Calculation of the confidence intervals for transformation parameters in the registration of medical images
Images from different individuals typically cannot be registered precisely because anatomical features within the images differ across the people imaged and because the current methods for image registration have inherent technological limitations that interfere with perfect registration. Quantifying the inevitable error in image registration is therefore of crucial importance in assessing the effects that image misregistration may have on subsequent analyses in an imaging study. We have developed a mathematical framework for quantifying errors in registration by computing the confidence intervals of the estimated parameters (3 translations, 3 rotations, and 1 global scale) for the similarity transformation. The presence of noise in images and the variability in anatomy across individuals ensures that estimated registration parameters are always random variables. We assume a functional relation among intensities across voxels in the images, and we use the theory of nonlinear, least-squares estimation to show that the parameters are multivariate Gaussian distributed. We then use the covariance matrix of this distribution to compute the confidence intervals of the transformation parameters. These confidence intervals provide a quantitative assessment of the registration error across the images. Because transformation parameters are nonlinearly related to the coordinates of landmark points in the brain, we subsequently show that the coordinates of those landmark points are also multivariate Gaussian distributed. Using these distributions, we then compute the confidence intervals of the coordinates for landmark points in the image. Each of these confidence intervals in turn provides a quantitative assessment of the registration error at a particular landmark point. Because our method is computationally intensive, however, its current implementation is limited to assessing the error of the parameters in the similarity transformation across images. We assessed the performance of our method in computing the error in estimated similarity parameters by applying that method to real world dataset. Our results showed that the size of the confidence intervals computed using our method decreased  i.e. our confidence in the registration of images from different individuals increased  for increasing amounts of blur in the images. Moreover, the size of the confidence intervals increased for increasing amounts of noise, misregistration, and differing anatomy. Thus, our method precisely quantified confidence in the registration of images that contain varying amounts of misregistration and varying anatomy across individuals.
similarity transformation confidence region nonlinear least-squares mutual information