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18
Kernel Density Estimation and Intrinsic Alignment for Knowledgedriven Segmentation: Teaching Level Sets to Walk
 International Journal of Computer Vision
, 2004
"... 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 fun ..."
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Cited by 111 (16 self)
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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 closedform 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.
Universal smoothing factor selection in density estimation: theory and practice (with discussion
 Test
, 1997
"... 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 fac ..."
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Cited by 31 (11 self)
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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.
A fuzzy, nonparametric segmentation framework for
 DTI and MRI analysis,” in Proc. Inf. Process. Med. Imag. (IPMl), 2007
"... Abstract—This paper presents a novel fuzzysegmentation method for diffusion tensor (DT) and magnetic resonance (MR) images. Typical fuzzysegmentation schemes, e.g., those based on fuzzy C means (FCM), incorporate Gaussian class models that are inherently biased towards ellipsoidal clusters charact ..."
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Cited by 22 (2 self)
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Abstract—This paper presents a novel fuzzysegmentation method for diffusion tensor (DT) and magnetic resonance (MR) images. Typical fuzzysegmentation schemes, e.g., those based on fuzzy C means (FCM), incorporate Gaussian class models that are inherently biased towards ellipsoidal clusters characterized by a mean element and a covariance matrix. Tensors in fiber bundles, however, inherently lie on specific manifolds in Riemannian spaces. Unlike FCMbased schemes, the proposed method represents these manifolds using nonparametric datadriven statistical models. The paper describes a statisticallysound (consistent) technique for nonparametric modeling in Riemannian DT spaces. The proposed method produces an optimal fuzzy segmentation by maximizing a novel informationtheoretic energy in a Markovrandomfield framework. Results on synthetic and real, DT and MR images, show that the proposed method provides information about the uncertainties in the segmentation decisions, which stem from imaging artifacts including noise, partial voluming, and inhomogeneity. By enhancing the nonparametric model to capture the spatial continuity and structure of the fiber bundle, we exploit the framework to extract the cingulum fiber bundle. Typical tractography methods for tract delineation, 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, such as the cingulum. The results demonstrate that the proposed method extracts this structure significantly more accurately as compared to tractography. Index Terms—Diffusion tensor imaging (DTI), fuzzy sets, image segmentation, information theory, magnetic resonance imaging (MRI), Markov random fields, nonparametric modeling, Riemannian statistics.
The double kernel method in density estimation
 Ann. Inst. Henri Poincar'e
, 1989
"... Abstract. Let f nh be the ParzenRosenblatt 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 ..."
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Cited by 9 (1 self)
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Abstract. Let f nh be the ParzenRosenblatt 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.
Multivariate highdimensional cortical folding analysis, combining complexity and shape, in neonates with congenital heart disease
 In IPMI
, 2009
"... Abstract. The paper presents a novel statistical framework for cortical folding pattern analysis that relies on a rich multivariate descriptor of folding patterns in a region of interest (ROI). The ROIbased approach avoids problems faced by spatialnormalizationbased approaches stemming from the ..."
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Cited by 6 (1 self)
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Abstract. The paper presents a novel statistical framework for cortical folding pattern analysis that relies on a rich multivariate descriptor of folding patterns in a region of interest (ROI). The ROIbased approach avoids problems faced by spatialnormalizationbased approaches stemming from the severe deficiency of homologous features between typical human cerebral cortices. Unlike typical ROIbased methods that summarize folding complexity or shape by a single number, the proposed descriptor unifies complexity and shape of the surface in a highdimensional space. In this way, the proposed framework couples the reliability of ROIbased analysis with the richness of the novel cortical folding pattern descriptor. Furthermore, the descriptor can easily incorporate additional variables, e.g. cortical thickness. The paper proposes a novel application of a nonparametric permutationbased approach for statistical hypothesis testing for any multivariate highdimensional descriptor. While the proposed framework has a rigorous theoretical underpinning, it is straightforward to implement. The framework is validated via simulated and clinical data. The paper is the first to quantitatively evaluate cortical folding in neonates with complex congenital heart disease. 1
On The Relationship Between Stability Of Extreme Order Statistics And Convergence Of The Maximum Likelihood Kernel Density Estimate
 Annals of Statistics
, 1989
"... this paper, we are concerned with the L 1consistency of f. It is known [Chow, Geman and Wu (1983) and Devroye and GySrfi [DG] (1985), pages 153154] that whenever f has compact support, then almost surely as n, , (1.5) IL(x) /(x)l 0 ..."
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Cited by 5 (4 self)
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this paper, we are concerned with the L 1consistency of f. It is known [Chow, Geman and Wu (1983) and Devroye and GySrfi [DG] (1985), pages 153154] that whenever f has compact support, then almost surely as n, , (1.5) IL(x) /(x)l 0
J.C.: Fuzzy nonparametric dti segmentation for robust cingulumtract extraction
 MICCAI (1). Volume 4791 of Lecture Notes in Computer Science
, 2007
"... Abstract. This paper presents a novel segmentationbased approach for fibertract extraction in diffusiontensor (DT) images. Typical tractography methods, incorporating thresholds on fractional anisotropy and fiber curvature to terminate tracking, can face serious problems arising from partial volu ..."
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Cited by 4 (0 self)
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Abstract. This paper presents a novel segmentationbased approach for fibertract extraction in diffusiontensor (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 segmentationbased method allows fuzzy class memberships to optimally extract information within partialvolumed voxels. Unlike typical fuzzysegmentation schemes employing Gaussian models that are biased towards ellipsoidal clusters, the proposed method models the manifolds underlying the classes by incorporating nonparametric datadriven 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
Clinical Neonatal Brain MRI Segmentation using Adaptive Nonparametric Data Models and Intensitybased Markov Priors
"... Abstract. This paper presents a Bayesian framework for neonatal braintissue segmentation in clinical magnetic resonance (MR) images. This is a challenging task because of the low contrasttonoise ratio and large variance in both tissue intensities and brain structures, as well as imaging artifacts ..."
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Cited by 4 (1 self)
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Abstract. This paper presents a Bayesian framework for neonatal braintissue segmentation in clinical magnetic resonance (MR) images. This is a challenging task because of the low contrasttonoise ratio and large variance in both tissue intensities and brain structures, as well as imaging artifacts and partialvolume effects in clinical neonatal scanning. We propose to incorporate a spatially adaptive likelihood model using a datadriven nonparametric statistical technique. The method initially learns an intensitybased prior, relying on the empirical Markov statistics from training data, using fuzzy nonlinear support vector machines (SVM). In an iterative scheme, the models adapt to spatial variations of image intensities via nonparametric density estimation. The method is effective even in the absence of anatomical atlas priors. The implementation, however, can naturally incorporate probabilistic atlas priors and Markovsmoothness priors to impose additional regularity on segmentation. The maximumaposteriori (MAP) segmentation is obtained within a graphcut framework. Cross validation on clinical neonatal brainMR images demonstrates the efficacy of the proposed method, both qualitatively and quantitatively. 1
Adaptive Nonparametric Markov Models and InformationTheoretic Methods for Image Restoration and Segmentation
, 2007
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Local Component Analysis
, 2011
"... Kernel density estimation, a.k.a. Parzen windows, is a popular density estimation method, which can be used for outlier detection or clustering. With multivariate data, its performance is heavily reliant on the metric used within the kernel. Most earlier work has focused on learning only the bandwid ..."
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Cited by 1 (0 self)
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Kernel density estimation, a.k.a. Parzen windows, is a popular density estimation method, which can be used for outlier detection or clustering. With multivariate data, its performance is heavily reliant on the metric used within the kernel. Most earlier work has focused on learning only the bandwidth of the kernel (i.e., a scalar multiplicative factor). In this paper, we propose to learn a full Euclidean metric through an expectationminimisation (EM) procedure, which can be seen as an unsupervised counterpart to neighbourhood component analysis (NCA). In order to avoid overfitting with a fully nonparametric density estimator in high dimensions, we also consider a semiparametric GaussianParzen density model, where some of the variables are modelled throughajointly Gaussian density, while others are modelled throughParzen windows. For these two models, EM leads to simple closedform updates based on matrix inversions and eigenvalue decompositions. We show empirically that our method leads to density estimators with higher testlikelihoods than natural competing methods, and that the metrics may be used within most unsupervised learning techniques that rely on local distances, such as spectral clustering or manifold learning methods. Finally, we present a stochastic approximation scheme which allows for the use of this method in a largescale setting. 1