We address the problem of image segmentation with statistical shape priors in the context of the level set framework. Our paper makes two contributions: Firstly, we propose to generate invariance of the shape prior to certain transformations by intrinsic registration of the evolving level set function. In contrast to existing approaches to invariance in the level set framework, this closed-form solution removes the need to iteratively optimize explicit pose parameters. Moreover, we will argue that the resulting shape gradient is more accurate in that it takes into account the e#ect of boundary variation on the object's pose.

In earlier work with Gabor Lugosi, we introduced a method to select a smoothing factor for kernel density estimation such that, for all densities in all dimensions, the L1 error of the corresponding kernel estimate is not larger than 3+e times the error of the estimate with the optimal smoothing factor plus a constant times Ov~--~-n/n, where n is the sample size, and the constant only depends on the complexity of the kernel used in the estimate. The result is nonasymptotic, that is, the bound is valid for each n. The estimate uses ideas from the minimum distance estimation work of Yatracos. We present a practical implementation of this estimate, report on some comparative results, and highlight some key properties of the new method.

Abstract. This paper presents a novel statistical fuzzy-segmentation method for diffusion tensor (DT) images and magnetic resonance (MR) images. Typical fuzzy-segmentation schemes, e.g. those based on fuzzy-C-means (FCM), incorporate Gaussian class models which are inherently biased towards ellipsoidal clusters. Fiber bundles in DT images, however, comprise tensors that can inherently lie on more-complex manifolds. Unlike FCM-based schemes, the proposed method relies on modeling the manifolds underlying the classes by incorporating nonparametric datadriven statistical models. It produces an optimal fuzzy segmentation by maximizing a novel information-theoretic energy in a Markov-randomfield framework. For DT images, the paper describes a consistent statistical technique for nonparametric modeling in Riemannian DT spaces that incorporates two very recent works. In this way, the proposed method provides uncertainties in the segmentation decisions, which stem from imaging artifacts including noise, partial voluming, and inhomogeneity. The paper shows results on synthetic and real, DT as well as MR images. 1

this paper, we are concerned with the L 1-consistency of f. It is known [Chow, Geman and Wu (1983) and Devroye and GySrfi [DG] (1985), pages 153-154] that whenever f has compact support, then almost surely as n-, , (1.5) IL(x) -/(x)l 0

Abstract. Let f nh be the Parzen-Rosenblatt kernel estimate of a density f on the real line, based upon a sample of n i.i.d. random variables drawn from f, and with smoothing factor h. Let g nh be another kernel estimate based upon the same data, but with a different kernel. We choose the smoothing factor H so as to minimize � |f nh − g nh|, and study the properties of f nH and g nH. It is shown that the estimates are consistent for all densities provided that the characteristic functions of the two kernels do not coincide in an open neighborhood of the origin. Also, for some pairs of kernels, and all densities in the saturation class of the first kernel, we show that lim sup n→∞ E � � |fnH − f|) � E � � � ≤ C, infh |fnh − f| where C is a constant depending upon the pair of kernels only. This constant can be arbitrarily close to one.

Abstract. This paper presents a novel segmentation-based approach for fiber-tract extraction in diffusion-tensor (DT) images. Typical tractography methods, incorporating thresholds on fractional anisotropy and fiber curvature to terminate tracking, can face serious problems arising from partial voluming and noise. For these reasons, tractography often fails to extract thin tracts with sharp changes in orientation, e.g. the cingulum. Unlike tractography—which disregards the information in the tensors that were previously tracked—the proposed method extracts the cingulum by exploiting the statistical coherence of tensors in the entire structure. Moreover, the proposed segmentation-based method allows fuzzy class memberships to optimally extract information within partial-volumed voxels. Unlike typical fuzzy-segmentation schemes employing Gaussian models that are biased towards ellipsoidal clusters, the proposed method models the manifolds underlying the classes by incorporating nonparametric data-driven statistical models. Furthermore, it exploits the nonparametric model to capture the spatial continuity and structure of the fiber bundle. The results on real DT images demonstrate that the proposed method extracts the cingulum bundle significantly more accurately as compared to tractography. 1

has been read by each member of the following supervisory committee and by majority vote has been found to be satisfactory.

Abstract: Let f, be the L, cross-validated kernel estimate of a univariate density f. We show that 1iminfE If,-fl>,l n-m / when K is a symmetric bounded unimodal density, f is a monotone density on [0, 00) and x3/4f(x) + 00 as x JO.

Kernel density estimation, pre-smoothed, cross-validation, optimal rates AMS 1980 Subject Classifications: Primary 62G05, Secondary 62G20.. In the problem of nonparametric density estimation, kernel estimators 4It with cross-validated bandwidths are considered. An example is given to show that, even in the case where both density function and kernel have compact support, ordinary cross-validation is sub-optimal. A "pre-smoothed " modification of the cross-validated technique is proposed. By techniques similar to those of Hall (Biometrika 69 (1982) 383-390) it is shown that this density estimator achieves the well-known asymptotically optimal rate of convergence. This estimator does not make use of the precise amount of smoothness that is assumed on the density, but it is required that the density be not too smooth

Nonparametric density estimation, kernel estimator, In the problem of nonparametric estimation of a probability density, kernel estimators are considered. Uniform consistency is established when the bandwidth is chosen by a version of cross-validation. This estimator has been shown in Marron (1983a) to have excellent mean integrated square error properties