Retrospective 3D Registration of MR Images

Retrospective 3D Registration of MR Images

Renjie He and Ponnada Narayana
Department of Radiology, University of Texas-Houston Medical School, Houston, TX

Purpose

Introduction

Conclusions

Acknowledgment

References

Purpose:

The purpose of these studies was to implement and evaluate a robust, fast, and automatic registration technique for retrospective alignment of 3D MR images.

Introduction:

Accurate and robust retrospective image registration is critical for the detection of subtle neuropathological changes with time and for accurate analysis of functional and diffusion MRI. Image registration involves minimization of cost function, based on a similarity measure. Mutual information (MI) is generally considered to be a robust similarity measure in the registration of medical images [1][2]. Image registration based on MI is essentially a nonlinear optimization problem. Various manipulations in the registration procedure, such as interpolation and missing data, generally introduce local minima in the cost function. Multiresolution paradigm, in which image registration is propagated from coarse to fine scale, is commonly used for improving the efficiency of registration. Our extensive simulations demonstrated that image sub sampling in multiresolution paradigms introduces distortions and shift in the MI peak. Therefore, conventional optimization techniques do not always lead to the correct global solution. Retrospective registration techniques, even with multiresolution strategies, generally tend to be slow near the neighborhood of optimum solution. Therefore, any registration strategy should pay particular attention to the computational efficiency. Here we describe a global optimization technique for image registration, based on MI, which can be used in conjunction with a multi-resolution paradigm that is fast and leads to global optimization. This technique combines genetic algorithm in continuous space (GACS) [3], a stochastic method and is very efficient in large search space, with dividing rectangle (DIRECT [4]), a deterministic method that theoretically guarantees global optimization and is efficient in small search space. In these studies we combined DIRECT with GACS to determine the global maximum of MI. Extensive simulations were performed to investigate the behavior of MI on multi echo MR images. The performance of GACS and DIRECT in large and small search spaces was evaluated and the suitable parameters were determined. This method was applied to register pre-contrast with pot-contrast T1-weighted images and multi-echo images of multiple sclerosis (MS) patients. Finally these results are compared with AIR3, a freeware package that is commonly used for image registration. This technique was applied to register magnetic resonance images of multiple sclerosis (MS) patient brains.

Simulations of Mutual Information using Multi-echo MR Images:

For two images A and B with intensities and respectively the MI can be written as:



								(1)

The marginal probability distributions			and	and the joint probability distributions
	are estimated by normalizing the marginal and joint histograms of the overlapping parts of
images A and B.

In image registration, a`i` and b`i` are related through the geometric transformation T`t,` and by adjusting the transformation
parameters, t,		can be maximized for image alignment (Figs. 1 and 2).



Fig. 1 Joint probability distributed as a function of rotation; top row: image B is generated by rotating image A between –25o to +25o around z axis; bottom row, the joint probability between image A and image B.



Fig. 2 MI as a function of shift	Fig. 3 Affirmative images

Most of the reported simulations in the published literature are based on a single translation or rotation along one direction. Such simulations do not always reveal the complex behavior of MI. In contrast, in the current studies we have performed extensive simulations, based on multi-echo affirmative images (5) to guide us identify appropriate strategies for global optimization. Affirmative imaging sequence is based on the fast spin echo (FSE) sequence and produces four images per slice. The first two are the FSE images acquired at 17 ms and 102 ms echo times. The third and fourth images are acquired at the same echo times, but incorporate radio frequency pulses for both CSF suppression and magnetization transfer. The four affirmative images (image matrix 256 X 256) from a single slice in a patient with multiple sclerosis (MS) are shown in Fig. 3. All the four images are acquired in an interleaved manner and are perfectly aligned with each other. Therefore, MI can be simulated by shifting and/or rotating one image relative to the others.

In these simulations, we converted the 2D multi-slice data to 3D data with isotropic voxel resolution, using trilinear interpolation. The size of the 3D data was 256 X 256 X 152.

Fig. 4 MI as a function of shifts in the x-y plane between affirmative images 3 and 4. For clarity, the behavior of MI (a), zoomed around the origin, is shown in (b).

As can be seen from these images, the interpolation effect manifests itself as ridges in MI (Fig. 4b). This is consistent with other published reports. However, for these relatively high-resolution images MI exhibits a well-defined peak (Fig. 4a) that corresponds to perfect image alignment, implying that the global registration can be reliably obtained. As indicated earlier, multiscale registration paradigm is generally employed for increasing the speed of registration. This involves sampling the image at coarse to fine levels. At each level, the global optimum of the cost function is determined and propagated to the next level. The number of points sampled in the coarse level is small. In this case the shape of MI tends to be irregular. In order to demonstrate this, we simulated MI as a function of rotation between the first and third affirmative images around the y and z axes. The results of these simulations are shown for fully sampled image (Figs. 5a and 5b) and for a sample image of size 64 X 64 X 38 (Figs. 5c and 5d).

Fig. 5 MI as function of rotations of affirmative image 3 relative image 1 around the y and z axes: original image (a), zoomed around the peak (b), sampled image (c), zoomed around the peak (d).

These simulations clearly demonstrate the irregular shape of MI, when the number of sampled points is small. More importantly, these results show that the peak of MI does not correspond to the correct alignment. The implementation of registration based on MI should take the irregularity of this function into consideration, and most importantly modify it so that the global maximum corresponds to the correct transformation. In practice there are two basic methods that can be used for modifying the cost function: data smoothing and the data rebinning. Both operations are helpful in improving the stability of statistics, especially in the coarse resolution or small sample cases. As can be seen from Fig. 5, the MI function for the sub sampled image exhibits a number of local minima. These simulations also show that that the peak of MI does not reflect correct alignment. While smoothing and intensity rebinning alleviate these problems to some extent, these operations do not completely eliminate them. This again points out to the need for global optimization technique.

In these studies, registration was based on rigid body transformation with three translations and three rotations. Extensive calculations and simulations were performed to determine the optimum values of smoothing and intensity rebinning parameters, the number of chromosomes, iterations, search range, and convergence behavior of GACS and DIRECT. To quantitatively evaluate the performance of this registration technique, we initially applied various predetermined transformations (rotations and translations) to one affirmative image relative to itself and the other three images. Then we applied the registration technique to calculate the transformation parameters for aligning the images and compared them with the deliberately applied transformation parameters.

This is a reasonable approach for quantitative evaluation, since the four affirmative images are in exact alignment with each other. In these calculations, the Gaussian blurring was set to 2.0 along x, y and z directions, images were rebinned to 256 gray level, and the sampling ratios of 81, 27, 9, 3, and 1 were used in the multiresolution paradigm. This is a reasonable approach for quantitative evaluation, since the four affirmative images are in exact alignment with each other. In these calculations, the Gaussian blurring was set to 2.0 along x, y and z directions, images were rebinned to 256 gray level, and the sampling ratios of 81, 27, 9, 3, and 1 were used in the multiresolution paradigm. The number of iteration for GACS was set to 40. Typical results of these calculations along with comparisons with AIR3 are summarized in Tables 1-3. The results of AIR are included in Tables 1 and 2. It can be seen that the results obtained with AIR were quite comparable with those obtained with our method. However, in a few instances, as shown in Table 3, AIR yielded inaccurate transformation parameters while our method yielded provided expected results. The reasons for the failure of AIR in these instances are not clear.

Table 1. Comparisons between actual parameter values and the average estimated results obtained using GACS, GACS+DIRECT, and AIR for the same affirmative image. SD= Standard Deviation, the statistics are based on 20 runs for each set of transformation parameter.

Table 2. Comparisons between actual parameter values and the average estimated results obtained using GACS, GACS+DIRECT, and AIR on different affirmative images. SD= Standard Deviation, the statistics are based on 20 runs for each pair of test and reference image

Table 3. Comparisons between actual parameter values and the average estimated results obtained using GACS+DIRECT and AIR on images from different images, where AIR fails to work correctly. SD= Standard Deviation, the statistics are based on 20 runs for each set of transformation parameter.

Fig. 6 Registration results. Four successive slices are displayed in each row. Reference post-contrast images (a), pre-contrast images prior to alignment (b), affirmative images prior to alignment (c), registered pre-contrast images (d), and registered affirmative images (e).

These images demonstrate, at least visually, the excellent quality of registration. Similar results were consistently observed in all the 10 patients that were part of these studies.

Conclusions:

Global optimization of cost function based on mutual information for image registration is presented. This method is based on the combination of stochastic (GACS) and deterministic (DIRECT) optimization. Extensive simulations were performed to demonstrate the complex behavior of MI. This method was evaluated by transforming one affirmative image relative to the other images. Comparisons of performance between AIR and our method demonstrated the advantages of our method. Finally this technique was applied to register MR images of MS patients acquired with different sequences at different times with excellent results.

Acknowledgment:

This work is supported by the National Institutes of Health grant NS31499. We thank Dr. Jerry Wolinsky for providing access to the images on the MS patients.

References:

1. M. Holden, et. Al., “Voxel similarity measures for 3-D serial MR brain image registration,” IEEE Trans Med Imaging,
	vol. 19, pp. 94-102, 2000.
2. J. West et al. “Comparison and evaluation of retrospective intermodality brain image registration techniques,”
	J Comp Assist Tomogr, vol. 21, pp. 554-66, 1997.
3. X. Qi and F. Palmieri, “Theoretical analysis of evolutionary algortihms with an infinite population size in continuous
	space, Part I: Basic properties of selection and mutation. Part II: Analysis of the diversification role of the crossover,” IEEE Trans Neural Networks, vol. 5, pp. 102-129, 1994.
4. D. R. Jones, C. D. Perttunen, and B. E. Stuckman, “Lipschitzian optimization without the Lipschitz constant,” JOTA,
	vol. 7, pp. 157-181, 1993
5. B. J. Bedell, P. A. Narayana, and J. S. Wolinsky, “A dual approach for minimizing false lesion classifications on
	magnetic resonance images,” Magn Reson Med, vol. 37, pp. 94-102, 1997.

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