From Feature to Transformation Uncertainty

Here, we assume that the matches are right. However, measurement errors on the features induce an estimation error on the transformation. We developed in [31,34] a method where we register the features, estimate the noise on the frames, and propagate this noise to estimate the uncertainty of the rigid transformation.

Feature Uncertainty

For extremal points (modeled as frames), we proposed a "compositive" model of noise. Basically, this means that the noise is assumed to be identical on all extremal points in the local frame (i.e., with respect to the surface normal and the principal directions). This has to be compared with the standard additive noise model on points where we assume an identical noise with respect to the image axes. In the case of the MR images of the next section, this leads to an interesting observation: We draw in Fig. 10 a graphical interpretation of the covariance matrix estimated on the extremal points after registration. We obtain an approximately diagonal covariance matrix with standard deviations at — a^ — 2 deg, an = 6 deg for the rotation vector (these are typical values of the angle of rotation around the corresponding axis) and ax —<ax — 0.8 mm, ax = 0.2 mm for the position. As far as the trihedron part is concerned, this means that the surface normal is more stable than the principal

FIGURE 10 Graphical interpretation of the "compositive" noise model estimated on extremal points. The uncertainty of the origin ( point X) is 4 times larger in the tangent plane than along the surface normal. The uncertainty of the normal is isotropic, whereas the principal directions t1 and t2 are 3 times more uncertain in the tangent plane.

FIGURE 9 Qualitative estimation of the number of false positives involving at least % matches in MR images of 2500 features. Comparison between frames and points: We need roughly 5 times more point matches than frame matches to obtain the same probability (10 frames and 56 point matches for a probability of 10~10 ).

FIGURE 10 Graphical interpretation of the "compositive" noise model estimated on extremal points. The uncertainty of the origin ( point X) is 4 times larger in the tangent plane than along the surface normal. The uncertainty of the normal is isotropic, whereas the principal directions t1 and t2 are 3 times more uncertain in the tangent plane.

FIGURE 11 Uncertainty induced on the point positions (image corners, left; some object points, right) by the transformation.

FIGURE 11 Uncertainty induced on the point positions (image corners, left; some object points, right) by the transformation.

directions, which is expected since the normal is a first-order differential characteristic of the image and the principal directions are second order ones.

For the position, the coordinate along the normal is once again more stable than in the tangent plane for the same reasons. The 3D standard deviation of the position is a = 1.04, which is in agreement with the additive noise model on points. However, for the additive noise model, the estimated covar-iance is isotropic. Thus, using an adapted model of noise on frames allows us to extract much more information about the feature noise. This constitutes an a posteriori validation of our "compositive" model of noise on extremal points.

Transformation Uncertainty

Now the problem is to propagate the feature uncertainty to the rigid transformation. Let /represent a rigid transformation and y the observed data. The optimal transformation f minimizes a given criterion C(f, y) (for instance, the least-squares or the Mahalanobis distance). Let O(f, y) = SC(f, y)/Sf. The characterization of an optimum is Q(f, y) = 0. Now, if the data are moving around their observed values, we can relate the new optimal parameters using a Taylor expansion. Let H = SO/Sf and JO = SO/Sy be the values of the second-order derivatives of the criterion at the actual values (y, /). We have

Thus, we can express (an approximation of) the covariance of the resulting transformation using the covariance on features and the criterion derivatives.

However, a covariance matrix on a rigid transformation is quite hard to understand, since it mixes angular values for the rotation and metric values for the translation. To characterize the transformation accuracy with a single number, we can compute the uncertainty (expected RMS error) induced on a set of representative points by the registration uncertainty alone (without the uncertainty due to the feature extraction).

In our case, two sets are particularly well suited: The position of the matched extremal point represents the localization of the object of interest, whereas the corners of the image symbolize the worst case (Fig. 11). In the following example, we find for instance a typical boundary precision around acorn = 0.11 mm and a typical object precision far below the voxel size: aobj = 0.05 mm for echo-1 registrations. The values are even a little smaller for echo-2 registrations: acorn = 0.10 and aobj = 0.045 mm.

Validation Index

Last but not least, we need to validate this whole chain of estimations to verify if our uncertainty prediction is accurate. We observed that, under the Gaussian hypothesis, the Mahalanobis distance between the estimated and the exact transformation (or between two independent estimations of the same transformation) should be y6 distributed (if the covariance matrix on the estimation is exact). To verify this, the idea is to repeat a registration experiment N times and to compute the empirical mean value I = Ji2 = N X) <"2 and the variance a2 of this Mahalanobis distance. The values for an exact y62 are respectively 6 and 12. We can also verify using the Kolmogorov-Smirnov test (K-S test) that the empirical distribution corresponds to the exact distribution. The validation index I [34] can be interpreted as an indication of how the estimation method underestimates (I >6) or overestimates (I < 6) the covariance matrix of the estimated transformation. It can also be seen as a sort of relative error on the error estimation.

We run several sets of tests with synthetic data and verify that our uncertainty estimations very perfectly validated for more than 15 extremal point matches. Now, the question we want to answer is: It is still valid for real data?

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