Supersymmetry, shape invariance and the Legendre equations

In three space dimensions, when a physical system possesses spherical symmetry, the dynamical equations automatically lead to the Legendre and the associated Legendre equations, with the respective orthogonal polynomials as their standard solutions. This is a very general and important result and appears in many problems in physics (for example, the multipole expansion etc). We study these equations from an operator point of view, much like the harmonic oscillator, and show that there is an underlying shape invariance symmetry in these systems responsible for their solubility. We bring out various interesting features resulting from this analysis from the shape invariance point of view.Comment: 4 pages, 1 figure; to appear in PL

Multi-set canonical correlation analysis for 3D abnormal gait behaviour recognition based on virtual sample generation

Small sample dataset and two-dimensional (2D) approach are challenges to vision-based abnormal gait behaviour recognition (AGBR). The lack of three-dimensional (3D) structure of the human body causes 2D based methods to be limited in abnormal gait virtual sample generation (VSG). In this paper, 3D AGBR based on VSG and multi-set canonical correlation analysis (3D-AGRBMCCA) is proposed. First, the unstructured point cloud data of gait are obtained by using a structured light sensor. A 3D parametric body model is then deformed to fit the point cloud data, both in shape and posture. The features of point cloud data are then converted to a high-level structured representation of the body. The parametric body model is used for VSG based on the estimated body pose and shape data. Symmetry virtual samples, pose-perturbation virtual samples and various body-shape virtual samples with multi-views are generated to extend the training samples. The spatial-temporal features of the abnormal gait behaviour from different views, body pose and shape parameters are then extracted by convolutional neural network based Long Short-Term Memory model network. These are projected onto a uniform pattern space using deep learning based multi-set canonical correlation analysis. Experiments on four publicly available datasets show the proposed system performs well under various conditions

Polynomial-based non-uniform interpolatory subdivision with features control

Starting from a well-known construction of polynomial-based interpolatory 4-point schemes, in this paper we present an original affine combination of quadratic polynomial samples that leads to a non-uniform 4-point scheme with edge parameters. This blending-type formulation is then further generalized to provide a powerful subdivision algorithm that combines the fairing curve of a non-uniform refinement with the advantages of a shape-controlled interpolation method and an arbitrary point insertion rule. The result is a non-uniform interpolatory 4-point scheme that is unique in combining a number of distinctive properties. In fact it generates visually-pleasing limit curves where special features ranging from cusps and flat edges to point/edge tension effects may be included without creating undesired undulations. Moreover such a scheme is capable of inserting new points at any positions of existing intervals, so that the most convenient parameter values may be chosen as well as the intervals for insertion. Such a fully flexible curve scheme is a fundamental step towards the construction of high-quality interpolatory subdivision surfaces with features control

Planar PØP: feature-less pose estimation with applications in UAV localization

© 20xx IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.We present a featureless pose estimation method that, in contrast to current Perspective-n-Point (PnP) approaches, it does not require n point correspondences to obtain the camera pose, allowing for pose estimation from natural shapes that do not necessarily have distinguished features like corners or intersecting edges. Instead of using n correspondences (e.g. extracted with a feature detector) we will use the raw polygonal representation of the observed shape and directly estimate the pose in the pose-space of the camera. This method compared with a general PnP method, does not require n point correspondences neither a priori knowledge of the object model (except the scale), which is registered with a picture taken from a known robot pose. Moreover, we achieve higher precision because all the information of the shape contour is used to minimize the area between the projected and the observed shape contours. To emphasize the non-use of n point correspondences between the projected template and observed contour shape, we call the method Planar PØP. The method is shown both in simulation and in a real application consisting on a UAV localization where comparisons with a precise ground-truth are provided.Peer ReviewedPostprint (author's final draft

Hierarchical Object Parsing from Structured Noisy Point Clouds

Object parsing and segmentation from point clouds are challenging tasks because the relevant data is available only as thin structures along object boundaries or other features, and is corrupted by large amounts of noise. To handle this kind of data, flexible shape models are desired that can accurately follow the object boundaries. Popular models such as Active Shape and Active Appearance models lack the necessary flexibility for this task, while recent approaches such as the Recursive Compositional Models make model simplifications in order to obtain computational guarantees. This paper investigates a hierarchical Bayesian model of shape and appearance in a generative setting. The input data is explained by an object parsing layer, which is a deformation of a hidden PCA shape model with Gaussian prior. The paper also introduces a novel efficient inference algorithm that uses informed data-driven proposals to initialize local searches for the hidden variables. Applied to the problem of object parsing from structured point clouds such as edge detection images, the proposed approach obtains state of the art parsing errors on two standard datasets without using any intensity information.Comment: 13 pages, 16 figure