Kluwer - Handbook of Biomedical Image Analysis Vol

13.2.1.3 K-means Clustering

In a digital image, regions with the same structure may have corresponding spikes in their intensity distributions. Intervals of the intensity distribution are more likely to have a higher variance if the same structure region is not in the same interval. For example, in standard CT or MR images, which in general contain different structure regions, such as background, cortical bone, white matter, gray matter, and cerebrospinal fluid (CSF). In the joint-histogram we would like to see the same structure is assigned to the same bin, i.e., for each bin to have a high probability of the same structure so that less dispersion in the joint-histogram is achieved. The method proposed here is to make all the voxels of background map to one bin by background segmentation using region growing, and have the remaining voxels map to the rest of bins (with a variable bin size for each bin) by k-means clustering, i.e., minimizing the variance of intensities within each bin [22].

Following is the k-means clustering algorithm used:

1. Initially partition the image voxels into k bins.

(1a) Put all the background voxels into bin 0.

(1b) Calculate the step size for the other k − l bins using

MaxIntensity−Minlntensity . Each bin will be assigned all voxels whose

k − 1

intensity falls within the range of its boundary. (1c) Calculate the centroid of each bin.

2.For each voxel in the image, compute the distances to the centroids of its current, previous, and next bin, if it exists; if it is not currently in the bin with the closest centroid, switch it to that bin, and update the centroids of both bins.

3.Repeat step 2 until convergence is achieved; that is, continue until a passthrough all the voxels in the image causes no new assignments, or until a maximum number of iterations is reached. The maximum number of iterations was set to 500.

13.2.1.4 Normalized Mutual Information

Mutual information (MI) can be thought of as a measure of how well one image

explains the other, and when maximized indicates optimal alignment.

Mutual information I( A, B) was proposed for intermodality medical image registration by several researchers [2, 3, 11] . The formula to compute mutual information is:

where PA(a), PB(b) denote the marginal probability distributions of the two random variables, A and B, and PAB (a, b) is their joint probability.

The formulation of normalized mutual information (NMI) as described by Studholme et al. [17] is used:

N M I( A, B) = H( A) + H(B)

H( A, B)

where

H( A) = − PA(a) log PA(a)

a

and

H( A, B) = − PAB(a, b) log PAB(a, b)

ab

H( A) and H(B) are the entropies of A and B, and H( A, B) is their joint entropy. NMI, devised to overcome the sensitivity of MI to change in image overlap, has been shown to be more robust than standard mutual information [17]. Image registration using NMI is performed in the following manner:

1.Take one of the two input images as floating image, the other as reference image.

2.Choose starting parameters for the transformation.

3.Apply the transformation to the floating image. Evaluate the NMI between reference image and transformed floating image.

4.Stop if convergence is achieved. If not, choose a new set of parameters, repeat steps 3 and 4.

When the registration parameters (three translations and three rotations) are applied to the floating image, the algorithm iteratively transforms the floating image with respect to the reference image while making the NMI measure calculated from the voxel intensities optimal. While all samples are taken at grid points of the floating image, their transformed position will, in general, not

coincide with a grid point of the reference image and interpolation of the reference image is needed to obtain the image intensity value at this point.

The NMI registration criterion states that the images are geometrically aligned by the transformation Tλ for which NMI(A, B) is maximal. (The registration parameters are denoted by λ and Tλ is the transformation based on the six registration parameters given above.) To decide if the NMI is optimal, algorithms to measure optimality are applied. If the NMI is not optimal, a new set of parameters will be chosen and evaluated. The Downhill simplex is used to determine optimality. This method, as shown by NeIder and Mead [19, 20], only requires evaluation of the cost function, the derivative computation need not be redone.

13.2.1.5 Two-Stage Multiresolution Registration

Multiresolution approach [7–10] is widely used in medical image registration due to the following two features:

1.Methods for detecting optimality cannot guarantee that a global optimal value will be found. Parameter spaces for image registration are usually not simple. There are often multiple optima within the parameter space, and registration can fail if the optimization algorithm converges to the wrong optimum.

2.Time to evaluate the registration criterion is proportional to the number of voxels: Medical images consist of a large number of voxels. During registration the main portion of computational time is consumed by the resampling voxels of the floating image with respect to the reference image according to actual geometrical transformation.

The idea of a multiresolution hierarchical approach is to register a coarse (low resolution) image first and then to use the result as the starting point for finer (high resolution) image registration, and so on [24]. In practice, the multiresolution approach proves to be helpful. It can improve the optimization speed, improve the capture range, and the algorithm is relatively robust.

In the first level of our two-stage multiresolution registration method, one of the three binning techniques, described above, is applied to the segmented images before they are down-sampled and the registration is performed on the

coarser level. Nearest neighbor interpolation method is used in this level. In the second level, the original images are registered and a trilinear partial volume interpolation method [18] is employed. This process is done for each of the three binning techniques.

13.2.2 Experiments and Discussion

13.2.2.1 Data Set

This study involved the data sets of seven patients, each consisting of CT and a subset of six Magnetic-Resonance (MR) volumes (spin-echo T1, PD, T2 and the rectified version of each of these three). The MR-T1-rectified image was not available for patient six. Thus, 41 image pairs were available to be registered. These images were provided by The Retrospective Registration Evaluation Project database maintained by J. Michael Fitzpatrick from Vanderbilt University, Nashville, TN, USA [26]. For each patient data set, all CT images were registered to the MR image using the MR image as the reference image.

All image pairs were registered using the maximization of NMI. Registration transformation was limited to six-parameter rigid-body type (three translations and three rotations) transformations. Registrations were conducted on a PC, having a 2.4 Ghz Intel Pentium 4 processor, and a 512MB DDR SDRAM memory.

The experiments were performed using a two-stage multiresolution approach described above. When the binarization approach was used, the foreground/foreground bin was given an additional weight when the joint histogram was computed in the first level. The additional weight was heuristically determined using the ratio of foreground to background voxels. The number of histogram bins used to compute the normalized mutual information criterion was set to 256 bins for both CT and MR.

The Downhill-simplex optimization method was used throughout the experiments. Optimization for each pair of images started from the same initial position with all three translation offsets set to zero millimeters and all three rotation angles set to zero degrees. Convergence was declared when the fraction difference between the lowest and the highest function value evaluated at the vertices of the simplex was smaller than 10−5.

Data Set Pairs

(a)

1800

1600

1400

1200

Data Set Pairs

( b)

Figure 13.5: Time required to perform registration for each of the 41 CT-MR

pairs: (a) the binarization approach and (b) the nonlinear binning method.

13.2.2.3 Accuracy

The accuracy of all 41 experiments with respect to the fiducial-mark-base gold standard can be found on the web (see http://www.vuse.vanderbilt.edu/ image/registration/).

In addition, we compare the results of our approaches to those of several other approaches published in the literature. The comparison is based on a methodology proposed by West and Fitzpatrick [21], who let selected researchers access a standard set of image volumes to be registered. They also act as a repository for the ideal registration transformations (gold standard) acquired by a prospective method using fiducial markers. These markers are removed before the volumes are disclosed to the investigators, who then face a retrospective blind registration task. After registration, they email back a set of transformation parameters that are compared to the gold standard.

Tables 13.1 and 13.2 show the median and maximum accuracy reached by the investigators taking part in that project [21]. All errors are in units of mm. The method using binarization approach is labelled LO1, while the method using nonlinear binning technique is labelled LO2. We observed that both techniques give impressive results for CT-MR registration.

13.2.2.4 Optimization Steps

In the downhill simplex optimization method, the number of optimization steps is measured by the number of times the cost function is called. The mean number and the standard deviation of optimization steps for the three binning techniques, for each patient data set, is compared for CT-MR registration in Table 13.3 and Table 13.4.

The binarization approach needs the least number of optimization steps in the first level (Table 13.3). The methods using linear binning and nonlinear binning need 30% to 102% more steps than the binarization approach. Of the three binning techniques, the binarization approach has the most stable performance for all the patients. The reason is that binarization can give an extreme blurring of the images and so eliminates local optima. The performances of linear and nonlinear binning are pretty much the same, while nonlinear binning is better in four out of seven patients.

Table 13.1: Median error for CT-to-MR registration in ‘mm’ between the prospective gold-standard and several retrospective registration techniques. The label “rect.” indicates that the MR image was corrected for geometrical distortion before registration. The result of the technique using the binarization approach is labelled LO1, while the one using the nonlinear binning method is labelled LO2.

Table 13.2: Maximum error for CT-to-MR registration. See notes in Table 13.1.

Table 13.3: Number of optimization steps used by each of the three

binning techniques for first level.