Non-linear registration of MRI acquisition of different subjects

Contact: Pierre Hellier , Christian Barillot,Etienne Memin, Patrick Perez

Purposes
Method
General formulation
Robust estimators
Multiresolution and multigrid approaches
Results
Experiment on simulated data
Experiment on real data
Experiment on a database of 18 subjects
Purposes
The ways of observing human brain have tremendously evolved during the past 10 years. Nowadays, physicians must face not only the huge volume of data, but also the complementarity between the different images. As a matter of fact, the different acquisitions are not redundant but complementary, and should not be neglected for the patient's health. Medical image registration has thus become a crucial issue.

The non-linear registration of brains from different subjects allows to build an anatomical atlas of the cortex. Some atlases (Ono, Talairach) already exist, but they appear to be inadequate, because they often lack legibility and capacity to evolve, and their interpretation is very difficult. The major problem in building an atlas is the important variability of the human brain. It has been clearly shown (by Ono in particular) that we cannot assume topological equivalence between two different brains. Considering the same sulcus of different subjects, one may find large differences of orientation, size, and even topology (one sulcus may be interrupted or absent for instance).

We designed a new method for medical image registration. The registration is formulated as a minimization problem involving robust estimators. We propose an efficient hierarchical optimization framework which is both multiresolution and multigrid in order to accelerate the algorithm and to improve the quality of the estimation. The adaptive partition, on wich the multigrid minimization is performed, allows to limit the estimation to the areas of interest, to accelerate the algorithm, and to refine the estimation in specified areas. The performances of this method are objectively evaluated on simulated data and its benefits are demonstrated on a large database of real acquisitions.
Method

We formulate the registration problem as the minimization of a cost function including two terms: a similarity measure (in the monomodal case we choose the optical flow), and a regularization term. The optical flow hypothesis, introduced by Horn et Schunck, assumes that the luminance of a physical point does not vary much between the two volumes to be registered. To regularize the deformation field, we introduce a quadratic difference of the deformation field between neighbours. We thus minimize the following cost function:
Where f stands for the luminance function, w is the estimated deformation field cd C is the set of neighbouring pairs. Furthermore, we incorporate robust estimators in the cost function whose utility is twofold: to limit the influence of the acquisition noise and to be able to estimate transformations that modify the topology of the structures. More precisely, we introduce robust M-estimators, leading to the minimization of the cost function:
This formulation is still quadratic with respect to the deformation field and is more robust. The minimization is now alternated (between updating the weights and estimating the deformation field). In the case of large displacement, we use a classical incremental multiresolution procedure (see figure). At the coarsest level, displacements are reduced, and the linearization hypothesis (linear expansion of the optical flow hypothesis) can be used.

Furthermore, at each level of resolution, we use a multigrid minimization based on successive partitions of the initial volume (see animation). At each grid level, corresponding to a partition of cubes, we estimate a 12-dimension parametric increment field for each cube of the partition. The energy is consequently smoother, and has fewer local minima. This results is a rough estimate of the desired solution, which is then used to initialize the next level. This optimization strategy improves the quality and the convergence rate as compared to standard iterative solvers (such as Gauss-Seidel). When we change from grid level, each cube is adaptively divided.

Example of multiresolution/multigrid minimization. For each resolution level (on the left), a multigrid strategy (on the right) is performed. For legibility reasons, the figure is a 2D illustration of a 3D algorithm with volumetric data.
Example of multiresolution/multigrid minimization. For each resolution level (on the left), a multigrid strategy (on the right) is performed. For clarity reasons, the figure is a 2D illustration of a 3D algorithm with volumetric data.

Results
The simulated data of the MNI (Montreal Neurological Institute) is used to evaluate the registration method. Data have been collected with 3 levels of noise and inhomogeneity. We designe a synthetic deformation field made up of a global affine field with large deformations combined with local stochastic perturbations. We do not try to build a ``realistic'' field, but rather a field with the following properties: large deformations and local perturbations that modify the topology of the structures, in order to validate the basic hypothesis of our work. We compare the multigrid method with a global affine registration method, in which a 12-parameter deformation is estimated for the entire volume.

As we have the binary classification of the phantom, we can asses the quality of the registration based on the overlap of two volumes: the first volume is the initial classification, i.e. a gold standard (grey matter/white matter), the second volume is the deformed classification, registered with the estimated deformation field. We then measure out overlapping ratios like the sensibility, the specificity, and the total performance. Results are presented in following table. We also compute the mean square error (MSE) which is an indicator of the quality of the registration. Despite the use of binary classes defined on a discrete grid, the resulting measures that we obtain are very satisfactory. In particular, the algorithm performs well even under the severist imaging conditions (9%noise and 40% inhomogeneity).


Numerical measures of the quality of the registration on simulated data. Specificity, sensibility and total performance measures are given for 3 levels of noise and 2 registration methods. We manage to recover up to 93% of the deformation even in presence of important noise (9%) and image intensity inhomogeneity (40%).


Results of the 3D method are presented in the next figure. Two 3D MRI-T1 volumes of two different subjects are registered. The reconstructed volume -with trilinear interpolation- is presented, computed with the target volume and the final displacement field. The quality of the registration is assed by comparing the reconstructed volume with the source volume. We also present two volumes of difference, one before and one after registration. The adaptive partition at grid level 2 (we do not present further grid level for readability reasons) is also presented.

The difference volumes must be interpreted carefully, since we get the superposition of two errors. The first is the registration error that the method could not account for. The second error is due to the variability of the acquisition process making the two original histograms different. The computation takes about 1:30 hour on an Ultra Sparc 30 (300 Mhz). The volumes are 256*256*200. We use 3 levels of resolution because the displacement amplitude may reach 30 voxels.


source volume reconstructed volume target volume
AVI (629Ko) or MPEG (1308 Ko) AVI (619Ko) or MPEG (2537 Ko) AVI (943Ko) or MPEG (2615 Ko)


initial difference final difference adaptive partition

Furthermore, for each grid and resolution level, we compute the reconstructed volume at the initial volume size. The next animation (avi) shows how the registration is refined in a hierarchical way. Large deformations are estimated at coarsest resolution levels (blurred reconstructed volumes).


hierachical estimation of the deformation field
AVI (124 Ko) or MPEG (627 Ko)

In order to validate the registration method on a larger database, we acquire MRI-T1 volumetric data of 18 patients (here you can see an animation -avi (174Ko) or mpeg (829Ko)- of sagittal slices of the 18 subjects). One subject was chosen as the reference subject. We then performe the registration between the reference volume (source) and each of the other subjects (target) using always the same set of parameters for the algorithm. Finally we get 17 reconstructed volumes that may be compared to the reference volume. We averaged the reconstructed volume in order to have a global overview of the quality of the method. Next figure presents the average volume computed over 17 patients after (a) global affine registration (b) robust multigrid registration. After global affine registration and averaging, the internal anatomical structures are blurred, because the registration is not precise enough. However, after a robust multigrid registration, we may distinguish precisely the contours of anatomical structures, such as ventricles, deep nuclei, white matter tracks, and even cortical regions (sylvian fissure and parietal region for instance). This demonstrates the robustness of the method over a realistic database of subjects.


averaging after global affine registration

averaging after robust multigrid registration

reference subject


References

  1. P. Hellier, C. Barillot, E. Mémin, P. Pérez. Medical image registration with robust multigrid techniques. Proc. of the 2nd Int. Conf. on Medical Image Computing and Computer-Assisted Intervention, MICCAI, C. Taylor, A. Colchester (eds.), Springer Verlag, Lecture Notes in Computer Science, Volume 1679, pages 680-687, Septembre 1999.
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Last modified: Wed Feb 2 10:24:54 MET 2000