Shape deformation analysis from the optimal control viewpoint

A crucial problem in shape deformation analysis is to determine a deformation of a given shape into another one, which is optimal for a certain cost. It has a number of applications in particular in medical imaging. In this article we provide a new general approach to shape deformation analysis, within the framework of optimal control theory, in which a deformation is represented as the flow of diffeomorphisms generated by time-dependent vector fields. Using reproducing kernel Hilbert spaces of vector fields, the general shape deformation analysis problem is specified as an infinite-dimensional optimal control problem with state and control constraints. In this problem, the states are diffeomorphisms and the controls are vector fields, both of them being subject to some constraints. The functional to be minimized is the sum of a first term defined as geometric norm of the control (kinetic energy of the deformation) and of a data attachment term providing a geometric distance to the target shape. This point of view has several advantages. First, it allows one to model general constrained shape analysis problems, which opens new issues in this field. Second, using an extension of the Pontryagin maximum principle, one can characterize the optimal solutions of the shape deformation problem in a very general way as the solutions of constrained geodesic equations. Finally, recasting general algorithms of optimal control into shape analysis yields new efficient numerical methods in shape deformation analysis. Overall, the optimal control point of view unifies and generalizes different theoretical and numerical approaches to shape deformation problems, and also allows us to design new approaches. The optimal control problems that result from this construction are infinite dimensional and involve some constraints, and thus are nonstandard. In this article we also provide a rigorous and complete analysis of the infinite-dimensional shape space problem with constraints and of its finite-dimensional approximations

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

Shape transition and oblate-prolate coexistence in N=Z fpg-shell nuclei

Nuclear shape transition and oblate-prolate coexistence in $N=Z$ nuclei are investigated within the configuration space ($2p_{3/2}$, $1f_{5/2}$, $2p_{1/2}$, and $1g_{9/2}$). We perform shell model calculations for $^{60}$Zn, $^{64}$Ge, and $^{68}$Se and constrained Hartree-Fock (CHF) calculations for $^{60}$Zn, $^{64}$Ge, $^{68}$Se, and $^{72}$Kr, employing an effective pairing plus quadrupole residual interaction with monopole interactions. The shell model calculations reproduce well the experimental energy levels of these nuclei. From the analysis of potential energy surface in the CHF calculations, we found shape transition from prolate to oblate deformation in these $N=Z$ nuclei and oblate-prolate coexistence at $^{68}$Se. The ground state of $^{68}$Se has oblate shape, while the shape of $^{60}$Zn and $^{64}$Ge are prolate. It is shown that the isovector matrix elements between $f_{5/2}$ and $p_{1/2}$ orbits cause the oblate deformation for $^{68}$Se, and four-particle four-hole ($4p-4h$) excitations are important for the oblate configuration.Comment: 6 pages, 5 figures, accepted for publication in Phys. Rev.

Shape analysis in shape space

This study aims to classify different deformations based on the shape space concept. A shape space is a quotient space in which each point corresponds to a class of shapes. The shapes of each class are transformed to each other by a transformation group preserving a geometrical property in which we are interested. Therefore, each deformation is a curve on the high dimensional shape space manifold, and one can classify the deformations by comparison of their corresponding deformation curves in shape space. Towards this end, two classification methods are proposed. In the first method, a quasi conformal shape space is constructed based on a novel quasi-conformal metric, which preserves the curvature changes at each vertex during the deformation. Besides, a classification framework is introduced for deformation classification. The results on synthetic and real datasets show the effectiveness of the metric to estimate the intrinsic geometry of the shape space manifold, and its ability to classify and interpolate different deformations. In the second method, we introduce the medial surface shape space which classifies the deformations based on the medial surface and thickness of the shape. This shape space is based on the log map and uses two novel measures, average of the normal vectors and mean of the positions, to determine the distance between each pair of shapes on shape space. We applied these methods to classify the left ventricle deformations. The experimental results shows that the first method can remarkably classify the normal and abnormal subjects but this method cannot spot the location of the abnormality. In contrast, the second method can discriminate healthy subjects from patients with cardiomyopathy, and also can spot the abnormality on the left ventricle, which makes it a valuable assistant tool for diagnostic purposes