Jean-Marie Morvan

Restricted delaunay triangulations and normal cycle

We address the problem of curvature estimation from sampled smooth surfaces. Building upon the theory of normal cycles, we derive a definition of the curvature tensor for polyhedral surfaces. This definition consists in a very simple and new formula. When applied to a polyhedral approximation of a smooth surface, it yields an efficient and reliable curvature estimation algorithm. Moreover, we bound the difference between the estimated curvature and the one of the smooth surface in the case of restricted Delaunay triangulations.

Free-Space Optical Communications: Capacity Bounds, Approximations, and a New Sphere-Packing Perspective

The capacity of the free-space optical channel is studied. A new recursive approach for bounding the capacity of the channel based on sphere-packing is proposed. This approach leads to new capacity upper bounds for a channel with a peak intensity constraint or an average intensity constraint. Under an average constraint only, the derived bound is tighter than an existing sphere-packing bound derived earlier by Farid and Hranilovic. The achievable rate of a truncated-Gaussian input distribution is also derived. It is shown that under both average and peak constraints, this achievable rate and the sphere-packing bounds are within a small gap at high SNR, leading to a simple high-SNR capacity approximation. Simple fitting functions that capture the best known achievable rate for the channel are provided. These functions can be of practical importance especially for the study of systems operating under atmospheric turbulence and misalignment conditions.

On the angular defect of triangulations and the pointwise approximation of curvatures

Let S be a smooth surface of E3, p a point on S, km, kM, kG and kH the maximum, minimum, Gauss and mean curvatures of S at p. Consider a set {pippi+1}i = 1,....,n of n Euclidean triangles forming a piecewise linear approximation of S around p--with pn+1 = p1. For each triangle, let γi be the angle ∠pippi+1, and let the angular defect at p be 2π - Σiγi. This paper establishes, when the distances ||ppi|| go to zero, that the angular defect is asymptotically equivalent to a homogeneous polynomial of degree two in the principal curvatures.For regular meshes, we provide closed forms expressions for the three coefficients of this polynomial. We show that vertices of valence four and six are the only ones where kG can be inferred from the angular defect. At other vertices, we show that the principal curvatures can be derived from the angular defects of two independent triangulations. For irregular meshes, we show that the angular defect weighted by the so-called module of the mesh estimates kG within an error bound depending upon km and kM.Meshes are ubiquitous in Computer Graphics and Computer Aided Design, and a significant number of papers advocate the use of normalized angular defects to estimate the Gauss curvature of smooth surfaces. We show that the statements made in these papers are erroneous in general, although they may be true pointwise for very specific meshes. A direct consequence is that normalized angular defects should be used to estimate the Gauss curvature for these cases only where the geometry of the meshes processed is precisely controlled. On a more general perspective, we believe this contributions is one step forward the intelligence of the geometry of meshes, whence one step forward more robust algorithms.

Expression-robust 3D face recognition via weighted sparse representation of multi-scale and multi-component local normal patterns

In the theory of differential geometry, surface normal, as a first order surface differential quantity, determines the orientation of a surface at each point and contains informative local surface shape information. To fully exploit this kind of information for 3D face recognition (FR), this paper proposes a novel highly discriminative facial shape descriptor, namely multi-scale and multi-component local normal patterns (MSMC-LNP). Given a normalized facial range image, three components of normal vectors are first estimated, leading to three normal component images. Then, each normal component image is encoded locally to local normal patterns (LNP) on different scales. To utilize spatial information of facial shape, each normal component image is divided into several patches, and their LNP histograms are computed and concatenated according to the facial configuration. Finally, each original facial surface is represented by a set of LNP histograms including both global and local cues. Moreover, to make the proposed solution robust to the variations of facial expressions, we propose to learn the weight of each local patch on a given encoding scale and normal component image. Based on the learned weights and the weighted LNP histograms, we formulate a weighted sparse representation-based classifier (W-SRC). In contrast to the overwhelming majority of 3D FR approaches which were only benchmarked on the FRGC v2.0 database, we carried out extensive experiments on the FRGC v2.0, Bosphorus, BU-3DFE and 3D-TEC databases, thus including 3D face data captured in different scenarios through various sensors and depicting in particular different challenges with respect to facial expressions. The experimental results show that the proposed approach consistently achieves competitive rank-one recognition rates on these databases despite their heterogeneous nature, and thereby demonstrates its effectiveness and its generalizability.

An efficient multimodal 2D + 3D feature-based approach to automatic facial expression recognition

We propose a feature-based 2D+3D multimodal facial expression recognition method.It is fully automatic benefit from a large set of automatically detected landmarks.The complementarities between 2D and 3D features are comprehensively demonstrated.Our method achieves the best accuracy on the BU-3DFE database so far.A good generalization ability is shown on the Bosphorus database. We present a fully automatic multimodal 2D + 3D feature-based facial expression recognition approach and demonstrate its performance on the BU-3DFE database. Our approach combines multi-order gradient-based local texture and shape descriptors in order to achieve efficiency and robustness. First, a large set of fiducial facial landmarks of 2D face images along with their 3D face scans are localized using a novel algorithm namely incremental Parallel Cascade of Linear Regression (iPar-CLR). Then, a novel Histogram of Second Order Gradients (HSOG) based local image descriptor in conjunction with the widely used first-order gradient based SIFT descriptor are used to describe the local texture around each 2D landmark. Similarly, the local geometry around each 3D landmark is described by two novel local shape descriptors constructed using the first-order and the second-order surface differential geometry quantities, i.e., Histogram of mesh Gradients (meshHOG) and Histogram of mesh Shape index (curvature quantization, meshHOS). Finally, the Support Vector Machine (SVM) based recognition results of all 2D and 3D descriptors are fused at both feature-level and score-level to further improve the accuracy. Comprehensive experimental results demonstrate that there exist impressive complementary characteristics between the 2D and 3D descriptors. We use the BU-3DFE benchmark to compare our approach to the state-of-the-art ones. Our multimodal feature-based approach outperforms the others by achieving an average recognition accuracy of 86.32%. Moreover, a good generalization ability is shown on the Bosphorus database.

Expression robust 3D face recognition via mesh-based histograms of multiple order surface differential quantities

This paper presents a mesh-based approach for 3D face recognition using a novel local shape descriptor and a SIFT-like matching process. Both maximum and minimum curvatures estimated in the 3D Gaussian scale space are employed to detect salient points. To comprehensively characterize 3D facial surfaces and their variations, we calculate weighted statistical distributions of multiple order surface differential quantities, including histogram of mesh gradient (HoG), histogram of shape index (HoS) and histogram of gradient of shape index (HoGS) within a local neighborhood of each salient point. The subsequent matching step then robustly associates corresponding points of two facial surfaces, leading to much more matched points between different scans of a same person than the ones of different persons. Experimental results on the Bosphorus dataset highlight the effectiveness of the proposed method and its robustness to facial expression variations.

Approximation of the Normal Vector Field and the Area of a Smooth Surface

Abstract This paper deals with the comparison of the normal vector field of a smooth surface S with the normal vector field of another surface differentiable almost everywhere. The main result gives an upper bound on angles between the normals of S and the normals of a triangulation T close to S. This upper bound is expressed in terms of the geometry of T, the curvature of S and the Hausdorff distance between both surfaces. This kind of result is really useful: in particular, results of the approximation of the normal vector field of a smooth surface S can induce results of the approximation of the area; indeed, in a very general case (T is only supposed to be locally the graph of a lipschitz function), if we know the angle between the normals of both surfaces, then we can explicitly express the area of S in terms of geometrical invariants of T, the curvature of S and of the Hausdorff distance between both surfaces. We also apply our results in surface reconstruction: we obtain convergence results when T is the restricted Delaunay triangulation of an ε-sample of S; using Chew’s algorithm, we also build sequences of triangulations inscribed in S whose curvature measures tend to the curvatures measures of S.

An automatic 3D expression recognition framework based on sparse representation of conformal images

We propose a general and fully automatic framework for 3D facial expression recognition by modeling sparse representation of conformal images. According to Riemann Geometry theory, a 3D facial surface S embedded in ℝ3, which is a topological disk, can be conformally mapped to a 2D unit disk D through the discrete surface Ricci Flow algorithm. Such a conformal mapping induces a unique and intrinsic surface conformal representation denoted by a pair of functions defined on D, called conformal factor image (CFI) and mean curvature image (MCI). As facial expression features, CFI captures the local area distortion of S induced by the conformal mapping; MCI characterizes the geometry information of S. To model sparse representation of conformal images for expression classification, both CFI and MCI are further normalized by a Mobius transformation. This transformation is defined by the three main facial landmarks (i.e. nose tip, left and right inner eye corners) which can be detected automatically and precisely. Expression recognition is carried out by the minimal sparse expression-class-dependent reconstruction error over the conformal image based expression dictionary. Extensive experimental results on the BU-3DFER dataset demonstrate the effectiveness and generalization of the proposed framework.

3D Facial expression recognition based on histograms of surface differential quantities

3D face models accurately capture facial surfaces, making it possible for precise description of facial activities. In this paper, we present a novel mesh-based method for 3D facial expression recognition using two local shape descriptors. To characterize shape information of the local neighborhood of facial landmarks, we calculate the weighted statistical distributions of surface differential quantities, including histogram of mesh gradient (HoG) and histogram of shape index (HoS). Normal cycle theory based curvature estimation method is employed on 3D face models along with the common cubic fitting curvature estimation method for the purpose of comparison. Based on the basic fact that different expressions involve different local shape deformations, the SVM classifier with both linear and RBF kernels outperforms the state of the art results on the subset of the BU-3DFE database with the same experimental setting.

Deformations of isotropic submanifolds in Kähler manifolds

We study the first and second variations of isotropic submanifolds which preserve the isotropy. In order to do so, we introduce the notions of harmonic, exact and isotropic variations and investigate basic properties of isotropic submanifolds which are minimal under such deformations. Many results in this respect are then obtained. In particular, we obtain a new characterization of Maslov class in terms of such deformations.