Morphing of Triangular Meshes in Shape Space

We present a novel approach to morph between two isometric poses of the same non-rigid object given as triangular meshes. We model the morphs as linear interpolations in a suitable shape space $\mathcal{S}$. For triangulated 3D polygons, we prove that interpolating linearly in this shape space corresponds to the most isometric morph in $\mathbb{R}^3$. We then extend this shape space to arbitrary triangulations in 3D using a heuristic approach and show the practical use of the approach using experiments. Furthermore, we discuss a modified shape space that is useful for isometric skeleton morphing. All of the newly presented approaches solve the morphing problem without the need to solve a minimization problem.Comment: Improved experimental result

Shape Space Methods for Quantum Cosmological Triangleland

With toy modelling of conceptual aspects of quantum cosmology and the problem of time in quantum gravity in mind, I study the classical and quantum dynamics of the pure-shape (i.e. scale-free) triangle formed by 3 particles in 2-d. I do so by importing techniques to the triangle model from the corresponding 4 particles in 1-d model, using the fact that both have 2-spheres for shape spaces, though the latter has a trivial realization whilst the former has a more involved Hopf (or Dragt) type realization. I furthermore interpret the ensuing Dragt-type coordinates as shape quantities: a measure of anisoscelesness, the ellipticity of the base and apex's moments of inertia, and a quantity proportional to the area of the triangle. I promote these quantities at the quantum level to operators whose expectation and spread are then useful in understanding the quantum states of the system. Additionally, I tessellate the 2-sphere by its physical interpretation as the shape space of triangles, and then use this as a back-cloth from which to read off the interpretation of dynamical trajectories, potentials and wavefunctions. I include applications to timeless approaches to the problem of time and to the role of uniform states in quantum cosmological modelling.Comment: A shorter version, as per the first stage in the refereeing process, and containing some new reference

Sobolev metrics on shape space of surfaces

Let $M$ and $N$ be connected manifolds without boundary with $\dim(M) < \dim(N)$, and let $M$ compact. Then shape space in this work is either the manifold of submanifolds of $N$ that are diffeomorphic to $M$, or the orbifold of unparametrized immersions of $M$ in $N$. We investigate the Sobolev Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions: G^P_f(h,k) = \int_{M} \g(P^f h, k)\, \vol(f^*\g) where \g is some fixed metric on $N$, f^*\g is the induced metric on $M$, $h,k \in \Gamma(f^*TN)$ are tangent vectors at $f$ to the space of embeddings or immersions, and $P^f$ is a positive, selfadjoint, bijective scalar pseudo differential operator of order $2p$ depending smoothly on $f$. We consider later specifically the operator $P^f=1 + A\Delta^p$, where $\Delta$ is the Bochner-Laplacian on $M$ induced by the metric $f^*\bar g$. For these metrics we compute the geodesic equations both on the space of immersions and on shape space, and also the conserved momenta arising from the obvious symmetries. We also show that the geodesic equation is well-posed on spaces of immersions, and also on diffeomorphism groups. We give examples of numerical solutions.Comment: 52 pages, final version as it will appea

Multilinear Wavelets: A Statistical Shape Space for Human Faces

We present a statistical model for $3$D human faces in varying expression, which decomposes the surface of the face using a wavelet transform, and learns many localized, decorrelated multilinear models on the resulting coefficients. Using this model we are able to reconstruct faces from noisy and occluded $3$D face scans, and facial motion sequences. Accurate reconstruction of face shape is important for applications such as tele-presence and gaming. The localized and multi-scale nature of our model allows for recovery of fine-scale detail while retaining robustness to severe noise and occlusion, and is computationally efficient and scalable. We validate these properties experimentally on challenging data in the form of static scans and motion sequences. We show that in comparison to a global multilinear model, our model better preserves fine detail and is computationally faster, while in comparison to a localized PCA model, our model better handles variation in expression, is faster, and allows us to fix identity parameters for a given subject.Comment: 10 pages, 7 figures; accepted to ECCV 201

Currents and finite elements as tools for shape space

The nonlinear spaces of shapes (unparameterized immersed curves or submanifolds) are of interest for many applications in image analysis, such as the identification of shapes that are similar modulo the action of some group. In this paper we study a general representation of shapes that is based on linear spaces and is suitable for numerical discretization, being robust to noise. We develop the theory of currents for shape spaces by considering both the analytic and numerical aspects of the problem. In particular, we study the analytical properties of the current map and the $H^{-s}$ norm that it induces on shapes. We determine the conditions under which the current determines the shape. We then provide a finite element discretization of the currents that is a practical computational tool for shapes. Finally, we demonstrate this approach on a variety of examples

3-D facial expression representation using statistical shape models

This poster describes a methodology for facial expressions representation, using 3-D/4-D data, based on the statistical shape modelling technology. The proposed method uses a shape space vector to model surface deformations, and a modified iterative closest point (ICP) method to calculate the point correspondence between each surface. The shape space vector is constructed using principal component analysis (PCA) computed for typical surfaces represented in a training data set. It is shown that the calculated shape space vector can be used as a significant feature for subsequent facial expression classification. Comprehensive 3-D/4-D face data sets have been used for building the deformation models and for testing, which include 3-D synthetic data generated from FaceGen Modeller® software, 3-D facial expression data caputed by a static 3-D scanner in the BU-3DFE database and 3-D video sequences captured at the ADSIP research centre using a 3dMD® dynamic 3-D scanner