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In this paper, we introduce a novel two‐step modeling method to generalize cubic Hermite interpolators on the space of probability measures P+(I)$$ {\mathcal{P}}_{+}(I) $$. First, we develop new approaches to capture the Riemannian geometric structure of P+(I)$$ {\mathcal{P}}_{+}(I) $$ when equipped with Fisher–Rao metric. Furthermore, we develop and detail all numerical tools on P+(I)$$ {\mathcal{P}}_{+}(I) $$, namely, Levi–Civita connection, minimal geodesics, parallel transport, exponential map, and logarithm map. Then, we demonstrate that preliminary analysis results yield significant benefits in constructing an optimal cubic Hermite spline on P+(I)$$ {\mathcal{P}}_{+}(I) $$ as a nonlinear Riemannian manifold, precisely where conventional numerical methods fail. [ABSTRACT FROM AUTHOR]

In this paper, we adopt a Bayesian point of view, based on Gaussian processes, for classifying high-dimensional data. Since computing the exact marginal likelihood remains difficult, if not impossible, for discrete likelihoods and high-dimensional inputs, we introduce two different methods to improve the efficiency and the scalability of Gaussian processes for classification: scalable Laplace approximation and scalable expectation propagation, together with an embedding for dimension reduction. The proposed methods are shown to handle multimodal posteriors and improve the prediction quality, jointly. Various numerical tests on simulated and real data provide a comparison with alternative Bayesian and non Bayesian predictors. More particularly, the comparison study confirms that the ability of our proposed methods to estimate the marginal likelihood efficiently yields better predictions, especially for complex tasks. [ABSTRACT FROM AUTHOR]

We present a new statistical framework for landmark ?>curve-based image registration and surface reconstruction. The proposed method first elastically aligns geometric features (continuous, parameterized curves) to compute local deformations, and then uses a Gaussian random field model to estimate the full deformation vector field as a spatial stochastic process on the entire surface or image domain. The statistical estimation is performed using two different methods: maximum likelihood and Bayesian inference via Markov Chain Monte Carlo sampling. The resulting deformations accurately match corresponding curve regions while also being sufficiently smooth over the entire domain. We present several qualitative and quantitative evaluations of the proposed method on both synthetic and real data. We apply our approach to two different tasks on real data: (1) multimodal medical image registration, and (2) anatomical and pottery surface reconstruction. [ABSTRACT FROM AUTHOR]

We introduce a new statistical model for clustering populations of functions and multidimensional curves where the domain is a real interval I. We consider that we are given a finite set of observations from a population of curves or functions with values in R d , for a fixed d ≥ 1 and arguments in I. We are interested in the population background where the clustering is the process of grouping curves into homogeneous sub-populations. In particular, we define a distribution function for each sub-population and use the statistical geometry of the the space of smooth densities to explore a Bayesian model with a spherical Gaussian process prior. We also give the expression of the log-posterior distribution on coefficients resulting from the expansion. Finally, the practical interest of the proposed method is illustrated on simulated and real datasets of multidimensional curves. [ABSTRACT FROM AUTHOR]

We propose a new method to approximate curves that interpolate a given set of time-labeled data on Riemannian symmetric spaces. First, we present our new formulation on the Euclidean space as a result of minimizing the mean square acceleration. This motivates its generalization on some Riemannian symmetric manifolds. Indeed, we generalize the proposed solution to the the special orthogonal group, the manifold of symmetric positive definite matrices, and Riemannian n-manifolds with constant negative curvature. By means of this generalization, we are able to prove that the approximates enjoy a number of nice properties: The solution exists and is optimal in many common situations. Several examples are provided together with some applications and graphical representations. [ABSTRACT FROM AUTHOR]

Abstract This paper introduces a new framework to fit a C 1 Bézier path to a given finite set of ordered data points on shape space of curves. We prove existence and uniqueness properties of the path and give a numerical method for constructing an optimal solution. Furthermore, we present a conceptually simple method to compute the optimal intermediate control points that define this path. The main property of the method is that when the manifold reduces to a Euclidean space or the finite dimensional sphere, the control points minimize the mean square acceleration. Potential applications of fitting smooth paths on Riemannian manifold include applications in robotics, animations, graphics, and medical studies. In this paper, we will focus on different medical applications to predict missing data from few ordered observations. [ABSTRACT FROM AUTHOR]

We study the problem of joint registration and deformation analysis of endometrial tissue using 3D magnetic resonance imaging (MRI) and 2D trans-vaginal ultrasound (TVUS) measurements. In addition to the different imaging techniques involved in the two modalities, this problem is complicated due to: 1) different patient pose during MRI and TVUS observations, 2) the 3D nature of MRI and 2D nature of TVUS measurements, 3) the unknown intersecting plane for TVUS in MRI volume, and 4) the potential deformation of endometrial tissue during TVUS measurement process. Focusing on the shape of the tissue, we use expert manual segmentation of its boundaries in the two modalities and apply, with modification, recent developments in shape analysis of parametric surfaces to this problem. First, we extend the 2D TVUS curves to generalized cylindrical surfaces through replication, and then we compare them with MRI surfaces using elastic shape analysis. This shape analysis provides a simultaneous registration (optimal reparameterization) and deformation (geodesic) between any two parametrized surfaces. Specifically, it provides optimal curves on MRI surfaces that match with the original TVUS curves. This framework results in an accurate quantification and localization of the deformable endometrial cells for radiologists, and growth characterization for gynecologists and obstetricians. We present experimental results using semi-synthetic data and real data from patients to illustrate these ideas. [ABSTRACT FROM PUBLISHER]

Given data points p,..., p on a closed submanifold M of ℝ and time instants 0= t< t<⋅⋅⋅< t=1, we consider the problem of finding a curve γ on M that best approximates the data points at the given instants while being as 'regular' as possible. Specifically, γ is expressed as the curve that minimizes the weighted sum of a sum-of-squares term penalizing the lack of fitting to the data points and a regularity term defined, in the first case as the mean squared velocity of the curve, and in the second case as the mean squared acceleration of the curve. In both cases, the optimization task is carried out by means of a steepest-descent algorithm on a set of curves on M. The steepest-descent direction, defined in the sense of the first-order and second-order Palais metric, respectively, is shown to admit analytical expressions involving parallel transport and covariant integral along curves. Illustrations are given in ℝ and on the unit sphere. [ABSTRACT FROM AUTHOR]

A statistical analysis of shapes of facial surfaces can play an important role in biometric authentication and other face-related applications. The main difficulty in developing such an analysis comes from the lack of a canonical system to represent and compare all facial surfaces. This paper suggests a specific, yet natural, coordinate system on facial surfaces, that enables comparisons of their shapes. Here a facial surface is represented as an indexed collection of closed curves, called facial curves, that are level curves of a surface distance function from the tip of the nose. Defining the space of all such representations of face, this paper studies its differential geometry and endows it with a Riemannian metric. It presents numerical techniques for computing geodesic paths between facial surfaces in that space. This Riemannian framework is then used to: (i) compute distances between faces to quantify differences in their shapes, (ii) find optimal deformations between faces, and (iii) define and compute average of a given set of faces. Experimental results generated using laser-scanned faces are presented to demonstrate these ideas. [ABSTRACT FROM AUTHOR]

We study shapes of facial surfaces for the purpose of face recognition. The main idea is to 1) represent surfaces by unions of level curves, called facial curves, of the depth function and 2) compare shapes of surfaces implicitly using shapes of facial curves. The latter is performed using a differential geometric approach that computes geodesic lengths between closed curves on a shape manifold. These ideas are demonstrated using a nearest-neighbor classifier on two 3D face databases: Florida State University and Notre Dame, highlighting a good recognition performance. [ABSTRACT FROM AUTHOR]

Recent developments in shape analysis and retrieval play an important role in a wide variety of applications that potentially require matching of objects with different geometries. In shape classification, there is no natural way to represent an object but the similarity measure, distance between representations or descriptors, depends heavily on the strategy of computing optimal correspondences. In this paper we introduce a new numerical method for registering surfaces. Thus, finding an optimal one-to-one correspondence between their shapes. Unfortunately, solving this type of optimization problem is generally hard because the solutions space is nonlinear with no natural manifold structure on it. To overcome such limitations we make use of recent methods to represent objects then we find optimal correspondences using a discretized approximation of the search space. The proposed method has the advantage of extending the Riemannian analysis of 3D curves in a natural way for surfaces. We demonstrate the proposed algorithms using different public datasets for 2D and 3D objects matching and classification. [ABSTRACT FROM AUTHOR]

This paper develops a new machine vision framework for efficient detection and classification of manufacturing defects in metal boxes. Previous techniques, which are based on either visual inspection or on hand-crafted features, are both inaccurate and time consuming. In this paper, we show that by using autoencoder deep neural network (DNN) architecture, we are able to not only classify manufacturing defects, but also localize them with high accuracy. Compared to traditional techniques, DNNs are able to learn, in a supervised manner, the visual features that achieve the best performance. Our experiments on a database of real images demonstrate that our approach overcomes the state-of-the-art while remaining computationally competitive. [ABSTRACT FROM AUTHOR]

Statistical shape analysis plays a key role in various medical imaging applications. Such methods provide tools for registering, deforming, comparing, averaging, and modeling anatomical shapes. In this work, we focus on the application of a recent method for statistical shape analysis of parameterized surfaces to simulation of endometrial tissue shapes. The clinical data contains magnetic resonance imaging (MRI) endometrial tissue surfaces, which are used to learn a generative shape model. We generate random tissue shapes from this model, and apply elastic semi-synthetic deformations to them. This provides two types of simulated data: (1) MRI-type (without deformation) and (2) transvaginal ultrasound (TVUS)-type, which undergo an additional deformation due to the transducer׳s pressure. The proposed models can be used for validation of automatic, multimodal image registration, which is a crucial step in diagnosing endometriosis. [ABSTRACT FROM AUTHOR]

We propose a new deformable slice-to-volume registration method to register a 2D Transvaginal Ultrasound (TVUS) to a 3D Magnetic Resonance (MR) volume. Our main goal is to find a cross-section of the MR volume such that the endometrial implants and their depth of infiltration can be mapped from TVUS to MR. The proposed TVUS-MR registration method uses contour to surface correspondences through a novel variational one-step deformable Iterative Closest Point (ICP) method. Specifically, we find a smooth deformation field while establishing point correspondences automatically. We demonstrate the accuracy of the proposed method by quantitative and qualitative tests on both semi-synthetic and clinical data. To generate semi-synthetic data sets, 3D surfaces are deformed with 4–40% degrees of deformation and then various intersection curves are obtained at 0–20° cutting angles. Results show an average mean square error of 5.7934 ± 0.4615 mm, average Hausdorff distance of 2.493 ± 0.14 mm, and average Dice similarity coefficient of 0.9750 ± 0.0030. [ABSTRACT FROM AUTHOR]

Abstract Morphometric assessments of the dentition have played significant roles in hypotheses relating to taxonomic diversity among extinct hominins. In this regard, emphasis has been placed on the statistical appraisal of intraspecific variation to identify morphological criteria that convey maximum discriminatory power. Three-dimensional geometric morphometric (3D GM) approaches that utilize landmarks and semi-landmarks to quantify shape variation have enjoyed increasingly popular use over the past twenty-five years in assessments of the outer enamel surface (OES) and enamel–dentine junction (EDJ) of fossil molars. Recently developed diffeomorphic surface matching (DSM) methods that model the deformation between shapes have drastically reduced if not altogether eliminated potential methodological inconsistencies associated with the a priori identification of landmarks and delineation of semi-landmarks. As such, DSM has the potential to better capture the geometric details that describe tooth shape by accounting for both homologous and non-homologous (i.e., discrete) features, and permitting the statistical determination of geometric correspondence. We compare the discriminatory power of 3D GM and DSM in the evaluation of the OES and EDJ of mandibular permanent molars attributed to Australopithecus africanus , Paranthropus robustus and early Homo sp. from the sites of Sterkfontein and Swartkrans. For all three molars, classification and clustering scores demonstrate that DSM performs better at separating the A. africanus and P. robustus samples than does 3D GM. The EDJ provided the best results. P. robustus evinces greater morphological variability than A. africanus. The DSM assessment of the early Homo molar from Swartkrans reveals its distinctiveness from either australopith sample, and the "unknown" specimen from Sterkfontein (Stw 151) is notably more similar to Homo than to A. africanus. [ABSTRACT FROM AUTHOR]