Chafik Samir

An Intrinsic Framework for Analysis of Facial Surfaces

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.

A Gradient-Descent Method for Curve Fitting on Riemannian Manifolds

Given data points p0,…,pN on a closed submanifold M of ℝn and time instants 0=t0<t1<⋅⋅⋅<tN=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 ℝn and on the unit sphere.

Elastic Shape Analysis of Cylindrical Surfaces for 3D/2D Registration in Endometrial Tissue Characterization

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.

Piecewise-Bézier C1 Interpolation on Riemannian Manifolds with Application to 2D Shape Morphing

We present a new framework to fit a path to a given finite set of data points on a Riemannian manifold. The path takes the form of a continuously-differentiable concatenation of Riemannian Bezier segments. The selection of the control points that define the Bezier segments is partly guided by the differentiability requirement and by a minimal mean squared acceleration objective. We illustrate our approach on specific manifolds: the Euclidean plane (for sanity check), the sphere (as a first nonlinear illustration), the special orthogonal group (with rigid body motion applications), and the shape manifold (with 2D shape morphing applications).

Fitting smooth paths on Riemannian manifolds: Endometrial surface reconstruction and preoperative MRI-based navigation

We present a new method to fit smooth paths to a given set of points on Riemannian manifolds using (C^1) piecewise-Bezier functions. A property of the method is that, when the manifold reduces to a Euclidean space, the control points minimize the mean square acceleration of the path. As an application, we focus on data observations that evolve on certain nonlinear manifolds of importance in medical imaging: the shape manifold for endometrial surface reconstruction; the special orthogonal group SO(3) and the special Euclidean group SE(3) for preoperative MRI-based navigation. Results on real data show that our method succeeds in meeting the clinical goal: combining different modalities to improve the localization of the endometrial lesions.

Elastic Morphing of 2D and 3D Objects on a Shape Manifold

We present a new method for morphing 2D and 3D objects. In particular we focus on the problem of smooth interpolation on a shape manifold. The proposed method takes advantage of two recent works on 2D and 3D shape analysis to compute elastic geodesics between any two arbitrary shapes and interpolations on a Riemannian manifold. Given a finite set of frames of the same (2D or 3D) object from a video sequence, or different expressions of a 3D face, our goal is to interpolate between the given data in a manner that is smooth. Experimental results are presented to demonstrate the effectiveness of our method.

Statistical Shape Model for Simulation of Realistic Endometrial Tissue

We propose a new framework for developing statistical shape models of endometrial tissues from real clinical data. Endometrial tissues naturally form cylindrical surfaces, and thus, we adopt, with modification, a recent Riemannian framework for statistical shape analysis of parameterized surfaces. This methodology is based on a representation of surfaces termed square-root normal fields (SRNFs), which enables invariance to all shape preserving transformations including translation, scale, rotation, and re-parameterization. We extend this framework by computing parametrization-invariant statistical summaries of endometrial tissue shapes, and random sampling from learned generative models. Such models are very useful for medical practitioners during different tasks such as diagnosing or monitoring endometriosis. Furthermore, real data in medical applications in general (and in particular in this application) is often scarce, and thus the generated random samples are a key step for evaluating segmentation and registration approaches. Moreover, this study allows us to efficiently construct a large set of realistic samples that can open new avenues for diagnosing and monitoring complex diseases when using automatic techniques from computer vision, machine learning, etc.

A Statistical Framework for Elastic Shape Analysis of Spatio-Temporal Evolutions of Planar Closed Curves

We propose a new statistical framework for spatiotemporal modeling of elastic planar, closed curves. This approach combines two recent frameworks for elastic functional data analysis and elastic shape analysis. The proposed trajectory registration framework enables matching and averaging to quantify spatio-temporal deformations while taking into account their dynamic specificities. A key ingredient of this framework is a tracking method that optimizes the evolution of curves extracted from sequences of consecutive images to estimate the spatio-temporal deformation fields. Automatic estimation of such deformations including spatial changes (strain) and dynamic temporal changes (phase) was tested on simulated examples and real myocardial trajectories. Experimental results show significant improvements in the spatio-temporal structure of trajectory comparisons and averages using the proposed framework.

A new curves-based method for adaptive multimodal registration

We propose a new multimodal image registration method when using Magnetic Resonance Imaging (MRI) and Transvaginal Ultrasound (TVUS) imaging separately. Given two sets of corresponding curves in the MRI and TVUS images, we estimate a deformation vector field that registers the TVUS image to the MRI in such a way that the corresponding curves match exactly and the deformation is sufficiently smooth over the whole image domain. Experimental results on real data show that the proposed method outperforms some of the state-of-the-art methods.

Cylindrical Surface Reconstruction by Fitting Paths on Shape Space

We present a differential geometric approach for cylindrical anatomical surface reconstruction from 3D volumetric data that may have missing slices or discontinuities. We extract planar boundaries from the 2D image slices, and parameterize them by an indexed set of curves. Under the SRVF framework, the curves are represented as invariant elements of a nonlinear shape space. Differently from standard approaches, we use tools such as exponential maps and geodesics from Riemannian geometry and solve the problem of surface reconstruction by fitting paths through the given curves. Experimental results show the surface reconstruction of smooth endometrial tissue shapes generated from MRI slices.