We present a new class of estimators for approximating the entropy of multi-dimensional probability densities based on a sample of the density. These estimators extend the classic "m-spacing" estimators of Vasicek and others for estimating entropies of one-dimensional probability densities. Unlike plug-in estimators of entropy, which £rst estimate a probability density and then compute its entropy, our estimators avoid the dif£cult intermediate step of density estimation. For £xed dimension, the estimators are polynomial in the sample size. Similarities to consistent and asymptotically ef£cient one-dimensional estimators of entropy suggest that our estimators may share these properties.
Coregistration of different modality imaging serves to increase the ease and accuracy of stereotactic procedures. In many cases, magnetic resonance (MR) stereotaxis is supplanting computerized tomography (CT). The advantages of increased anatomical detail and multiplanar imaging afforded by MR, however, are offset by its potential inaccuracy as well as the more cumbersome and less available nature of its hardware. A system has been developed by one of the authors by which MR imaging can be performed separately without a stereotactic fiducial headring. Then, immediately prior to surgery, a stereotactic CT scan is obtained and software is used to coregister CT and MR images anatomically by matching cranial landmarks in the two scans. The authors examined this system in six patients as well as with the use of a lucite phantom. After initially coregistering CT and MR images, six separate anatomical (for the patients) and eight artificial (for the phantom) targets were compared. With coregistration, in comparison to CT fiducial scans, errors in each axis are less than or equal to 1 mm using the Cosman-Roberts-Wells system. In fact, the coregistered images are more accurate than MR fiducial images, in the anteroposterior (p = 0.001), lateral (p < 0.05), and vertical (p < 0.03) planes. Three-dimensional error was significantly less in the coregistered scans than the MR fiducial images (p < 0.005). The coregistration procedure therefore not only increases the case of MR stereotaxis but also increases its accuracy.
Abstract. Assessment of normal and abnormal anatomical variability requires a coordinate system enabling inter-subject comparison. We present a binary minimum entropy criterion to assess affine and nonrigid transformations bringing a group of subject scans into alignment. This measure is a data-driven measure allowing the identification of an intrinsic coordinate system of a particular group of subjects. We assessed two statistical atlases derived from magnetic resonance imaging of newborn infants with gestational age ranging from 24 to 40 weeks. Over this age range major structural changes occur in the human brain and existing atlases are inadequate to capture the resulting anatomical variability. The binary entropy measure we propose allows an objective choice between competing registration algorithms to be made.
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