Validation with Real Data

The experiment is performed using multiple 2D contiguous magnetic resonance images (MRI) that constitute a 3D

Ring Enhancing Lesion

FIGURE 12 (Left) Example of MS images. The same slice of one acquisition in echo-1(le/t) and echo-2 (right). (Right) evolution of an image row going through a lesion across 24 time points over a year. Left: without registration; Right: after registration and intensity correction. Original 3D images courtesy of Dr. Charles Guttmann and Prof. Ron Kikinis from the Brigham and Woman's Hospital (Harvard Medical School, Boston).

FIGURE 12 (Left) Example of MS images. The same slice of one acquisition in echo-1(le/t) and echo-2 (right). (Right) evolution of an image row going through a lesion across 24 time points over a year. Left: without registration; Right: after registration and intensity correction. Original 3D images courtesy of Dr. Charles Guttmann and Prof. Ron Kikinis from the Brigham and Woman's Hospital (Harvard Medical School, Boston).

representation of the head. The images are part of an extensive study of the evolution of the multiple sclerosis (MS) disease performed at the Brigham and Woman's Hospital (Harvard Medical School, Boston) by Dr. Guttmann and Prof. Kikinis. Each patient underwent a complete head MR examination several times during one year (typically 24 different 3D acquisitions). The aim is to register precisely all the images acquired at multiple time points in order to segment the lesions and evaluate very accurately their evolution (Fig. 12).

Each acquisition provides a first echo image and a second echo image (typically 256 x 256 x 54 voxels of size (0.9375 x 0.9375 x 3 mm). The two images represent the same slice of T2 weighted signal imaged at different echo times. Thus, they are expected to be in the same coordinate system. This protocol was designed to optimize the contrast in the two channels for easier tissue segmentation. Considering two acquisitions A and B, the registrations of echo-1 images (A1 to B1) and echo-2 images (A2 to B2) give two relatively independent estimates of the genuine transformation from A to B. The comparison of these two transformations using the Mahalanobis distance gives a real validation index that can be tested for the accuracy of the uncertainty estimation.

In this experiment, the images being close enough, we used the iterative closest feature algorithm. Typically, we matched 1000 extremal points out of the about 3000 extracted with a residual mean square error (RMS) of about 1 mm.

Direct Validation Shows Biases With n different acquisitions, we can run n * (n — 1)/2 registrations per echo. In a first experiment, we compared directly the registrations between the corresponding echo-1 and echo-2 images. The resulting validation index clearly indicates that the transformations do not agree (p = I>50 instead of 6). However, our registration method cannot detect systematic biases.

To discover the biases, we ran a series of experiments where we repeated the same registration while varying the algorithm parameters. This confirms that the observed uncertainty is similar in size and shape to the predicted one. Moreover, other experiments show that the inter-echo-1 and the inter-echo-2 registrations are compatible but the two groups significantly differ (Fig. 13). Thus we concluded that there was a systematic bias between echo-1 and echo-2 registrations. Additional experiments showed that the bias was different for each registration.

FIGURE 13 This diagram represents three acquisitions A, B, and C with the three echo-1 images (Aj, Bj, Q) and the three echo-2 images (A2, B2, C2). The echo-1 and echo-2 registrations are significantly different

(P2(fab1, /ab2 ^ P2(fac1 , /ac2 ), P2(fbc1, /bc2 ) > 50) but the mtra-ech°-1 and intra-echo-2 registrations are compatible (p2(fBC1 ofAB1,fACl)—6 and H2(fBCi o fAB , fACi) —6). This led us to assume a global bias for each acquisition between echo-1 and echo-2 images, represented here by the transformations fA, fB, and fC.

FIGURE 13 This diagram represents three acquisitions A, B, and C with the three echo-1 images (Aj, Bj, Q) and the three echo-2 images (A2, B2, C2). The echo-1 and echo-2 registrations are significantly different

(P2(fab1, /ab2 ^ P2(fac1 , /ac2 ), P2(fbc1, /bc2 ) > 50) but the mtra-ech°-1 and intra-echo-2 registrations are compatible (p2(fBC1 ofAB1,fACl)—6 and H2(fBCi o fAB , fACi) —6). This led us to assume a global bias for each acquisition between echo-1 and echo-2 images, represented here by the transformations fA, fB, and fC.

Estimation of the Biases To estimate the biases, we first observed that the transformation from image A1 to image B2 can be written ^A1b2 = fB ° fAB1 = fAB2 ° Ia. If measurements where perfect, the bias fA could be expressed for any other image Z: fA = fA^ ° fz ° fAz1. Since measurements are noisy, we obtain an estimator of the bias fA by taking the Frechet mean value [29]:

In this formula, each acquisition bias depends upon the others. Thus, we begin with null biases (identity transformations) and iteratively estimate each bias until convergence.

We effectively obtain a different bias for each acquisition that significantly differs from the identity. However, from a more global point of view, all the biases could be modeled as an "additional" noise on the transformation with an identity mean and standard deviations of < = 0.06 deg on the rotation (not significantly different from 0) and <jx = 0.09, ay = 0.11 and < = 0.13 mm on the translation (significantly different from 0). Very similar values were observed for other patients.

Validation with Bias Although the biases appear very small, they are sufficient to explain the previous errors in the registration accuracy prediction. Indeed, taking the biases into account, the real validation index between acquisition A and B becomes lAB = M2( fB ° fAB,, fAB2 ° fA) .

Since the biases are estimated from the registration values, using their uncertainties in this formula would bias the validation index toward low values. Thus, we consider them as deterministic. The mean value and standard deviation of this new index across all registrations are now very close to their theoretical value (see Table 1).

Origin of the Bias Most of the extremal points we match are situated on the surface of the brain and the ventricles. These surfaces appear differently in echo-1 and echo-2 images because of the difference in contrast. Other artifacts such as chemical shift or susceptibility effects (see for instance [18]) may also account for the observed bias as they influence the detection of extremal points. Indeed, the two echoes are acquired with different receiver RF bandwidth to improve the signal/noise ratio [19]. Therefore, the chemical shift and the susceptibility effect are different in the two echoes.

We plan to correlate the biases with diverse quantities in the images in order to understand their origin. Ultimately, we would like to predict the biases in each acquisition before registration. This would allow the definite validation of the registration accuracy prediction.

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