to-A CT registration using the unidirectional and ICC algorithms, respectively, for the data sets shown in Fig. 4. Columns (c) and (d) show the absolute magnitude transitivity error for the A-to-B-to-C-to-A CT registration using the unidirectional and ICC algorithms, respectively. The rows from top to bottom show coronal, sagittal, and transverse views.

A-B-A Mag. ICC Error (unidirectional)	A-B-A Mag. ICC Error (ICC)	A-B-C-A Mag. Trans. Error (unidirectional)	A-B-C-A Mag. Trans. Error (ICC)
$\includegraphics[width=15cm]{trans_paper01.figs/skulls/skull_inv_trans_3x4}$
(a)	(b)	(c)	(d)

Figure 10: Columns (a) and (b) show the absolute magnitude ICC error for the A-to-B-to-A MRI registration using the unidirectional and ICC algorithms, respectively, for the data sets shown in Fig. 5. Columns (c) and (d) show the absolute magnitude transitivity error for the A-to-B-to-C-to-A MRI registration using the unidirectional and ICC algorithms, respectively. The rows from top to bottom show coronal, sagittal, and transverse views.

A-B-A Mag. ICC Error (unidirectional)	A-B-A Mag. ICC Error (ICC)	A-B-C-A Mag. Trans. Error (unidirectional)	A-B-C-A Mag. Trans. Error (ICC)
$\includegraphics[width=15cm]{trans_paper01.figs/brains/brains_inv_trans_3x4}$
(a)	(b)	(c)	(d)

These experiments demonstrate that there are two primary sources of inverse consistent correspondence error. The first source of error occurs at the image edges and is due to the inability of the registration algorithm to completely match the borders. The second source of error occurs away from the image edges and is due to the linear-elastic regularization. The regularization error is much larger than the edge matching error and is reduced significantly by the constraining the forward and reverse transformations to be inverses of each other.

The average and maximum inverse consistency errors for the phantom, CT, and MRI experiments are tabulated in Tables 1 and 2. Twenty additional MRI data sets were used to compile the values in Table 2. The results are reported for the whole domain of the images for each experiment and for specific regions of interest (ROI). The ROI for the CT experiment corresponded to voxels in the head above the Frankfort Horizontal (FH) and the ROI for the MRI experiment corresponded to the brain voxels. The CT ROI is important for pre-operative planing and post-operative evaluation for corrective craniofacial surgery. The MRI ROI was chosen because it is not important what the error is outside of the brain.

Table 1 shows that there was a 46 and 60 times improvement with respect to the average and maximum inverse consistency error, respectively, when comparing the inverse consistent algorithm to the unidirectional algorithm for the phantom experiments. Likewise, the average and maximum inverse consistency error improved 8.6 and 24 times, respectively, using the inverse consistent constraint for the CT data.

Table 2 shows both the ICC error and the squared intensity error between the deformed and target images. This table shows that the squared intensity error was similar (on average 6% less) between the unidirectional (UD) and the inverse consistent (IC) algorithms with the UD algorithm having slightly lower values than the IC algorithm. However, the maximum inverse consistency error over the entire image domain was on average 205 times smaller for the inverse consistent algorithm. These result demonstrates that there is a trade-off between reducing the registration error measured by the absolute intensity difference and the registration error measured by the inverse consistency error. In the current case, we were able to reduce the maximum inverse consistency error on average by 205 times while only increasing the squared intensity error on average by 6%. 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 inverse consistency error.

**Table 1:** Inverse Consistency and Transitivity Experimental Errors for algorithms using the inverse consistency constraint (w/ ICC) and without (unidirectional).
Data	ROI	Max/Ave	Inv. Consist. Error			Transitivity Error
			unidir.	w/ ICC	Ratio	unidir.	w/ ICC	Ratio
			(voxels)	(voxels)		(voxels)	(voxels)
Phantom	whole domain	Max.	1.2	0.026	46	2.7	1.7	1.6
Phantom	whole domain	Ave.	0.36	0.006	60	1.0	0.68	1.5
CT	whole domain	Max.	12	1.4	8.6	15	11	1.4
CT	whole domain	Ave.	1.0	0.014	71	1.4	0.88	1.6
CT	head above FH	Max.	12	1.4	8.6	13	10	1.3
CT	head above FH	Ave.	2.1	0.086	24	2.35	1.4	1.7

**Table 2:** Maximum Inverse Consistency Error and Squared Intensity Error for 3D MRI Experiment over the Entire Domain for algorithms using the inverse consistency constraint (w/ ICC) and without (unidirectional).
Exp.	Max. Inv. Consist. Error			Sq. Int. Error
	unidir.	w/ ICC	Ratio	unidir.	w/ ICC
	(voxels)	(voxels)
01-02	0.46	0.0020	229	275	287
02-03	0.22	0.0014	160	270	292
03-04	0.28	0.0014	192	321	339
04-05	0.38	0.0019	201	365	398
05-06	0.35	0.0017	206	325	340
06-07	0.47	0.0026	185	330	366
07-08	0.32	0.0020	158	319	344
08-09	0.35	0.0023	155	382	412
09-10	0.61	0.0031	198	374	405
10-11	0.42	0.0025	170	326	347
11-12	0.23	0.0012	197	245	258
12-13	0.26	0.0013	206	251	270
13-14	0.22	0.0013	170	260	273
14-15	0.24	0.0012	201	216	226
15-16	0.40	0.0013	309	260	278
16-17	0.55	0.0015	372	319	343
17-18	0.33	0.0018	188	312	327
18-19	0.34	0.0017	197	289	306
19-20	0.31	0.0015	203	279	302
Average	0.35	0.0018	205	N/A	N/A
Sd. Dev	0.11	0.00053	52	N/A	N/A